Facade design is a key component of architectural expression, and increasingly a key design consideration due to growing importance of factors like: energy efficiency, occupant comfort, cultural contextualization, and architectural intent. With the rapid rise in computing power, engineers and architects are able to create wide ranges of patterns quickly using parametric modeling (e.g., Grasshopper). However, this increase in modeling ability can leave designers without baseline guidance and design benchmarks. While lattices, tilings and tessellations have long been a significant part of architectural expression, there is no cohesive epistemology and taxonomy for geometric patterns as architectural forms. As a first step in addressing this dearth of systemic organization, variations on regular polygons (i.e., bending the edges) and the patterns generated by edge-to-edge repetition, based on regular and semi-regular tilings are presented. A simple set of rules and parameters for manipulating polygon edges to generate geometrically and visually interesting infinite lattices (and transformations) is methodically introduced. The resulting geometric patterns can be viewed through an architectural engineering lens to develop a systematic taxonomy of Platonic and Archimedean patterns.
This paper focuses on reifying otherwise abstract concepts while simultaneously tying together disparate and relatively inchoate concepts scattered around the literature. While the concept as currently presented appears theoretical, lattices can provide direct solutions to design problems, or inspiration for further exploration. These particularly can be used to address critical issues in design today, with great potential in areas such as interactive and responsive facades. Several architectural projects are cited to provide examples of how this methodology has been used to generate useful facade designs. The organization presented herein serves as an initial study into development of a taxonomy of architectural patterns.
With the rapid expansion of parametric modeling software, development of new patterns and geometric forms has never been easier. However, the rise of parametric modelling necessitates development of a language
“Tiles” and tilings are ubiquitous in our built environment. Examples of regular tilings in a plane, composed of regular polygons (“tiles”), exist in nature, and have been known for centuries
This section presents the rules governing edge-bending of polygons (regular and semi-regular) and their subsequent assembly into infinite tilings. To date, there does not exist in the literature a consistent
A holistic geometrical framework for manipulation of polygons based on an extremely simple rule of bending the edges, which generates infinite sub-sets of new polygons is presented. These polygons generate
The authors gratefully acknowledge Skidmore Owings and Merrill for the use of images.
Aurenhammer, Franz, Rolf Klein, and Der-Tsai Lee. Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Company, 2013.
Beghini, Alessandro, Lauren L. Beghini, Joshua A. Schultz, Juan Carrion, and William F. Baker. "Rankine’s theorem for the design of cable structures." Structural and Multidisciplinary Optimization 48, no. 5 (2013): 877-892.
Coxeter, Harold Scott Macdonald. Regular polytopes. Courier Corporation, 1973.
Ghosh, S., and R. L. Mallett. "Voronoi cell finite elements." Computers & Structures 50, no. 1 (1994): 33-46.
Grünbaum, Branko, and Geoffrey C. Shephard. "Tilings by regular polygons." Mathematics Magazine 50, no. 5 (1977): 227-247.
Grünbaum, Branko, and Geoffrey Colin Shephard. Tilings and patterns. Freeman, 1987.
Kaplan, Craig S. "Computer generated islamic star patterns." Bridges (2000): 105-112.
Kaplan, Craig S. "Islamic star patterns from polygons in contact." In Proceedings of Graphics Interface 2005, pp. 177-185. Canadian Human-Computer Communications Society, 2005.
Kaplan, Craig S., and David H. Salesin. "Islamic star patterns in absolute geometry." ACM Transactions on Graphics (TOG) 23, no. 2 (2004): 97-119.
Lalvani, Haresh. "Crescent-shaped polygonal tiles." U.S. Patent 4,620,998, issued November 4, 1986.
Lalvani, Haresh. 1997. “Visual Morphology of Space Labyrinths: A Source for Architecture and Design.” In Beyond the Cube: The Architecture of Space Frames, edited Jean-Francois Gabriel. New York: John Wiley & Sons, Inc. 409-426.
Neyrinck, Mark. "The Cosmic Spiderweb and General Origami Tessellation Design." arXiv preprint arXiv:1809.00015 (2018).
Pietroni, Nico, Davide Tonelli, Enrico Puppo, Maurizio Froli, Roberto Scopigno, and Paolo Cignoni. "Voronoi grid-shell structures." (2014).
Pottman, Helmut. "Geometry and new and future spatial patterns." Architectural Design 79, no. 6 (2009): 60-65.
Pottmann, Helmut, Qixing Huang, Bailin Deng, Alexander Schiftner, Martin Kilian, Leonidas Guibas, and Johannes Wallner. "Geodesic patterns." In ACM Transactions on Graphics (TOG), vol. 29, no. 4, p. 43. ACM, 2010.
Schultz, Joshua and and Neil Katz. “Origami-Inspired Facade Design:Parametric Studies for Architectural and Structural Efficiency.” In Proceedings of FTI 2nd World Congress 2018