# Mucube

Mucube
Rank3
SpaceEuclidean
Notation
Bowers style acronymMuc
Schläfli symbol${\displaystyle \{4,6\mid 4\}}$ ${\displaystyle \left\{4,{\frac {6}{1,3}}\mid 4\right\}}$
Elements
Faces3N squares
Edges6N
Vertices2N
Vertex figureSkew hexagon
Petrie polygonsTriangular helices
Holes3N squares
Measures (edge length 1)
Dihedral angle${\displaystyle 90^{\circ }}$
Related polytopes
DualMuoctahedron
Petrie dualPetrial mucube
HalvingHalved mucube
φ 2 Cube
κ ?Petrial mutetrahedron
Convex hullCubic honeycomb
Abstract & topological properties
Schläfli type{4,6}
OrientableYes
Genus
Properties
SymmetryR4×2
ConvexNo
Dimension vector(2,1,2)
History
Discovered byJohn Flinders Petrie[2]
First discovered1926[1]

The mucube, short for multiple cube, is a regular skew apeirohedron in Euclidean 3-space. Its faces are squares, with 6 meeting at each vertex.

The mucube can be constructed from the cubic honeycomb. Its faces are a subset of the faces of the cubic honeycomb, but with some removed (square holes) such that each set of coplanar faces turns into a checkerboard pattern. It can also be constructed as the first-order kappa κ 1  of the cube.

It can also be formed as a quotient of the order-6 square tiling by identifying every 4th vertex on each hole.

The mucube may at first appear to have two types of edges, "convex" and "concave". However, unlike traditional polyhedra, the mucube does not have an "interior" or "exterior"; rather its surface divides space into two congruent sections which are indistinguishable under the its symmetry group. Thus the two apparent types of edges are not actually distinct, and the mucube is in fact isotoxal like all other regular polyhedra.

## Vertex coordinates

Coordinates for the vertices of a mucube with unit edge length are given by (i, j, k) , where i , j , k  take on any integer value.

## Related polytopes

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

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

Each of the latter appears as a pseudocell of the corresponding former. This is a part of a general correspondence where {n,p} appears as a pseudocell of {4,2pn}.