# Muoctahedron

Muoctahedron
Rank3
SpaceEuclidean
Notation
Bowers style acronymMuo
Schläfli symbol{6,4∣4}, ${\displaystyle \left\{6,{\frac {4}{1,2}}\mid 4\right\}}$
Elements
Faces4N  hexagons
Edges12N
Vertices6N
Vertex figureSkew square
Petrie polygonsTriangular helices
Holes3N squares
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\dfrac {1}{3}}\right)\approx 109.47122^{\circ }}$
Related polytopes
ArmyBatch
RegimentBatch
DualMucube
Petrie dualPetrial muoctahedron
SkewingSkewed muoctahedron
κ ?Petrial muoctahedron
Convex hullBitruncated cubic honeycomb
Abstract & topological properties
Schläfli type{6,4}
OrientableYes
Genus
Properties
SymmetryR4×2
ConvexNo
Dimension vector(2,1,2)
History
Discovered byJohn Finders Petrie[2]
First discovered1926[1]

The muoctahedron or muo, short for multiple octahedron, is a regular skew apeirohedron in Euclidean 3-space. Its faces are hexagons, with 4 meeting at each vertex.

Note that on visual inspection, the muoctahedron 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 muoctahedron is in fact isotoxal like all other regular polyhedra.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the bitruncated cubic honeycomb.

## Constructions

As with other regular polyhedra with Schläfli types of the form {p,4}, the muoctahedron can also be constructed as the rectification of a polyhedron with the type {p,p}, in this case the halved mucube.

The muoctahedron can be constructed from the bitruncated cubic honeycomb. Its faces are the hexagonal faces of the bitruncated cubic honeycomb.

Abstractly it can also be formed as a quotient of the order-4 hexagonal tiling by identifying every 4th vertex on each hole. It can also be constructed as the first-order kappa κ 1  of the octahedron.

## Related polytopes

{6,4∣3} is one of several skew polyhedra of the form {6,4∣n}.

There is a direct correspondence between these and the regular polyhedra with triangular faces.