Geometric Patterns as Architectural Forms

A Case-Study in Development of a Taxonomy of Tilings



Photo of Joshua Schultz, PhD, PE, LEED AP

Joshua Schultz, PhD, PE, LEED AP

Assistant Professor

Gonzaga University

Photo of Neil Katz, AIA

Neil Katz, AIA


Skidmore, Owings & Merrill



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

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“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

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Methodology and Examples

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

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Conclusions and Future Work

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

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The authors gratefully acknowledge Skidmore Owings and Merrill for the use of images.

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