Tetrahelical triangular tiling

Tetrahelical triangular tiling
Rank	3
Space	Euclidean
Notation
Schläfli symbol	$\{\infty ,3\}^{(a)}$, $\left\{{\frac {3}{1,0}},3:{\frac {4}{1,0}}\right\}$
Elements
Faces	∞ triangular helices
Edges	∞
Vertices	∞
Vertex figure	Triangle
Petrie polygons	∞ square helices
Related polytopes
Army	Octet
Regiment	Trihelical square tiling
Petrie dual	Trihelical square tiling
Abstract & topological properties
Schläfli type	{∞,3}
Properties
Chiral	Yes
History
Discovered by	Grünbaum
First discovered	1975

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. 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:

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

where $i,j,k$ range over the integers.

External links

• jan Misali (2020). "there are 48 regular polyhedra".

Bibliography

• Grünbaum, Branko (1975), "Regular polyhedra - old and new" (PDF), Aequationes Mathematicae
• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.