PROGRAMMABLE FOLDING: COMPUTATIONAL GEOMETRY WITH EMBEDDED ASSEMBLY LOGIC

Annie Locke Scherer Integrative Technologies M.Sc. Candidate M. Arch University of Michigan B.S. Arch University of Virginia

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ITECH M.Sc. Programme ICD Institute for Computational Design ITKE Institute for Building Structures and Structural Design Thesis Advisors Prof. AA Dipl.(Hons.) Arch. Achim Menges Prof. Dr.-Ing. Jan Knippers Dipl.-Ing. Oliver David Krieg M.Arch, B.Arch.Sci David Correa Dipl.-Ing. Benjamin Felbrich 2014-2015

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To mom, Danny, Grandad and Pat: Thank you for your unending love and support, that made this all possible.

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CONTEXT INTRODUCTION.........................................08 AIM............................................................15 CONTEXT...................................................19 STATE OF THE ART.....................................25 METHODS..................................................33 DESIGN RESEARCH DEVELOPMENT........41 RESEARCH PROPOSAL.............................85 DISCUSSION..............................................91 OUTLOOK..................................................95

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INTRODUCTION CHAPTER 01

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INTRODUCTION One does not normally design leftover waste material. More commonly, one designs and what is left over is discarded. Programmable Folding is centered around strategic cuts. Careful design of removed material has the ability to imbue a flat sheet with intrinsic curvature. The aim of this thesis is to create parametrically-derived folding patterns that approximate irregular, doubly-curved surfaces when assembled. Looking at fundamentals of cut patterns, this thesis investigates the relatively unexplored world of origamiâ€™s lesser-known cousin: kirigami. This, like origami, is the art of folding sheets but also utilizes simple cuts to program inherent curvature into a flat material. Programmable Folding analyzes the basic geometrical rules of kirigami folding, identifying which parameters are flexible and which ones must be followed. The resulting geometrical pattern encodes a sheet material with an inherent assembly logic and provides the necessary information to fold the material into a complex, 3-dimensional surface. This project builds upon research from much of the origami world, particularly referencing the work of Tomohiro Tachi (University of Tokyo), Daniel Piker (Foster + Partners), and Toen Castle (University of Pennsylvania). Grasshopper and Kangaroo for Rhino are the primary design tools to realize the geometrical complexities within double-curvature folding. These tools allow for easy manipulation of areas and degrees of curvature, quick generation of a cut and tabbing pattern, and simulation of the final folded geometry. Programmable Folding explores design possibilities of geometry on multiple levels and delivers a final product whose design is embedded in the generated folding pattern. This thesis simplifies complex geometrical problems of approximating an irregular surface, while simultaneously investigating typology, assembly logic and structural performance.

â€œAll designers fold. That is, all designers crease, pleat, bend, curve or wrap twodimensional sheets of material, and by these processes of folding, create three-dimensional objects... Since almost all objects are made from sheet materials (such as fabric, plastic, sheet metal or cardboard), or are fabricated from components used to make sheet forms (such as bricks - a brick wall is a sheet form), folding can be considered one of the most common of all design techniques.â€? -Paul Jackson, origamist

ORIGAMI When most people think of origami, the first image that comes to mind is that of a folded paper crane or a childâ€™s â€œcootiecatcher.â€? While these are the most simple and common uses for folding paper, the same fundamental geometrical concepts can be applied in a much more complex manner. With careful programming, it is possible to achieve irregular surfaces and forms that could be applied in an architectural context. While a great deal of intricacy can be achieved in origami surfaces, Programmable Folding examines the inherent geometric complexities of folding and demonstrates that origami does not allow for easy asymmetrical manipulation of patterns. While some patterns such as Figure 01.01 have the freedom to bend and form around sub-structures, it is not possible to create a rigid surface without additional support.

Figure 01.01 Work by Thomas Diewald, applying Ron Reschâ€™s classic origami pattern to create architectural surfaces.

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KIRIGAMI Deriving inspiration from the University of Pennsylvania Astronomy & Physics Department, this research builds on the question of how to create three-dimensional forms from flat sheet materials. Try to wrap a sphere, and soon the geometrical problems become increasingly complicated. No amount of folding can change a flat sheet of paper to have more â€œintrinsicâ€? curvature. Kirigami, like origami, is the art of folding paper, but with strategic cuts. This small addition has allowed artists, scientists, and designers to create complex curvature and shapes that have never before been possible with a single sheet. Kirigami is an elegant solution to imbue surfaces with more or less curvature based on simple geometric principles. As seen in Figure 01.02, by removing material from a sheet and folding it, it is possible to fold a flat sheet and achieve zero, positive, or negative Gaussian curvature. These fundamentals will later be extrapolated to create more complex curvature and surface approximations.

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Figure 01.02 Simple kirigami

切り紙 Kirigami: “kiru” = to cut, “kami” = paper

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AIM

CHAPTER 02

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AIM Architecture mainly consists of planar elements, with sheet materials being the most common base of building design. In the past, architecture has kept a clear division between architectural skin and structure. The substructure is independent from the cladding, and the two do not necessarily have a relationship. The aim of this thesis is to bridge the gap between skin and structure. It investigates the potentials of a light-weight, flat-pack, rigid panel system that has structural and spatial capabilities. This type of system would have a differentiated pattern with an embedded assembly technology. This thesis investigates the fundamentals of kirigami patterns, seeing what parameters are flexible and which rules must not be broken. Connections between folds are programmed within the plastic sheets, so no added materials are needed for the joints. By programming the underlying geometry and intrinsic curvature, the resulting kirigami patterns are used to investigate potential design systems that apply this logic to architectural design.

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Figure 2.01 Kirigami fold pattern for monkey saddle geometry

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CONTEXT CHAPTER 03

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ORIGAMI FOLDING In order to create a bespoke folding system, it is first necessary to understand classic origami folds and their architectural capabilities. Ron Reschâ€™s patterns give a high degree of variability as well as potential structural capacity, while folds like those of Yoshimura and Miura Ori have the ability to flat pack in multiple configurations. Although many of these folding patterns have interesting architectural implications formally, most of the patterns are impossible to manipulate asymmetrically for they must retain symmetry in order for the global system to function.

Miura Ori

Ben Parker

Yoshimura

Paul Jackson

Hans Buri

Ron Resch

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John Mckeever

Ron Resch

Paul Jackson

Ben Parker

Folds Flat Variability Deployability Rigidity

Miura Ori

Figure 03.01 Common origami folding patterns

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KIRIGAMI With the simple addition of a few strategic cuts, kirigami allows the generation of intricate fold patterns with substructure bases of triangles, rectangles, or even hexagons. While it is possible to create objects or flat patterns, Programmable Folding focuses on the possibilities of approximating complex curvature. The process of manipulating kirigami geometry patterns will be explored further in design research development.

objects

triangle base

quad base

hex base

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3-D surface approximation

Figure 03.02 A wide array of kirigami variations, as explored by Mike Tanis

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STATE OF THE ART CHAPTER 04

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FOLDING IN ARCHITECTURE Many architectural works have been inspired by origami, whether in shape or in function. In looking at precedents, The operability, asymmetry, and potential reconfigurability of the systems are related to one another in Figure 04.01. Note that most are symmetrical, becasue of the difficulty of generating variable patterns.

Manual

Canary Warf Kiosk | Make Architecture

Operable

Static

Evolution Door | Klemens Torggler

Actuated

Hoberman Arch | Salt Lake City

Bowoos Pavilion

Figure 04.01 Origami Inspired Precedents

Actuated Rigid Plates | Erik Hull

Shenzhen International Airport

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Reconfigurable

Novi Sad | Origami Forum

Resonant Chamber | RVTR

Enoc Armegnol Folded Chair

Woodskin | MammaFotogramma

Ron Resch Origami

Cruise Terminal Yokohama

Appended Space | Mohamad Al Khayer

Fold Plate Hut | Osaka

Asymmetrical Panels

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ASYMMETRIC FOLDING PATTERNS & CORRUGATION PARAMETRIC MIURA ORI

Because of the geometrical complexities inherent in origami, it is extremely difficult to manipulate a symmetrical pattern locally without making an unfoldable pattern. Here, Tachi looks at asymmetrically adjusting patterns to create complex curvature. Figure 04.02 shows a simple manipulation of the classic Miura Ori fold pattern to create asymmetrical double-curvature. Because of the Kawasaki-Justin Theorem, the sum of the mountain and valley folds around a vertex must equal zero to make a flat folding pattern. This severely limits the geometrical possibilities for asymmetrically manipulating classic origami folds.

θ1

θ2 θ4

θ3

FREE FORM ORIGAMI TESSELLATION

Tomohiro Tach is one of the only origamists to research complex, asymmetrical forms from single sheets. As illustrated in Figure 04.02, based on generalizations of Ron Resch’s patterns, Tachi demonstrates how to fold a sheet of stainless steel into a freeform origami tessellation.

ORIGAMI CORRUGATION

Tessellated Group develops mechanically folded sheet material and utilizes them as structure for double-layer systems. This corrugation improves the typical performance while reducing manufacturing cost and waste. While none of their designs accommodate asymmetrical patterns, Tessellated Group’s products can be applied in a multitude of industries, ranging from architecture to aerospace to construction and packaging.

Figure 04.02 Pattern for variable curvature

Kawasaki-Justin Theorem

θ1 + θ3 = θ2 + θ4

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Figure 04.03 Free-form origami tessellation

Figure 04.04 Industrial origami

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KIRIGAMI RESEARCH IN PHYSICS Physicists Randall Kamien and Toen Castle at the University of Pennsylvania are investigating how to approximate a curved surface with kirigami. Their work is based on French researchers’ study of how sunflower seeds cover the dome of the sunflower. The seeds have a precise pattern of 5, 6, or 7 sided; this natural variation allows for the seeds to pack evenly around the dome. While origami uses the technique of “tucking” to hide excess material in the final shape, kirigami addresses these limits and removes the material completely, allowing folding without any excess material. Kamien and Castle have simplified these principles into unique patterns, each with their own spatial and structural potentials. By introducing a step in the modules, they can approximate curves much like that of a voxel.

Figure 04.05 Sunflower Lattice of Seeds. Blue, red, and green cells are pentagons, hexagons and heptagons, respectively

Figure 04.06 Variation within single 5/7 kirigami module

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Figure 04.07 Projected stepping pattern of a ziggurat, along with cut pattern

Figure 04.08 Left: A duopotent lattice of sixons in the flat state, where grey hexagons denote excised regions of paper. Right: the two folded state configurations of the duopotent sheet

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METHODS

CHAPTER 05

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METHODS COMPUTATIONAL KIRIGAMI

Programmable Folding develops a robust computational model that uses the principles of Gaussian curvature as a substructure to kirigami cut patterns in order to approximate curvature. Strategic, computationally determined cuts imbue the pattern with intrinsic curvature that will allow the flat pattern to take its pre-determined shape once assembled.

FABRICATION

Programmable folding utilizes 0.5mm x 800mm x 1200mm plastic sheets to express the full geometrical possibilities of computational kirigami. These plastic sheets allow enough flexibility for incremental folding for assembly. Additionally, the plastic is structural enough to withstand the plastic deformation along the dashed fold lines. A flexible material was specifically chosen as rigid kirigami would require simultaneous actuation of all folds. Furthermore, a rigid structure would be comprised of multiple materials within the joints, which requires more assembly instruction and materials. The ZĂźnd cut plotter allows Programmable Folding to be realized at a real architectural scale. Because of the sheet limitations of the laser cutter, the cut plotter is the best option for taking advantage of full polypropylene sheets with as little waste as possible.

FOLDING

Instead of adding additional material to join the kirigami edges, Programmable Folding takes advantage of the kirigami cut outs and programs the tabs into the material that would normally be removed as waste. One set of tabs closes the holes, while the other reciprocally locks each module into the other to create a more rigid structure.

Figure 05.01 Zero, positive, and negative Gaussian curvature

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1m²

0.5mm Polypropylene Sheets: 800x1200mm

4.5m²

Bed size: 1500mmx3000mm

Zünd Knife Cutter Universal Cutting Tool: cuts material up to 5mm

Figure 05.02 Fabrication tools and flexible material

Figure 05.03 Tabbing and reciprocally interlocking modules

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GAUSSIAN CURVATURE Gaussian curvature is the basis for programming material with intrinsic curvature. It is the product of both principle curvatures at any given point, which is equal to the intrinsic curvature. By looking into typology of curvature with tessellation and adding or subtracting the number of vertices around a node, it is very simple to manipulate and program a surface’s resulting Gaussian curvature. If the vertex has 5 instead of 6 vertices, the corresponding surface has positive Gaussian curvature while adding a 7th results in a saddle-like negative curvature shape. This basic theorem is the underlying logic for parametrically manipulating generic kirigami folding patterns. After understanding the different typologies that can be created with simple manipulation of node vertices and developing a tool to easily do so, it is possible to use this logic as a sub-structure and generate more controlled, geometrically complex kirigami.

{3;5}

(+) Gaussian curvature Σ Θ > 360

{3;7}

no Gaussian curvature Σ Θ = 360

(-) Gaussian curvature Σ Θ < 360

Figure 05.04 Positive and negative Gaussian curvature by adding and removing vertices

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(+) gaussian curvature

no gaussian curvature

(-) gaussian curvature

Figure 05.05 Positive and negative Gaussian curvature applied to kirigami patterns

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DIGITAL METHODS Programmable Folding utilizes a robust computational model to analyze global formations and irregular surfaces. This model takes an irregular input surface, approximates its underlying Gaussian curvature, and creates a flat sub-structure for the kirigami based on the required intrinsic curvature. After running the kirigami pattern through a kangaroo script, the necessary cut outs are determined and a flat-pack folding pattern can be generated. All folding and cut lines, along with tabbing, are pre-programmed and the final pattern is ready for production.

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COMPUTATIONAL MODEL

:

base geometry for global curvature

program local curvature

OUTPUT

triangulated surface approximation & percent deviation

flat kirigami fold pattern

sheet specs required for construction

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DESIGN RESEARCH DEVELOPMENT

CHAPTER 06

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SIMPLE KIRIGAMI MODULES I explored a few simple modules of kirigami and parametrized the variables. It is important to note that not all variables can be changed without inserting additional folds. These are some of the study models I fabricated in an effort to have a better understanding of the folds, cuts, and relationship between panels, and how to produce more or less curvature.

w2

w2

w3 h1

h2

h2

Module 1

h1 w1

w3

w1

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SIMPLE KIRIGAMI MODULES This series of experiments investigates triangular cuts and â€œglideâ€? modules. Here I discovered that not all parameters can be modified without folding implications. Some of the tests show gaps in the folded module and begin to reveal the parameters of kirigami that cannot as easily be manipulated without consequences.

h3 h3 h2

d1

h2

d1 d2

h1

d2

h1 w1 Module 2

w2 w1

w2

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SIMPLE KIRIGAMI MODULES These hexagonal cuts are a variation of Module 1, with two additional “relief” folds on each side to allow synclastic or anticlastic bending between modules.

d1

Module 3

d1

h3

h3

h2

h2

h1

h1

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KIRIGAMI FUNDAMENTALS After numerous studies with the basic kirigami cuts and folds, some key principles of kirigami can be abstracted. For example, a right angle between a cut will produce a vertical panel, while making it larger or smaller will change the relationship to the lower panel. Changing width between mountain and valley folds will influence a moduleâ€™s height, while manipulating the width of the cut will simply change the dimensions between modules. Finally, changing angle between the mountain and valley folds to be smaller or larger will result in an downward or upward curvature (respectively) between modules.

acute slit

right angle slit

obtuse slit

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wide slit

tapering edges

widening edges

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MANIPULATING KIRIGAMI MODULES SYMMETRICAL QUAD PATTERN

These modules are arranged in a simple quad pattern with slight variations in order to understand their relationship to one another. These studies look at the resolution of panelization and experiment with changing single folds from mountain to valley. A single fold can change the global pattern from flat, to cylindrical, to undulating.

ASYMMETRICAL QUAD PATTERN

Taking some of the knowledge from module placement manipulation, the next step is to move around and rotate the modules. While moving a module up and down is simple and successful, module rotation is less so as it creates many unforeseen issues that are much more mathematically complicated to solve. The diagrams on the next page explain these limits.

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MODULE MANIPULATION Further investigation in global configurations requires a more indepth look at translation of modules. While moving a module up, down, and side to side, the kirigami cut remains unchanged because all edge cut sides remain equal. However, if a rotation is made, the cut edges are no longer equal and must be adjusted to compensate for the additional length. This also involves the addition of another crease to alleviate any tension between plates and subsequently complicates the kirigami pattern.

module translation

module rotation

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unlimited translation

unequal edge lengths need to be compensated for

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SIXTON PATTERN Hexagonal patterns allow much more freedom when designing a global pattern as the geometry of triangles is embedded in the global pattern as well. These studies attempt to examine at synclastic and anticlastic surfaces, and investigate how one could potentially approximate curvature and sustain variation within the patterns.

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SURFACE DECONSTRUCTION These surfaces have the opposite approach of previous tests, as they begin with the surface design, decompose the curvature and slope, and apply the learned fundamentals of kirigami modules. After the surface is decomposed, a kirigami cut pattern is arranged with rigid plates sandwiching a thin sheet of flexible plastic to act as a hinge. The mitred joints inform the angle between two adjacent plates, and rubber bands act as the actuation to pull the structure into its final configuration.

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mitre joint for flexibility with stops held in place with tension

wood sandwich panels polypropelyene for live hinge

Figure 06.01 Joint possibility with multi-material kirigami

Global Form Deconstruction

Figure 06.02 Sandwich assembly with rigid kirigami

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RIGID KIRIGAMI This prototypical model is constructed of two layers of wood panels, sandwiching a sheet of polypropylene plastic between them. Mitred joints, similar to the Woodskin project, allow folding between panels with a specific programmed angle. The final configuration is held in place with rubber bands to simulate tension cables and small bolts.

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Figure 06.03 Folding and unfolding of rigid kirigami

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KIRIGAMI TRIPLET COMBINATION Kirigami cuts can also be combined into double-layered structures in many base forms: triangles, squares, and hexagons. These â€œtripletsâ€? result in more complex forms that create an interesting sandwiching structure. The structural capabilities of these modules have enormous amount of potential for thick plate systems, and they can be varied to approximate curvature. After extensive research into the kirigami modules and potentials, this Programmable Folding focuses on the triangle triplets, as they allow easy surface approximation and eliminate planarity issues of quads and hexagons.

section and axon of ki

formulating kirigami triplet

section and axon of kirigami triplet double-layer structure

Figure 06.02 Kirigami double-layer structural potentials

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Figure 06.01 Potential kirigami triplet combinations

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MANIPULATING TRIANGLE RATIOS One of the two methods for creating curvature in kirigami is to take advantage of the triangle relationship of each side of the sandwich structure. By subtly adjusting these ratios of upper and lower triangles, one can control the intensity and location of curvature. Because the triangles are being adjusted symmetrically around their center, there is very little additional computation that needs to be applied to create a working kirigami pattern.

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: Figure 06.02 Changing top and bottom triangle ratios to achieve varying levels of curvature

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:

:

:

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APPROXIMATING OVERALL CURVATURE Based on the relationship between the top triangle, bottom triangle, and depth of the pattern, one can calculate the curvature at any specified point. This subdivision is important to the resolution and aesthetic of the final design. These simple angle calculations are embedded within Programmable Folding’s computational design tool so that geometrical constraints predict unfoldable kirigami patterns.

W1

H

θ

W2

W1 -W 2

W1

θ

H

θ

H

W

2

W

1

θmax=

tan

-1

-W

2

( ) ΔH ΔW

θ

total

= 180 - 2 θ

Figure 06.03 Calculating curvature at a specific point

CALCULATING THE ANGLE BETWEEN SEGMENTS: Based on the relationship between the top triangle, bottom triangle, and depth of the pattern, one can calculate the curvature at any specified point.

θ = 180 - 2θ max

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100

100

50

θ

max

= 233°

100

25

θ

max

= 208°

10

θ

max

= 191°

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1D CURVATURE All of the previous kirigami studies have investigated double curvature (dome) and zero curvature (flat surfaces). In manipulating kirigami to curve in one direction, it is impossible to do so without scaling the triangles unevenly. Figure 06.05 and Figure 06.06 demonstrate the only ways to create a 1D curvature with equilateral triangle kirigami. Figure 06.04 explains the geometrical problems with 1D kirigami curvature. The desired inside and outside layers are represented by the cylinders nested inside each other. However, the inside cylinder must be scaled in three dimensions instead of only two. While the kirigami works for the first few layers, one moves up the cylinder, the connection between inside and outside triangles become more and more skewed. Furthermore, unrolling the geometry (Figure 06.05) creates a pattern that requires additional joining. Figure 06.06 utilizes truncation as an alternative solution, although the truncation forms additional holes within the folded pattern that cannot be closed. These experiments were a crucial turning point in the research for Programmable Folding, as it quickly became evident that a much more robust computational tool was necessary to program the required kirigami cuts.

connection panels become increasingly skewed

Figure 06.04 Diagram of internal scaling that must happen in order to achieve one dimensional curvature

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Figure 06.05 Approximating 1 dimensional curvature with without truncation

Figure 06.06 Approximating 1 dimensional curvature with truncation

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MANIPULATING TRIANGLE RATIOS As seen in the previous research development and in Figure 06.07, it is possible to create double curvature by having a consistent ratio of top to bottom triangles in a kirigami sandwich structure. The next step is to program local curvature by asymmetrically manipulating the top and bottom sides. In Figure 06.09, the top side of the triangles are scaled from one end of the pattern to the other. This creates zero curvature on one end, and gradually transitions into double curvature. Figure 06.08 uses the same approach, but scales the triangles on each side inversely to each other, creating a simple double curvature. While triangle scaling to achieve curvature works when applied gradually, further computational tools are required when working with more complex surfaces.

:

:

Figure 06.07 Consistent triangle ratios

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:

:

Figure 06.08 Change triangle ratios with indirect proportions

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:

:

:

Figure 06.09 Change triangles with single attractor on one side

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ASYMMETRICAL MANIPULATIONS After thoroughly investigating symmetrical scaling of triangles to create curvature in kirigami sandwich structures, Programmable Folding looks to asymmetrical manipulation. Because adjacent, connecting edges of kirigami have to be equal, it is necessary to relax modified geometry to create a foldable pattern. Remember the complex problems with one dimensional curvature? By asymmetrically scaling one side of the sandwich structure and relaxing the geometry with kangaroo, it is possible to create a one dimensional curvature from one sheet without truncation.

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: Figure 06.10 Asymmetrical manipulation and edge relaxation

:

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KIRIGAMI RULES After achieving a wide range of variability in kirigami, there are two rules which cannot be broken. As shown in Figure 06.11, the edge lengths must be equalized after asymmetrically manipulating the triangles. This is to ensure that the kirigami pattern can fold and have all the seams match. Figure 06.12 shows the geometric limits of manipulating the pattern, and how the scaling of triangles past the limits of adjacent triangles is not possible.

Figure 06.11 Edge length correction to ensure foldability

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Figure 06.12 Over-manipulation of kirigami pattern

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SADDLE SHAPE After experimenting with manipulating kirigami in one dimension, I applied the same logic to scale the pattern in two dimensions. This scaling, after cut out lengths are equalized, exhibits negative Gaussian curvature and folds into a saddle shape.

Figure 06.13 Manipulating the pattern in two dimensions

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Figure 06.15 Saddle simulation. Highlighted: direction of triangle scaling

Figure 06.14 Simple saddle prototype

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MANIPULATING KIRIGAMI SUB-STRUCTURE Apart from manipulating kirigami triangle ratios, it is possible to determine general forms by modifying the hexagonal substructrue. This catalogue shows the wide range of curvature possible by simply changing a simple vertex. The research proposal will use ths as one of the methods for approximating curvature.

1x_octagon_lowres

1x_quad, 1x_octagon_lowres

1x_heptagon, 1xpentagon_lowres

Figure 06.16 Mesh substructure catalogue and resulting fold geometry

1x_octagon_highres

1x_quad, 1xoctagon_highres

1x_heptagon_highres

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1x_quad_lowres

1x_quad_highres

1x_decagon_lowres

1x_decagon_highres

1x_dodecagon_lowres

3x_octagon_highres

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MANIPULATING KIRIGAMI SUB-STRUCTURE Building on the research of kirigami sub-structures, this monkey saddle is formed by replacing the central hexagon of a pattern with an octagon. The extra two vertices program the flat pattern with negative Gaussian curvature. This rest of the kirigami triangles are applied normally, relaxed into a foldable pattern, and assembled.

Figure 06.17 Monkey saddle base pattern

Figure 06.18 Flat folding pattern folding into a monkey saddle

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Figure 06.19 Monkey saddle folding simulation

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JOINT DETAIL TESTS The joint details went through a number of iterations beginning with the paper models. A system of tabs was developed out of necessity to assemble intricate models more quickly. After experimenting with different scales and secondary materials, I settled on a 2-2-tabbing joint system. The first “2” refers to the number of tabs per side, to add stability to the edges. The second “2” refers to the number of adjacent modules that each module is connected to. After the module interlocks its two open sides together, it locks into the next module. The second one, in turn, is locked into the third and the third is locked back to the first. This reciprocal tabbing system balances structural stability and assembly time.

Figure 06.20 2-2 tabbing system

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Figure 06.21 Single, large tabs

Figure 06.23 Zip tie joining experiments

Figure 06.22 Two tabs per side (final design)

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SURFACE RESOLUTION One interesting discovery during the design research phase was playing with the scale and resolution of kirigami fold patterns. The monkey saddle pattern used slightly less than a full sheet of polypropylene (because of its symmetric nature and asymmetry of the plastic sheet). The next prototype takes 1/6th of the monkey saddle pattern and scales the resolution up. Despite the difference in triangle numbers, both prototypes take up approximately the same area, only with differing thicknessess.

400m

m

monkey saddle prototype average triangle length: 40mm

m

0m

40

large prototype average triangle length: 120mm m

0m

400m

m

60

original polypropylene sheet m

m

00

12 800m

m

Figure 06.24 Size comparison of sample prototypes and their volume

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Figure 06.25 Monkey saddle pattern (400mm x 400mm x 40mm)

Figure 06.26 1/6th of monkey saddle pattern (400mm x 600mm x 120mm)

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RESEARCH PROPOSAL CHAPTER 07

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DESIGN GENERATION The overall design is generated from the two driving factors that have been explored in the design research phase: approximating curvature from bespoke mesh sub-structures and manipulating local curvature with top and bottom triangle ratios. First, the desired surface is approximated and unrolled into a flat pattern. Here we apply the first shape determining method: a hexagonal mesh is generated with an octagon at the curvature centroid to create negative curvature. Then the mesh is relaxed and the user chooses the boundary conditions of the final form. After the base mesh is finalized, a basic kirigami triangle pattern is applied. From there, the second tool for manipulating curvature is applied. Based on attractor points, one can program in local positive or negative curvature. Next, the pattern is run through kangaroo to equalize neighboring edge lengths. Once complete, the lines are connected to make the familiar kirigami pattern. After applying dashing and tabbing, the pattern is ready for production. Figure 07.01 Site planes generate design

ceiling-wall stair-ceiling central vertex stair-wall

wall-wall

unrolled site conditions

user-determined boundary condition

mesh pattern based on desired general form

generate kirigami triangles from mesh base

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locally manipulate curvature

check line lengths

generate kirigami pattern

generate kirigami pattern from base triangles

equilaze lines

cut sheet specs

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INSTALLATION PROPOSAL The final architectural installation of Programmable Folding exhibits the control that is achievable by asymmetrically manipulating kirigami fold patterns. It will comprise of ~25 sheets of 0.5mm polypropylene, which, when assembled, has a volume of over 2 mÂ˛. The design exhibits a response to pre-existing site conditions, filleting the wall corner and making a nice transition from the half-floor staircase into a wall element. Its specific curvature is possible by both manipulation of kirigami geometry substructure, as well as programmed local curvature based on triangle ratios. This prototype showcases the full control that Programmable Folding has achieved and provokes further questions about design opportunities and architectural space potentials of kirigami.

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DISCUSSION

CHAPTER 08

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DISCUSSION GEOMETRICAL CONSTRAINTS

As seen in the mesh base catalogue, there are a few geometries that are not compatible with kirigami triangle patterns. The next step in this research would be to branch out, investigating other geometries and understanding the full constraints. With a wider range of geometries, a more detailed catalogue can be developed and then the full understanding of architectural implications of kirigami would be realized.

JOINTS

Although it is not a problem in larger sheets, smaller folding patterns do not have the tabbing strength to combat the internal forces. To the right is an example from the monkey saddle prototype: even though the geometry works to keep all folds planar, the tabs are simply not strong enough at the center vertex where all the material forces are concentrated. It is possible, here, to introduce apertures to release the stress. This method acts as a design driver, and allows the general viewer to understand the base mesh logic with a cursory glance.

1x_quad, 1x_octagon_lowres

1x_heptagon_highres

Figure 08.01 Problematic mesh bases, usually caused by odd numbered vertices or the addition of too many verticies next to each other

3x_octagon_highres

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Figure 08.02 Center vertex of monkey saddle prototype

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OUTLOOK CHAPTER 09

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EMBEDDED ASSEMBLY INTELLIGENCE Although Programmable Foldingâ€™s final installation is incrementally folded, one possible avenue for further research is different kinds of actuation. Figures 9.01-9.04 demonstrate the possibilities of actuation explored in various engineering, architecture, and biology research. These actuators range from heat, robots, electricity or light. Although Programmable Folding utilizes a flexible plastic because of itsâ€™ ability to incrementally fold, it is extremely relevant to investigate thick, rigid materials when applying kirigami to larger, architectural scales. These would require some kind of simultaneous actuation, as they are not flexible enough for incremental folding. This design opportunity serves as a driver to further embed the assembly process within the programmed fold patterns. Integrating such intelligence would challenge the traditional design process and existing tools, and speculates on the potential of large scale deployable, folded structures.

Figure 09.01 Self-deploying origami stent grafts

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Figure 09.02 Robotic folding

Figure 09.03 Electrical programmable matter

Figure 09.04 Origami-enabled deformable silicon solar cells

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ARCHITECTURAL SYSTEM & FUNCTION As an outlook, I would like to consider the design implications of kirigami in a varying range of architectural systems. Whether applied to industrial design, temporary shelters, or full-scale permanent structures, kirigamiâ€™s spatial and structural capabilities can be implemented across multiple scales. More complex fold patterns, a higher degree of control, and varying thicknesses are all topics to be further explored. Using kirigami as a design driver for localized system differentiation, one can discover new functional roles and material possibilities of deployable sheet material in design and architecture.

industrial design

temporary shelters

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permanent, structural design space

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ACKNOWLEDGMENTS: Tutors: David Correa Oliver David Krieg Benjamin Felbrich Achim Menges Jan Knippers University of Pennsylvania Physics & Astronomy Department: Toen Castle, Randall Kamien RoboFold : Ema & Gregory Epps Design & Development: Djordje Stanojevic Boyan Milhalov Kenryo Takahashi Emily Scoones Maria Yablonina Yuliya Baranovskaya Georgi Kaslachev Fabrication: Colin Oâ€™Keefe Josh Few Sasha Mballa Bruno Knychalla Leonard Balas Becca Jaroszewski My family: Donna Stamps Danny Noneaker Don Stamps Pat Stamps

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REFERENCES N. M. Benbernou, E. D. Demaine, M. L. Demaine, and A. Ovadya (2011). “Universal Hinge Patterns for Folding Orthogonal Shapes in Origami.” A K Peters Ltd., pp. 405-419. H. Buri, Y. Weinand (2010). “Origami: Folded Plate Structuctures.”Laboratory for timber constructions IBOIS, - Ecole Polytechnique Fédérale Lausanne, EPFL Lausanne, Switzerland . A. Byoungkwon, N. Benbernou, E. Demaine, D. Rus (2011). “Planning to Fold Multiple Objects from a Single Self-Folding Sheet.” T. Castle et al. (2014). “Making the Cut: Lattice Kirigami Rules.” Phys. Rev. Lett. 113, 245502. G. Epps et al (2015). “Foldable = Makable.” www.curvedfolding.com. B. Felbrich. (2013). “Bionics in Architecture: Experiments with Multi-Agent Systems in Irregular Folded Structure.” Tutors: S. Wiesenhütter, D. Lordick J. Gattas, W. Wu, Z. You. “Miura-Base Rigid Origami Parameterizations of First-Level Derivative and Piecewise Geometries.” Department of Engineering Science: University of Oxford. F. Gioia, D. Dureisseix, R. Motro, B. Maurin (2014). “Design and Analysis of a Foldable/Unfoldable Corrugated Architectural Curved Envelope.” Università Politecnica delle Marche E. Hawkes, B. An, N. M. Benbernou, H. Tanaka, S. Kim, E. D. Demaine, D. Rus, and R. J. Wood (2010). “Programmable Matter by Folding.” Proceedings of the National Academy of Sciences. Hoberman, C. (1988). “Reversibly Expandable Three-Dimensional Structure.” United States Patent, No. 4,780,344. K. Kuribayashi, K. Tsuchiya, Z. You, D. Tomus, M. Umemoto, T. Ito, M. Sasaki (2005). “Self-Deployable Origami Stent Grafts as a Biomedical Application of Ni-rich TiNi Shape Memory Alloy Foil.” Y. Klein, E. Efrati, and E. Sharon (2007). “Shaping of Elastic Sheets by Prescription of Non-Euclidean Metrics,” Science 315 11161120. P. Jackson(2011). “Folding Techniques for Designers: From Sheet to Form.” Laurence King Publishing Ltd. London, United Kingdom. R.J. Lang (1996). A Computational Algorithm For Origami Design.” Proceedings of the Twelfth Annual Symposium on Computational Geometry (ACM, New York), pp. 98–105. L. Mahadevan and S. Rica (2005). “Self-Organized Origami.” Science 307, 1740. S. Miyashita, D. Rus (2013). “Multi-Crease Self-Folding by Uniform Heating.” Computer Science and Artificial Intelligence Laboratory: MIT. B. Mihaylov (2014). “Encoded Surfaces: Topological Design of Woven Hybrid Textiles.” ICD Diploma McNeel. “Grasshopper – Generative Modelling for Rhino.” http://grasshopper.rhino3d.com/ K. Miura (1985). “Method of Packaging and Deployment of Large Membranes in Space.” Proceedings of the 31st Congress of the International Astronautical Federation, IAF-80-A 31, (American Institute for Aeronautics and Astronautics, New York, 1980), pp. 1-9. J.H. Na et al. (2014). “Programming Reversibly Self-Folding Origami with Micro-patterned Photo Crosslinkable Polymer Trilayers.” Adv. Materials 27, 79-85. C. Onal, R. Wood, D. Rus (2011). “Towards Printable Robotics: Origami-Inspired Planar Fabrication of Three-Dimensional Mechanisms.” 2011 IEEE International Conference on Robotics and Automation. J. Paulose, B.G. Chen, and V. Vitelli (2015). “Topological Modes Bound to Dislocations in Mechanical Metamaterials.” Nature Phys. 11, 153-156. Piker, Daniel (2009). “Origami Electromagnetism.” https://spacesymmetrystructure.wordpress.com/2009/03/24/origamielectromagnetism/ H. Pottmann, A. Asperl, M. Hofer, A. Killian, D. Bentley (2007). “Architectural Geometry.” Bentley Institute Press: Exton, PA. J.F. Sadoc, J. Charvolin, and N. Rivier (2013). “Phyllotaxis on Surfaces of Constant Gaussian Curvature.” J. Phys. A: Math. Theor. 46, 295202. M. Schenk (2011). “Folded Shell Structures.” University of Cambridge: Doctoral Thesis. Supervisor: S.D. Guest M. Schenk, S. D. Guest. (2011). “Origami Folding: A Structural Engineering Approach.”

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J. L. Silverberg et al. (2014). “Using Origami Design Principles to Fold Reprogrammable Mechanical Metamaterials.” Science 345 647-649 (2014). J.L. Silverberg, et al. (2015). “Origami Structures with a Critical Transition to Bi-stability Arising from Hidden Degrees of Freedom.” Nature. Mat. 14, 389-393. D. Sussman, Y. Cho, T. Castle, T. Gong, E. Jung, S. Yang, R. Kamien (2015). “Algorithmic Lattice Kirigami: A Route to Pluripotent Materials.” R. Tang, H. Huang, H. Tu, H. Liang, M. Liang, Z. Song, Y. Xu, H. Jiang, H. Yu (2014). Origami-enabled deformable silicon solar cells. Applied Physics Letters 104. Tachi (2009). “Generalization of Rigid-Foldable Quadrilateral-Mesh Origami.” Journal of the International Association for Shell and Spatial Structures. T. Tachi (2013). “Freeform Origami Tessellations by Generalizing Resch’s Patterns.” Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. University of Tokyo. T. Tachi (2013). “Interactive Form-Finding of Elastic Origami.” University of Tokyo. T. Tachi (2009). “Simulation of Rigid Origami.” In Origami: The Fourth International Conference on Origami in Science, Mathematics, and Education. T. Tachi (2010). Freeform Variations of Origami , J. Geometry and Graphics 14, 203-215. M. Tolley, S. Felton, S. Miyashita, L. Xu, B. Shin, M. Zhou, D. Rus, R. Wood (2013). “Self-Folding Shape Memory Laminates for Automated Fabrication.” Z. You (2014). “Folding Structures Out of Flat Materials: Reconfigurable machines and internally structured materials can be created through folding.” Materials Design: Vol. 345, Issue 6197.

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LIST OF FIGURES Figure 01.01 http://thomasdiewald.com/blog/?p=743 Figure 03.01 Folding Techniques for Designers: From Sheet to Form https://spacesymmetrystructure.files.wordpress.com/2009/01/foldingpatterns.pdf Folding Techniques for Designers: From Sheet to Form Folding Techniques for Designers: From Sheet to Form https://spacesymmetrystructure.files.wordpress.com/2009/01/foldingpatterns.pdf Folding Techniques for Designers: From Sheet to Form http://www.detail.de/architektur/themen/origami-faltkunst-fuer-tragwerke-000497.html https://farm8.staticflickr.com/7028/6684347075_b272005316.jpg https://spacesymmetrystructure.files.wordpress.com/2009/01/foldingpatterns.pdf https://spacesymmetrystructure.files.wordpress.com/2009/01/foldingpatterns.pdf http://blog.novedge.com/2014/05/the-edge-michele-calvano-and-the-architecture-of-folded-surfaces.html Figure 03.02 https://instagram.com/hyperqbert/ Figure 04.01 http://www.evolo.us/architecture/bowoos-bionic-research-pavilion-is-inspired-by-marine-biodiversity/ http://laughingsquid.com/wp-content/uploads/2014/02/the-evolution-door-a-clever-door.jpg http://www.a10.eu/news/headlines/sculptural_object_novi_sad.html http://www.archdaily.com/362951/woodskin-the-flexible-timber-skin/ http://2.bp.blogspot.com/_Je8B4--Ky9E/S52NTUSBoII/AAAAAAAAADI/5bzAN2vZ_sE/s1600-h/olafur_eliasson2-1.jpg https://vimeo.com/89379866 www.rvtr.com http://www.eric-hull.com/appended-space/ http://www.evolo.us/architecture/bowoos-bionic-research-pavilion-is-inspired-by-marine-biodiversity/ http://thedesigninspiration.com/articles/shenzhen-baoan-international-airport-expansion-t3/ http://www.eikongraphia.com/?p=324 http://www.pleatfarm.com/2009/12/21/folded-plate-hut-in-osaka/ Figure 04.03 T. Tachi (2013). “Freeform Origami Tessellations by Generalizing Resch’s Patterns.” Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. University of Tokyo. Figure 04.02 https://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/ Figure 04.04 http://www.tessellated.com/ Figure 04.05 http://www.wired.com/2015/01/quanta-curves-from-flatness-kirigami/ Figure 04.06, - Figure 04.08 D. Sussman, Y. Cho, T. Castle, T. Gong, E. Jung, S. Yang, R. Kamien (2015). “Algorithmic Lattice Kirigami: A Route to Pluripotent Materials.” Figure 05.04 https://en.wikipedia.org/wiki/Gaussian_curvature Figure 09.01 K. Kuribayashi, K. Tsuchiya, Z. You, D. Tomus, M. Umemoto, T. Ito, M. Sasaki (2005). “Self-Deployable Origami Stent Grafts as a Biomedical Application of Ni-rich TiNi Shape Memory Alloy Foil.” Figure 09.02 www.robofold.com Figure 09.03 E. Hawkes, B. An, N. M. Benbernou, H. Tanaka, S. Kim, E. D. Demaine, D. Rus, and R. J. Wood (2010). “Programmable Matter by Folding.” Proceedings of the National Academy of Sciences. Figure 09.04 R. Tang, H. Huang, H. Tu, H. Liang, M. Liang, Z. Song, Y. Xu, H. Jiang, H. Yu (2014). Origami-enabled deformable silicon solar cells. Applied Physics Letters 104.

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ABSTRACT: The aim of this thesis is to bridge the gap between skin and structure. Programmable Folding explores computationally manipulating the geometry of kirigami folding patterns in order to approximate complex double curvature. These differentiated patterns can be translated to flat-pack sheets with an embedded assembly logic. By programming the underlying geometry and intrinsic curvature, the resulting kirigami patterns investigate potential design systems of how this logic can be applied to architectural design. KEY WORDS: folding – pattern – flat-pack – double-layer – kirigami - computational geometry – plastic – corrugation

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