2021 Joint Mathematics Meetings

# Debora Coombs and Duane Bailey

## Statement

I am curious about quasicrystals; how they might grow naturally, and their relationship to higher dimensions. Penrose tiling is a precise mathematical projection of a five-dimensional cubic matrix. It also correlates with the structure of quasicrystals, providing a rich source for my work. These sculptures, made in collaboration with computer scientist Duane Bailey, lift and skew Penrose’s rhombs into crystalline surfaces. I’ve also made tilings in glass and mirror to project light; and from individual rhombic blocks of wood embedded with magnets to mimic bonds between particles. I use my lifelong experience as an artist and deep love of geometry to ask questions. To build, observe and explore a tiny, fascinating corner of Nature.

## Artworks

This sculpture is a patch of Penrose tiling raised into the third dimension. It is built from a single shape, a golden rhombus orientated according to its equivalent self in two-dimensions. If the tiling were to begin growing with the small pink and green cluster of tiles (lower center) the orientation of all dark tiles could be predicted to infinity. The orientation of the lighter colored tiles cannot. This is based upon the work of Laura Effinger-Dean who, in 2006, developed algorithmic methods for determining which tiles may be predicted to infinity. Viewed obliquely, the tiling appears disordered. Stereopsis makes it hard to see the classic Penrose. But shifting viewpoints reveal new sightlines and rhythmic sequences of tiles.
This sculpture is an oblique slice through a five-dimensional cubic matrix projected onto three dimensions. It was made by folding thousands of small rectangular cards into identical rhombs and gluing them together. Each rhombus is a projection of the square face of a hypercube. The skewed faces that are visible show up in ten different orientations and the sculpture’s color palette encodes this. The rhombs occur aperiodically positioned at different heights, adding random highlights and shadow while pearlescent surfaces allow colors to blend and scatter. These optical properties become even more fascinating at a very small scale where scientists are exploring their potential as an invisibility shield, or cloaking device.