Tetrahelical triangular tiling

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Tetrahelical triangular tiling
Schläfli symbol[1][note 1],
Facestriangular helices
Vertex figureTriangle
Petrie polygonssquare helices
Related polytopes
RegimentTrihelical square tiling
Petrie dualTrihelical square tiling
Abstract & topological properties
Schläfli type{∞,3}
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[edit | edit source]

The vertex coordinates of a tetrahelical triangular tiling of edge length 1 are given by all permutations of:

  • ,
  • ,

where range over the integers.

Notes[edit | edit source]

  1. This symbol is ad hoc. There is no general meaning to other than to distinguish it from other polytopes of the same Schläfli type.

External links[edit | edit source]

References[edit | edit source]

Bibliography[edit | edit source]

  • 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.