Trihelical square tiling

The trihelical square tiling or facetted halved mucube is a regular skew apeirohedron that consists of square helices, with three meeting at each vertex. All square helices in the trihelical square tiling that share a vertex are perpendicular to each other.

The trihelical square tiling is the Petrie dual of the tetrahelical triangular tiling.

Vertex coordinates
The vertex coordinates of a trihelical square tiling of edge length 1 are given by


 * $$(2 \sqrt{2} i, 2 \sqrt{2} j, 2 \sqrt{2} k)$$
 * $$(2 \sqrt{2} i+ \sqrt{2}, 2 \sqrt{2}j+ \sqrt{2}, 2 \sqrt{2}k+ \sqrt{2})$$
 * $$(2 \sqrt{2} i, 2 \sqrt{2} j+ \frac{\sqrt{2}}{2}, 2 \sqrt{2} k+\frac{\sqrt{2}}{2})$$
 * $$(2 \sqrt{2} i+ \sqrt{2}, 2 \sqrt{2}j-\frac{\sqrt{2}}{2}, 2 \sqrt{2}k-\frac{\sqrt{2}}{2})$$
 * $$(2 \sqrt{2} i+ \frac{\sqrt{2}}{2}, 2 \sqrt{2} j+ \frac{\sqrt{2}}{2}, 2 \sqrt{2} k+ \sqrt{2} )$$
 * $$(2 \sqrt{2} i-\frac{\sqrt{2}}{2}, 2 \sqrt{2}j-\frac{\sqrt{2}}{2}, 2 \sqrt{2}k)$$
 * $$(2 \sqrt{2} i+ \frac{\sqrt{2}}{2}, 2 \sqrt{2} j, 2 \sqrt{2} k - \frac{\sqrt{2}}{2})$$
 * $$(2 \sqrt{2} i - \frac{\sqrt{2}}{2}, 2 \sqrt{2}j+ \sqrt{2}, 2 \sqrt{2}k+ \frac{\sqrt{2}}{2})$$

where $$i,j,k$$ range over the integers.

External references

 * jan Misali (2020). "there are 48 regular polyhedra".
 * jan Misali (2020). "there are 48 regular polyhedra".