# Mutetrahedron

Mutetrahedron
Rank3
SpaceEuclidean
Notation
Schläfli symbol${\displaystyle \{6, 6 \mid 3\}}$
Elements
FacesN hexagons
Edges3N
VerticesN
Vertex figureSkew hexagon
Related polytopes
ArmyQuarter cubic honeycomb
RegimentQuarter cubic honeycomb
DualMutetrahedron
Petrie dualPetrial mutetrahedron
Abstract properties
Schläfli type{6,6}
Topological properties
OrientableYes
Genus
Properties
ConvexNo

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.