Mutetrahedron

Mutetrahedron
Rank3
SpaceEuclidean
Notation
Bowers style acronymMut
Schläfli symbol
• {6,6∣3}
• ${\displaystyle \left\{6,{\frac {6}{1,3}}\mid 3\right\}}$
Elements
FacesN  hexagons
Edges3N
VerticesN
Vertex figureSkew hexagon
Petrie polygonsTriangular helices
Holes2N  triangles
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left({\dfrac {1}{3}}\right)\approx 70.52877}$
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.