Jared Pincus

Statement

I am a PhD student in computer science, an aspiring professor, and a dabbler in recreational math, game design, and graphics. I learned of the Trihelical Square Tiling (TST) via Mitch Halley, who also coined its name [youtu.be/_hjRvZYkAgA]. Mesmerized by the TST's structure, I sought to visualize it with a physical model. The TST, formally $\{\infty,3\}^{(b)}$, is a regular apeirohedron comprised of square helices spiraling in 3 orthogonal directions. Its Petrial, $\{\infty,3\}^{(a)}$, consists of triangular helices spiraling in 4 directions. These 7 axes match the 7 pairs of faces of a cuboctahedron. Certain crystal lattices, namely strontium silicide, naturally conform to the TST’s structure [arxiv.org/abs/1403.0045].

Artworks

Displayed is a finite section of the TST, built modularly from 3D-printed pieces. The red and blue axes indicate the axes of the square and triangular helices respectively. Each vertex of the TST has 3 coplanar edges. Each piece of the model corresponds to a vertex, with 3 “spokes” that connect to other pieces to form full edges. Spokes slot together at an angle, to offset their pieces by $2\arctan{\frac{1}{\sqrt{2}}}\approx 70.5°$. Due to the symmetries of the TST, it is impossible to connect any pieces "incorrectly". The pieces are also designed to fit together snugly, though some have been glued together for the sake of their display.