Mutetrahedron

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Mutetrahedron
Rank3
SpaceEuclidean
Notation
Bowers style acronymMut
Schläfli symbol
  • {6,6∣3}
Elements
FacesN  hexagons
Edges3N 
VerticesN 
Vertex figureSkew hexagon
Petrie polygonsTriangular helices
Holes2N  triangles
Measures (edge length 1)
Dihedral angle
Related polytopes
ArmyQuarter cubic honeycomb
RegimentQuarter cubic honeycomb
DualMutetrahedron
Petrie dualPetrial mutetrahedron
φ 2 Tetrahedron
κ ?Petrial mucube
Abstract & topological properties
Schläfli type{6,6}
OrientableYes
Genus
Properties
SymmetryP4×2
ConvexNo
History
Discovered byH.S.M. Coxeter[1]

The mutetrahedron or mut, short for multiple tetrahedron, is a regular skew apeirohedron in Euclidean 3-space. Its faces are hexagons, with 6 meeting at each vertex.

The mutetrahedron is based on the cyclotruncated tetrahedral-octahedral honeycomb. Its faces are the hexagonal faces of the cyclotruncated tetrahedral-octahedral honeycomb, each trihexagonal tiling plane in it turns into a checkerboard pattern.

It can also be formed as a quotient of the order-6 hexagonal tiling by identifying every 3rd vertex on each hole, or as the first-order kappa κ 1  of the tetrahedron.

Note that on visual inspection, the mutetrahedron may at first appear to have two types of edges, "convex" and "concave". However, its "interior" and "exterior" are congruent and not distinguished by the polyhedron's symmetry group, so the two apparent types of edges are not actually distinct, and the mutetrahedron is in fact isotoxal like all other regular polyhedra.

External links[edit | edit source]

References[edit | edit source]

Bibliography[edit | edit source]

  • Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. pp. 333–335. ISBN 9781439864890.
  • Coxeter, H.S.M. (1999). The Beauty of Geometry: Twelve essays. Dover Publications, Inc. pp. 157 ff. ISBN 9780486409191.
  • McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.