Trihelical square tiling

Trihelical square tiling
Rank	3
Space	Euclidean
Notation
Schläfli symbol	$\{\infty,3\}^{(b)}$
Elements
Faces	N square helices
Edges	3M×N
Vertices	2M×N
Vertex figure	Triangle
Related polytopes
Regiment	Trihelical square tiling
Petrie dual	Tetrahelical triangular tiling
Abstract properties
Schläfli type	{∞,3}

The trihelical square tiling or petrial facetted halved mucube is a regular skew apeirohedron that consists of square helices, with three meeting at each vertex. All adjacent 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. It also is the second-order facetting of the petrial halved mucube, so the edges and vertices of the trihelical square tiling are a subset of those found in the halved mucube.

Vertex coordinates

The vertex coordinates of a trihelical square tiling of edge length 1 are given by all permutations of:

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

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

External links

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

Bibliography

• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.