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A402a PATTERN OPERABILITY Daniel Nguyen | Neil Leach, Biayna Bogosian

Table of Contents Introduction.....................................................................................................................................................................1 Islamic Tiling and Muqarnas Dome Studies..................................................................................................................2-7 Biological Systems and Scale Efficiency.....................................................................................................................8-11 Modeling, Material Studies.......................................................................................................................................12-26 Final Design.............................................................................................................................................................27-58 Conclusion...............................................................................................................................................................59-60 Bibliography..................................................................................................................................................................61


The basis of this research is to study a set of patterns found in Islamic tiling architecture, then begin to either contradict that investigation with biological systems, and ultimately demonstrate how these logics can be analyzed and interpreted to be used for architectural purposes. This process will develop a strong argument for implementation in an installation that is computational based on parametricism and pattern logic.


Muqarnas vault studies are examples of static structures that create vaults for buildings with sophisticated patterns, but are unable to change according to new circumstances. Biological and computational studies will demonstrate how systems can scale through growth or movement in order to adapt to conditions through their inherent material logic, but still obtain the same initial pattern.


Examine systems/logics that allow them to perform these actions, and introduce those into an installation that would serve as a prototype for a larger object.





Girih Tiles Girih tiles are a set of five tiles that were used in the creation of tiling patterns for decoration of buildings in Islamic architecture



Regular Decagon


Elongated (Irregular Convex) Hexagon

Source(s): “Ancient Islamic Penrose Tiles | Numbers | Science News.� Ancient Islamic Penrose Tiles | Numbers | Science News.


Patterns Used in Islamic Vaults

Diamond St. Andrew’s Cross Oblique Octangular Star

Diamond Greek Cross

Oblique Octangular Star

Greek Cross

Cruciform Octangular Star

Core Octagon Perpendicular Octangular Star Core Circle Cruciform Octangular Star Basic Octagon Peripheral Circle

The diagrams above consist of a series of common pattern logics that are implemented in Islamic tiling as well as vaulting mechanisms in order to provide a packing logic. In addition, it demonstrates how a series of vectors can be extrapolated to create larger geometries while simply extending the initial linework.

Source(s): Pereira, JosĂŠ. The Sacred Architecture of Islam. New Delhi: Aryan International, 2004. Print.


Islamic Muqarnas

The Arabic word for stalactite vault, it is an architectural ornament developed around the middle of the 10th century in north eastern Iran and simultaneously in central North Africa involves 3D architectural decorations of niche-like elements arranged in tiers. Muqarnas take the form of small pointed niches, stacked in tiers projecting beyond those below and can be constructed in brick, stone, stucco or wood. The two-dimensional projection of Muqarnas vaults consists of a small variety of simple geometrical elements.

Source(s): Grabar, Oleg. Muqarnas: An Annual on Islamic Art and Architecture. New Haven: Yale UP, 1983. Print.


Muqarnas Assembly Types

Passage from Square to Circle

Rectilinear Muqarnas

Muwarnas Niche

Passage from Square to Octagon

Curvilinear Muqarnas

Muqarnas Forming Stalactites




The diagrams above illustrate the varying types of Muqarnas, which is a decorative corbel commonly used in Islamic architecture. Their pattern logic allow them to create vaulted ceilings and conceal a considerable amount of structure within the domes and vaults. The illustration to the left shows the utilization of squinches in a dome’s ceiling as a construction filling, which would ultimately allow for a base to receive a spherical or octagonal dome.

Source(s): Pereira, JosĂŠ. The Sacred Architecture of Islam. New Delhi: Aryan International, 2004. Print.


Geometric Pattern Logic

Architectural Tiling Implementations

Gaining an understanding of the underlying pattern logic utilized in these ancient muqarnas domes demonstrates how tiles or cells can come together and can be assembled as a whole aggregated. Further investigation into how these tiles and patterns can be implemented into three dimensional space by variegation their connection points and interrelationships will suggest a more adaptable pattern logic than is available in these static muqarnas domes. Source(s): Brend, Barbara. Islamic Art. Cambridge, MA: Harvard UP, 1991. Print. Pereira, JosĂŠ. The Sacred Architecture of Islam. New Delhi: Aryan International, 2004. Print.





Fibonacci Numbers / Sequences

Fibonacci numbers are generated from a function such that adding preceding terms will allow for an increasing set of integers. Functions: 0, 1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, ... Resulting Integers: 0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... Introducing this logic into a series of squares that scale up in size according to the fibonaacci numbers will create what is considered the golden spiral, and it is implemented in several natural systems in order to act as an efficience packing strategy for cells.

2 3 1




Source(s): Dunlap, R. A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997. Print. â&#x20AC;&#x153;Flowers and Fibonacci.â&#x20AC;? Flowers and Fibonacci. N.p., n.d. Web. 25 Nov. 2012. < rpamaths/rpampages/sunflower.html>.


Scale Packing Strategies Using Fibonacci Series in Nature

Pine Cone Scales

Sunflower Florets

Geometric Pattern

Geometric Pattern





The Fibonacci series is utilized in natural occurrences like the geometry of a pine cone’s scales or florets that grow on a sunflower because it is considered the most efficient packing methodology. In these two instances, there is noticeably 8 scales rotating counterclockwise toward the edge and 13 rotating clockwise, while the sunflower features 13 counterclockwise cells rotating to the exterior edge and 21 rotating clockwise.

Source(s): Dunlap, R. A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997. Print. “Flowers and Fibonacci.” Flowers and Fibonacci. N.p., n.d. Web. 25 Nov. 2012. < rpamaths/rpampages/sunflower.html>.


Skin Studies

An analysis of overlapping and shifting skin cell conditions in biological systems demonstrates how inherent geometries in biological skin cells or “scutes” allows the best possible surface pattern in order for skin to be shed and a underlying layer to be exposed for further growth. Studying tiling pattern logic in natural skin systems provided a larger understanding of how scales come together to assemble flexible vaulting and skin systems that could adapt to varying conditions.

Source(s): “The Leading Provider of Science Images and Footage.” Science Photo Library. N.p., n.d. Web. 26 Nov. 2012. <http://www.>. 11



Precedent Studies







Examples were studied that show how a series of geometries could be expanded and contracted through a mechanism that is allowed by their pattern. These systematic approaches would suggest a logic to open and close; however, these models only promote a two dimensions of expansion and contraction. These were simply studied to see how a packing pattern logic could be used to open and close a geometry.




Source(s): â&#x20AC;&#x153;Home Geometric Toy Tile Magic-5.â&#x20AC;? Home Geometric Toy Tile Magic-5. N.p., n.d. Web. 25 Nov. 2012. <http://>.


Experimenting with Definitions in Rhinocerous / Grasshopper


Utilizing a definition that implements a series of vector poitns in Rhinocerous, sliders were utilized to adjust the scale and step size of the geometry. These parameters allowed for the initial geometry to be adjusted to the desired size while the f(x) and f(y) function propoagate further geometries outward from the center.

Source(s): â&#x20AC;&#x153;Grasshopper - Generative Modeling for Rhino.â&#x20AC;? Grasshopper - Generative Modeling for Rhino. N.p., n.d. Web. 26 Nov. 2012. <>. 14

Experimenting with Definitions in Rhinocerous / Grasshopper


Analyzing the pattern logic used in the muqarnas tiling as well as the biological studies, a composition of the research resulted in a geometry with a central attractor point whereby the geometrical logic would be harnessed from. The “scales” in the installation would presumably be capable of sliding past one another, and the initial geometric pattern developed by a fibonacci series would make this possible.

Source(s): “Grasshopper - Generative Modeling for Rhino.” Grasshopper - Generative Modeling for Rhino. N.p., n.d. Web. 26 Nov. 2012. <>.


Step / Scale Width

Riser / Scale Thickness

Run / Scale Length

angle of rotation

In a spiral staircase, there is a primary system that the steps align with while a secondary system acts as a series of points that the steps are arrayed out to that form a consistency between all the steps; the mechanism by which the installation would operate, would function similarily in how the scales overlap


Moiré Effect

Variating / Scales variatingCells cells/scales

connectionPoints points Connection

A series of overlapping transparent scales would enable a filtered light effect and playing with transparency through thickness and porosity of material

Phsyical study conducted to achieve the moiré effect found in overlapping grids done with varying colored cells. Cells could shift, overlap, and rotate in order to distort the appearance of the surface.

Source(s): Oster, Gerry, and Yasunori Nishijima. Moiré Patterns,. San Francisco: W.H. Freeman, 1963. Print.


Chuck Hoberman With a background in design and engineering, Chuck Hoberman is known for several installations that allow a set geometry to be able to perform several actions and adapt to circumstances when forces are applied. He is widely known for the Hoberman sphere, an object that a group of the students in the studio examined in order to comprehend how similar performance could be duplicated in a pattern or material logic. The experiments conducted in studio thereafter were used to determine a pattern logic that would facilitate similar flexibility and allow for an object that could be rapidly installed and generate a space.

Hoberman Sphere

Chuck Hoberman. New York: Museum of Modern Art, 1994. Print.


Three Formal Studies

Studies were conducting using the golden spiral generated by the fibonacci series that created a pattern that could be extruded and extended in order to translate from a strictly 2D pattern to a 3D form. This pattern would provide a much more flexible system that could be rearranged according to its desired use; furthermore, it would allow the lattice to be stacked and packed away when no longer needed.

This system largely focused on the idea of extruding the existing spiral pattern that could perhaps be stacked as a primary system, while it would be suspended by a secondary system using rope or wire.


This study was performed in order to understand how two of the same spirals could be connected from end to end in order to continue the form of this golden spiral while simply aggregating one unit continuously. In addition, it proposed the idea of surrrounding and engulfing the user within the space.

Lastly, this study was a summation of the previous studies in that we utilized a spiraling set of diamonds that would allow not only for a more defined geometry, but give the spiraling legs a more crystalline formation, providing space to play with transparency and opaqueness of the material.


Further studies were conducted in order establish more efficient connections between the units. Tape and rubber bands were utilized for the connections between the cells. This methodology of connections allowed the sequence of cells to snap back; therefore, the usage of rubber bands is prevalent in future studies in order to achieve this same effect.


Individual Unit Assembly

Connection Points

Folding Lines



Bent Scheme A

Bent Scheme B This study model was constructed to demonstrate how connection points could allow a fairly rigid cell to snap back and forth when applying pressure to areas where four separate cells meet. This would allow operability for the apertures in the vault and allow a varying surface condition. 22




This study was conducted to determine whether a simplified set of triangles could be generated the golden spiral and still assembled to create a vaulted dome. The golden sprial pattern was overlaid with a series of vectors connecting interlocking points, which is where the connection points would be established to hold the units together. Unfortunately, the usage of simple rubber band connections was not sufficient in holding the structure together, and it required a rigid wire at the base and top in order to retain its form, so ultimately it became more a study of flexible and rigid connection points in the unit aggregation.

Source(s): Jean, Roger V. Phyllotaxis: A Systemic Study of Plant Pattern Morphogenesis. Cambridge [England: Cambridge UP, 1994. Print.Cambridge UP, 1994. Print. index.html>. 23



Contracted Considering that the research focused largely on pyramid, conical forms, and the research focused on an operable unit that was capable of being installed on a site and being adaptable, a study model was built to determine the variability a subdivided cube could provide. It was ideal that the user of the deployable tiles could bring this object to the site and install it according to the desired use in mind. In this study, the corners were the points of operabilty and pyramid cells could be turned on an axis from that point. In addition, the rubber band connections allow a recoiling behavior in the object so that one could open up the object, but its inherent behavior would force it to snap back into the cube form.

Deployed 24

Cube - Enclosed

Cube - Unrolled


Pentagon - Triangle Connections

Pentagon - Pentagon Connections Additional study models were built to see if a vaulting system could be constructed using solely pentagons and triangular shaped cells, and they were successful in creating arches that would span above the user; however, they were no more operable than a static muqarnas dome; therefore, further studies would have to be applied in ordert obtain a unit aggregate that would allow an operable tesselation of similar units. 26





Building off the previous subdivided cube study, this was a culmination of the previous investigations and the resultant was essentially a dodecahedron that would allow a immense operability while utilizing little volume prior to installaiton in the space. Considering that the installation would provide little time to set the proposed object in the space, the goal was to generate a vaulting mechanism that could be introduced rapidly, and have the connection points provide considerable operability. The dodecahedron is made up of several pentagonal crystals, while two of them are divided into five smaller triangular edged crystals. The rationality behind the subdivision of these two points is to create a female connection for additional dodecahedrons to mount onto the object and allow the object to not be just solitary vault, but a larger aggregate that could span larger spaces.








Process of Assembly


2 Once we had the foundation for the object we began material studies in order to determine the potential of playing with transparency and irridescence of the material utilized on the final implementation. Studies performed with 3M irridescent adhesive film applied to the plexi would create a intriguing crystalline mass, and would illustrate a curious field effect when implemetned at full scale. The user would be able to inhabit the space, and the variation between outside and inside would compell the inhabitants of the space to explore the interior and exterior of the inverted installation. The investigative study would suggest that we had developed some sort of inverted muqarnas vaulting mechanism, that was not only compact when necesesary, but ultimately, an expansive vaulting scheme that could ultimately provide shelter, while the openings in the pentagonal shaped crystalls would distort and refract light into the space.





Individual Unit from Proposed Installation



Building Types / Unit Assemblages

Initial design proposed at a larger scale that would adequate for human inhabitation and utilization. This scheme suggested a logic that would allow people to inhabit the spaces carved out by the vaults and domes.


Building Types / Unit Assemblages

In addition, a secondary scheme was developed that would encourage the developed inverted muqarnas components to stack and arch over over spaces. The initial intended use case was sculptural components that would work alongside the dome scheme; however, it yielded very little interactivity for the user.


Assembly Methodologies: Cluster

Unit Aggregates

Units Assembled


Assembly Methodolgies: Tier

Fourth Tier

Third TierThird


Second Tier Second Tier

First Tier

First Tier


Edge Details








1 Pressure Plate Anchor 2 Tempered Double Pane Glazing with embedded 3M Radiant film 3 Silicon Strip Sealant 4 Rotating Steel Hinge 5 Inner Aluminum Extrusion 6 Outer Clamp Extrusion 7 Magnetic Locking Notch


Installation Model Interlocking Joint Connections

Static Edges

Rotational Joints

Interlocking Joints

For the purposes of the physical installaitno we would install in Lower Rosendin we considered either CNC milling or lasercutting grooves and joints as part of the fabrication components that would allow these units to share edge to edge connections. Some of the early studies like the interwoven had effective results; therefore, they were utilized in the final installation.


Full Scale Interlocking Joint Connections




This simple diagrams explains the process when two edge conditions meet and how the adaptive hinges allow latches on to interlock.


Joint Flexibility

The joints allow for considerable operability since they feature a rotational aspect to


Building Types / Unit Assemblages

Tower A 594 Edge Connections / 297 of Units / 1782 Faces


Building Types / Unit Assemblages

Tower B 565 Edge Connections / 113 of Units / 678 of Faces


Building Types / Unit Assemblages

Tower C 1188 Edge Connections / 594 Units / 7128 of Faces


Building Types / Unit Assemblages

Tower D 460 Edge Connections / 230 Units / 1380 Faces


Building Types / Unit Assemblages

Vault A 164 Edge Connections / 82 Units / 492 Faces


Building Types / Unit Assemblages

Vault B 112 Edge Connections / 56 Units / 336 of Faces


Building Types / Unit Assemblages

Vault C 450 Edge Connections / 150 Units / 900 of Faces


Building Types / Unit Assemblages

Vault D 92 Edge Connections / 46 Units / 276 Faces


Building Types / Unit Assemblages

Building Skin A 248 Edge Connections / 124 Units / 744 of Faces


Building Types / Unit Assemblages

Building Skin B 160 Edge Connections / 80 Units / 480 of Faces


Building Types / Unit Assemblages

Building Skin C 292 Edge Connections / 146 Units / 876 Faces


Building Types / Unit Assemblages

Building Skin D 234 Edge Connections / 117 of Units / 702 Faces


Building Types / Unit Assemblages

Dome A 104 Edge Connections / 52 Units / 312 Faces


Building Types / Unit Assemblages

Dome B 130 Edge Connections / 65 Units / 390 Faces


Building Types / Unit Assemblages

Half Dome 84 Connections / 42 Units / 252 Faces


Building Types / Unit Assemblages

Dome C 42 Edge Connections / 21 of Units / 126 of Faces


The realized design would suggest areas for the users to occupy as well as interact with one another. The idea was that these units would allow for temporary installation to be used for varying purposes, and then when the object was no longer needed it could be disassembled and removed from the site.


This installation served as a working prototype that would illustrate how these units could be put together and used to compose the larger aggregate suggested in the drawings. 3M radiant film was applied to 1/16th inch plexi, and the patterns were laser cut. Zip ties were then used to support the triangular units together. To limit the amount of weight in the units, each unit would have three faces that were open to limit the amount of material weight. The arrived design would allow for rapid installation in varying locations and depending on the desired use case could serve as a adaptable scheme that could be applied anywhere. The investigation conducted in the studio provided a developed understanding of the varying flexibility in a pattern logic while also demonstrating the ephemeral light quality a material can have on a space. 57






Through the investigation of Islamic tiling patterns, biological pattern studies, computational and physical modeling, a developed understanding was yielded that demonstrates how a simple pattern assembled using a series of similar units can be assembled to create a larger aggregate. The methods used in ancient Islamic Muqarnas and tiling patterns are reintroduced into these modern systems that are active and operable, while requiring a small footprint for storage of the units themselves. Similar to the Islamic tiling patterns and Muqarnas domes and vaults, a simple geometry allow for a greater aggregate whole, which was reproduced in our final proposal. The Biological studies served as an additional investigation into understanding how scales are used in skin systems to adapt to change in the specimenâ&#x20AC;&#x2122;s change in size over time through shedding systems or the overlapping of new scales as old ones decay. This process in dynamic skin systems discussed the idea of overlapping, sliding and shifting, which differentiated from the static system of pattern logic in a Islamic tiling system that resisted any movement. The physical model studies served to develop an assembly unit that could be aggregated in a large scale system that could easily be packed away and removed from the site, but also efficient in its utilization of connections. The dodecahedron provides variability in its inherent geometry while allowing for hundreds of different methods for assembly, and it was the driving force towards the end result. In addition, a focus of the studio was material intelligence and the effect it can have on a space; the 3M radiant film applied to the surface of the acrylic and the material effect it would propose at a larger scale would present an ephemeral light quality about this object and would invite users to interrogate the spaces and volumes.


Bibliography “Ancient Islamic Penrose Tiles | Numbers | Science News.” Ancient Islamic Penrose Tiles | Numbers | Science News. Brayer, Marie-Ange. Biothing, Alisa Andrasek. Orléans: HYX, 2009. Print. Brend, Barbara. Islamic Art. Cambridge, MA: Harvard UP, 1991. Print. Chuck Hoberman. New York: Museum of Modern Art, 1994. Print. Dunlap, R. A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997. Print. “Flowers and Fibonacci.” Flowers and Fibonacci. N.p., n.d. Web. 25 Nov. 2012. < uk/rpamaths/rpampages/sunflower.html>. “Grasshopper - Generative Modeling for Rhino.” Grasshopper - Generative Modeling for Rhino. N.p., n.d. Web. 26 Nov. 2012. <>. “Home Geometric Toy Tile Magic-5.” Home Geometric Toy Tile Magic-5. N.p., n.d. Web. 25 Nov. 2012. <>. “Islamic Textiles.” Islamic Textiles. N.p., n.d. Web. 11 Sept. 2012. < islamicT.html>. Kheiri, Sattar. Islamic Architecture. London: J. Tiranti, 1923. Print. Meinecke, Michael. Patterns of Stylistic Changes in Islamic Architecture: Local Traditions versus Migrating Artists. New York: New York UP, 1996. Print. Miller, Susan Gilson., Mauro Bertagnin, Emily Gottreich, and William Granara. The Architecture and Memory of the Minority Quarter in the Muslim Mediterranean City. Cambridge, MA: Harvard Univ Graduate School of Design, 2010. Print. Moussavi, Farshid, and Daniel Lopez. The Function of Form. Barcelona: Actar, 2009. Print. Oster, Gerry, and Yasunori Nishijima. Moiré Patterns,. San Francisco: W.H. Freeman, 1963. Print. Pereira, José. The Sacred Architecture of Islam. New Delhi: Aryan International, 2004. Print. Robinson, Chase F. A Medieval Islamic City Reconsidered: An Interdisciplinary Approach to Samarra. Oxford: Oxford UP, 2001. Print.



A402a_Fall 2012  

Research documentation compiled in my Fall 2012 architectural studio.