Trihelical square tiling

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Trihelical square tiling
Rank3
SpaceEuclidean
Notation
Schläfli symbol[1][note 1]
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[edit | edit source]

The vertex coordinates of a trihelical square 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.