# Mucube

Mucube Rank3
SpaceEuclidean
Notation
Schläfli symbol{4, 6 | 4}
Elements
Faces3N squares
Edges6N
Vertices2N
Vertex figureSkew hexagon
Related polytopes
DualMuoctahedron
Petrie dualPetrial mucube
Convex hullCubic honeycomb
Abstract properties
Schläfli type{4,6}
Topological properties
OrientableYes
Genus
Properties
ConvexNo

The mucube or muc, short for multiple cube, is one of the three regular skew apeirohedra in Euclidean 3-space. It's an infinite polyhedron that consists solely of squares, with 6 meeting at each vertex.

The mucube is based on 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. In fact, when the cubic honeycomb is being given as x4o3o4x, then the here solely being used squares are the ones of type x . . x.

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

## 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.