Mucube
| Mucube | |
|---|---|
 | |
| Rank | 3 |
| Space | Euclidean |
| Notation | |
| Schläfli symbol | {4, 6 | 4} |
| Elements | |
| Faces | 3N squares |
| Edges | 6N |
| Vertices | 2N |
| Vertex figure | Skew hexagon |
| Related polytopes | |
| Dual | Muoctahedron |
| Petrie dual | Petrial mucube |
| Convex hull | Cubic honeycomb |
| Abstract properties | |
| Schläfli type | {4,6} |
| Topological properties | |
| Orientable | Yes |
| Genus | ∞ |
| Properties | |
| Convex | No |
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[edit | edit source]
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.
Derivatives[edit | edit source]
External links[edit | edit source]
- Wikipedia Contributors. "Regular skew apeirohedron".
- Klitzing, Richard. "Skew polytopes".
- Klitzing, Richard. "hisquat".
References[edit | edit source]
- Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. pp. 333–335.
- Coxeter, H.S.M. (1999). The Beauty of Geometry: Twelve essays. Dover Publications, Inc. pp. 157 ff.
- McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space". Discrete Computational Geometry. 17: 449–478.
- jan Misali (2020). "there are 48 regular polyhedra".