Mutetrahedron

Mutetrahedron
Rank	3
Space	Euclidean
Notation
Schläfli symbol	$\{6, 6 \mid 3\}$
Elements
Faces	N hexagons
Edges	3N
Vertices	N
Vertex figure	Skew hexagon
Related polytopes
Army	Quarter cubic honeycomb
Regiment	Quarter cubic honeycomb
Dual	Mutetrahedron
Petrie dual	Petrial mutetrahedron
Abstract properties
Schläfli type	{6,6}
Topological properties
Orientable	Yes
Genus	∞
Properties
Convex	No

The mutetrahedron or mut, short for multiple tetrahedron, is one of the three regular skew apeirohedra in Euclidean 3-space. It's an infinite polyhedron that consists solely of 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 modwrap of the order-6 hexagonal tiling by identifying every 3rd vertex on each hole.

External links

• jan Misali (2020). "there are 48 regular polyhedra".
• Klitzing, Richard. "hihexat".

• Klitzing, Richard. "Skew polytopes".
• Wikipedia Contributors. "Regular skew apeirohedron".

References

• 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.