# Tetrahelical triangular tiling

Tetrahelical triangular tiling
Rank3
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{\infty ,3\}^{(a)}}$[1][note 1], ${\displaystyle \left\{{\frac {3}{1,0}},3:{\frac {4}{1,0}}\right\}}$
Elements
Facestriangular helices
Edges
Vertices
Vertex figureTriangle
Petrie polygonssquare helices
Related polytopes
ArmyOctet
RegimentTrihelical square tiling
Petrie dualTrihelical square tiling
Abstract & topological properties
Schläfli type{∞,3}
Properties
ChiralYes
History
Discovered byGrünbaum
First discovered1975

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[2], so the edges and vertices of the tetrahelical triangular tiling are a subset of those found in the halved mucube. The tetrahelical triangular tiling is a chiral polyhedron; its helices are either all clockwise or all counterclockwise.

## Vertex coordinates

The vertex coordinates of a tetrahelical triangular 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.

## Notes

1. This symbol is ad hoc. There is no general meaning to ${\displaystyle (a)}$ other than to distinguish it from other polytopes of the same Schläfli type.