# Trihelical square tiling

Trihelical square tiling
Rank3
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{\infty,3\}^{(b)}}$[1]
Elements
FacesN square helices
Edges3M×N
Vertices2M×N
Vertex figureTriangle
Related polytopes
RegimentTrihelical square tiling
Petrie dualTetrahelical 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[2], 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:

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

where ${\displaystyle i,j,k}$ range over the integers.