Trihelical square tiling

Trihelical square tiling
Rank3
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{\infty ,3\}^{(b)}}$[1][note 1] ${\displaystyle \left\{{\frac {4}{1,0}},3:{\frac {3}{1,0}}\right\}}$
Elements
FacesN  square helices
Edges3M×N
Vertices2M×N
Vertex figureTriangle
Petrie polygonsTriangular helices
Related polytopes
RegimentTrihelical square tiling
Petrie dualTetrahelical triangular tiling
Abstract & topological properties
Schläfli type{∞,3}
Properties
ChiralYes
History
Discovered byBranko Grünbaum
First discovered1975

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 a chiral polyhedron; its helices are either all clockwise or all counterclockwise.

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.

Notes

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