MANIFOLD

Manifold Integrated Design Strategies for Honeycomb Systems Andrew Kudless

Manifold Integrated Design Strategies for Honeycomb Systems Andrew Kudless MA Dissertation Emergent Technologies + Design Architectural Association October 2004

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Contents 4 Overview Introduction 5 Hypothesis 6 Scope 7 Organization 9 10 Strategies 18 Tactics Production Logics 20 Parametric Matrix 36 Performative Relationships 66 82 Resources Experiments 84 References 162 180 Status 183 Appendix Bibliography 184 Index 186 Image Credits 187 Acknowledgements 188

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Project Overview

Project Overview: Introduction The central aim of the research is the development of a material system with a high degree of integration between its design and performance. This integration is inherent to natural material systems for they have been developed through evolutionary means which intricately tie together the form, growth, and behaviour of the organism. In industrial material systems, the level of integration is far lower resulting in wide and potentially problematic gaps between how the system is made, what it looks like, and how it actually performs in an environment. This research explores strategies of integration for a particular industrially produced material system for use in architectural applications.

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Project Overview: Hypothesis This research develops a honeycomb system that is able to adapt to diverse performance requirements through the modulation of the system’s inherent geometric and material parameters while remaining within the limits of available production technologies. The project is based on the desire to form an integrated and generative design strategy based on a biomimetic approach to architectural design and fabrication.

The system developed in this research is not meant as a replacement to the designer nor an automation of the design process, but rather it presents an open framework through which the designer can work, enabling a more integral relationship between the various conflicting and overlapping issues in the development of an architectural project. The research represents a tool, waiting to be actively used with specific project data and embedded in a built artifact.

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Project Overview: Scope The scope of the research is split between the initial development of the honeycomb system as a general design tool and the subsequent utilization of this tool in the design of a prototype with specific performative requirements.

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Within this framework, the scope is further defined by a desire to work in an opportunistic way with existing hardware and software technologies rather than rely on production techniques and materials yet to be realized or proprietary software packages outside the means of most designers. Too often in architecture, and specifically digitally generated architecture, there is a disjunction between the scale of the concept and the scale of the built artifact. Models, both digital and physical, are more often based on visual representation rather than on a particular material or production logic. The problems that arise from such a strategy are numerous and include a highly inefficient design development process where time, labour, and often the original concepts are lost in reworking models to fit a potential construction strategy.

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Project Overview: Scope (cont.)

This research attempts to avoid these issues by limiting the scope of potential CAD/CAM strategies to those which have a clear correspondence to large-scale technologies. Not all materials and modelling techniques have the same ability to scale up or respond to change easily. This research will focus on developing tactics that work within the limits of existing technologies while at the same time pushing these limits in innovative directions. For example, the laser cutter will be used not simply because it cuts quickly and accurately, but because it has a clear analogue in complex large-scale construction such as ship-building.

Likewise, the folding of sheet materials such as paper is used as an analogy to sheet-metal bending processes that are industrially common.

However, the scope of the work is not limited to only an exploration of multi-scalar production techniques. There is also the desire to study the entire scope of an architectural project, from early analysis, to form generation, through to production. Through the inclusion of all phases of an architectural project in the scope of the work, it is hoped that a more comprehensive design process can be explored. The research develops tactics for creating a networked infrastructure that will acknowledge issues relating to site, program, material, structure, budget, and production through the development of the large-scale prototype.

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Project Overview

Introduction

Hypothesis

Production: Overview

Primary Logics

Pre-Fabrication Tactics

Surface Topology

Equipment Parameters

Surface Geometry

Unfolding

Surface Skin

Nesting Scope

Organization

Secondary Logics Digital Fabrication

Identity Logistics

Strategies

Digital Generation Parametric Matrix

Biomimetic Research Geometric Parameters Abstraction + Translation Material Parameters Parametric Generation Performance: Overview

Skin Fenestration

Specification

Layer Quantity

MEL Script

Layer Topology

Structure

Tactics

Multi-Parametric Design

Surface Parameters

Layer Scale

Grid Parameters

Layer Binding

Cell Parameters

Grid Geometry

Thickness Weight

Grid Uniformity

Elasticity

Cell Connectivity

Transparency

Cell Edge Orienation

Surface Geometry

Production Logics

Thermal Conductivity

Non-Mechanical Performance

Cell Fill

Grid Geometry

000_Base

Grid Uniformity

Resources

Performative Relationships

Resources: Overview

Cellular Solids: Form-Finding Honeycombs

Architecture Experiments

003_Layer Scale 004_Deformed Grid

Honeycombs: Overview Honeycomb Models

References Nature

Project Status Status: Overview

006_Parallel 007_Self-Similar

Serpentine Pavilion 2002

008_Closed

Prada LA Panelite

009_Sphere 010_Polar

Simmons Hall

011_Orientation

Honeycomb Sandwich Panels

012_Skin 1

CNC Cutting Radiolaria

013_Skin 2 014_Prototype Model

Honeycombs + Hives

015_Syn/Anticlastic

Kelvin and Plateau

016_Curvature Depth

Voronoi Algorithms UV Coodinate System

Math + Science

005_Pocket

Bruges Pavilion 2002

Sheet Metal Bending Industry

002_Multiple Layers

Cell Fill

Mashrabiya Bibliography

001_Cell Depth

Cell Depth

Honeycomb Prototype Parametric Matrix

Cell Depth

Layer Binding

Cell Edge Orientation Tactics: Overview

Grid Denisty

Bending Radius

Grid Density Light and Vision

Layer Connectivity

Layer Parameters

Surface Skin

Emergent Performance

Skin Triangulation

Honeycomb Algorithm

Growth Algorithms Strategies: Overview

Project Overview:

Skin-Fabrication Tactics

017_Cell Curvature

Organization The research is organized into three inter-related categories: Strategies, Tactics, and Resources. Borrowed from military terminology, these words are often used within architecture without a true understanding of how they enable a integrated organization of a project.

To review their meanings, strategy is defined as the global methods and processes that are established with respect to a given long-term goal. Where strategy deals with the top-level organization of a system, tactics are the opposite. Tactics are the low-level actions that attempt to aline with and fulfil the strategic demands. Resources are the materials at hand with which these tactical processes can be accomplished.

Within the framework of this research project, these terms provide a convenient way to structure a complex and inter-related set of information. The project strategy can be defined as the set of operations that need to occur in order to satisfy the stated aim of the research: to create an integrated architectural system that is able to adapt to multiple performance criteria.

These strategies are then deployed through the tactical use of parametric design and it’s potential implications on performance as well as the logics of production. These tactics are explored through the available resources of experiments and external references. Finally, there is a small section devoted to the current status of the project. This section will provide information on the current areas of development as well as point at further areas of development in a post-disseration phase of the research.

018_Skin 3

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Strategies

Project Overview

Introduction

Hypothesis

Biomimetic Research Research into the geometric and performative properties of the material group know as Cellular Solids.

Scope

Organization

Abstraction + Translation Strategies: Overview

Strategies Biomimetic Research

The translation of the biomimetic reserach into a hybrid industrial production logic involving large-scale CNC cutting, braking, and honeycomb panel assembly techniques.

Abstraction + Translation

Tactics

Parametric Generation

Parametric Generation

Emergent Performance

The development of a generative matrix of parameters based on the inherent geometric constriants of the hybird production logic through the use of numerous modeling experiments.

Multi-Parametric Design

Tactics: Overview

Emergent Performance Production Logics

A study of the emergent relationships between geometric parameters and specific performance characteristics.

Parametric Matrix

Resources

Performative Relationships

Resources: Overview

Experiments

References

Project Status Status: Overview

Multi-Parametric Design The parametric matrix is made operative through project information relating to site, program and budget.

Strategies: Overview The project strategy is composed of five related operations. Although in general the different operations work linearly, that is, one operation informs the sequential operation, there are also feedback loops between the operations that cause a non-linear effect. Information generated from one operation often feeds the redevelopment of earlier operations which in turn re-informs the entire process.

The initial operation in this strategy is based on biomimetic research and form-finding processes. Studying natural systems and their means of self-organized performance provides potential design strategies. However, the means by which this performance is achieved must be abstracted to conform to the constraints of industrial production logics. Unlike natural systems which have gone through millions of years of development within their own morphogenetic logics to reach their current state of performance, man-made systems must work with the much younger and less sophisticated production logics of industrial manufacturing. From this process of abstraction into technologically feasible means, the next operation is to uncover the inherent parameters of the system, creating a generative matrix of geometric and material properties.

Through an iterative process whereby different combinations of parameters are tested in experimental models, it is then possible to study the emergent performance qualities of a particular set of parameters. Once these relationships are known between the system’s parameters and its performance, these relationships can be reversed such that for a desired performance the correct set of parameters are made operative.

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Stategies: Biomimetic Research Until recently the material sciences were dominated by specialists in specific materials rather than specific properties. That is, metallurgists studied metals, polymer scientists studied plastics, but the general properties that traversed a particular material category and were exhibited by a highly diverse range of materials, both natural and man-made, were left unexamined. In the last 20 years, one of these trans-material groups has increasingly become a prominent and highly successful area of material experimentation and research. Looked at as a group that share many properties spread over varied materials, cellular solids represent an interesting case study of the value of research that focuses on broad general properties rather than specific material categories. In the simplest definition, cellular solids are materials that are an assembly of cells. Thus, many materials that one would normally think of as very different can be categorized together according to their geometric and mechanical properties rather than with reference to their chemical or biological origin. Cellular solids range from materials with biologic origins such as wood, bone, and sponges to industrially created materials such as foamed aluminium, expanded plastics, and honeycomb sandwich panels. The group even includes materials that traditional fall outside of the spectrum of material science such as foods like bread, potato crisps, and meringue.

The usefulness of grouping together materials Related Topics: Cellular Solids: Form-Finding Nature: Radiolaria Nature: Bee’s Honeycombs Math + Science: Kelvin + Plateau

85 173 175 176

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Stategies: Biomimetic Research (cont.) as diverse as these is that they share many of the same geometric and mechanical properties and thus it becomes easier to understand their behaviour as a group than if the knowledge remained locked within a particular material investigation. In addition, the ability to translate specific properties from one material to another is facilitated by a common geometric and mechanical understanding.

The aim of this research is to provide a general background on the geometric, mechanical, and morphological properties and processes of cellular solids. The desire to study cellular solids stems from the fact that not only are cellular solids relatively unknown among architects, but that they provide a potentially important material and technological innovation through their ability to integrate multiple performance properties into one lightweight and energy efficient system. The research is split into two sections. Section One focuses on the basic geometric and mechanical properties of cellular solids with an attempt to trace a history of the ideas that inform the research. It also deals with the morphology and functions of specific examples from the natural and industrial worlds respectively. This section took the form of a written document, parts of which appear in the reference section of this multimedia document. The second section took the for of various experiments that attempted to get a more empirical understanding of the self-organizing geometries of cellular solids.

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Stategies:

hexagon

reinforced hexagon

overexpanded

square cell

Abstraction + Translation The integration between form, growth, and behaviour exemplified in natural cellular solids is impressive, however, for designers to take advantage of this level of integration new industrial methods must be developed. Thus the task is abstracting the geometric and material properties of cellular solids into industrial production logics. This process of abstraction and translation in this project begins with a study of existing man-made cellular solids and their fabrication. Honeycomb sandwich panels are one of the most widely used industrial cellular solids. Primarily they are used the aerospace industry, where high strength to weight ratios are key, but are beginning to be used in architecture as lightweight wall partitions.

Based on the way in which these panels are fabricated, they have certain constraints relating to the amount and type of curvature that they can handle. In addition, they are very limited in the amount of variability in cell sizes and depth across any one panel, producing a homogeneous performance despite potentially heterogeneous requirements. By looking at other technologies and fabrication processes not normally associated with honeycomb sandwich panels, it is possible to enlarge their range of performative and formal possibilities.

Related Topics: underexpanded

flex-core

Industry: Honeycomb Sandwich Panels Industry: CNC cutting Industry: Sheet Folding Primary Production Logic

170 171 172 22

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surface parameters

layer parameters

grid parameters

Stategies:

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

no triangulation

no openings

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

unfilled

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

surface parameters surface topolopy

surface geometry

surface skin

layer parameters skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

horizontal

parallel to surface

vertical

constrained by vertices

grid parameters layer binding

layer scale

grid geometry

grid density

Parametric Generation The strategic use of parametric generation is comprised of two operations. First, a matrix of all the known variables effecting the geometric and material definitions of the system are organized into a clear relational diagram. With this matrix, the task is to explore it through modelling experiments.

cell parameters grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

Related Topics: Parametric Matrix: Overview Experiments: Overview Honeycomb Models: Overview

36 84 87

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Stategies: Emergent Performance It is essential to the development of a new material system to understand the relationship between specific geometrical parameters and observed performance. The strategic use of modelling experiments and simulations adds to this understanding.

Related Topics: Parametric Matrix: Overview Performative Relationships: Overview Experiments: Overview

36 66 84

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Stategies: Multi-Parametric Design The aim of the multi-parametric design strategy is to make the relationships between performance and parameters operative in a design process. Based on specific performance criteria, a suitable set of parameters can be selected and modified to fit the design intent.

During design development, it is possible to quickly rework the design in response to new criteria by modifying the related parameters.

This strategy allows for an open and generative design process that is intricately linked with the design’s means of production, its geometric definition, and its performance requirements.

Related Topics: Honeycomb Prototype: Overview

145

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Tactics

Project Overview

Introduction

Hypothesis

Tactics:

Production Logics A brief description of some of the issues that informed the development of the production logic as well as the digital workflow and scripting processes.

Scope

Organization

Strategies: Overview

Strategies Biomimetic Research

Abstraction + Translation

Parametric Generation

Parametric Matrix An in-depth look at the various geometric and material parameters effecting the development of the honeycomb system.

surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

ray

slots

bifurcated

partial

linear

differential

polar

closed 2 axes

anticlastic curvature

complex

zero gaussian curvature

closed 2 sides

centered

horizontal

parallel to surface

vertical

constrained by vertices

material parameters

Emergent Performance

Tactics

material thickness

material weight

material bending radius

material elasticity

insignificant

insignificant

low range

low range

material transparency

low

significant

significant

high range

high range

high

Multi-Parametric Design

Tactics: Overview

Production Logics

Performative Relationships Parametric Matrix

Resources

Performative Relationships

A study of some of the interesting relationships between specific parameters and observable performative characteristics that emerged from the experimental models.

Plan

Plan

cell fill

Overview The tactics are divided into three sections: Parameters, Performance, and Production.

In contrast to the strategies of the research, these tactics deal primarily with factors that emerge out of the inherent form, behaviour, and growth of the system. They are opportunistic of available means and resources; making do within the imposed strategic set-up. As Michel de Certeau describes tactics: “Tactics are procedures that gain validity in relation to the pertinence they lend to time - to the circumstances which the precise instant of a intervention transforms into a favourable situation, to the relations among successive moments in an action, to the possible intersections of durations and heterogeneous rhythms, etc..” (de Certeau, p.38) Thus, these tactics were developed on a ad hoc basis as new situations developed which required new methods of working or new avenues of exploration. For example, some parameters in the matrix were obvious from the start, others emerged only through months of testing and research. Likewise, although the strategy set out a general production logic that looked promising, the ways in which this logic was informed by the local conditions of specific digital fabrication equipment and software packages was a product of many small-scale tactical decisions.

Resources: Overview Elevation

Section

Elevation

Single-sided Load Test

Double-sided Load Test

Unloaded

Unloaded

Loaded (1:21 ratio)

Loaded (1:10 ratio)

Section

Loaded (1:20 ratio)

Experiments

Related Topics: References

Project Status Status: Overview

Organization Production Logics: Overview Parametric Matrix: Overview Performative Relationships: Overview

9 20 36 66

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Project Overview

Introduction

Hypothesis

Production: Overview

Primary Logics

Skin-Fabrication Tactics Pre-Fabrication Tactics

Surface Topology

Equipment Parameters

Surface Geometry

Unfolding

Surface Skin

Nesting Scope

Organization

Secondary Logics Digital Fabrication

Identity Logistics Honeycomb Algorithm

Skin Fenestration

Specification

Layer Quantity

MEL Script

Layer Topology

Growth Algorithms Strategies: Overview

Strategies Biomimetic Research

Digital Generation Parametric Matrix

Geometric Parameters Abstraction + Translation Material Parameters Parametric Generation Performance: Overview

Structure

Tactics

Multi-Parametric Design

Surface Parameters

Layer Scale

Grid Parameters

Layer Binding

Cell Parameters

Grid Geometry

Thickness Weight

Grid Uniformity

Elasticity

Cell Connectivity

Transparency

Cell Edge Orienation

Surface Geometry

Production Logics

Thermal Conductivity

Non-Mechanical Performance

Cell Fill

Grid Geometry

000_Base

Grid Uniformity

Resources

Performative Relationships

Resources: Overview

Cellular Solids: Form-Finding Honeycombs

Architecture Experiments

003_Layer Scale 004_Deformed Grid

Honeycombs: Overview Honeycomb Models

References Nature

Project Status Status: Overview

Serpentine Pavilion 2002

008_Closed

Panelite

009_Sphere 010_Polar

Simmons Hall

011_Orientation

Honeycomb Sandwich Panels

012_Skin 1

CNC Cutting Radiolaria

013_Skin 2 014_Prototype Model

Honeycombs + Hives

015_Syn/Anticlastic

Kelvin and Plateau

016_Curvature Depth

Voronoi Algorithms UV Coodinate System

Math + Science

006_Parallel 007_Self-Similar

Prada LA

Production Logics: Overview The research into the production logics of the honeycomb system is divided into 4 categories. The first is the Primary Production Logic, which sets out the main techniques used in the actual physical manufacturing of the honeycomb. However there are other, less central production techniques and these are discussed in the next category, Secondary Production Logics. The third category deals with some of the particular issues related to Digital Fabrication such as specific equipment parameters and some of the other techniques used to prepare the digital files for CNC output. The last category groups together some of the concepts and techniques of the Digital Generation process. This involves a description of the algorithms used to construct the honeycomb including a short outline or specification of the custom-designed script and the transfer of information between applications.

005_Pocket

Bruges Pavilion 2002

Sheet Metal Bending Industry

002_Multiple Layers

Cell Fill

Mashrabiya Bibliography

001_Cell Depth

Cell Depth

Honeycomb Prototype Parametric Matrix

Cell Depth

Layer Binding

Cell Edge Orientation Tactics: Overview

Grid Denisty

Max Bending Radius

Grid Density Light and Vision

Layer Connectivity

Layer Parameters

Surface Skin

Emergent Performance

Skin Triangulation

Production Logics:

017_Cell Curvature

Related Topics: Primary Production Logic Digital Generation: Honeycomb Algorithm

22 30

018_Skin 3

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Production Logics: Production Logics: Overview This diagram illustrates the basic production process, from the initial digital starting point to the final physical honeycomb. Each of these operations will be detailed in the following sections, however below is a quick description of each of these processes.

1

2

3

4

5

6

Digital Generation: 1. Given surface 2. Grid is mapped across surface 3. Growth algorithm connects grid points 4. Other points on surface are connected 5. Points indicating cell depth are connected 6. Polylines are lofted to create strips Digital Fabrication: 7. Strips are triangulated 8. Single strip is selected 9. And unfolded and labelled 10. And then nested onto a given sheet size

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8

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Physical Fabrication: 11. Strips are laser-cut and organized 12. Single strip is selected 13. And folded 14. And then assembled into honeycomb

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Related Topics:

Production Logics: Overview Digital Generation: Honeycomb Algorithm Honeycomb Prototype: Pre-Fabrication

11

12

13

14

20 30 156

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Production Logic: Primary Production Logic The primary production logic developed in this research combines three industrially common, yet usually unrelated, techniques to form a hybrid production logic. Using honeycomb panels as the starting point, this hybrid logic uses large scale CNC laser or plasma cutting machines to produce non-uniform strips which are then folded with metal braking machines. Through the use of the CNC cutting process the assembled honeycomb is able to have a greatly increased range of properties.

Where the standard honeycomb is fixed to one global cell size and only a limited range of curvature, the hybrid production process enables each cell to be unique in size and shape which, in turn, enables a much larger range of curvature. This variation in the honeycomb has significant performance consequences.

The honeycomb can now respond to specific structural, thermal, and other forces not only globally, but locally across the surface.

Step 1: Cut -manual: knife, torch -cnc: laser, water-jet, plasma cutting machine

Step 2: Fold -manual: hand, press brake -cnc: press brake

Step 3: Assemble -mechanical connections -welded connections -glued connections

Related Topics: Production Logics: Overview Industry: Honeycomb Sandwich Panels Industry: CNC cutting Industry: Sheet Folding

20 170 171 172

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Heated Press Technique: Used for flat sheets

Matched Mould Technique: Used for mass-production of identical panels

Vacuum Bag Technique: Used for smaller-run and custom formed sheets

Production Logic: Secondary: Skin Fabrication Tactics Typically in the production of honeycomb panels, a rigid (non-elastic) skin is used to fix the honeycomb into a particular geometry. Most often this is a flat sheet however the panels have a limited amount of flexibility, depending on the cell type, enabling them be forced into other curvatures. There are different techniques used to do this but all involve the use of a mould over which the honeycomb is compressed and a skin is applied.

Within the context of this research, new skin tactics were developed to address two issues.

First, as the honeycombs are assembled, they already have an inherent curvature due to their non-uniform strip geometry. So the question is not how to force the honeycomb into a new geometry but how to keep the honeycomb in the assembled geometry. The second issue results from the scalar differences and more customized nature of an architectural project compared to a typical honeycomb sandwich panel.

At the scale of a building, it would be difficult to produce and apply a skin that covered the entire honeycomb in one sheet. In addition the mould used would be a large structure in itself, creating a huge amount of waste material.

Related Topics: Industry: Honeycomb Sandwich Panels

Three Common Techniques for Skin Application in Traditional Honeycomb Panels

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Production Logic: Overlap cells between top and bottom skins in orange

Diagramatic section through cells

Top Layer

Secondary: Skin FabricationTactics (cont.)

With these issues in mind, there are two ways in which this project addresses the skin issue.

These two tactics have fundamentally different production techniques as well as performative results.

Essentially, both attempt to break down the scale of the skin into more manageable parts, however they differ in the time of application.

Due to the flexible nature of the honeycomb prior to the skin application, the time in which the skin is applied is central to the performance of the system.

In the first tactic, the skin is applied in parallel with the assembly of the honeycomb strips.

As each strip is applied, a folded rigid material is attached to either the top or bottom of the honeycomb. These strips can be pre-cut and keyed to the exact shape of each cell. The top and bottom skin strips can be offset from each other such that they tie the honeycomb structure together. This integration of skin geometry and strip geometry greatly increases the strength of the honeycomb system.

The second tactic also increases the strength

Bottom Layer

Related Topics: Exploded axon of model_018 showing skin strips spanning across honeycomb cells

Surface Parameters: Skin Surface Parameters: Triangulation Surface Parameters: Fenestration Model_018_Skin 3

42 43 45 141

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Production Logic: Secondary: Skin FabricationTactics (cont.) of the honeycomb but it does so after the honeycomb has been assembled. Due to the flexibility of the cells before the application of a skin, the structure will have shifted according to the load paths travelling through it. Applying a skin at this point allows the structure to first find the natural displacement of its cells and then it fixes them in place. Each cell face (both inside and outside) is fitted with a shrink wrap skin, similar to the films used in the packaging industry. The air is then evacuated from the interior of the cell, creating a vacuum within.

This negative pressure and the skin would keep the cell from shifting in shape. The details of how this can be done at a large scale are not yet developed.

There are advantages to both tactics. The parallel assembly tactics can be done off-site under more controlled conditions. It is also more efficient since the skin can be fabricated at the same time as the strips. In addition, the skin can be fabricated in such a way as to add a pre-stressed property to the overall honeycomb. On the other hand, the post-assembly tactic allows the honeycomb to adjust to loading forces that may have been missed in structural simulations.

Related Topics: Structure: Surface Skin

69

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Production Logic: Digital Fabrication: Equipment Parameters It is often easy when using digital fabrication technologies to forget that they are simply tools and not some sort of magical device which produces the work for you. As tools, they have their own set of parameters and limits just like any other tool. By investing time in understanding the craft of these machines, the designer greatly adds to his ability to produce work that not only avoids potential conflicts, but lives up to the craft tradition within architecture.

Within this project, the parameters of the laser-cutting machine were explored to provide a new way to quickly fold paper strips. Normally, if a fold is needed in paper and the cutting is done by hand, one can simply score the material on the side opposite the direction of the fold. By varying the speed and power settings of the laser cuttter, scores are easy to produce. However, in this project scores were needed on both sides to allow the strips to be folded in both directions. By manipulating the third variable, frequency, in laser cutting machines it was possible to produce a perforated line. Frequency is the number of pulses of light that are beamed onto the material per inch.

By varying the number of pulses per inch (ppi) a line that was drawn as a solid line can be cut as a perforated one.

Related Topics: Industry: CNC cutting

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Production Logic: Digital Fabrication: Unfolding The folding of sheet-metal and paper is an industrially common process to form all types of parts, from cardboard boxes to metal brackets. In recent years, the increase in digitally designed parts has required the use of unfolding algorithms to flatten a 3D digital object into 2D lines, ready to be printed as a template or sent directly to a braking machine.

This reverse origami-like process of unfolding geometry has been integrated into very few software packages. Often it is seen only in specialized software meant for the sheet-metal industry. However, the 3D modeller, formZ, does have the abilty to unfold geometry and it was used to unfold all the honeycombs in this project.

These diagrams illustrate the power of the unfold tool to produce highly complex 3D geometry into easily cut 2D patterns. The upper diagrams show two variations on the unfolding of a geodesic sphere. Depending on which segment on the sphere is chosen as the starting point for the algorithm, different fold patterns emerge whcih could be strategically used. The diagram at upper right shows the addition of connector tabs.

The bottom diagram shows the unfolding of one strip from the honeycomb model_008.

Related Topics: Industry: Sheet Folding

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Production Logic: Digital Fabrication: Nesting The nesting of components onto sheets is an important part of the digital fabrication process. Nesting is essentially a way to save time, labor, and money during the cutting process.

The more parts that can be put onto one sheet the few number of total sheets, which means less loading and loading of the machine, and less cost in material expenditure. For example, this sheet is, by area, 64% efficient. That is, 36% of the material of the sheet is waste. A highly complex mathematical procedure, nesting algorithms are most often outside of offthe-shelf CAD software packages. However, many plug-ins and other specialist softwarte packages exist to handle this important function. Unfortunetly, these applications are also very expensive. Due to this, all the nesting in this project was done manually, without the help of any software. This dramatically increased the amount of labor needed to complete the preperation of files as well as inefficiency of the cutting process. Hopefully as CNC cutting becomes more standard within the architectural martket, more applications will support nesting algorithms.

Related Topics: Honeycomb Prototype: Fabrication Sheet Honeycomb Prototype: Nested Sheet

154 155

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Production Logic: Digital Fabrication: Identity Logistics One of the more difficult tasks when dealing with systems with a high number of unique components is identifying and tracking each part. Without a rigorous and logical system of notation it is nearly impossible to tell one component from another and even harder to make sure the right component is where it is needed at a specific time.

Front Layer

The system developed for this project is as simple as possible. First each strip is marked with a “F” or “B” indicating its position on either the Front or Back layer. Next, a two digit code gives the module name on that layer. The first digit refers to the column letter and second digit is the row number. Finally, the last two digits indicate the strip number on that module.

4 3 C

2 1

Row C, Column 1

B

C

A

B

A

4

3

2 1

E2

Folded Strip #05 E1

1

4

3

2

6

5

7

8

10

9

11

13

12

14

15

16

17

18

19

20

22

23

24

Related Topics:

Unfolded Strip #05 1 2

Honeycomb Prototype: Modules Honeycomb Prototype: Fabrication Sheet Honeycomb Prototype: Nested Sheet 8

153 154 155

23

24

21

22

20

17

16

18 19

14

15

13

11 12

10

9

7

5 6

4

3 E2

E1

F - C1 - 05

21

In addition, systems for how the fabricator should position the unfolded strip in respect to his/her body were developed to facilitate the correct folding of each strip. If a strip was folded incorrectly it could lead to a host of new problems. By standardizing the manufacturing logic in relation to the individual’s body, it was easy for the fabricator to know which direction to fold the strips. For example, one of the rules was to always have the end with the identity label on your right hand side. Then rotate the strip into the position shown in the Folded Strip diagram. Then begin folding accordingly.

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Production Logic: Digital Generation: Honeycomb Algorithm The digital generation process is comprised by the following 6 basic steps. Within each of these steps many variations exist.

1. A surface is created according to the desired global geomtry of the honeycomb.

2. Points are mapped across this surface defining the eventual verticies of the honeycomb strips.

3. An algorithm is used to connect points into a polyline based on the established production logic of folded strips.

1

2

3

4. This algorithm is looped across all other points on the surface to form the wireframe honeycomb mesh.

5. This process is repeated for corresponding points above the surface.

6. The honeycomb strips are created between the lower and upper polylines.

The two most important things in this process are the inputs for the algorithm: the given surface which determines the global performance of the honeycomb and the grid geometry that determines the cell density.

4

5

6

Related Topics:

Digital Generation: Basic MEL Script Digital Generation: Growth Algorithms Cell Parameters: Connectivity Project Status: Overview

32 35 56 181

30

Production Logic:

Input: source surface

Based on these inputs find the U and V coordinates for each grid interval on the surface and save these values in an array

Draw polylines connecting the appropriate grid points based on the established drawing procedures

If target surface exists and has not been mapped yet

Else, finished

DVDH Procedure Input: # of grid divisions in the U and V axes

HDVD Procedure

Attachment Procedure

Procedure for finding the intersection points between the source surface’s normals and the target surface

Digital Generation: Script Specification This diagram provides a basic explaination of how the custom-design script in Maya works.

The script was designed to be modular so that as new functionality was needed, new script modules could be added. For example, at first the script only generated the honeycomb mesh on a source surface but with the addition of a “if, else” conditional statement it is able to adapt to the existance of a target surface, generating the top and bottom chords of each honeycomb strip in one run. The Drawing Procedures (DVDH) are the ways in which the grid points are connected to each other. Other drawing procedures could be added to vary the style of honeycomb mesh. For detailed decriptions of each section of this specification see the following MEL code.

VDHD Procedure DHDV Procedure

Input: target surface

Related Topics: Digital Generation: Basic MEL Script

32

31

Production Logic: //This script will generate a honeycomb surface based on a source surface’s UV //points and a target surface’s geometry.

drawVDHD (($u+2), $v, $uVal, $vVal, $uDiv, $vDiv, $surfSrc, $surfTrgt); drawHDVD (($u+2), $v, $uVal, $vVal, $uDiv, $vDiv, $surfSrc, $surfTrgt);

//Code by Andrew Kudless. This code is open source and free to use, copy, and modify. } //Enter the source and target surface’s names //Procedure for drawings DVDH-type curves string $surfSrc = “surface1”; string $surfTrgt = “surface2”;

proc drawDVDH (int $u, int $v, float $uVal[], float $vVal[], int $uDiv, int $vDiv, string $surfSrc, string $surfTrgt){

//Create the array that holds the U and V values for the given divisions.

float $coordStrt[] = pointOnTarget ($uVal[$u], $vVal[$v], $surfSrc, $surfTrgt); if (($u+1) <= $uDiv && ($v+1) <= $vDiv){ float $coordEnd[] = pointOnTarget ($uVal[$u+1], $vVal[$v+1], $surfSrc, $surfTrgt); string $curveStrt = drawSegment1 ($coordStrt, $coordEnd); print (“DVDH: “ + $curveStrt + “ is being drawn.” + “\n”); $coordStrt = $coordEnd;

float $uDiv = 20; float $vDiv = 10; float $maxDiv = max($uDiv, $vDiv); float $uVal[], $vVal[];

while ($u <= ($uDiv) | $v <= ($vDiv)){ if (($v+2) > $vDiv)break; $coordEnd = pointOnTarget ($uVal[$u+1], $vVal[$v+2], $surfSrc, $surfTrgt); $curveStrt = drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

for ($i = 0; $i <= $uDiv; $i++){ $uVal[$i] = 1/$uDiv * $i; print (“The value of U at “ + $i + “ is: “ + $uVal[$i] + “\n”);

if (($u+2) > ($uDiv) | ($v+3) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+2], $vVal[$v+3], $surfSrc, $surfTrgt); $curveStrt = drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

} for ($i = 0; $i <= $vDiv; $i++){ $vVal[$i] = 1/$vDiv * $i; print (“The value of V at “ + $i + “ is: “ + $vVal[$i] + “\n”); }

if (($u+3) > $uDiv)break; $coordEnd = pointOnTarget ($uVal[$u+3], $vVal[$v+3], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

print(“U array has a size of “ + size($uVal) + “.\n” ); print(“V array has a size of “ + size($vVal) + “.\n” );

if (($u+4) > ($uDiv) | ($v+4) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+4], $vVal[$v+4], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

//Begin for loop that draws curves for first half of surface for ($i = 0; $i <= $maxDiv; $i = $i+3){ int $u = 0; int $v = $i; drawDVDH ($u, $i, $uVal, $vVal, $uDiv, $vDiv, $surfSrc, $surfTrgt); if (($v+2) > $vDiv)break; drawHDVD ($u, ($v+2), $uVal, $vVal, $uDiv, $vDiv, $surfSrc, $surfTrgt);

Digital Generation: Basic MEL Script The following sections show the actual code that was used in Maya to generate the honeycomb mesh. The code was written in the MEL scripting language. It is included here as open source code, free to use and modify. For a digital copy of the code, go to the project website.

$u = $u+3; $v = $v+3; } } } //Procedure for drawings HDVD-type curves

drawVDHD ($u, ($v+2), $uVal, $vVal, $uDiv, $vDiv, $surfSrc, $surfTrgt); proc drawHDVD (int $u, int $v, float $uVal[], float $vVal[], int $uDiv, int $vDiv, string $surfSrc, string $surfTrgt){ } //Begin for loop that draws curves for second half of surface for ($i = 0; $i <= $maxDiv; $i = $i+3){ int $u = $i; int $v = 0; drawDHDV ($u, $v, $uVal, $vVal, $uDiv, $vDiv, $surfSrc, $surfTrgt); if (($u+2) > $uDiv)break;

float $coordStrt[] = pointOnTarget ($uVal[$u], $vVal[$v], $surfSrc, $surfTrgt); if (($u+1) <= $uDiv){ float $coordEnd[] = pointOnTarget ($uVal[$u+1], $vVal[$v], $surfSrc, $surfTrgt); string $curveStrt = drawSegment1 ($coordStrt, $coordEnd); print (“HDVD: “ + $curveStrt + “ is being drawn.” + “\n”); $coordStrt = $coordEnd; while ($u <= ($uDiv) | $v <= ($vDiv)){ if (($u+2) > ($uDiv) | ($v+1) > ($vDiv))break;

Related Topics: Digital Generation: Script Specification

31

32

Production Logic: Digital Generation: Basic MEL Script (cont.) $coordEnd = pointOnTarget ($uVal[$u+2], $vVal[$v+1], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd; if (($v+2) > $vDiv)break; $coordEnd = pointOnTarget ($uVal[$u+2], $vVal[$v+2], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

$coordStrt = $coordEnd; $u = $u+3; $v = $v+3; } } } //Procedure for drawings DHDV-type curves

if (($u+3) > ($uDiv) | ($v+3) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+3], $vVal[$v+3], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

proc drawDHDV (int $u, int $v, float $uVal[], float $vVal[], int $uDiv, int $vDiv, string $surfSrc, string $surfTrgt){ float $coordStrt[] = pointOnTarget ($uVal[$u], $vVal[$v], $surfSrc, $surfTrgt); if (($u+1) <= $uDiv && ($v+1) <= $vDiv){ float $coordEnd[] = pointOnTarget ($uVal[$u+1], $vVal[$v+1], $surfSrc, $surfTrgt); string $curveStrt = drawSegment1 ($coordStrt, $coordEnd); print (“DHDV: “ + $curveStrt + “ is being drawn.” + “\n”); $coordStrt = $coordEnd;

if (($u+4) > $uDiv)break; $coordEnd = pointOnTarget ($uVal[$u+4], $vVal[$v+3], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd; $u = $u+3; $v = $v+3; }

while ($u <= ($uDiv) | $v <= ($vDiv)){ if (($u+2) > $uDiv)break; $coordEnd = pointOnTarget ($uVal[$u+2], $vVal[$v+1], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

} } //Procedure for drawings VDHD-type curves

if (($u+3) > ($uDiv) | ($v+2) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+3], $vVal[$v+2], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

proc drawVDHD (int $u, int $v, float $uVal[], float $vVal[], int $uDiv, int $vDiv, string $surfSrc, string $surfTrgt){ float $coordStrt[] = pointOnTarget ($uVal[$u], $vVal[$v], $surfSrc, $surfTrgt); if (($v+1) <= $vDiv){ float $coordEnd[] = pointOnTarget ($uVal[$u], $vVal[$v+1], $surfSrc, $surfTrgt); string $curveStrt = drawSegment1 ($coordStrt, $coordEnd); print (“VDHD: “ + $curveStrt + “ is being drawn.” + “\n”); $coordStrt = $coordEnd;

if (($v+3) > $uDiv)break; $coordEnd = pointOnTarget ($uVal[$u+3], $vVal[$v+3], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd; if (($u+4) > ($uDiv) | ($v+4) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+4], $vVal[$v+4], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd;

while ($u <= ($uDiv) | $v <= ($vDiv)){ if (($u+1) > ($uDiv) | ($v+2) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+1], $vVal[$v+2], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd; if (($u+2) > $uDiv)break; $coordEnd = pointOnTarget ($uVal[$u+2], $vVal[$v+2], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd; if (($u+3) > ($uDiv) | ($v+3) > ($vDiv))break; $coordEnd = pointOnTarget ($uVal[$u+3], $vVal[$v+3], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt); $coordStrt = $coordEnd; if (($v+4) > $vDiv)break; $coordEnd = pointOnTarget ($uVal[$u+3], $vVal[$v+4], $surfSrc, $surfTrgt); drawSegment($coordStrt, $coordEnd, $curveStrt);

$u = $u+3; $v = $v+3; } } } //Procedure for drawing single segment and attaching it to last segment proc string drawSegment1 (float $coordStrt[], float $coordEnd[]){ float $xSrt = $coordStrt[0]; float $ySrt = $coordStrt[1]; float $zSrt = $coordStrt[2]; float $xEnd = $coordEnd[0]; float $yEnd = $coordEnd[1]; float $zEnd = $coordEnd[2];

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Production Logic: Digital Generation: Basic MEL Script (cont.) string $curveStrt = `curve -d 1 -p $xSrt $ySrt $zSrt -p $xEnd $yEnd $zEnd`; return $curveStrt; } //Procedure for drawing single segment and attaching it to last segment proc string drawSegment (float $coordStrt[], float $coordEnd[], string $curveStrt){ float $xSrt = $coordStrt[0]; float $ySrt = $coordStrt[1]; float $zSrt = $coordStrt[2]; float $xEnd = $coordEnd[0]; float $yEnd = $coordEnd[1]; float $zEnd = $coordEnd[2]; string $curve1 = `curve -d 1 -p $xSrt $ySrt $zSrt -p $xEnd $yEnd $zEnd`; attachCurve $curveStrt $curve1; delete $curve1; return $curveStrt; }

string $cutCrv[]; string $crv[] = eval(“ls -type nurbsCurve `listRelatives -s`”); string $surf[] = eval(“ls -type nurbsSurface `listRelatives -s`”); if (size($crv)==1&&size($surf)==1){ string $iSurf[] = `extrude -ch 0 -et 0 -l 0.001 $crv[0]`; if (catch(`intersect -ch 0 -cos 0 -fs 1 -tol 0.001 $iSurf[0] $surf[0]`)){ delete `ls -sl`; print “\n\n”; error “there is no intersection point to return...”; } else { $cutCrv = `ls -sl`; if (size(`listRelatives -c $cutCrv`)>1) { delete $cutCrv; delete $iSurf; error “pretty likely there is a double intersection.”; } delete $iSurf; select -r $crv[0] $cutCrv[0]; intersectCrvPreset 1 0.001 6 1 0 0 2; string $intPnt[] = `ls -sl`; float $pnt[] = `getAttr ($intPnt[0] + “.wp”)`; delete `listRelatives -p $intPnt`; delete $cutCrv; return $pnt; } }else{error “ *** select just one nurbs curve and one nurbs surface ***”;} }

//Procedure for finding the point of intersection between the a source surface’s normal and another (target) surface. proc float[] pointOnTarget (float $uVal, float $vVal, string $surfSrc, string $surfTrgt){ float $coordSrc[] = `pointOnSurface -u $uVal -v $vVal -p $surfSrc`; float $xSrc = $coordSrc[0]; float $ySrc = $coordSrc[1]; float $zSrc = $coordSrc[2]; float $coordNrml[] = `pointOnSurface -u $uVal -v $vVal -n $surfSrc`; float $xNrml = $coordNrml[0]; float $yNrml = $coordNrml[1]; float $zNrml = $coordNrml[2]; string $lineNrml = `curve -d 1 -p 0 0 0 -p $xNrml $yNrml $zNrml`; move $xSrc $ySrc $zSrc $lineNrml; string $lineExtend[] = `extendCurve -s 2 -d 500 $lineNrml`; select $lineExtend $surfTrgt; float $coordIntsct[] = `crvSrfIntPnt`; delete $lineExtend; return $coordIntsct; } //Procedure for finding the intersection point between a surface and a curve, returned in world coord. global proc float[] crvSrfIntPnt() {

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Production Logics: Digital Generation: Growth Algorithms There are two ways in which the algorithm used to grow the honeycomb geometry across the surface can be modified. The first is through changing the grid from which the algorithm takes input points. At the moment, most of the research has used the inherent U and V isoparms of the surface as the origins of these girds. That is, the U and V lines that go from one side of the surface to the other are a factor of the actual geometric definition of the surface. It would be possible to not base the algorithm on these direct isoparms, but one a external data source. For example, the grid points could be based on FEM analysis of stress concentrations in the surface. These load paths would provide an alternative grid from which points could be drawn.

The second way in which the growth algorithm could be modified is through the use of different drawing procedures. For example, at the moment the script relies on a loop of drawing a diagonal, a vertical, another diagonal, and finally a horizontal segment. This could be modified in numerous ways to provide new honeycomb mesh patterns. One such modification can be seen on the “Cell Connectivity” parameter page.

Related Topics: Cell Parameters: Connectivity Project Status: Overview

56 181

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Project Overview

Introduction

Hypothesis

Production: Overview

Primary Logics

Skin-Fabrication Tactics Pre-Fabrication Tactics

Surface Topology

Equipment Parameters

Surface Geometry

Unfolding

Surface Skin

Nesting Scope

Organization

Secondary Logics Digital Fabrication

Identity Logistics Honeycomb Algorithm

Skin Fenestration

Specification

Layer Quantity

MEL Script

Layer Topology

Growth Algorithms Strategies: Overview

Strategies

Digital Generation Parametric Matrix Overview

Biomimetic Research Geometric Parameters Abstraction + Translation Material Parameters Parametric Generation Performance: Overview

Structure

Tactics

Multi-Parametric Design

Surface Parameters

Layer Scale

Grid Parameters

Layer Binding

Cell Parameters

Grid Geometry

Thickness Weight

Grid Uniformity

Elasticity

Cell Connectivity

Transparency

Cell Edge Orienation

Surface Geometry

Cell Fill

Grid Geometry

000_Base

Cell Edge Orientation Tactics: Overview

Production Logics

Thermal Conductivity

Non-Mechanical Performance

Resources

Performative Relationships

Resources: Overview

Cellular Solids: Form-Finding Honeycombs

Architecture Experiments

003_Layer Scale 004_Deformed Grid

Honeycombs: Overview Honeycomb Models

References Nature

Project Status Status: Overview

006_Parallel 007_Self-Similar

Serpentine Pavilion 2002

008_Closed

Prada LA Panelite

009_Sphere 010_Polar

Simmons Hall

011_Orientation

Honeycomb Sandwich Panels

012_Skin 1

CNC Cutting Radiolaria

013_Skin 2 014_Prototype Model

Honeycombs + Hives

015_Syn/Anticlastic

Kelvin and Plateau

016_Curvature Depth

Voronoi Algorithms UV Coodinate System

Math + Science

005_Pocket

Bruges Pavilion 2002

Sheet Metal Bending Industry

002_Multiple Layers

Cell Fill

Mashrabiya Bibliography

001_Cell Depth

Cell Depth

Honeycomb Prototype Parametric Matrix

Cell Depth

Layer Binding

Grid Uniformity

Parametric Matrix: Overview The material system is understood to be composed of two types of parameters: geometric and material. By uncovering the variables which each parameter, the ability of the system to provide a wide range of performative properties is increased. The following pages will describe in depth each of the parameters.

Grid Denisty

Bending Radius

Grid Density Light and Vision

Layer Connectivity

Layer Parameters

Surface Skin

Emergent Performance

Skin Triangulation

Parametric Matrix:

017_Cell Curvature 018_Skin 3

Related Topics: Organization Strategies: Parametric Generation Honeycomb Models: Overview

9 15 87

36

Parametric Matrix: Surface Parameters surface parameters

LayerlayerParameters parameters

Grid grid Parameters parameters

Cell cell Parameters parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

no triangulation

no openings

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

unfilled

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Geometric Parameters The geometric parameters are divided into 4 sub-categories: surface, layer, grid, and cell.

Some of the parameter values exist in binary relationships, that is, they either are present in the system or not. Other parameter values are more relatively related. For example, Grid Density is measure in relative terms, high and low where as layer quantity is measured in absolute, binary terms, either single layer or multiple layers.

Through the development of this matrix of parameters, it is possible to map out the different options for a design as well as relate two seemingly unrelated designs through their geometric definition rather than only a visual appearance.

Related Topics:

Geometric Parameters: Surface Geometric Parameters: Layers Geometric Parameters: Grid Geometric Parameters: Cell

38 47 51 55

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Parametric Matrix:

surface parameters surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

open 2 axes

complex curvature

open 2 sides

no triangulation

no openings

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

Geometric Parameters: Surface The first sub-category of geometric parameters arelayer those that deal with the surface layer of the system.

Here, the term surfacetopology is used in two quantity ways. The first meaning refers to the source surface on which the honeycomb is generated.

This surface provides the largest influence on the global character and performance of the honeycomb. The way in which it is geometrically and topologically defined determine many performative effects.

bifurcated

Related Topics:

zero gaussian curvature

la conn

single layer n/a meaning of the term surface is in The second reference to the skin of the honeycomb. Traditionally, a rigid skin is applied to honeycomb panels to produce a sandwich panel system where the upper and lower skins act as the chords of a beam and the honeycomb is the web of a beam. This explanation of the honeycomb’s depth will be developed further in the parameter dealing multipleand layerperformance sections laminated with cell depth as well as in the honeycomb panel reference section. However, it is important to understand how the skin can not only provide added rigidity but also thermal and light conductivity. How this skin is geometrically defined will be explored in the three sections on skin parameters.

Parametric Matrix: Overview Geometric Parameters complex

layer pa

p

36 37

centered

38

surface parameters

layerMatrix: parameters Parametric

surface topolopy

(0,1) surface geometry

(1,1) surface skin

skin triangulation

skin fenestration

layer quantity

open 2 axes

complex curvature

open 2 sides

no triangulation

no openings

single layer

(0,0)

(1,0)

(0,1), (1,1)

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

closed 2 sides

ray

slots

(0,0), (1,0) (0,1) - (1,1) closed 2 axes

anticlastic curvature

multiple layer

Surface Parameters: Topology The surface topology has a large effect on the layer layer layer way in which the rest of the parameters are topology mapped across connectivity the surface. Since many ofbinding the parameters are based on the surface’s local coordinate system (UV points), an open surface will act very differently from a closed surface. There is the additional issue of how the system deals with the module of the pattern on a surface where its minimum U or V coordinates are coincident with its maximum n/acoordinates. For example, n/a on a surface with n/a one or two axes closed, the honeycomb algorithm must take into account the total number of strips, verifying that this total is divisible by the pattern module (usually 3) otherwise the strips will misalign at the axis origin.

There is the additional issue with surfaces with both axes closed that, at the poles, all laminated full distributed strips meet at one point. This may be a mathematical possibility however it is not possible when dealing with physical materials. The cells become too small and the ratio of cell size to material thickness increases to the point that it is impossible to make the cells.

bifurcated

partial

linear

Related Topics:

complex

zero gaussian (0,0) - (1,0) curvature

centered

Math + Science: UV Coordinate System Model_008_Closed Model_009_Sphere

179 113 117

39

surface parameters surface topolopy

surface geometry

surface skin

layer parameters Parametric Matrix: skin triangulation

skin fenestration

layer quantity

layer topology

open 2 axes

complex curvature

open 2 sides

no triangulation

no openings

single layer

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

Surface Parameters: Geometry This parameter measures the type of surface layer layer curvature as a way to make some broad catconnectivity bindinggeometries. The egories of different surface reason curvature is used as a metric of geometry is that in the case of many structures, and particularly this honeycomb structure, curvature has wide reaching effects in both the performance and production of the system.

On way of measuring curvature is called Gaussian Curvature Analysis.n/a It is defined as the n/a n/a product of the two principle curvatures at any point on a surface. The principle curvatures for a surface point are the maximum and minimum normal curvatures at that point. The normal curvatures are defined by the radius of circle that best aligns with a section taken through the surface at that point. Surfaces that have positive Gaussian curvature everywhere full distributed matched are called synclastic, where as surfaces with negative Gaussian curvature are called anticlastic. Surfaces can have zero curvature, and thus be flat, or complex or mixed curvature, having some areas that are synclastic and others that are anticlastic. In the diagrams, the colour red represent relatively high positive Gaussian curvature, green is zero, and blue is negative. partial linear differential The performative aspects of these different curvatures will be explored in the “Surface Geometry Performance” section, however there are significant aspects of curvature that relate Related Topics:

Structure: Surface Geometry Model_015_Syn/Anticlastic complex

zero gaussian curvature

67 133

centered

Source Surface

Gaussian Curvature Analysis

Honeycomb

layer scale

40

Parametric Matrix: Surface Parameters: Geometry (cont.) to the production of the honeycomb strips.

The diagrams show the relationship between different curvatures and the unfolding strip pattern. The interesting thing here is that the strips of the anticlastic honeycomb are nearly flat compared to the synclastic honeycomb.

The significance of this is the amount of waste material that is generated when the strips of the synclastic honeycomb are nested.

One other interesting property that came out of this exploration is the relationship between surfaces that have a constant curvature (of any type) compared to their ability to stack after being cut. If a surface has constant Gaussian curvature the strips will perfectly align on top of each other, making them much easier to bundle and transport.

41

surface parameters

layer parameters

urface ometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

complex curvature

open 2 sides

no triangulation

no openings

single layer

n/a

n/a

synclastic curvature

closed 1 side

anticlastic curvature

closed 2 sides

ro gaussian curvature

Open Both Sides: transverse

holes

-Flexible -High Air Flow

ray

Closed 1 Side: multiple layer

laminated

-More Rigid -Weather Barrier

slots

bifurcated

Parametric Matrix: Surface Parameters: Skin Honeycomb panels gain much of their strength layer layer grid through the addition of rigid skins on both binding geometry sides, however therescale are other performative properties that skins add other than structural. Some of these are discussed in the Performance section, but in terms of geometry, the skin issue becomes quite difficult when the surface geometry is not flat. Various techniques have been developed that address these issues and are discussed in the “Skin n/aFabrication” section. n/a quadrilateral

Closed full Both Sides: distributed -Most Rigid -Low Thermal Conductivity

partial

linear

matched

equalateral

differential

polar

Related Topics:

centered

Model_000_Base Model_012_Skin 1 Model_018_Skin 3 Industry: Honeycomb Sandwich Panels

89 124 141 170

42

ce parameters

surface skin

layer parameters skin triangulation

skin fenestration

Parametric Matrix:

open 2 sides

no triangulation

no openings

layer topology

layer connectivity

layer binding

single layer

n/a

n/a

n/a

multiple layer

laminated

full

distributed

matched

equalateral

bifurcated

partial

linear

differential

polar

Actual Honeycomb cell below skin

closed 1 side

transverse

holes

Surface Parameters: Triangulation There are three primary ways in which the hexlayer grid grid agonal cells can be triangulated to allow flat scale geometry sheet material to cover them. These triangu-density lation methods can be mixed on each side to produce added structural and aesthetic value.

For example, in model_018, the “ray” triangulation pattern was used on both side however the point from which the line originated was flipped. This overlapping of triangulation patterns would potentially increase the stiffness of the honeycomb. Itquadrilateral also adds an interesting low n/a visual pattern to the structure.

layer quantity

Top hexagon triangulation

grid parameter

high

Bottom hexagon triangulation

closed 2 sides

ray

slots

Related Topics: Secondary: Skin Fabrication Tactics Model_018_Skin 3

23 141

centered

43

Parametric Matrix: Surface Parameters: Triangulation (cont.)

These drawings show the same basic honeycomb cell geometry with different variations on the tiling of the triangulation patterns for just one side.

1. Base 1 tile set, Ray 2. Base 1 tile set, Ray, alternate origin 3. Base 4 tile set, Ray, 4 different origins 4. Base 1 tile set, Transverse 5. Base 1 tile set, Centred 6. Base 2 tile set, Centred, 1 tile rotated

1

2

3

4

5

6

44

layer parameters

gridMatrix: parameters Parametric

skin angulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

o triangulation

no openings

single layer

n/a

n/a

n/a

n/a

transverse

holes

ray

slots

multiple layer

Surface Parameters: Fenestration (cont.)

There are numerous ways in which the skin grid grid grid can be fenestrated while preserving the strucgeometry density uniformity of the methods used tural integrity of it. Two in this research use the space within each triangular face to place openings. This allows each hexagon to remain triangulated and rigid while the areas of skin that are doing the least structural work are removed, providing light and air to pass through the honeycomb.

The first skin example uses quadrilateral low a “ray” triangula- uniform tion of the hexagons, reversed on the opposite side, to produce a lattice-like structure. The visual complexity of the skin plays against the typical honeycomb look of the structures. That is, by manipulating the skin in these ways, it is possible to enrich the structure’s aesthetic potential beyond just the hexagonal geometry of its inner core.

laminated

full

distributed

matched

equalateral

bifurcated

partial

linear

differential

polar

high

non-uniformity

Related Topics:

centered

Model_018_Skin 3 Material Parameters: Transparency Structure: Surface Skin

141 65 69

45

Parametric Matrix: Surface Parameters: Fenestration (cont.)

The second skin fenestration example is similar to the first. Instead of irregular holes, standard width slots are cut from the centre of each triangular face. The slots could be filled with glass or polycarbonate inserts. Due to the regularity of each slot width, the inserts could be mass-produced to that width and then cut to fit the custom length of each slot.

one skin strip

46

ters

layer parameters

Parametric Matrix:grid parameters Geometric Parameters: Layers The addition of multiple layers in a honeycomb grid grid system is one of the central differences bedensityhonthis research and the traditional tween geometry eycomb system. Typically, one would simple increase the cell depth if a stronger honeycomb was needed. By using two or more layers though, each with its own “grain” or anisotropic behaviour, a much better performance can be obtained. The parameters shown here develop this idea through the way the layers geometrically relate to each other. low quadrilateral

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

no triangulation

no openings

single layer

n/a

n/a

n/a

n/a

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

ray

slots

bifurcated

partial

linear

differential

polar

high

no

Related Topics: Parametric Matrix: Overview Geometric Parameters

un

36 37

centered

47

layer parameters

skin estration

layer quantity

layer topology

layer connectivity

grid parameters Parametric Matrix: layer binding

layer scale

grid geometry

Layer Parameters: Quantity, Topology, and Connectivity

grid grid cell The first three layer parameters are closely density uniformity connectivity related and quite simple to describe geometrically.

Layer Quantity This is a simple parameter relating to the number of layers stacked on top of each other.

o openings

holes

slots

single layer

multiple layer

n/a

n/a

n/a

n/a

quadrilateral

low

Layer Connectivity If there are multiple layers that are also lamidescribes the relation- diagonal 2 highnated, this parameter non-uniformity ship between the layers either fully laminated or partially laminated.

laminated

full

distributed

matched

equalateral

bifurcated

partial

linear

differential

polar

Top Right: Interior of Model_14 showing partial layer connectivity.

Bottom Right: Exterior views of Model_14.

uniform

Layer Topology If a structure has multiple layers, this parameter describes if the layers are actually separate layers laminated together or just bifurcated sections of the same base layer.

diagonal 1

horizontal

Related Topics:

Model_002_Multiple Layers Model_005_Pocket Model_014_Installation Test Honeycomb Prototype: Regional Geometry

95 105 128 vertical 147

48

arameters

ayer nectivity

grid parameters

cell parametersMatrix: Parametric

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

partial

linear

differential

polar

layer Parameters: Binding As multiple layers are introduced into the syscell tem, the way in which cell edge cell they are bound to each depthother becomesorientation a central concern. For exam- fill ple, two layers of veneer in a sheet of plywood only increase the stiffness of the plywood because they are glued together. This may seem obvious, however their are many different ways in which two layers of honeycomb can be connected. Based on the different cell connectivity patterns (see section on this parameter) different connections between the layers areunfilled constant normal to surface possible. The diagrams illustrate some of these “bindings”. In the last column the shaded red areas indicate planes that are shared by both the upper and lower honeycomb layers. It can be seen that some layering patterns produce a series of shared planes in a linear arrangement where as others produce a more distributed patterns. These differences in layer bindvariable normal to filled applied to a honeycomb ings can be tactically arbitrary to produce a desiredsurface structural behaviour.

horizontal

parallel to surface

Related Topics:

vertical

Cell Parameters: Connectivity Structure: Layer Binding Model_014_Installation Test constrained by vertices

56 70 128

49

grid parameters

cell parameters

ayer nding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

n/a

n/a

quadrilateral

low

uniform

diagonal 1

tributed

linear

matched

equalateral

high

non-uniformity

diagonal 2

Parametric Matrix:

layer Parameters: Scale Although most honeycombs with multiple laycell cell edge cell ers pursued a tactic where both layers had the depth orientation fill scale, there are precsame, or matched, layer edents in the natural world where differentially scaled layers provide added functionality to the material system. Some radiolaria have a larger primary cellular skeleton and a much finer cellular structure that spans across the large cells. The fine structure acts as a series of valves controlling the flow of material skin and the larger celconstant normalacross the organism’s unfilled to surface lular layer providing the structural support of 20 x 20 (D1) + 20 x 10 (D1) that skin. (Otto, IL 38, p.90)

variable

The large diagram at left shows three differentially scaled honeycomb meshes. The medium-scaled layer is 3 times smaller than the primary layer and the smallest layer is 9 times smaller. This drawings was produced within normal to filled seconds through the use of the custom-dearbitrary surfacesigned script.

20 x 10 (D1) + 20 x 20 (D2) differential

polar

horizontal

parallel to surface

Related Topics:

vertical

10 x 10 (D1) + 30 x 30 (D1) + 90 x 90 (D1)

Digital Generation: Honeycomb Algorithm Model_003_Layer Scale Nature: Radiolaria constrained by vertices Radiolaria with multiple layer scales

30 99 173

50

layer parameters

grid parameters

Parametric Matrix:

cell parameter

Geometric Parameters: Grid The Grid Parameter sub-category is mainly cell cell concerned with the geometric arrangement connectivity of points which define the vertices depth of the honeycomb mesh. After the actual surface geometry itself, the grid parameters have the largest influence on how the honeycomb is defined and performs. A major investigation of any further research will be on increasing the amount of control and coordination of the grid parameters.

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

bifurcated

partial

linear

differential

polar

c or

horizontal

Related Topics: Parametric Matrix: Overview Geometric Parameters

36 37

c

vertical

51

grid parameters

cell parameters

Parametric Matrix:

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

Quadrilateral Grid (90 degrees) matched

equalateral

differential

polar

high

non-uniformity

Heterogeneous Honeycomb diagonal 2

variable

horizontal

normal to arbitrary surface

Grid Parameters: Geometry The first grid parameter is related to the bacellsic geometrical organization of the points. fillThere are numerous grids that could be used to describe these arrangements, two of them developed in the diagrams. The importance of this parameter is that it essentially determines the shape of each cell, which in turn informs the performative properties of the overall honeycomb system. For example, when a 90o (quadrilateral) gird is used, a much more heterogeneous cell shape emerges compared unfilled to a 60o (equilateral) grid. This heterogeneous cell shape has important implications in the structural performance of the system, which is detailed in the “Grid Geometry Performance” section.

In this research the primary grid geometry used was the 90o grid for two reasons. First, filled as was stated above, interesting structural behaviours emerged from this gird. Second, the 90o grid aligned well with the geometric definition of NURBS surfaces which use a 2 axes coordinate system. See the reference section on the UV coordinate system for more details on this topic.

parallel to surface

Related Topics: Structure: Grid Geometry Math + Science: Voronoi Algorithms

Equilateral Grid (60 degrees)

vertical

Homogeneous Honeycomb

73 177

constrained by vertices

52

grid ometry

drilateral

ualateral

grid parameters grid density

low

cell parameters grid uniformity

uniform

cell connectivity

non-uniformity

polar

constant

diagonal 1

10 x 10 (D1) high

cell depth

Parametric Matrix:

cell edge orientation

normal to surface

30 x 30 (D1)

horizontal

unfilled

90 x 90 (D1) variable

diagonal 2

cell fill

normal to arbitrary surface

filled

Grid Parameters: Density The grid density parameter is a more relative description of the total number of grid intervals in a certain area rather than an absolute description. That is, there is no absolute meaning to “high density” or “low density” with out reference to a particular surface and scale.

Within this framework, it is then possible to define the density of the honeycomb by the number of grid intervals along the two primary axes, U and V. As the diagrams illustrate, these interval values can be equal (10 x10) or unequal (10 x 30). However, even if an equal grid interval is used for both axes, that does not mean that the density will be uniform on the surface nor that the cells will be a certain shape because the spacing of the intervals relies on the actual length of the axes. That is, the same grid interval will generate very different patterns on a square surface compared to a more elongated rectangular surface. The differences become even more pronounced when the length of the isoparms (lines parallel with the axes in the UV coordinate system) are unequal, as is the case with curved surfaces.

parallel to surface

Related Topics: 10 x 20 (D1)

10 x 30 (D1) vertical

10 x 90 (D1) constrained by vertices

Structure: Grid Density Model_004_Deformed Grid Honeycomb Prototype: Cell Denisty

74 102 152

53

parameters

cell parameters

Parametric Matrix:

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

low

uniform

diagonal 1

constant

normal to surface

unfilled

high

non-uniformity

diagonal 2

Non-uniform grid lines horizontal

variable

normal to arbitrary surface

parallel to surface

filled

Grid Parameters: Uniformity It is possible to use grids which have intervals which are not equal to generate non-uniform honeycomb meshes. For example, instead of each interval point being described as a multiple of a base unit (x, 2x, 3x, Nx....), the intervals can be described more complexly (x, x+1, x+3, x+5, x+N...). The possible ways in which the grid could be desrbied mathematically are endless. Therefore, it becomes important to select certain grid algorithms based on some external criteria such as performance requirements.

For example, in the honeycomb Prototype (see section), the grid organization was based on the desire to have a higher density of cells at the bottom than at the top in order to increase the weight, and thus the stability, at the base. Creating links between the mathematical description of the grid and particular performance criteria is one of the central areas of further development in this research.

Resultant honeycomb mesh

Related Topics:

vertical

constrained by vertices

Digital Generation: Growth Algorithms Structure: Grid Uniformity Model_004_Deformed Grid Honeycomb Prototype: Local Geometry

35 75 102 149

54

grid parameters

cell parameters

Parametric Matrix:

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

unfilled

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

polar

horizontal

Geometric Parameters: Cell The final sub-category of geometric parameters deals with various properties of the cell.

Although the cell is the basic unit of the honeycomb system, many of its parameters are actually set by higher level parameters such as the grid parameters.

parallel to surface

Related Topics: Parametric Matrix: Overview Geometric Parameters vertical

36 37

constrained by vertices

55

cell parameters

grid niformity

cell connectivity

cell depth

Parametric Matrix:

cell edge orientation

D

cell fill

H

D

2H 2V

V DVDH Connectivity

uniform

on-uniformity

diagonal 1

constant

D

D2VD2HDVDH Connectivity

normal -D to surface -V

D H V D

D

unfilled

-D -H

D

2H 2V

D

D H V D

D

3H

3V

diagonal 2

-D-V-D-H Connectivity

variable

normal to arbitrary surface

filled

D3VD3HD2VD2HDVDH Connectivity

D

D

2H 2V

H H

horizontal

HDH-D Connectivity

D

Cell Parameters: Connectivity The cell connectivity parameter describes the way in which the cellular pattern is grown from individual segments. This growth algorithm is composed of a repeating series of what are called “grammars”, which describe the direction of movement from a grid definition point. The number of different grammars is limitless and this research concerned itself with only two, so there are many potentially interesting grammars yet to be explored. The two grammars used in this research are the DVDH grammar and it’s negative opposite, D-V-D-H. It is interesting to note that despite only using two simple grammars, a relatively large and complex honeycomb system has been developed. The inclusion of even more complex grammars, or potentially a search algorithm which would search through a list of grammars suitable for a particular requirement could greatly add to both the depth and functionality of the honeycomb system.

D D H V

-D

parallel to surface

DVDH2DVDHD2VD2H Connectivity

D H V D

2D

D V

vertical

V-DVD Connectivity

constrained by vertices

2H 2V

D

2D H D

-D V DV2DHD2VD2H Connectivity

Related Topics: Digital Generation: Honeycomb Algorithm

30

V D

56

cell nectivity

agonal 1

agonal 2

orizontal

vertical

cell parameters cell depth

Parametric Matrix:

cell edge orientation

cell fill

Section through Model_008

Source surface

constant

normal to surface

unfilled

Outside edge of cell determined by the curvature analysis of source surface

Lines normal to source curve determine the cell wall angles variable

normal to arbitrary surface

filled

Cell Parameters: Depth The cell depth parameter is one of the most important differences between the honeycomb system developed in this research and the traditional honeycomb. Although traditional honeycombs can have different depths, within any one panel, all of the cells will have the same depth unless machined down in some way.

The ability to vary the depth of the cell greatly increases the range of performances that the honeycomb can satisfy.

One of the areas in which the variable cell depth was explored was in the relationship between cell depth and the local surface curvature. In Model_008 the generating surface was sampled at each grid point for its curvature information and this was used to determine the cell depth at that point. The resultant effect was to create thicker, rib-like structures where the surface had the highest curvature and thin cells where the surface was close to being flat.

Relatively flat curvature yields thinner cells

parallel to surface

Relatively high curvature yields thicker cells

Related Topics:

constrained by vertices

Structure: Cell Depth Model_001_Cell Depth Honeycomb Prototype: Regional Geometry

77 92 148

57

cell parameters

Parametric Matrix:

cell depth

cell edge orientation

constant

normal to surface

variable

normal to arbitrary surface

cell fill

Axonometric - Normal to Surface unfilled

Plan - Normal to Surface

Elevation - Normal to Surface

The normal to surface option is the most commonly used relationship in this research. Geometrically, the cell edges are always normal to the source surface. If the target surface (the surface that defines the depth of the honeycomb) is offset uniformly from the source surface, then the cell edges will also be normal to the target surface.

filled

Axonometric - Normal to XY Plane

Cell Parameters: Edge Orientation There are three primary relationships between the cell edges and the generating surface(s): normal to the source surface, normal to an arbitrary surface, or constrained by both the source surface and target surface. A fourth relationship, parallel to surface, is considered a distant cousin of these three primary relationships because it is so different in both its geometrical description and the process of digital generation.

Plan - Normal to XY Plane

parallel to surface

In contrast, if the honeycomb mesh is extruded normal to a surface or plane that is not the Elevation - Normal to XY Plane source surface, it is described as “normal to arbitrary surface”. As can be seen most clearly in the plan view, the cell edges are parallel to each other. The different behaviours of these two cell edge orientations (normal or parallel) is detailed in the “Cell Edge Orientation Performance” section in this document.

The “constrained by vertices” relationship was not developed in this research.

Related Topics:

constrained by vertices

Axonometric - Constrained by Vertices

Plan - Constrained by Vertices

Elevation - Constrained by Vertices

Structure: Cell Edge Orientation Model_011_Orientation Model_017_Cell Curvature Honeycomb Prototype: Regional Geometry

76 122 139 147

58

Parametric Matrix:

ell edge entation

cell fill

normal o surface

unfilled

normal to arbitrary surface

parallel to surface

nstrained by vertices

Rigid Plate Fill

filled

Cell Parameters: Fill The Cell Fill parameter is fairly self-explanatory, however the pattern and quantity of filled cells as well as the resultant behaviour of the cells is less clearly understood. Due to the non-triangulated geometry of the hexagonal geometry, the cells are quite flexible without some sort of interior “fill” or the application of an exterior skin. The nature of this fill element can come in very different forms, from a simple rigid plug or plate, or a more complex pneumatic chamber. The performance properties of these different options are detailed in the “Cell Fill Performance” section as well as under the “Thermal Performance” section.

As the diagram shows, it is possible, and po+ + + tentially more efficient, to fill only certain cells + + instead of every cell. These filled cells provide + + of rigidity that could be strategically + + + areas linked with particular structural forces. The + + + + MIT Simmons Hall (see reference section) is precedent of how this cell fill param+ + + + aetergood was used on a large scale to stabilize the Positive Pressure Fill

- - - - - - - - -- Negative Pressure Fill

structure while still allowing light, air, and view through the cellular facade.

Related Topics:

Surface Parameters: Skin Structure: Cell Fill Thermal Conductivity Architecture: Simmons Hall, MIT

42 78 80 169

59

Parametric Matrix: material parameters material thickness

material weight

material bending radius

material elasticity

material transparency

insignificant

insignificant

low range

low range

low

significant

significant

high range

high range

high

Parametric Matrix: Material Parameters Although many of the honeycomb parameters can be reduced into simple geometric relationships, there exists a higher level of parametric understanding informed by the actual material characteristics of the honeycomb. For example, although a certain curvature may be possible geometrically, when it is put in context of the actual material to be used the material’s elasticity may limit the curvature to a much higher degree. The list of material parameters shown here is a preliminary list of some of the factors that have effected the research thus far. It is clear that further research must enlarge the understanding of how particular material properties relate to the geometric and performative definitions of the system.

Related Topics: Parametric Matrix: Overview Project Status: Overview

36 181

60

material parameters

Parametric Matrix:

material thickness

material weight

material bending radius

material elasticity

material transparency

insignificant

insignificant

low range

low range

low

significant

significant

high range

high range

high

The second issue relates to the difference in thickness between cell walls that are attached to an adjacent strip and those that are unattached. The geometric result is that each hexagonal cell has 4 single thickness walls and 2 double thickness walls. The structural implication of this is that the honeycomb will be stronger in the direction parallel with the double thickness cell walls.

x x

Material Parameters: Thickness There are two primary issues relating to the material thickness. First, at certain dimensions, the material thickness can be considered insignificant to the global calculations of the structure. However, the majority of the time this is not the case for even a small thickness when multiplied dozens or hundreds of times for each cell in a honeycomb will result in significant amounts of material that needs to be accounted for in the geometric definition of the honeycomb. The difficulty in writing a script that can accommodate this parameter prevented material thickness to be calculated into any of the models developed in this research. The result of treating a significant geometric and material factor as negligible can be seen in the diagram. If the honeycomb has a fixed boundary that it must fit into, the extra material provided by the cell wall thickness must go somewhere, often deforming regions of cells beyond what could be called their normal range of movement.

2x

Related Topics:

2x x x

Digital Generation: Script Specification Structure: Grid Geometry Honeycomb Prototype: Final Exhibition

31 73 158

61

material parameters

Parametric Matrix:

aterial ckness

material weight

material bending radius

material elasticity

material transparency

ignificant

insignificant

low range

low range

low

gnificant

significant

high range

high range

high

Material Parameters: Weight The material weight is another parameter that has a direct relationship with the scale of the honeycomb. In the early experimental models the weight of the paper used to construct the honeycomb was insignificant, but when the much larger prototype was made the effect of the material weight was obvious. In the photo and diagram it can bee seen the digital, unweighted geometry in comparison to the actual built honeycomb prototype. The cells are much more elongated in the physical honeycomb due to the great loading stress at the base of the structure. D’Arcy Thompson, in his book “On Growth and Form” talks about the importance of scale to a physical system, “In physical science the scale of absolute magnitude becomes a very real and important thing; and a new and deeper interest arises out of the changing ratio of dimensions when we come to consider the inevitable changes of physical relations with which it is bound up. The effect of scale depends not on a thing in itself, but in relation to its whole environment or milieu; it is in conformity with the thing’s ‘place in Nature’, its field of action and reaction in the Universe” (Thompson, p.17) Thus, when this system is scale up even further to building size proportions, this material parameter will become even more of a concern, ultimately setting a limit for its maximum size.

Related Topics:

Project Status: Overview

181

62

aterial weight

ignificant

gnificant

material parameters material bending radius

Parametric Matrix: material elasticity

material Neutral Line (NL) transparency

Neutral Line (NL)

A A

low range

low range

low

R

high range

high range

Material Parameters: Bending Radius When bending or folding materials that have a significant material thickness, the bending radius must be entered into the calculations.

During the bending process the material on the inside of the fold is compressed where as the material on the outside is stretched. In order to keep an accurate measurement of the length of the member, this bending deformations must be accounted for and planned into the cutting of the flat honeycomb strip. This true length exists somewhere inside of the material thickness and is called the Neutral Line.

The location of the neutral line depends on the hardness of the material and the sharpness of the bending radius.

a

Although these calculations were not performed for the models in this research, larger scale honeycombs would require a more precise understanding of how the material’s T bending radius contributes to the total material length.

high T C

B

B Sharp Right Angle Bend

Radius Bend

NL location is 0.2 x T in from the inside fold edge for ductile, non-ferrous metals, 0.3 for harder metals

NL location is 0.5 x T in from the edges

NL length = (A + B) - ((2 x 0.8) x T)

NL length = (A + B) + ((p2(R+(T/2))/(360/a))

Related Topics:

Industry: Sheet Folding

172

63

parameters

Parametric Matrix:

aterial ending adius

material elasticity

material transparency

low range

low range

low

high range

high range

high

Material Parameters: Elasticity When the honeycomb is mapped across a curved surface nearly all of the faces on each strip will have a small amount of anticlastic curvature. If the material used to construct the honeycomb has a small amount of elasticity then this non-planer geometry is not a problem. In fact, by pushing and pulling the material into place, a certain amount of pre-stress is introduced into the honeycomb. However, there is a limit to the amount the material strips can be stretched beyond which they fail. The task is to find the right negotiation between the material’s elasticity and the desired geometry.

Form the material side, if the elasticity is too large, the pre-stress will be lost. If the elasticity is too little, the desired form may be impossible.

Above: (top) Rendering of Model_014, (bottom) Gaussian curvature analysis of Model_014 with essentially flat areas shown in red and non-planar faces in green or blue. In these honeycombs, all faces will either be flat or anticlastic since each face is defined by two straight segments.

It is interesting to briefly look at some specific examples. The Gaussian curvature analysis of model_15 shows that the faces that are the most non-planer are those at the centre of the saddle shape. In addition, there is a large difference in face curvature between the faces running parallel to the cell orientation compared to those which are perpendicular. It is unclear of the significance of this, but it is interesting to note the spiral pattern that emerges (seen best in the perspective in red).

Left: Perspective and plan of Model_015 with a Gaussian curvature analysis color coding the same as described above.

Related Topics:

Model_015_Syn/Anticlastic Model_016_Cell Depth/Curvature Model_017_Cell Curvature

133 136 139

64

Parametric Matrix:

material elasticity

material transparency

low range

low

high range

high

Material Parameters: Transparency The material that is used as a skin has the possibility adding more than just structural stiffness to the honeycomb system. Although the folded skin concept was originally developed with a sheet material like thin aluminium in mind, the tactic would work just as well with a non-elastic plastic or fibreglass sheet material. These fabric-like sheets could just as easily be laser cut and attached to the cell walls.

Related Topics: Secondary: Skin Fabrication Tactics

23

65

Project Overview

Introduction

Hypothesis

Production: Overview

Primary Logics

Pre-Fabrication Tactics

Surface Topology

Equipment Parameters

Surface Geometry

Unfolding

Surface Skin

Nesting Scope

Organization

Secondary Logics Digital Fabrication

Identity Logistics

Strategies

Digital Generation Parametric Matrix

Biomimetic Research Geometric Parameters Abstraction + Translation Material Parameters Parametric Generation Performance: Overview

Skin Fenestration

Specification

Layer Quantity

MEL Script

Layer Topology

Structure

Tactics

Multi-Parametric Design

Surface Parameters

Layer Scale

Grid Parameters

Layer Binding

Cell Parameters

Grid Geometry

Thickness Weight

Grid Uniformity

Elasticity

Cell Connectivity

Transparency

Cell Edge Orienation

Surface Geometry

Cell Fill

Grid Geometry

000_Base

Cell Edge Orientation Tactics: Overview

Production Logics

Thermal Conductivity

Non-Mechanical Performance

Resources

Performative Relationships

Resources: Overview

Cellular Solids: Form-Finding Honeycombs

Architecture Experiments

003_Layer Scale 004_Deformed Grid

Honeycombs: Overview Honeycomb Models

References Nature

Project Status Status: Overview

006_Parallel 007_Self-Similar

Serpentine Pavilion 2002

008_Closed

Prada LA Panelite

009_Sphere 010_Polar

Simmons Hall

011_Orientation

Honeycomb Sandwich Panels

012_Skin 1

CNC Cutting Radiolaria

013_Skin 2 014_Prototype Model

Honeycombs + Hives

015_Syn/Anticlastic

Kelvin and Plateau

016_Curvature Depth

Voronoi Algorithms UV Coodinate System

Math + Science

005_Pocket

Bruges Pavilion 2002

Sheet Metal Bending Industry

002_Multiple Layers

Cell Fill

Mashrabiya Bibliography

001_Cell Depth

Cell Depth

Honeycomb Prototype Parametric Matrix

Cell Depth

Layer Binding

Grid Uniformity

Performative Relationships: Overview The final tactic within this research is the development of the performative relationships seen in the experimental models. The task of this section is to understand how particular geometric parameters lead to specific behaviours. The research has primarily focused on the structural relationships, however the honeycombs exhibit many other interesting performative properties which are briefly touched on here but will be researched further in future work.

Grid Denisty

Bending Radius

Grid Density Light and Vision

Layer Connectivity

Layer Parameters

Surface Skin

Emergent Performance

Skin Triangulation

Honeycomb Algorithm

Growth Algorithms Strategies: Overview

Performative Relationships:

Skin-Fabrication Tactics

Related Topics: Organization Strategies: Emergent Performance

9 16

017_Cell Curvature 018_Skin 3

66

Performative Relationships: Positive Gaussian Curvature (Synclastic Surface)

Negative Gaussian Curvature (Anticlastic Surface) Perspectives

Zero Gaussian Curvature (Flat Surface)

Curvature Analysis

Model

Curvature Analysis

Plans

Model

Front Elevation

Curvature Analysis

Side Elevation

Model

Structure: Surface Geometry The relationship between an individual cell’s geometry and the surface geometry it was generated from was studied in order to understand how local and global structural rigidity emerged in the honeycomb system. Three test were carried out using the digital and physical models created during the experiment Model_17. Starting with the most basic set-up, a cell based on the zero gaussian curvature surface was itself analysed for curvature changes. Since the generating surface was flat, so were all the cell’s faces. The effect of this is that the cell has no in-built pre-stress holding it in a particular geometry. Therefore, the cell can be compressed completely flat. On the other hand, both of the cells based on the positive and negative gaussian curvature generating surfaces have significant non-planer faces. In order to physically build these models, the faces needed to be twisted into alignment, pre-stressing the cell into a more fixed geometry. When load is applied on these cells they flex slightly but can not completely compress due to their geometric limits unlike the zero gaussian curvature cell.

Note: in the diagrams red indicates flat or zero gaussian curvature, blue represents high negative gaussian curvature and green low gaussian curvature. The cell faces never have positive gaussian curvature.

Related Topics: Surface Parameters: Geometry Model_017_Cell Curvature

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Performative Relationships: Structure: Surface Geometry (cont.)

To the left are tests showing the comparable rigidity of cells generated from three different surfaces: flat, synclastic, and anticlastic. Note how the cell in the top row is the only one that can completely compress due to the lack of any pre-stress in the cell faces.

Unloaded cell based on zero gaussian curvature generating surface

Loaded cell based on zero gaussian curvature generating surface

Unloaded cell based on positive gaussian curvature generating surface

Loaded cell based on positive gaussian curvature generating surface

Unloaded cell based on negative gaussian curvature generating surface

Loaded cell based on negative gaussian curvature generating surface

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Performative Relationships: Unloaded honeycomb without skin.

Loaded honeycomb without skin; cells are free to deform.

Structure: Surface Skin The addition of a rigid skin greatly increases the structural performance of a honeycomb.

Without a skin, each cell is able to deform when the honeycomb is put under pressure. When a skin is added, the cell’s shape becomes more fixed. This arrangement combines the lightness of a honeycomb core with the stiffness of a rigid skin. The combination acts very similarly to a truss or wide flange beam where material is concentrated away from the neutral axis, providing material where it can best resist bending forces.

Unloaded honeycomb with skin.

Loaded honeycomb without skin; cells are fixed and skin carries most of the axial loading and bending stresses.

Related Topics: Secondary: Skin Fabrication Tactics Surface Parameters: Skin Model_018_Skin 3

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Performative Relationships: Structure: Layer Binding A number of loading tests were carried out on Model_014 in order to understand the degree to which the binding of two layers affects the rigidity of the entire structure. The model was tested both as a single layer and as a double layer structure. The bottom row shows the single-layer, unloaded condition. The next row up shows the deformation in both the photograph and a diagram based on digitized data gather during the loading process. Under a load to weight ratio of 1:21, i.e. 21 times the self-weight of the single layer honeycomb, the upper edge of the honeycomb deflected an average of 22%. When the second layer was added to the model and a similar load to weight ratio was applied, the structure only deflected 2.7%.

Test 5 -Double-Sided -Load Ratio 1:20 -Vertical Deflection: 2.7%

Test 4 -Double-Sided -Load Ratio 1:10 -Vertical Deflection: 1%

Test 3 -Double-Sided -Unloaded

Test 2 -Single-Sided -Load Ratio 1:21 -Vertical Deflection: 20%

Test 1 -Single-Sided -Unloaded

Related Topics:

Diagrams based on digitized data taken during experiment

Elevation photos taken during loading tests

Layer Parameters: Binding Model_002_Multiple Layers Model_014_Installation Test Honeycomb Prototype: Modules

49 95 128 153

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Performative Relationships:

Plan

Plan

Elevation

Structure: Layer Binding (cont.)

A section of the model was digitized with a higher resolution to more clearly understand the movement of the structure under load.

The first set of diagrams show the single-layer deflection of the honeycomb versus the double layer deflection. In the single-sided test it is clear that the structure is slumping in the direction of the cell orientation. When a second layer is added the structure resists slumping in either direction and only moves down slightly.

Section

Elevation

Single-sided Load Test

Double-sided Load Test

Unloaded

Unloaded

Loaded (1:21 ratio)

Loaded (1:10 ratio)

Section

Loaded (1:20 ratio)

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Performative Relationships:

Plan

Plan

Elevation

Structure: Layer Binding (cont.)

The final diagrams show the paths of movement during the loading procedure. The horizontal movement shown in the double-sided structure is more a factor of the experimental set-up than an accurate description of the load vectors. The top of the model was free to move and thus horizontal movement has a higher margin of error than vertical movement.

Section

Elevation

Single-sided Load Test

Double-sided Load Test

Load Vectors (from 1:0 to 1:21 load ratio)

Load Vectors (from 1:0 to 1:10 load ratio)

Section

Load Vectors (from 1:10 to 1:20 load ratio)

72

Performative Relationships: Structure: Grid Geometry Based on the cell orientation, which is a product of the grid geometry, the honeycomb exhibits anisotropic behaviour. When force is applied perpendicular to the primary cell orientation, the honeycomb is quite flexible. However, when force is applied parallel to the cell orientation, the honeycomb moves very little.

This knowledge can be used to link information gained from structural analysis software with algorithms that construct the grid geometry on the surface to form a cell orientation that is local to the specific load paths.

force up force up no force

no force

force down force down

Honeycomb is flexible to bending forces perpendicular to the cell orientation

Related Topics: Honeycomb is resistant to bending forces parallel with the cell orientation

Surface Parameters: Geometry Grid Parameters: Geometry

40 52

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Performative Relationships: Linear Elasticty Region (Cell walls begin to bend)

Diagrams redrawn from “Cellular Solids: Structures and Properties”

by Lorna J. Gibson and Michael F. Ashby

Linear Elasticty Region (Axial Compression of cell walls)

Densification Region

Densification Region

Increasing relative cell density

Plateau Region (Elastic Buckling, Plastic Bending, or Brittle Fracture)

(Cell walls begin to touch)

Stress

Stress

(Cell walls begin to touch)

Increasing relative cell density

Plateau Region (Elastic or Plastic Buckling, or Brittle Crushing)

0

0.25

0.50

0.75

Strain Schematic Stress/Strain Diagram for a Honeycomb loaded in compression IN PLANE

1.0

0

0.25

0.50

0.75

1.0

Strain Schematic Stress/Strain Diagram for a Honeycomb loaded in compression OUT OF PLANE

Structure: Grid Density The grid density is the controlling parameter in the relative density of the honeycomb cells.

Relative density is the ratio of a cellular solid’s weight compared to an equal volume of the same material without cells. This ratio ranges from 0.001 (an ultra-low density foam) to 0.400 for some softwoods (Gibson and Ashby, 2). At this higher ratio it is usually more appropriate to describe the material as a solid with some holes in it rather than a solid that is completely composed of holes and the material connecting them. Gibson and Ashby give the general equation of ρ*/ρs for the relative density of a cellular solid where ρ* is the density of the cellular solid and ρs is the density of the solid material found in the cell walls. This concept is important because it enables one to quickly compare the amount of material being used in accomplishing certain tasks such as structural loading or thermal insulation.

In the schematic stress/strain diagrams, it can be seen that as relative density increases the honeycomb has a longer period of linear elasticity where the cell walls begin to bend but do not buckle, fracture, or crush. As more load is applied the stress/strain curve plateaus while the cell walls undergo elastic or plastic deformations or fractures. Finally, when the load is high the cell walls begin to touch each other, densifying to the point where it is no longer a “cellular” solid but simply a solid.

Related Topics: Grid Parameters: Density Model_004_Deformed Grid

53 102

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Performative Relationships: Structure: Grid Uniformity The structural performance of the honeycomb is directly related to its relative density.

As was detailed in another topic, relative density is determined by comparing the amount of material in a given area to the same amount of material is the structure was solid and not cellular. Because of this relationship, changing the grid uniformity essentially changes the relative density across the honeycomb, which in turn creates non-uniform performative effects. By strategically locating areas of high grid density where the largest load paths exist and decreasing the density in areas of little load, the grid automatically becomes non-uniform and responsive to the local forces acting on the structure.

Related Topics: Digital Generation: Growth Algorithms Grid Parameters: Uniformity Model_004_Deformed Grid Project Status: Overview

35 54 102 181

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Performative Relationships: Structure: Cell Edge Orientation As was detailed in the “Cell Edge Orientation” parameter section, the cell wall edges were determined through two methods in this research. The first methods is to project perpendicular lines out from the generating surface.

These lines, called normals, define the cell wall edges and can be seen in diagram 2. The other method used is to define an arbitrary plane or surface from which normals are projected out. Where these normal lines intersect the honeycomb surface planes, the cell edges are established (diagram 1). The performative difference between these two methods can be seen in diagram 3. Where the arbitrary plane methods produces a globally smooth transition from cell edge to cell edge, the other method produces cell edges that are more locally oriented. It is suspected that the more local orientation of each cell edge would contribute to a more rigid structure because the cells would be less able to collapse. The less parallel two cell edges are, the more pre-stress has been built into the cell providing more stiffness. See the “surface geometry” performance section for more details on this relationship.

1. Cell Wall Edges are perpendicular to a curved arbitrary plane between the two layers

2. Cell Wall Edges are perpendicular to the inside surfaces of the two layers

3. Composite drawing showing the difference between the two methods of defining the Cell Edge Orientation

Related Topics: Cell Parameters: Edge Orientation Model_011_Orientation Honeycomb Prototype: Regional Geometry

58 122 147

76

Performative Relationships:

Generating surface Mean Curvature analysis Perspective view

Structure: Cell Depth Similar to the depth of a beam or truss, cell depth has a large effect on the structural performance of honeycombs. In an experiment to test the relationship between cell depth and surface curvature it was found that higher curvature requires higher cell depth in order to resist the bending forces within the cell faces. That is, when the strips are pushed into position and glued, a certain amount of force is required to keep the strips in that position. If the cell depth is too shallow the pre-stress put into the cell shape dissipates out.

Generating surface Mean Curvature analysis Elevation view

Honeycomb A: High curvature: High Cell Depth Perspective view

Honeycomb A: High curvature: High Cell Depth Section view X=1

Honeycomb A: High curvature: High Cell Depth Elevation view

Honeycomb B: High curvature: Low Cell Depth Perspective view

Honeycomb B: High curvature: Low Cell Depth Section view X=1

Honeycomb B: High curvature: Low Cell Depth Elevation view

In the experiment a surface with variable curvature was mapped with two different honeycomb meshes. The first model had higher cell depth where higher mean curvature existed on the generating surface. The other model had higher cell depth where there was the least amount of mean surface curvature. When the Honeycomb A: High curvature: High Cell Depth desired height of the arch is measured against Elevation view of physical model the result height in the physical models it is clear that higher cell height resists the bendX=0.87 ing forces better and keeps the honeycomb in a form closer to the digital form.

Honeycomb B: High curvature: Low Cell Depth Elevation view of physical model X=0.77

Related Topics: Cell Parameters: Depth Model_001_Cell Depth Model_008_Closed Model_016_Cell Depth/Curvature

57 92 113 136

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Performative Relationships:

Triangulated shapes provide rigidity, however they have very brittle and quick failures.

Structure: Cell Fill Filling the honeycomb cells dramatically increases the stiffness of the system however it also negates one of the best features of the honeycomb: to adapt to loads through the deformation of cell shape. Potentially this negation could cause undesired failure in the structure.

Although done for a different reason, one solution for this problem is seen in Steven Holl’s Simmons Hall at MIT. In order to stiffen the exterior cellular structure of the building select cells were filled in. This strategy could be used in a honeycomb system such that only certain key cells would be filled to stiffen the structure, but the rest would be left open to adapt to the applied loads.

Unfilled hexagons are very ductile to applied forces. They will have a larger range of elastic deformation before failure.

Filled hexagons can handle more load but will fail similar to a triangulated shape, violently.

Related Topics: Cell Parameters: Fill Architecture: Simmons Hall, MIT

59 169

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Performative Relationships: Light and Vision Although this research has not focused on the performative relationships between light and vision and the geometric parameters, many interesting relationships exist and can be briefly described. Nearly all of the geometric parameters effect the way light passes through the honeycomb. The primary geometric parameters involved in light filtration are: Surface Geometry, Skin Fenestration, Grid Density, Grid Uniformity, Cell Edge Orientation, and Cell Depth. Through negotiating between these parameters and the desired lighting conditions and visual transparency, a wide range of honeycomb configurations is possible.

Related Topics:

Honeycomb Prototype: Visual Testing Project Status: Overview

151 181

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Performative Relationships: Thermal Conductivity One of the primary uses for cellular solids is as thermal insulators. This can be seen in foam coffee cups to foam insulation sheets for buildings. Their low thermal conductivity is a factor of their low relative density. The more air or other trapped gas within a material structure, the lower its thermal conductivity.

Although no tests were done in this research to empirically understand the thermal properties of the honeycomb system, some properties can be abstracted out from cellular solids in general. The primary factor in having a low thermal conductivity is trapping air. Thus, the most important geometric parameter is how the cell is covered or filled. In addition, the volume of trapped air is important, so the cell depth parameter would be a factor.

However, even without covering the cells the honeycomb system would significantly contribute to the thermal environment through the way it modulates light. By using passive solar techniques to inform the cell density, orientation, and depth, dramatic thermal effects could be accomplished.

Related Topics: Cell Parameters: Depth Cell Parameters: Fill Project Status: Overview

57 59 181

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Performative Relationships:

Aesthetic Requirements Financial Requirements

Programmatic Requirements

Site Requirements

Geometric + Material Parameters

Lighting Requirements

Manufacturing + Construction Requirements

Non-Mechanical Performance Although most of the performative relationships researched thus far are based on understanding the flow of mechanical forces through the honeycomb, many other flows, behaviours, and other performative properties that are non-mechanical effect the development of the honeycomb system. For example, each of the parameters has a direct relationship with the amount of material needed to build the honeycomb. Changing this amount through a small modification to a geometric parameter may have a large effect on the financial and manufacturing requirements of the structure. If cell density is increased in response to a desired light condition, this could vastly increase the amount of material needed, the amount of labour and time needed to manufacture it, and the amount of money needed to pay for all of it.

The goal is to understand how to work with systems which have complex interconnected relationships such as this. Although this research has focused on geometric and structural relationships, acknowledging that the system exists in a much larger network of forces is the first step towards creating a integrated design strategy.

Structural Requirements Thermal Requirements

Related Topics: Project Status: Overview

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Resources

Project Overview

Introduction

Hypothesis

Production: Overview

Primary Logics

Skin-Fabrication Tactics Pre-Fabrication Tactics

Surface Topology

Equipment Parameters

Surface Geometry

Unfolding

Surface Skin

Nesting Scope

Organization

Secondary Logics Digital Fabrication

Identity Logistics Honeycomb Algorithm

Skin Fenestration

Specification

Layer Quantity

MEL Script

Layer Topology

Growth Algorithms Strategies: Overview

Strategies

Digital Generation Parametric Matrix

Biomimetic Research Geometric Parameters Abstraction + Translation Material Parameters Parametric Generation Performance: Overview

Structure

Tactics

Multi-Parametric Design

Surface Parameters

Layer Scale

Grid Parameters

Layer Binding

Cell Parameters

Grid Geometry

Thickness Weight

Grid Uniformity

Elasticity

Cell Connectivity

Transparency

Cell Edge Orienation

Surface Geometry

Cell Fill

Grid Geometry

000_Base

Grid Uniformity

Production Logics

Thermal Conductivity

Non-Mechanical Performance

Resources

Performative Relationships

Resources: Overview

Cellular Solids: Form-Finding Honeycombs

003_Layer Scale 004_Deformed Grid

Honeycomb Models

Architecture Experiments

References Nature

Project Status Status: Overview

Serpentine Pavilion 2002

008_Closed

Prada LA Panelite

009_Sphere 010_Polar

Simmons Hall

011_Orientation

Honeycomb Sandwich Panels

012_Skin 1

CNC Cutting Radiolaria

013_Skin 2 014_Prototype Model

Honeycombs + Hives

015_Syn/Anticlastic

Kelvin and Plateau

016_Curvature Depth

Voronoi Algorithms UV Coodinate System

Math + Science

006_Parallel 007_Self-Similar

Sheet Metal Bending Industry

005_Pocket

Bruges Pavilion 2002

Mashrabiya Bibliography

002_Multiple Layers

Cell Fill

Honeycomb Prototype Parametric Matrix

001_Cell Depth

Cell Depth Honeycombs: Overview

Overview The resources are divided into two main areas based on resources that are internal or external to the research. Internal resources are those things that must be created in order for the research to progress. These are called Experiments. External resources are things which others have already done which have certain associated meanings with this research. These References are sometimes architectural precedents, although they can also be mathematical concepts or biological research.

Cell Depth

Layer Binding

Cell Edge Orientation Tactics: Overview

Grid Denisty

Bending Radius

Grid Density Light and Vision

Layer Connectivity

Layer Parameters

Surface Skin

Emergent Performance

Skin Triangulation

Resources:

017_Cell Curvature 018_Skin 3

Related Topics: Organization Experiments: Overview References: Overview

9 84 162

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Resources: Experiments: Overview The experiments are divided into three main sections which follow roughly a chronological order. Prior to the establishment of the dissertation topic on honeycomb structures, a series of experiments studying the self-organizing geometry of packed cellular bodies were done. Using form-finding methods, these models lead to an interest in cellular solids and honeycombs specifically.

A much larger series of experiments were then done exploring the various geometric parameters of honeycombs. Through and iterative process of digital and physically model building, testing, and analysis, a more holistic understanding of the integrated relationships between the parameters and performance was gained.

With this information the final experiment was undertaken in the form of a large prototype for the AA Project’s Review.

Related Topics:

Cellular Solid Models

Honeycomb Models

Honeycomb Prototype

Strategies: Parametric Generation Parametric Matrix: Overview Honeycomb Models: Overview Honeycomb Prototype: Overview

15 36 87 145

84

Experiments:

1. Cellular bodies (balloons) are tightly packed within a fixed boundary.

2. Plaster is poured into the interstitial spaces between the cells.

3. After the plaster is dry, the balloons are removed leaving a stable model which can be analyzed.

3

4

4

3

5

5 4

6 6

6

5 6

6

5 6

5 6 5

5. Diagram showing the connectivity between cells. A line is drawn between any two cells whose walls touched.

6 6 6

4

5 6 5 5

6 5

4

5 5

6 6 6 6

6

5 6

6 6

6

5 6

6 6

5 6

6

5 5

5

6

6 6

6

4

6 6

6

6

5

6

6

6

6

4

6

6

5

4. A schematic diagram of the cell positions.

6

4

5

5 6

6

6

4

5

6

5

5

Cellular Solids: Form-Finding In order to understand the morphogenetic qualities of Cellular Solids, a series of experiments were completed using balloons (filled with air or water) and plaster. Groups of balloons were allowed to self-organize within a confined space and then plaster was applied to fix the system in place. After the plaster was dry the balloons were removed and the resultant model of the space between the balloons was analysed for geometric patterns of development. Although these pattern emerged in all the models, the clearest example of self-organized structure emerged in one of the models that had only one layer of balloons. Under pressure from the boundary conditions, the balloons formed into a tight hexagonal pattern similar to a bee’s honeycomb. After further research into the geometry of packed bodies, it was discovered that this geometric property had be under scientific study for hundreds of years. The most significant work was done by Lord Kelvin and later Plateau with their work on the geometry of soap bubbles. Another interesting reference related to this experiment is Voronoi algorithms which can describe the geometric packing order of many diverse systems with the same set of rules.

4

5 6

4

6. When the number of connections of each cell is counted, the majority of cells form into hexagonal geometries.

Related Topics: Strategies: Biomimetic Research Nature: Bee’s Honeycombs Math + Science: Kelvin + Plateau

12 175 176

85

Experiments: Cellular Solids: Form-Finding 2 A second group of form-finding experiments was done with a non-rigid boundary and different numbers of cells. These experiments geometrically had more to do with the formation of foams rather than honeycombs. Balloons were inflated inside a larger balloon and then plaster was inserted into the interstitial spaces. The average number of neighbours each cell touched was counted for each model. As more cells were added there were more cells that did not touch the exterior boundary and were completely surrounded by other cells.

The goal of the experiment was to understand the geometry of these interior cells, however, unlike the 2 dimensional cellular arrangements in the earlier experiments, it was quite difficult to access the interior cells without destroying the model.

3 cells; Average Number of Neighbours: 3

3 cells; Average Number of Neighbours: 3

4 cells; Average Number of Neighbours: 4

8 cells; Average Number of Neighbours: 5.125

13 cells; Average Number of Neighbours: 6.308

21 cells; Average Number of Neighbours: 7.143

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Experiments: surface parameters surface topolopy

surface geometry

surface skin

layer parameters skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

grid parameters layer binding

layer scale

grid geometry

grid density

cell parameters grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

Honeycomb Models: Overview Starting with model_000 as the base model, the second group of experiments focused on creating a family of models that would be related to model_000 through the majority of parameters, but with small variations such that new geometric and performance qualities could be explored in a rigorous and structured way. The diagram illustrates the resultant 18 models plus the large scale installation that followed from this iterative proliferation of the base model.

The following sections contain more detailed information about each of the models parameters and information pertaining to the creation of the digital models as well as discuss the issues relating to the translation of these digital models into physical models. Not all the digital models were chosen for fabrication due to both limited time and resources, and sometimes the knowledge that a certain combination of variables was not worth pursuing further.

Related Topics: Digital Generation: Honeycomb Algorithm Parametric Matrix: Overview Model_000_Base

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Experiments: Model_000_Base Model_001_Cell Depth

Honeycomb Models: Overview 2 This chart shows the chronological order of development for the experiments as well as indicates which digital models were also physically fabricated.

Model_002_Multiple Layers Model_003_Layer Scale Model_004_Deformed Grid Model_005_Pocket Model_006_Parallel Model_007_Self-Similar Model_008_Closed Model_009_Sphere Model_010_Polar Model_011_Orientation Model_012_Skin 1 Model_013_Skin 2 Model_014_Installation Test Model_015_Syn/Anticlastic Model_016_Curvature Depth Model_017_Cell Curvature Model_018_Skin 3 Prototype 88

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_000_Base Model_000 represents the base set of parameters from which many of the subsequent models were created. Its form is nothing special in particular, however it was designed such that the surface would have complex curvature.

This one parameter change is in response to the aim of creating a highly adaptable honeycomb system. The body of research dealing with flat honeycomb systems and their limited uses in non-flat applications like aircraft design through vacuum-forming techniques is well established. However, this use is constrained by the geometry of the flat honeycomb to minimal and fairly simple curvatures. By starting with a complexly curved surface, the project already pushes the boundaries of honeycomb systems and creates a much more adaptable set of relationships between geometric parameters and performance.

Related Topics: Digital Generation: Honeycomb Algorithm Honeycomb Models: Overview

30 87

89

Experiments: Model_000_Base Cutting Pattern

90

Experiments: Model_000_Base Physical Model

91

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_001_Cell Depth This model’s primary goal was to test the possibility of varying cell depth across the surface. From the base surface (the same as in model_000, as will be the case for most of the models), a typical pattern of hexagons was mapped across the surface. However, instead of extruding each cell up to the same height, the cells were extruded such that their top members touched an arbitrary surface above.

This second surface had no special relationship with the base surface, i.e. the resultant cell height was not actually related to any inherent geometric or performance related issue, as will be seen in model_008 and model_016.

Related Topics: Honeycomb Models: Overview Model_000_Base Cell Parameters: Depth Structure: Cell Depth

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92

Experiments: Model_001_Cell Depth Cutting Pattern

93

Experiments: Model_001_Cell Depth Physical Model

94

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_002_Multiple Layers Based on quick and intuitive behavioural experiments with the first two models, it was obvious that the honeycomb structure was quite strong in relation to out-of-plane forces (e.g. pushing on them from above), but very flexible to in-plane forces. Pushing on the honeycomb from the sides caused a great amount of elastic deformation in the structure. Although this could be useful in many applications, this model was based on the hypothesis that a honeycomb could be made quite strong without the traditional industrial technique of applying a rigid skin.

This concept of using two honeycombs with opposite “grain” patterns is the basis for many of the other models as well as the large prototype for the AA Projects Review.

Related Topics: Honeycomb Models: Overview Model_005_Pocket Model_014_Installation Test Honeycomb Prototype: Regional Geometry

87 105 128 147

95

Experiments: Model_001_Cell Depth Cutting Patterns for the top (left) and bottom (right) layers. Notice the tabs coming out of the bottom layer’s strips. These tabs connect to the coincident faces on the top layer.

96

Experiments: Model_001_Cell Depth Physical Model

97

Experiments: Model_001_Cell Depth Physical Model from above. Note the new pattern that emerges from the overlapping layers: Instead of hexagons a series of octagons and squares fill the surface.

98

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_003_Layer Scale As an outgrowth of the research being done in model_002, this model explored two changes to the multiple layer parameters found in that model. First, the two layers were arranged in the same orientation and second, the two layers were designed at different scales at a ratio of 2:1. After constructing this model it was clear that the multiple layer scale parameter added very little performance to the system and no subsequent models in this vein were pursued. However, it could be that the difference in scales was too little, their performance too similar to view any significant emergent properties. There are examples in the uni-cellular sea creatures called diatoms of self-similar structures at multiple scales performing related, yet distinct functions.

Related Topics: Layer Parameters: Scale 50 Layer Parameters: Quantity, Topology, + Connectivity 48 Honeycomb Models: Overview 87

99

Experiments: Model_003_Layer Scale Cutting Pattern

100

Experiments: Model_003_Layer Scale Physical Model

101

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_004_Deformed Grid Returning to the single layer honeycomb, the main interest of this experiment was the possibility of varying the grid from which the hexagon cells were created. The grid attempted to create a concentration of cells towards the centre of the model and gradually increase the cell size towards the surface’s corners.

Although in this particular experiment the grid was hand designed and the honeycomb members were hand drafted in CAD, subsequent models explored the way in which the non-uniformity of the grid would emerge out of inherent geometric properties, such as high curvature. Later models also explored the automation of constructing non-uniform grids through the scripted use of the surface’s UV coordinate system.

Related Topics: Grid Parameters: Uniformity Structure: Grid Density Honeycomb Prototype: Local Geometry

54 74 149

102

Experiments: Model_004_Deformed Grid Cutting Pattern

103

Experiments: Model_004_Deformed Grid Physical Model

104

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_005_Pocket Interested in the exploring the multiple layer model area further, this model focused on creating partial connections between the layers, thus forming interior pockets of space between the layers. Although the visual result was satisfying, the geometric problems in such a system were large and further study was required in later models. For example, as the second surface’s control points were pulled to create the pocket, all the interior (boundary) UV points on the surface changed, sometimes in very small, but significant ways such that the joinery between the layers became often impossible. One solution to this problem was used in the AA Projects Review Installation, however it would be useful to have a way of fixing certain points on a surface as source points which are constrained according to two or more surfaces.

Related Topics: Layer Parameters: Quantity, Topology, + Connectivity 48 Structure: Layer Binding 70 Honeycomb Prototype: Overview 145

105

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Experiments:

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106

Experiments: Model_005_Pocket Physical Model

107

Experiments: Model_005_Pocket Physical Model

108

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_006_Parallel One of the more radical departures from the other models, this model focuses on creating a more three dimensional honeycomb structure by turning the cells parallel to the surface. Although it produced an interesting structure, the amount of material used is far larger for the same size the other honeycomb models. There is around twice as many areas were members overlap. If this provided more strength, it could be valuable, however, it does not appear to be any stronger than the other models. No other models were built exploring this parameter for this reason.

Related Topics:

Cell Parameters: Edge Orientation

58

109

Experiments: Model_006_Parallel Cutting Pattern

110

Experiments: Model_006_Parallel Physical Model

111

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_007_Self-Similar Building on the work of model_005, this model attempted to make a more uniform, partially laminated, and multiple layer system whereby there existed a self-similarity at various scales.

As each of the individual honeycomb layers stack next to each other, a larger honeycomb structure appears.

Related Topics: Layer Parameters: Quantity, Topology, + Connectivity 48 Structure: Layer Binding 70 Model_002_Multiple Layers 95 Model_014_Installation Test 128

112

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_008_Closed This model was the first to deal with the complications of having a closed surface as well as trying to create a operative relationship between a surface property (surface curvature) and a cell parameter (cell height). In order to have the honeycomb wrap around on itself, it was necessary to determine the way in which the pattern repeats itself and then loop the pattern such that no gap would occur. The surface was also mapped for local surface curvature and a method was created to relate these measurements to cell height.

The result was a structure that had thick skin where the surface curved the most and nearly even cell height where the surface was close to planer. Since the curved areas were inherently stronger to begin with, this process reinforced this structural relationship to create spine-like areas across the surface.

Related Topics: Surface Parameters: Topology Cell Parameters: Depth Structure: Cell Depth Model_016_Cell Depth/Curvature

39 57 77 136

113

Experiments: Model_008_Closed Cutting patterns nested onto two sheets.

OT B

56

55 54

30

29

31

53

28

32 52

27 TOP

60

51

33 59

20

19 21 18

58

22

17 26

57

25

16

24

23

15

46

47

14

45

44

48

13

12

49

11 10

43

08

50

09

34

01

42

07

02

35

41

40

36 37

39

38 06

03 05 04

114

Experiments: Model_008_Closed Detail of physical model

115

Experiments: Model_008_Closed Physical model

116

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_009_Sphere Similar to model_008, this model primary studied the issues surrounding a closed surface topology in relation to the honeycomb mapping. As one can see from the images, at the surface’s poles (V min and V max) the cells edges get closer together to the point that at the actual pole they would be coincident. As it is impossible to have a completely closed surface made entirely from hexagons (as seen in Fuller’s geodesic domes), other cell geometries need to be introduced.

Related Topics: Surface Parameters: Topology Model_008_Closed Math + Science: UV Coordinate System

39 113 179

117

Experiments: Model_009_Sphere Cutting Pattern

118

Experiments: Model_009_Sphere Interesting issues emerged in the construction of the model. As can be seen in the model photos, the resultant physical model is not a sphere. Two material relationships are contributing to this situation. First, in the digital model the thickness of the material is not factored into the algorithm which generates the honeycomb. This lack of correspondence between the digital and the physical model causes a small, but significant accumulation of material to occur that effects the global form.

The second material factor is that the ratio between the cell size, the material thickness, and the surface curvature was past the allowable range for this particular material. That is, the paper could not elastically bend enough to take on the required surface curvature. The equator of the sphere contained the most non-planer faces so this is the area that was beyond the material’s limit, causing the sphere to have a flatter curvature at the equator.

119

Experiments: Model_009_Sphere Light test with physical model

120

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_010_Polar This model was a quick, and as yet, unpromising, experiment into non-quadrilateral grids.

As the polar grids points get further away from their centre, the distance between neighbouring points on the gird also increases. This increase does not allow the honeycomb members further from the centre to have the same surface articulation and thus the structure quickly becomes only a very rough approximation of the generating surface.

Related Topics: Grid Parameters: Geometry

52

121

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_011_Orientation Again, as a quick digital experiment, this model explored the idea of not relating the cell edges to the generating surface but rather to an arbitrary plane. In this case the XY plane was used as the axis of extrusion for the hexagon map on the surface. One of the problems with this technique is that it limits the amount of global curvature is possible. At certain curvatures in reference to the arbitrary plane, the extruded form would start intersecting itself. However, this parameter was modified in the Projects Review Installation to constrain the grids on two target surfaces to the same source surface, allowing the binding of the surfaces. One inherent performance characteristic of this parameter is that the cells tend to be much more flexible than in models where the cell edges are normal to the local geometry.

Related Topics: Cell Parameters: Edge Orientation Structure: Cell Edge Orientation Honeycomb Prototype: Regional Geometry

58 76 147

122

Experiments:

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Model_011_Orientation Cutting Pattern (top) Physical Model (bottom)

123

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_012_Skin 1 This is the first model to deal with the application of a skin to the honeycomb system. Although this is relatively easy to do digitally, it is quite difficult to do in the physical world due to material constraints. Starting from the established method used within the aeronautics industry for skinning honeycomb panels, this experiment used a similar vacuum-forming process to warp a material to the desired curvature. However, this technique only works for relatively small panels and requires the fabrication of large moulds against which the panel can be pressed. Despite these problems, it was clear that the addition of a skin greatly added to the structural capacity of the honeycomb and thus the skin issue was pursued further in other models.

Related Topics: Secondary: Skin Fabrication Tactics Surface Parameters: Skin Structure: Surface Skin Industry: Honeycomb Sandwich Panels

23 42 69 170

124

Experiments: Model_012_Skin 1 Vacuum-formed physical model

125

Experiments: Model_012_Skin 1 Detail of physical model

126

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_013_Skin 2 A digital extension of the previous model, this model’s goal was to create a skinned honeycomb that not only allowed air to pass through the structure, but to develop multiple ways in which these openings would not conflict with the structural performance of the skin. By having two different skin patterns with alternating areas of openings, it is hypothesized that the skin would still provide a significant amount of structural integrity to the entire sandwich panel. As the difficulty of fabricating the anticlastic skin presented itself, the actual testing of this hypothesis has not yet been tested. However, an alternate tactic is used in model_018 to get around the geometric difficulty in this model.

Related Topics: Secondary: Skin Fabrication Tactics Surface Parameters: Skin Surface Parameters: Fenestration Structure: Surface Skin

23 42 45 69

127

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_014_Installation Test Model_14 builds on the previous multiplelayer models. The focus of this model is the arrangement of connection nodes between the two layers to create a large double-layer system. The nodes alternate between being connected at both top and bottom and being connected only in the middle. This pattern creates a undulating surface that gains rigidity through both the increased amount of curvature and the number of connection nodes in comparison with previous models.

Related Topics: Layer Parameters: Quantity, Topology, + Connectivity 48 Layer Parameters: Binding 49 Structure: Layer Binding 70 Honeycomb Prototype: Overview 145

128

Experiments: Model_014_Installation Test Detail views of digital model

129

Experiments: Model_014_Installation Test Cutting pattern for both layers. The model was roughly 70cm x 30cm, or 4 times the size of the previous models. The decision to increase the scale was based on the need to do loading tests as well as the desire to begin dealing with more complex logistical issues in preparation of the large prototype.

Related Topics: Digital Fabrication: Nesting

28

130

Experiments: Model_014_Installation Test Side and top views of physical model.

131

Experiments: Model_014_Installation Test Physical Model

132

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_015_Syn/Anticlastic This experiment was a simple exploration of surface curvature. Most of the models made up to this point had complex surface curvatures and it was determined that generating honeycombs with more constant curvatures could inform the research into the specific forces at play in the honeycombs. With the same two curves two surfaces were made; one with synclastic curvature and the other with anticlastic curvature.

One of the interesting results of this experiment relates to the geometric properties of the actual unfolded strips. Typically the unfolded strips are each unique, but in this experiment the strips were nearly identical. The significance of this relates to how production costs of larger strips could be reduced based on the amount of strips which were identical due to either better nesting or mass-production of each strip.

Related Topics: Surface Parameters: Geometry Structure: Grid Geometry

Synclastic Model

Anticlastic Model

40 73

133

Experiments: Model_015_Syn/Anticlastic Cutting Pattern: Anticlastic Model (top) Synclastic Model (bottom)

134

Experiments: Model_015_Syn/Anticlastic Physical models: Anticlastic model (left) Synclastic model (right)

135

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_016_Cell Depth/Curvature This experiment further explored the relationship between surface curvature and cell depth. Two models were created off of the same source surface. On one model the areas of highest surface curvature lead to high cell height whereas the other model reversed this relationship with high curvature leading to low cell height. As was shown in the section on performative relationships, the model with the higher cell depth at the high curvature was better able to resist the bending forces within each cell face.

Related Topics: Surface Parameters: Geometry Cell Parameters: Depth Model_001_Cell Depth Model_008_Closed

40 57 92 113

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Experiments: Model_016_Cell Depth/Curvature Physical Models: High Curvature : Low Cell Depth (left) High Curvature : High Cell Depth (right)

138

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_017_Cell Curvature In an attempt to better understand the forces at work at the level of the individual cell, a number of digital and physical models were produced. Essentially surfaces with different kinds and degrees of curvature were created and one cell was generated on it. Based on these cell models the individual faces of the cells were tested for their own curvature characteristics. Unsurprisingly, the cell faces generated from a flat surface had no curvature. However, when synclastic or anticlastic cells were tested there were some interesting properties that emerged. For example, in the faces that spanned across vertices that were different in both u and v values the amount of cell face curvature was the highest. What is important about this is that the stresses in the honeycomb cell faces are not only unequal globally but at a local level also. These stress distributions must be factored into the material selection process.

Related Topics: Surface Parameters: Geometry Structure: Surface Geometry

40 67

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Experiments: Model_017_Cell Curvature Physical models: Anticlastic Cell (top) Zero Gaussian Curvature Cell (middle) Synclastic Cell (bottom)

140

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Model_018_Skin 3 This experiment rethought the skin fabrication issue through applying the logic of the honeycomb strips to the actual skin material. The skin is made of a series of folded strips which allow the structure to be triangulated, and this stiffened, while remaining within the established production logic. One of the benefits of this tactic is that the honeycomb strips and the skin strips can be manufactured in parallel instead of sequentially. This allows the actual fabrication of the strips and skin to happen in tandem, building up the rigidity of the structure as each new strip of skin is attached. This is in contrast to applying a skin post-construction where the structure remains flexible and relatively difficult to work with. Applying a skin in this case would require some type of adaptive material which could adjust to the shifting structure. This model avoids these difficulties by treating the skin as integral with the underlaying honeycomb.

Another interesting benefit of this skin technique is that it allows for numerous fenestration patterns to be generated and fabricated during the laser-cutting process. This could be done in parallel with a water-jet cutting process to fabricate glass inserts for each opening.

Related Topics: Secondary: Skin Fabrication Tactics Surface Parameters: Triangulation Surface Parameters: Fenestration Model_013_Skin 2

23 43 45 127

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Experiments: Model_018_Skin 3 Slot fenestration style (left, top) Hole fenestration style (left, bottom) Rendering of large scale structure using the hole fenestration style. The advantage of this type of skin is that the different fenestration styles can all be applied in a 2D file after the strips have been unfolded, reducing the amount of time needed to prepare the files.

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Experiments: Model_018_Skin 3 Four views of the physical model

144

Experiments: surface parameters

layer parameters

grid parameters

cell parameters

surface topolopy

surface geometry

surface skin

skin triangulation

skin fenestration

layer quantity

layer topology

layer connectivity

layer binding

layer scale

grid geometry

grid density

grid uniformity

cell connectivity

cell depth

cell edge orientation

cell fill

open 2 axes

complex curvature

open 2 sides

n/a

n/a

single layer

n/a

n/a

n/a

n/a

quadrilateral

low

uniform

diagonal 1

constant

normal to surface

n/a

open 1 axis

synclastic curvature

closed 1 side

transverse

holes

multiple layer

laminated

full

distributed

matched

equalateral

high

non-uniformity

diagonal 2

variable

normal to arbitrary surface

filled

closed 2 axes

anticlastic curvature

closed 2 sides

ray

slots

bifurcated

partial

linear

differential

polar

complex

zero gaussian curvature

centered

horizontal

parallel to surface

vertical

constrained by vertices

Honeycomb Prototype This prototype was developed for the AA Project’s Review and sited in the AA’s Bar. This section will give a overview of the main issues behind the design and fabrication of the prototype. Based on the series of experimental models and performance tests, the prototype exhibits the highest level of integration between production logics, geometric and material parameters, and performance criteria. Unlike the models, the prototype had a site, a program, a budget, a schedule, and many other new criteria which could inform the design in a more in-depth way, testing the hypothesis that the honeycomb system could provide a high degree of integration between all levels of design.

Related Topics: Strategies: Multi-Parametric Design Layer Parameters: Binding Structure: Layer Binding Model_014_Installation Test

17 49 70 128

145

Experiments: Honeycomb Prototype: Global Geometry The global geometry of the prototype was determined in response to the spatial constraints of the site. The bar is heavily used, especially during the opening night, and a wide circulation space needed to be kept clear. Along the back walls of the bar a series of presentation boards were hung and a space in front of them was needed. Between these two constraints existed a narrow slot of space which formed the boundaries of the prototype.

7543mm

Wall Exhibition Area

Wall Exhibition Area

5295mm

Circulation Flow

Bar Front Members Room

Back Members Room

Dn

Up

Dn

36 Bedford Square 1st Floor Exhibition Plan for 2004 AA Projects Review

Related Topics:

Surface Parameters: Geometry Non-Mechanical Performance

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146

Experiments:

A

B

Honeycomb Prototype: Regional Geometry At the regional level the geometry was based on the earlier Model_014’s use of alternating rigid double-layered areas with flexible singlelayer areas. The pockets that formed between the single-layer areas were originally going to be used to display other models, however, as the design was developed, this function was dropped from the programme. However, the pockets did serve two structural functions.

First, they added curvature to the structure which increased it’s rigidity. Second, the pockets at the base provided the structure with a wider base, increasing the prototypes lateral stability.

C

Rigid multi-layer region

Internal Pocket

External Pocket and rigid multi-layer region

Related Topics: Wide Base

Section A

Section B

Rigid multi-layer region

Section C

Layer Parameters: Quantity, Topology, + Connectivity 48 Cell Parameters: Edge Orientation 58 Structure: Cell Edge Orientation 76

147

Experiments: 50mm

60mm

80mm + 2200mm

+ 2000mm

+ 1800mm

+ 1600mm

+ 1400mm

+ 1200mm

+ 1000mm

Honeycomb Prototype: Regional Geometry These diagrams of plan cuts taken every 200mm shows a second areas of development in relation to the prototype’s regional geometry. As the relationships between material, production time, labour, and cost as well as aesthetic concerns were better understood, the prototype went through a process of geometric refinement. Starting at 200mm, the base cell depth was changed to 100mm to reflect a better balance between strength and cost. In addition, the number of pockets was decreased making the structure smoother overall geometrically.

One of the constant relationships through these changes was the idea of relating cell depth to vertical height. As the structure grew taller, the cell depth was decreased from 100mm to only 50mm. This provided both more weight and more rigidity at the base where there were the highest stress concentrations.

+ 800mm

+ 600mm

+ 400mm

+ 200mm

100mm

Generation 3: - top cell depth: 50mm Prototype 1: - bottom cell depth: 100mm - minimum cross-sections of pocketsof pockets - 3 rows, -53 rows columns

100mm

Generation 2: - top cell depth: 60mm Prototype 1: - bottom cell depth: 100mm - thinner cross-sections of pocketsof pockets - 5 rows, -55 rows columns

200mm

Generation 1: - top cell depth: 80mm Prototype 1: - bottom cell depth: 200mm - thick base - 5 columns rows of pocketsof pockets - 5 rows, 5

+ 0mm

Plan Cuts

Related Topics:

Cell Parameters: Depth Structure: Cell Depth Model_001_Cell Depth Model_008_Closed

57 77 92 113

148

Experiments: Honeycomb Prototype: Local Geometry On the local geometric level, a new digital process was used to create a non-uniform cell distribution across the two layers. By using a custom parametrisized surface (i.e., one where isoparms were actively placed instead of passively created) the two layers had a much more controlled level of cell density.

Although not completely worked out, this process allowed there to be a higher cell density at the base than at the top, again adding weight to the base of the structure.

Back Layer Cell Orientation

Back Layer Elevation

Low Density Cells at Base

Related Topics: High Density Cells at Base

Front Layer Elevation

Front Layer Cell Orientation

Layer Parameters: Binding Cell Parameters: Connectivity Model_002_Multiple Layers Model_014_Installation Test

49 56 95 128

149

Experiments:

Back Layer Outer Honeycomb

Back Layer Inner Honeycomb Source Surface

Front Layer Inner Honeycomb

Front Layer Outer Honeycomb

Honeycomb Prototype: Local Geometry The process whereby the cells were generated relied on a much more complex system than in previous models. In order for the cells on each layer to match up at the correct spots a single source surface was used to generate the cells for both layers. Lines normal to this surface were projected out through the 4 target layers. Wherever these normal lines intersected with a target layer a grid definition point was created. Once all of the meshes were created, the correct polylines could be selected and lofted between to create the actual honeycomb strips.

Back Layer Outer Target Surface

Back Layer Inner Target Surface

Front Layer Inner Target Surface

Front Layer Outer Target Surface

Related Topics:

Digital Generation: Script Specification 31 Digital Generation: Basic MEL Script 32 Layer Parameters: Quantity, Topology, + Connectivity 48 Grid Parameters: Uniformity 54

150

Experiments: Honeycomb Prototype: Visual Testing Based on this cell density a series of renderings were done to test the visual transparency of the prototype. At this point it was decided that the cell density could be reduced by 10% to save labor time and money. Due to the parametric set-up, this reduction was easily accomplished by re-running the customdesigned script.

Related Topics: Grid Parameters: Density Light and Vision Non-Mechanical Performance

53 79 81

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Experiments: Honeycomb Prototype: Cell Density These two elevations of the front outer layer show the slight visual difference between the initial 100x40 grid that was used to generate the honeycomb and the final 90x36 grid. In addition the cell orientation was also switched at this point.

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Related Topics:

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Grid Parameters: Density Light and Vision Non-Mechanical Performance

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Experiments: Honeycomb Prototype: Modules The prototype was divided into 24 modules, 12 on each layer. The size of the module was controlled by three factors. First, the material size was fixed at 1900mm x 960mm. Second, the maximum available laser-cutting machine had a bed of 1480mm x 1000mm. So between these two factors, the sheet size was limited to 1480mm x 960mm. This essentially controlled the maximum height of each module.

The length of the modules was controlled by the third factor: ease of transportation. Since the modules were being fabricated off-site they needed to be small enough for a person to carry. Each module was thus roughly 2m long and 1m high.

Full prototype

Front Modules

A series of connector plates were fabricated to join the two layers together at the appropriate cells.

Related Topics: Digital Fabrication: Equipment Parameters Layer Parameters: Binding Digital Fabrication: Identity Logistics

Connecting Plates

Back Modules

26 49 29

153

Experiments:

F-B3

1 2 3 4 5

F-C4 6

F-C3

7 8

F-C2

9

F-B4

F-C1

10 11

F-B2

Honeycomb Prototype: Fabrication Sheet A fabrication sheet was created for each module to guide the team members through the process of folding and assembly. The large diagram shows the modules location relative to its neighbours and the two other sets of diagrams explain how each strip should be folded. In addition to this sheet, a series of instructions were developed to help orientate the team member’s positions with the strips such that each strip was folded in the correct direction. For example, when an unfolded strip was picked up the fabricator would rotate it so that the labelled end was always top up and in his/her left hand. They would then rotate it to the angle shown in the lower diagram and then begin folding.

12

F-B1

F-A4 13

F-A3

14

F-A2

15

F-A1

16 17 18 19 20 21

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Related Topics:

Digital Fabrication: Unfolding Digital Fabrication: Identity Logistics Honeycomb Prototype: Pre-Fabrication

27 29 156

27

154

Experiments: Honeycomb Prototype: Nested Sheet The unfolded strips were nested on to 81 sheets of 2.7mm paper board. Each sheet measured 1480mm x 960mm. Each strip was given a unique label, identifying its location in the prototype. In addition, holes were located and marked for the mechanical connections between modules.

960 mm

1480 mm

Related Topics:

Unique indentification label on each strip

Label indicating faces which need to be connected

Pre-cut holes for connector plates between modules

Digital Fabrication: Nesting Digital Fabrication: Identity Logistics Honeycomb Prototype: Fabrication Sheet

28 29 154

155

Experiments: Honeycomb Prototype: Prefabrication After the strips were laser cut and brought back to the fabrication studio, they were folded, glued, and then stored until the site was available for assembly.

1. Laser cut strips are delivered and organized

2. The strips are folded according to the notations on the fabrication sheets

Related Topics: Primary Production Logic Architecture: Bruges Pavilion 2002

3. The folded strips are glued together and clamped until dry

4. The assembled modules are carried to the site

22 163

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Experiments: Honeycomb Prototype: On-site Assembly On site, the modules were attached together, building up the rigidity of the structure as the two layers were pulled into alignment with each other and joined with the connector plates.

Related Topics: Layer Parameters: Binding Structure: Layer Binding Honeycomb Prototype: Modules

49 70 153

157

Experiments: Honeycomb Prototype: Final Exhibition Fully assembled prototype.

Related Topics: Strategies: Multi-Parametric Design

17

158

Experiments: Honeycomb Prototype: Final Exhibition Detail of connector plates (left) Photo showing the variable transparency of the structure.

Related Topics: Layer Parameters: Binding Light and Vision Honeycomb Prototype: Modules

49 79 153

159

Experiments: Honeycomb Prototype: Final Exhibition Detail of the base condition showing the increased cell density at the base.

Related Topics: Grid Parameters: Density Grid Parameters: Uniformity Honeycomb Prototype: Local Geometry

53 54 149

160

Experiments: Honeycomb Prototype: Final Exhibition Overall view of the prototype from above.

161

Project Overview

Resources:

Bruges Pavilion 2002

Introduction

References: Overview

Architecture Hypothesis

Serpentine Pavilion 2002

Scope Prada LA Organization Mashrabiya

Strategies: Overview

Strategies Biomimetic Research

Abstraction + Translation

Panelite

Industry Simmons Hall

Parametric Generation Honeycomb Sandwich Panels

Emergent Performance

Tactics

Multi-Parametric Design

CNC Cutting

Tactics: Overview Sheet Metal Bending Production Logics

Parametric Matrix

Resources

Nature

Performative Relationships

Radiolaria

Honeycombs + Hives

Resources: Overview Kelvin and Plateau Experiments

Related Topics:

Voronoi Algorithms

References Project Status

u=1 v=1 n=0

u=1 v=1 n=1

Status: Overview

Math + Science

Organization Resources: Overview u=0 v=1 n=0

UV Coodinate System

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u=0 v=1 n=1

0.9 0.8

u=1 v=0 n=1

u=1 v=0 n=0

0.7 0.6 0.5 0.4 0.3

0.9 0.8 0.2

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References: Architecture: Bruges Pavilion 2002 The Bruges Pavilion by Toyo Ito represents a good cese study for this research. It is the only known project to use a honeycomb structural system at a large scale relative to their traditional use in sandwich panels. Ito was drawn to the honeycomb as a way to filter light and effect visual transparency. In order to stiffen the structure it was partially covered with patches of aluminum sheets. Although honeycombs are usually fully covered with a skin, Ito and his engineers realized that a more local application of the skin patches could still dramatically increase the rigidity of the structure.

The honeycomb was pre-fabricated off site in modules that were then transported to the site and assembled together. This reduced the amount of time the public square was closed .

The pavilion is essentially a square tube with the two walls and the roof being made of aluminum honeycomb and the floor being a composite polycarbonate honeycomb. These floor honeycomb sheets have closed cells which means that there is a large amount of trapped air within them. The pavilion was surrounded by a shallow pool and the floor was able to float on it due to the trapped air within the polycarbonate panels.

Related Topics:

Surface Parameters: Skin Structure: Surface Skin Light and Vision

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References: Architecture: Bruges Pavilion 2002 Far Left: Transverse section through pavilion Near Left: Design and Construction Sequence 1. Comparison tests between non-skinned and partially skinned honeycomb.

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2. Diagram of skin and structure concept 3. Each aluminum strip is pressed into shape 4. The strips are locally clamped together 5. The entire module is globally clamped

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6. The strips are welded together.

7. The Skin is welded to the honeycomb strips 8. The modules are prepared for transport 9. The modules are delievered to the site

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10. The modules are quickly assembled

Related Topics: Honeycomb Prototype: Modules Honeycomb Prototype: Pre-Fabrication

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References: Architecture: Serpentine Pavilion 2002 The Serpentine Pavilion in London by Toyo Ito and Arup was of interest to this research for the way in which the conceptual design was implimented through a iterative digital process involving custom scripts and the way in which the production logic was informed by this process. Under a tight design and construction schedule the team needed to develop a tool which enabled quick and parametric changes to occur. The script that written had 4 parameters: the source shape, the degrees of rotation and amount of scale change between each iteration, and the center of each rotation. Thus by entering in different values for these parameters they could quickly produce new variations of the concept.

Once one of these variations was settled on the next step involved rationalizing the seemling chaotic structure of the pavilion. This was done by creating a heirarchy of modules based on the original order of iterations. Thus the members generated during the first iteration became the primary structure and modules.

Each new iteration then became a smaller module that attached to the primary modules.

The secondary members were also reduced in height slighly to allow for the glass and metal panels to be inset.

Related Topics: Digital Generation: Growth Algorithms Honeycomb Prototype: Modules

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References: Architecture: Prada LA Epicenter The Prada LA “epicenter” or flagship store

by Rem Koolhaas, OMA makes large use of a custom designed cellular solid. Inspired by the light filtration of small cleaning sponges, a process developing a large scale version was initiated. Hundreds of small scale, hand-crafted protypes were develpoed to determine the correct amount of light penetration, color, depth, and other variables. This process was then standardized digitial fabrication machines such as CNC and stereolithography were used to produce the final built components.

The foam wall offers not only light filtration but a means to hang clothes. This ad hoc arrangement of clothes enables the employees to rearrange the displays without the need for locating new hooks or repairing old holes.

Related Topics: Cellular Solids: Form-Finding Math + Science: Kelvin + Plateau

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References: Architecture: Mashrabiya Traditional Islamic architecture utilizes a multi-parametric device that reponds to multiple perfromance criteria in the form of the mashrabiya. The mashrabiya is a wooden lattice screen that has five functions: controlling the light penetration, controlling air flow, reducing air temperature, increasing air humidity, and creating a space of privacy. These functions are accomplished through a complex inter-related set of variables such as opening area, the dimensions and shape of the individual wooden lattice members, and the height from the floor. Often the screens modulate the percentage of opening from ground to ceiling to ensure the flow of air without decreasing privacy. One of the most interesting features of the mashrabiya is that it is able to passively control to humidity level in the space by releasing water vapor during the day that has been naturally collected during the cooler night. The mashrabiya also reduces the amount of glare from the incoming relfected light through the use of curved profiles on the lattice members.

The curved sections soften the reflected light compared to flat brise-soleil devices. This can be seen in the relatively soft edges in the shadows in the photograph.

Related Topics: Grid Parameters: Geometry Grid Parameters: Uniformity Light and Vision Thermal Conductivity

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References: Architecture: Panelite Panalite, a New York based material research and supply company, has been one of the pioneers in the development of honeycomb sandwich panels for the architectural market. Although the panels have been produced by various industrial manufactures for years, the focus had been on the aerospace market.

Panelite has worked with designers to expand the honeycomb panel beyond these industrial origins. They provide a range of cell sizes, cell shapes, and skin colors. At the moment the panels are primarily used as lightweight partitions, however they are begining to be used in more structurally intensive areas such as floors.

Related Topics: Strategies: Abstraction + Translation Structure: Surface Skin Light and Vision Industry: Honeycomb Sandwich Panels

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References:

Structural diagram of south elevation showing filled cells in red.

Architecture: Simmons Hall, MIT Simmons Hall at MIT is of interest to this research due to the way in which the cellular facade responds to specific structural forces.

Designed by the architect Steven Holl and the structural engineer Guy Nordenson in 2002, the skin is composed of perforated prefabricated concrete panels. Based on structural analysis of the cellular skin, the cells at critically over-stressed areas were filled. Nordenson explains, “The location of those filled panels was determined iteratively, i.e. by filling one and analysing the effects before adding another. The resulting pattern is a unique instance of non-linear or chaos patterns in a building structure”. In addition the stresses at each cell were coded into the building’s colour scheme.

Each window frame is painted according to the size of steel reinforcing bars cast within the concrete panels. Red is the largest diameter bars (and thus highest stress) and uncoloured is the smallest diameter bars.

Related Topics:

Digital Fabrication: Identity Logistics Cell Parameters: Fill Structure: Cell Fill

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References: 1. Skin Failure: Initial failure may occur in either compression or tension face. Caused by insufficient panel thickness, skin thickness, or skin strength.

2. Transverse Shear Failure: Caused by insufficient core strength or panel thickness

3. Flexural Crushing of Core: Caused by insufficient core flatwise compressive strength or excessive beam deflection.

4. Local Crushing of Core: Caused by low core compression strength.

5. General Buckling: Caused by insufficient panel thickness or insufficient core shear rigidity.

6. Shear Crimping: Sometimes occurs following, and as a consequence of, general buckling. Caused by low core shear modulus or low adhesive shear strength.

7. Skin Wrinkling: Skin buckles as a plate on an elastic foundation. It may buckle inward or outward, depending on relative strengths of core in compression and adhesive in flatwise tension.

Industry: Honeycomb Sandwich Panels Honeycombs, as industrially produced structures, first appeared in the mid 20th century.

Quickly their value to the emerging aircraft industry was acknowledged and the technology was widely researched and applied to planes during WWII in planes such as the British Mosquito. There were two main advantages that honeycomb served in aeronautical engineering. First, their strength to weight ratio was essential for the amount of lift that the wing design generated to support its weight. Second, honeycombs provide a high resistance to deformation in the out-of-plane orientation and this is useful in keeping the large relatively flat areas of wings from deforming under wind loads.

The most common honeycomb materials are aluminum or paper, although nearly any thin sheet material can be used. The most common fabrication technique is a process whereby thin sheets of material are glued together in alternating strips. These sheets are then cut into strips, perpendicular to the strips of glue, and then pulled apart to form an expanded sheet of hexagons. The process of pulling the material can be varied to produce both under-expanded honeycombs with each cell having a long thin diamond-like shape, and over-expanded honeycombs with each cell being nearly rectilinear. The honeycomb is then glued to a top and bottom rigid sheet material to produce an extremely strong structure.

Related Topics:

8. Intracell Buckling: Occurs with very thin skins and large core cells. This effect may cause failure by propagating across adjacent cells, thus inducing skin wrinkling.

Strategies: Abstraction + Translation Primary Production Logic Structure: Surface Skin Architecture: Panelite

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References: Industry: CNC Cutting Computer Numerically Controlled (or CNC) cutting machines were introduced in the early 1970’s and were quickly intergrated into various industries from large scale shipbuilding to the smallest microprocessors fabrication.

Some of these machines use new cutting techniques such as lasers or high-pressure water to do the actual cutting however others use more traditional plasma cutters. By being controlled directly from a digital file the fabrication of the parts can be performed much faster and more accurately than by manual methods. It also puts the designer in direct contact with the means of production.

1. CNC plasma cutters at a shipyard

2. Largest laser cutter in the world: 260m long x 4m wide; used for fabric

This research explored different cutting techniques. The project began with manual knife cuts but quickly moved on to small scale laser cutting. During the prototype phase, a larger and more sophisticated laser cutter was used.

Subsequent experiments proposed using plasma cutters for metal cutting and water-jet cutting for some of the glass skin inserts.

Related Topics:

3. Water-cooled plasma cutter: The metal is below water while cutting

4. Water-jet cutting: used for stone, glass and metal

Strategies: Abstraction + Translation Digital Fabrication: Equipment Parameters Digital Fabrication: Nesting Honeycomb Prototype: Nested Sheet

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References:

Original digital model of sheet metal part before unfolding operations

CNC Press Brake

Industry: Sheet Folding Although the folding of sheet materials has been around for a long time and can be seen in traditional crafts such as the japanese origami, several digital applications have been developed to facilitate the design and fabrication of these folded structures. Sheet-metal parts and cardboard boxes are the two most common uses for these softwares. As parts become more customized and are produced in shorter runs the need to quickly be able to calculate the cutting pattern for an unfolded part becomes increasingly important. Although there are some commonly used softwares like FormZ that is able to unfold geometry, most of the software is highly specialized. These applications are able to take into account the specific bending parameters and thicknesses of materials making the unfolding process much more accurate.

In order for the current research to continue, use of one of these more specialized softwares would be necessary, or algorithms used in these softwares would have to be integrated into the existing custom script.

Related Topics: Screen shots from the sheetmetal module of the SolidWorks software package showing the various operations that can be simulated and used to prepare the digital model for actual production.

Production Logics: Overview Digital Fabrication: Unfolding Honeycomb Prototype: Pre-Fabrication Architecture: Bruges Pavilion 2002

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References: Nature: Radiolaria The sea-creatures known as radiolaria serve as a good example of the morphogenesis of natural cellualr solids and provide the smallest known instance of a skeletal system, for radiolaria are only unicellular organisms. It is not clear what purpose these skeletons serve, but the logic of their development combines the geometry of bubbles with the process of mineralization. All organisms take in non-organic compounds, such as calcium salts, into their body when they eat and one of the main uses for them is the growth of structure. In the case of small organisms like radiolaria, where they are mainly governed by surface tension, these calcium salts tend to deposit at the boundaries between bubble-like cells. If one imagines two adjoining bubbles and a collection of small particles floating freely on their surfaces, it is clear that as these particles reach the valley-like edge that divides the bubbles the particles will have less freedom of movement and will essentially be stuck along the edge. Overtime, the accumulation of these non-organic particles will form bone-like structures. However, unlike the bones of humans, which provide a frame work on which hangs the rest of the body, it is unclear if radiolarian’s skeletons serve a function or are simply the result of the mineral deposits along the geometrically arranged cells.

Related Topics: Strategies: Biomimetic Research Cellular Solids: Form-Finding

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References: Nature: Radiolaria 1. Callimitra Agnesae 2. Ethmosphaera Conosiphonia 3. Actinomma Arcadophorum 4. Reticulm Plasmatique 5/6. Electron scanning microscope images of two radiolaria

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References:

1. Plan and section of bee’s honeycomb showing how two opposing cells meet.

2. Wax Honeycomb

Nature: Bee’s Honeycombs The bee’s honeycomb is the clearest instance of Plateau’s laws of cellular geometry. To quickly re-summarize Plateau’s work with bubbles, if each bubble has equal internal pressure, two bubbles will meet at a flat plane, three meet at a straight line, and four at a point. No more or less than these will form stable configurations. That is, if four bubbles are place together in an apparently stable configuration the bubbles will quickly reform such that only three meet at a point.

Furthermore, if all the bubbles are the same size, the three angles at an intersection will all equal 1200 . Based on these concepts, the development of the very regular bee’s honeycomb becomes clear. As bees of the same caste tend to be roughly the same size, they can be thought of as taking up the same area as they make an individual cell. As these cells come in contact with each other, their side walls quickly take on the regular hexagonal organization. However, as the hive is divided into different areas for different castes, the regularity breaks down at the boundaries between these areas. Figure 3 shows such a situation in the division between the cells of worker bees and the larger drone bees.

Related Topics:

3. Smaller cells for worker caste and large cells for drone caste.

4. Bees at work on honeycomb

Grid Parameters: Uniformity Cellular Solids: Form-Finding Math + Science: Kelvin + Plateau

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References: Math + Science: Kelvin + Plateau The first significant geometrical investigation directly related to cellular solids was that of Plateau’s study of soap bubbles in 1873 and D’Arcy Thompson’s elaboration of it in 1917.

Plateau was concerned with the geometry of bodies governed by surface tension. His theories can be visualized in figure 10 and summarized in these 4 ideas: (R. Phelan, 125)

stable 4 cells

unstable 4 cells Basic geometric relationship between two bubbles

Uniform cells packed together produce hexagons

The formation of hexagonal geometries through stable configurations

1. The faces of the cells, or films, have a mean curvature related to the pressures of the adjoining cells by the equation: Dp=2sK where Dp is the difference in pressure, s the amount of surface tension, and K the mean curvature of the face.

2. The meeting of three surfaces forms a line 3. The meeting of four surfaces forms a point 4. At these lines the symmetry is threefold (1200), and at the vertices it is fourfold (roughly 1090) also called tetrahedral.

Lord Kelvin, aka Sir William Thomson, famous for numerous other scientific ideas such as his namesake, absolute zero, worked on one of the first and most longstanding problems relating to the packing of cells governed by surface tension. Working with wire models, Kelvin posited the tetrakaidecahedron as the shape with the most minimal surface that still fills space. This shape is a truncated octahedron composed of 6 squares and 8 slightly curved hexagons.

Related Topics:

Photos of simple bubbble models demonstating Plateau’s rules.

More complex bubble configuration showing polyhedral cells

Strategies: Biomimetic Research Cellular Solids: Form-Finding Nature: Radiolaria Nature: Bee’s Honeycombs

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References: Math + Science: Voronoi Algorithms Irregularity in honeycombs and foams is a common occurrence. Although Kelvin and Plateau’s theorems help to describe how one can arrive at topological descriptions of cellular solids, it does not explain the how the irregularity was created or how to generate a particularly desired irregularity. The Voronoi diagram can help both mathematically describe and visually illustrate the importance of growth in the structural organization. R. Kusner and J.M. Sullivan give this definition of a Voronoi cell: “Given a collection of sites in space, we define the Voronoi cell for each site to be the region consisting of points closer to that site than to any other.” (Weaire, 72) These sites should be thought of as points of origin for the growth of cells. For instance, in the production of many foamed materials, a foaming agent is mixed into a liquid material, such as molten aluminum. When the mixture is properly adjusted, these foaming agents will release gas, forming gas bubbles in the molten metal. These bubbles are then trapped within the aluminum, forming a solid foamed material. In this example, the locations of the foaming agent particles are random and their growth is linear and parallel in time. As the spherical bubbles start to touch each other, polyhedral faces form. Thus, areas that have a lower density of foaming agent particles will have the largest cells since they have nothing Related Topics: Digital Generation: Growth Algorithms Project Status: Overview

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References: Math + Science: Voronoi Algorithms to stop them from growing.

However, these diagrams only begin to resemble a naturally or artificially produced cellular solid. As the limits imposed above are removed, or other factors dealing with material properties such as the maximum size of cell, the diagram starts to resemble known cellular solids. For example, in the case of bees, the cell will not be any bigger than the average size of the bees in that hive. If this limit is taken into account with the same set of random points but with all of the points that are closer than a minimum distance removed, the Voronoi diagram appears to resemble a natural honeycomb although still slightly angular and not quite as regular. More parameters, such as surface tension, non-linear growth, and other forces, such as loading, need to be added for the Voronoi diagram to fully resemble a bee’s honeycomb.

Although Voronoi diagrams are not by themselves able to describe the multitude of parameters that inform the various types of cellular solids, they are interesting as a way to geometrically construct organizations of space around given points. These organizations have many practical uses which can demonstrate the non-intuitive cellular relationships between things such as satellite navigation, data compression, and the habitat ranges of many species, and dried mud patterns (Gutzburg).

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References: u=1 v=1 n=0

u=1 v=1 n=1

u=0 v=1 n=0

Math + Science: UV Coordinate System The UV coordinate system is a common local coordinate system used in many 3D modelling programs. It is very similar to the traditional Cartesian coordinate system (xyz or world system) although it most often applies to a specific surface and not an entire model space.

The U and V axes are analogous to the X and Y axes in Cartesian space althougth U and V axes can be curved, intersect, and even wrap back on themselves. The direction perpedicular to a specific UV point is called the N axis in u=0 most programs for “Normal” although some v=1 call it the W axis. Thus each point on a curved n=1 surface has a different N axis.

0.9 0.8

u=1 v=0 n=1

u=1 v=0 n=0

0.7 0.6 0.5 0.4 0.3

0.9 0.8 0.2

0.7 0.6

0.1 0.5

0.4

0.3

u axis

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u=0 v=0 n=1

v axis Related Topics: Digital Generation: Honeycomb Algorithm Digital Generation: Script Specification Digital Generation: Basic MEL Script Surface Parameters: Topology

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Project Status

Project Status: Overview The hypothesis on which this research was based was that a honeycomb system could adapt to diverse performance requirements through the modulation of its inherent parameters while remaining within the limits of available production technologies.

I believe I have proven this hypothesis through my research. However, the research has also shown that there is much more work to be done to accomplish the desired high degree of integration between design and performance.

I see three areas that deserve further research towards this goal. First, an improved growth algorithm is necessary to provide a more specific relationship between the material and geometric parameters and the local performance requirements. For example, a link could be made between the output from structural analysis software or lighting studies and the input for the growth algorithm. In addition a evolutionary algorithm could be used to search through the great number of growth algorithms, selecting ones that best fit the design requirements.

The second area of further research is to develop a stronger understanding of the material parameters and how they relate to the geometric limits of the system. For example, it would be interesting to begin to test this system using a material that is not paper-based, Related Topics: ntroduction Hypothesis Scope Honeycomb Prototype: Overview

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Project Overview

Introduction

Hypothesis

Skin-Fabrication Tactics

Production: Overview

Primary Logics

Pre-Fabrication Tactics

Surface Topology

Equipment Parameters

Surface Geometry

Unfolding

Surface Skin

Nesting Scope

Organization

Identity Logistics

Secondary Logics

Honeycomb Algorithm

Digital Fabrication

1

Strategies

Digital Generation

Layer Quantity

MEL Script

Layer Topology

Parametric Matrix

Geometric Parameters

Layer Scale

Grid Parameters

Layer Binding

Cell Parameters

Grid Geometry

Parametric Generation

Bending Radius

Grid Uniformity

Elasticity

Cell Connectivity

Transparency

Cell Edge Orienation

Surface Geometry

Performance: Overview

Structure

Layer Binding

Cell Fill

Grid Geometry

000_Base

Grid Density Light and Vision

Grid Uniformity

Non-Mechanical Performance

Parametric Matrix

Cellular Solids: Form-Finding

008_Closed

Honeycombs: Overview Honeycomb Models Honeycomb Prototype

Performative Relationships

Resources: Overview

Honeycombs

Bibliography

Architecture Experiments

Prada LA Mashrabiya Panelite

References Nature Status: Overview

009_Sphere 010_Polar 011_Orientation

Honeycomb Sandwich Panels

012_Skin 1

CNC Cutting Radiolaria

013_Skin 2 014_Prototype Model

Related Topics:

Honeycombs + Hives

015_Syn/Anticlastic Digital Generation: Honeycomb Algorithm

Kelvin and Plateau

016_Curvature Depth Digital Generation: Growth Algorithms

Voronoi Algorithms UV Coodinate System

Math + Science

003_Layer Scale

Simmons Hall

Sheet Metal Bending Industry

Project Status

3

002_Multiple Layers

Serpentine Pavilion 2002

Cell Fill Production Logics

001_Cell Depth

And finally, the research has primarily been involved in structural performances, although cellular solids (and this honeycomb system in particular) have the potential to be useful in relation to many different performance criteria such as thermal conductivity and light and air control. Cellular solids are extremely common in nature because they are so adept at handleing multiple performance requirements. As designers begin to recognize the complexities of the systems they are working with, the use of cellular systems will greatly enable a more integrated approach to design.

Bruges Pavilion 2002

Cell Depth

Thermal Conductivity

2

Overview potentially making the system a more viable option for designers. The material properties of metal, for instance, could inform the amount of curvature allowed in each cell face as well as inform the structural loading analysis.

Further research in all these areas would increase the honeycomb system’s ability to pro004_Deformed Grid vide an integrated design framework, open 005_Pocket and responsive to various performance criteria while intricately tied to available means of 006_Parallel production.

007_Self-Similar

Cell Edge Orientation

Resources

Cell Depth

Surface Skin

Emergent Performance

Tactics: Overview

Grid Denisty

Weight

Material Parameters

Multi-Parametric Design

Layer Parameters

Thickness

Abstraction + Translation

Tactics

Layer Connectivity

Surface Parameters

Biomimetic Research

Skin Fenestration

Specification Growth Algorithms

Strategies: Overview

Skin Triangulation

Project Status:

017_Cell Curvature 018_Skin 3

Parametric Matrix: Material Parameters Performative Relationships: Overview

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Appendix

Appendix: Ashby, Michael F. and Gibson, Lorna J., Cellular Solids: Structures and Properties, Cambridge University Press, Cambridge, UK,1997.

Ashby, M.F.; Evans, A.G.; Fleck, N.A; Gibson, L.J.; Hutchinson, J.W.; Wadley, H.N.G., Metal Foams: A Design Guide, Butterworth-Heinemann, Oxford, UK, 2000.

Bibliography These sources have been used as tutorials and case studies in some of the conceptual, practical, and scientific issues discussed in this document.

Balmond, Cecil, Algorithm: Serpentine Gallery Pavilion 2002, in Verb:Matters, Actar, Barcelona, 2004.

Beukers, Adriaan and van Hinte, Ed, Lightness: The Inevitable Renaissance of Minimum Energy Structures, 010 Publishers, Rotterdam, 1998 Bitzer, Tom, Honeycomb Technologies: Materials, Design, Manufacturing, Applications and Testing, Chapman and Hall, London, 1997. de Certeau, Michel, The Practice of Everyday Life, University of California Press, Berkeley, 1984.

Darwin, Charles, The Origin of Species, Book Club Associates, London, 1973.

Delanda, Manuel, Intensive Science and Virtual Philosophy, Continuum, London and New York, 2002.

Delanda, Manuel, A Thousand Years of Nonlinear History, Swerve Editions, New York, 1997.

Doyle, Richard, On Beyond Living: Rhetorical Transformations of the Life Sciences, Stanford University PressStanford, California, 1997.

Fathhy, Hassan, Natural Energy and Vernacular Architecture, The University of Chicago Press, London, 1986.

Foreign Office Architects, Phylogenesis: FOA’s Ark, Actar, Barcelona, 2003.

Garofalo, Francesco, ed., Steven Holl, Thames and Hudson, London, 2003.

Gould, Stephen Jay, The Structure of Evolutionary Theory, Harvard University Press, Cambridge, Massachusetts, 2002.

Gunzburg, Max, Centroidal Voronoi Tessellations, found at: http://www.math.iastate.edu/gunzburg/voronoi.html Hensel, Michael; Menges, Achim; Weinstock, Michael; Emergence: Morphogenetic Design Strategies, Architectural Design, Vol. 74, No. 3 May/June 2004.

Holland, John H., Emergence: From Chaos to Order, Oxford University Press, Oxford, 1998.

Ito, Toyo, Bruges Pavilion, in Verb:Matters, Actar, Barcelona, 2004.

Johnson, Steven, Emergence: The Connected lives of Ants, Brains, Cities and Software, Penguin Books, London, 2001.

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Appendix: Bibliography Kelly, Kevin, Out of Control: The New Biology of Machines, Social Systems, and the Economic World, Perseus Books, Cambridge, Massachusetts, 1994.

Kohanski, Daniel, The Philosophical Programmer: Reflections on the Moth in the Machine, St. Martin’s Press, New York, 1998.

Leach, Neil; Turnbull, David; Williams, Chris, eds.; Digital Tectonics, Wiley-Academy, West Sussex, UK, 2004.

Ortega y Gasset, Jose, History as a System and Other Essays Toward a Philosophy of History, W.W. Norton and Co., New York, 1941.

Otto, Frei, ed., IL 28: Diatoms I: Shells in Nature and Technics, Institute for Lightweight Structures, Stuttgart, 1984.

Otto, Frei, ed., IL 33: Radiolaria: Shells in Nature and Technics II, Institute for Lightweight Structures, Stuttgart, 1990.

Otto, Frei, ed., IL 38: Diatoms II: Shells in Nature and Technics, Institute for Lightweight Structures, Stuttgart, 2004.

Picon, Antonie and Ponte, Alessandra, eds., Architecture and the Sciences: Exchanging Metaphors, Princeton Architectural Press, New York, 2003.

Perkowitz, Sidney, Universal Foam: From Cappuccino to the Cosmos, Walker and Company, New York, 2000.

Sande, Hera Van, Toyo Ito builds the Bruges 2002 Pavilion, Stichting Kunstboek, Oostkamp, 2002.

SHoP, eds., Versioning: Evolutionary Techniques in Architecture, Architectural Design, Vol. 72 No. 5 Sept/Oct 2002.

Spiller, Neil, The Cyber Reader: Critical Writings for the Digital Era , Phaidon, London, 1999.

Stattmann, Nicola, Ultra Light - Super Strong: A New Generation of Design Materials, Edition Form, Berlin, 2003.

Steele, James, An Architecture for People: The Complete Works of Hassan Fathy, Thames and Hudson, London, 1997.

Thompson, D’Arcy, On Growth and Form, Cambridge University Press, Cambridge, UK,1961. van Duijn, Chris, Material Research at OMA, in Verb:Matters, Actar, Barcelona, 2004.

Wakeford, R.E., Sheet Metal Work, Herts, England, Argus Books, 1985.

Weaire, Denis, ed. The Kelvin Problem: Foam Structures of Minimal Surface Area. Taylor and Francis, London, 1996.

Zijlstra, Els, Future Materials for Architecture and Design, Materia, Rotterdam, 2002.

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Appendix: Index Abstraction + Translation 14, 168, 170, 171 Acknowledgements 3, 188 Algorithm 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Bee’s Honeycombs 12, 85, 175, 176 Biomimetic 12, 13, 85, 173, 176 Bruges Pavilion 156, 163, 164, 172, 184, 187 Cell Cell Connectivity 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Cell Depth 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Cell Edge Orientation 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Cell Fill 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Cellular Solids 12, 85, 86, 166, 173, 175, 176, 184 Cell Depth 57, 64, 77, 79, 92, 93, 94, 96, 97, 98, 113, 136, 137, 138, 148 Cell Edge Orientation 58, 76, 122, 147 Cell Fill 59, 78, 169 CNC Cutting 171 Emergent Performance 16, 66 Equipment Parameters 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Fenestration 24, 45, 46, 79, 127, 141 Form-Finding 12, 85, 86, 166, 173, 175, 176 Geometric parameters 37, 38, 55, 66, 79, 84, 89, 181 Grid Density 53, 74, 102 Grid Geometry 52, 61, 73, 133 Grid Uniformity 54, 75, 79 Growth algorithm 35, 56, 181 Honeycomb Prototype 17, 21, 28, 29, 48, 53, 54, 57, 58, 61, 70, 76, 79, 84, 95, 102, 105, 122, 128, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 164, 165, 171, 172, 181

Identity Logistics 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Kelvin + Plateau 12, 85, 166, 175, 176 Layer Binding 49, 70, 71, 72, 105, 112, 128, 145, 157 Layer Connectivity 48 Layer Quantity 48 Layer Scale 50, 99, 100, 101 Layer Topology 48 Light and Vision 79, 151, 152, 159, 163, 167, 168 Mashrabiya 167 Material system 5, 16, 36, 50 MEL 30, 31, 32, 33, 34, 150, 179 Model_000_Base 42, 87, 89, 90, 91, 92 Model_001_Cell Depth 57, 77, 92, 93, 94, 96, 97, 98, 136, 148 Model_002_Multiple Layers 48, 70, 95, 112, 149 Model_003_Layer Scale 50, 99, 100, 101 Model_004_Deformed Grid 53, 54, 74, 75, 102, 103, 104 Model_005_Pocket 48, 95, 105, 106, 107, 108 Model_006_Parallel 109, 110, 111 Model_007_Self-Similar 112 Model_008_Closed 39, 77, 113, 114, 115, 116, 117, 136, 148 Model_009_Sphere 39, 117, 118, 119, 120 Model_010_Polar 121 Model_011_Orientation 58, 76, 122, 123 Model_012_Skin 1 42, 124, 125, 126 Model_013_Skin 2 127, 141 Model_014_Installation Test 48, 49, 70, 95, 112, 128, 129, 130, 131, 132, 145, 149 Model_015_Syn/Anticlastic 40, 64, 133, 134, 135 Model_016_Cell Depth/Curvature 64, 77, 113, 136, 137, 138 Model_017_Cell Curvature 58, 64, 67, 139, 140 Model_018_Skin 3 24, 42, 43, 45, 69, 141, 142, 143, 144 Module 29, 39, 153, 154, 164, 165, 172

Multi-Parametric Design 17, 145, 158 Nesting 28, 133 Non-Mechanical Performance 81, 146, 151, 152 Panelite 168, 170 Parametric Generation 15, 36, 84 Performance 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Plateau 12, 85, 166, 175, 176, 177 Prada LA 166 Pre-Fabrication 21, 154, 156, 164, 172 Production Logics 3, 19, 20, 21, 22, 35, 172 Radiolaria 12, 50, 173, 174, 176, 185 Script 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Serpentine Pavilion 165 Specification 30, 35, 48, 49, 56, 99, 105, 112, 128, 147, 149, 150 Surface curvature 40, 57, 77, 113, 119, 133, 136 Surface Geometry 40, 67, 68, 79, 139 Surface Skin 25, 45, 69, 124, 127, 163, 168, 170 surface topology 39, 117 Thermal Conductivity 42, 59, 80, 167 Triangulation 24, 43, 44, 141 Unfolding 27, 41, 172 UV Coordinate System 39, 117, 179 Voronoi Algorithms 52, 177, 178

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Appendix: pg5 pg8 pg12 pg13 pg14 pg23 pg50 pg74 pg80 pg158 pg159 pg160 pg163-4 pg165 pg166 pg167 pg168 pg169 pg170 pg171

pg172 pg173-4 pg175

pg176 pg178

(Bee’s Honeycomb): Trinty College Dublin website: http://www.tcd.ie/Physics/Foams/solid.html Photo by Francis Ware (Plastic Foam) on pg27 of: Otto, Frei, ed., IL 28: Diatoms I: Shells in Nature and Technics, Institute for Lightweight Structures, Stuttgart, 1984.

(Bee’s Honeycomb): http://www.svsu.edu/profsci/Anmin021.JPG (Common honeycomb patterns) on pg19 in: Bitzer, Tom, Honeycomb Technologies: Materials, Design, Manufacturing, Applications and Testing.

(Forming Techniques): http://www.hexcelcomposites.com/Markets/Products/Honeycomb/Sand_design_tech/p21.htm (Radiolaria) on p108 of Otto, Frei, ed., IL 38: Diatoms II: Shells in Nature and Technics, Institute for Lightweight Structures, Stuttgart, 2004.

(Cell Density Graphs) from pg99-100 of: Ashby, Michael F. and Gibson, Lorna J., Cellular Solids: Structures and Properties, Cambridge University Press, Cambridge, UK,1997.

(Microscopic image of Styrofoam): http://www.microscopy-uk.org.uk/mag/artoct02/amgallery.html Photo by Francis Ware Left photo by Sue Barr Photo by Francis Ware (Bruges Pavilion) from: Sande, Hera Van, Toyo Ito builds the Bruges 2002 Pavilion, Stichting Kunstboek, Oostkamp, 2002.

(Serpentine Pavilion) from: http://www.serpentinegallery.org/pavillionpress/index.html and: Balmond, Cecil, Algorithm: Serpentine Gallery Pavilion 2002, in Verb:Matters, Actar, Barcelona, 2004.

(Prada LA) from: van Duijn, Chris, Material Research at OMA, in Verb:Matters, Actar, Barcelona, 2004. and: Floto+Warner, Lydia Gould, OMA, Prada on http://www.europaconcorsi.com/db/pub/scheda.php?id=2719 (Mashrabiya)in: Fathhy, Hassan, Natural Energy and Vernacular Architecture, The University of Chicago Press, London, 1986.

(Panalite): www.e-panelite.com (Simmons Hall): Garofalo, Francesco, ed., Steven Holl, Thames and Hudson, London, 2003.

(Honeycomb Panels) in: Bitzer, Tom, Honeycomb Technologies: Materials, Design, Manufacturing, Applications and Testing, Chapman and Hall, London, 1997.

(CNC Cutting) http://www.bwcutters.com/html/news/bw_cargolifter.html and: http://www.shipbuilding.com.ua/pict/galery/17_pict_big_gal_Cutting-machines.JPG and: http://www.cfsb.com.tw/cnce.html and: http://www.ifw-jena.de/strahl/maschine.htm (Solidworks): http://www.solidworks.com/swexpress/sept03/200308_techtip_02.cfm and (Press Brake): http://www.lakeshoremachinetool.com/brakes.html (Radiolaria) in: Otto, Frei, ed., IL 33: Radiolaria: Shells in Nature and Technics II, Institute for Lightweight Structures, Stuttgart, 1990. and: Thompson, D’Arcy, On Growth and Form, Cambridge University Press, Cambridge, UK,1961.

(Bee’s Honeycomb) pg109: Thompson, D’Arcy, On Growth and Form, Cambridge University Press, Cambridge, UK,1961. and pg34 in: Weaire, Denis, ed. The Kelvin Problem: Foam Structures of Minimal Surface Area. Taylor and Francis, London, 1996. and: http://www.turtletrack.org/Issues01/Co10062001/CO_10062001_Beetree.htm (Bubble Diagrams) on pg100 and pg103 in: Thompson, D’Arcy, On Growth and Form, Cambridge University Press, Cambridge, UK,1961. and (photos) on pg225 and pg237 in: Otto, Frei, ed., IL 33: Radiolaria: Shells in Nature and Technics II, Institute for Lightweight Structures, Stuttgart, 1990.

(Salt Flats): http://hebb.mit.edu/people/ben/trips/DeathValley/DeathValley2.png

Image Credits All images in this document with the exception of those listed to the left have been produced

by Andrew Kudless.

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Appendix: Acknowledgements There are numerous people that have helped me throughout the year with my dissertation.

Foremost I would like to thank the directors of the Emtech programme, Michael Weinstock and Michael Hensel, as well as studio master Achim Menges in providing me both with the conceptual and technical knowledge to pursue my work and with the opportunity to develop my research on a large scale during the AA Projects Review. Their relentless commitment to the programme was an inspiration for all of the students. I look forward to collaborating with them in the future.

I would also like to thank my fellow students in the EmTech programme for their time and energy spent helping with the installation as well as their friendship and support throughout the year.

There were many people involved in the development of the Projects Review installation. I would like to thank Mohsen Mostafavi for his generous financial support. In addition, I am grateful for the technical support that Michael Weinstock and the Technical Studies department of the AA have provided. Terri Cummings of the PaperMarc Corporation generously donated the paper for this installation and Capital Models performed an amazing amount of work in a very short time. Martin Hemburg provided both technical and conceptual support in developing the scripts used in the research.

Lastly, I would like to thank my family and friends for their love and support during this difficult yet rewarding year.

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