Jared Pincus
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].
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.