# Tetrahelical triangular tiling

Tetrahelical triangular tiling
Rank3
SpaceEuclidean
Notation
Schläfli symbol$\{\infty,3\}^{(a)}$ Elements
Facestriangular helices
Edges
Vertices
Vertex figureTriangle
Related polytopes
RegimentTrihelical square tiling
Petrie dualTrihelical 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.