Cubic honeycomb
The cubic honeycomb, or chon, is the only regular honeycomb or tesselation of 3D Euclidean space. 8 cubes join at each vertex of this honeycomb. It is also the 3D hypercubic honeycomb.
Cubic honeycomb | |
---|---|
Rank | 4 |
Type | Regular |
Space | Euclidean |
Notation | |
Bowers style acronym | Chon |
Coxeter diagram | |
Schläfli symbol | {4,3,4} |
Elements | |
Cells | N cubes |
Faces | 3N squares |
Edges | 3N |
Vertices | N |
Vertex figure | Octahedron, edge length √2 |
Measures (edge length 1) | |
Vertex density | |
Dual cell volume | |
Related polytopes | |
Army | Chon |
Regiment | Chon |
Dual | Cubic honeycomb |
Petrie dual | Mucubic honeycomb |
Conjugate | None |
Abstract & topological properties | |
Orientable | Yes |
Properties | |
Symmetry | R_{4} |
Convex | Yes |
This honeycomb can be alternated into a tetrahedral-octahedral honeycomb, which is uniform.
Vertex coordinatesEdit
The vertices of a cubic honeycomb of edge length 1 are given by
- in which .
RepresentationsEdit
A cubic honeycomb has the following Coxeter diagrams:
- (regular)
- (as expanded cubic honecyomb)
- (S4 symmetry)
- (various square prismatic honeycombs)
- (various apeirogonal triprismatic honeycombs)
- qo3oo3oq3oo3*a&#zx (as hull of two alternate tetrahedral-octahedral honeycombs)
GalleryEdit
Related polytopesEdit
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Cubic honeycomb | chon | {4,3,4} | ||
Truncated cubic honeycomb | tich | t{4,3,4} | ||
Rectified cubic honeycomb | rich | r{4,3,4} | ||
Bitruncated cubic honeycomb | batch | 2t{4,3,4} | ||
Rectified cubic honeycomb | rich | r{4,3,4} | ||
Truncated cubic honeycomb | tich | t{4,3,4} | ||
Cubic honeycomb | chon | {4,3,4} | ||
Small rhombated cubic honeycomb | srich | rr{4,3,4} | ||
Great rhombated cubic honeycomb | grich | tr{4,3,4} | ||
Small rhombated cubic honeycomb | srich | rr{4,3,4} | ||
Great rhombated cubic honeycomb | grich | tr{4,3,4} | ||
Small prismated cubic honeycomb = Cubic honeycomb | chon | t_{0,3}{4,3,4} | ||
Prismatorhombated cubic honeycomb | prich | t_{0,1,3}{4,3,4} | ||
Prismatorhombated cubic honeycomb | prich | t_{0,1,3}{4,3,4} | ||
Great prismated cubic honeycomb | gippich | t_{0,1,2,3}{4,3,4} |
External linksEdit
- Klitzing, Richard. "chon".
- Wikipedia Contributors. "Cubic honeycomb".
- Binnendyk, Eric. "Category 1: Primaries" (#1).