Concept

First appearing around 700 AD, patterns known as “girih,” Persian for “knot,” became a dominant design element across Western Asia. By the 15th century, girih had been reimagined as tiles that could be combined to form quasi-crystalline aperiodic motifs, 500 years before their description by modern mathematics. In 1982, aperiodic diffraction patterns were observed in the atomic lattice of aluminum alloys, affirming their existence in nature. In addition to their mathematical and materials science applications, the large corpus of aperiodic tilings begs artistic study with rich creative value in textile design, sonification, texture synthesis, digital sequencing, and algorithmic art.

This work explores aperiodic pattern making by illustrating the base rule of a tiling as it aligns to each successive generation, its internal rhythms and harmonies revealed through increasingly granular subdivisions.

Development

These tilings are based on the substitution rules found in the Tilings Encyclopedia. Using Houdini, the prototiles were created from trigonometric definitions in the Encylopedia and scaled, rotated, and transformed to form each rule. The rules were defined as functions that recursively fed back into themselves, eventually growing into a sprawling patchwork of undulating rhythms and interconnected polygons.

In some cases, surfaces are perturbed by blending the colors of shared vertices, creating a bas-relief foundation for refraction. In others, overlapping tubular geometry from increasing substitution iterations illustrate the parameterization and affine transformations of the prototiles while highlighting self-similarity.

Images were created at extremely high resolution using photorealistic rendering techniques and printed on a large format, high-DPI printer. Careful attention was paid to drawing the viewer in and rewarding them for close inspection. When exhibited together as a set, each work contributes to the complete rule description, encouraging a longer duration of study.

Exhibitions

2017  The MOXI Museum. Santa Barbara, CA
2017  Rehabituation. Santa Barbara, CA
2016  Computation and Expression. Santa Barbara, CA
2016  Experience Workshop. Finland and Hungary
2016  Bridges Mathematical Art Conference. Jyvӓskylӓ, Finland
2016  White Noise. Santa Barbara, CA

Prototiles and their associated rules for the Watanabe Ito Soma 12-fold tiling.

Prototiles and their associated rules for the Watanabe Ito Soma 12-fold tiling.

 
Subspaces , The Wolf Museum of Exploration and Innovation (MOXI). Santa Barbara, CA. June 2017.

Subspaces, The Wolf Museum of Exploration and Innovation (MOXI). Santa Barbara, CA. June 2017.

Subspaces , Rehabituation: MAT End of Year Show. UC Santa Barbara. May 2017.

Subspaces, Rehabituation: MAT End of Year Show. UC Santa Barbara. May 2017.