Tetrahelical triangular tiling

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

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

Vertex coordinates

The vertex coordinates of a tetrahelical triangular 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.