Rectified cubic honeycomb
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Rectified cubic honeycomb | |
---|---|
Rank | 4 |
Type | Uniform |
Space | Euclidean |
Notation | |
Bowers style acronym | Rich |
Coxeter diagram | o4x3o4o () |
Elements | |
Cells | N octahedra, N cuboctahedra |
Faces | 8N triangles, 3N squares |
Edges | 12N |
Vertices | 3N |
Vertex figure | Square prism, edge lengths 1 (base) and √2 (sides) |
Measures (edge length 1) | |
Vertex density | |
Dual cell volume | |
Related polytopes | |
Army | Rich |
Regiment | Rich |
Dual | Joined cubic honeycomb |
Conjugate | None |
Abstract & topological properties | |
Orientable | Yes |
Properties | |
Symmetry | R_{4} |
Convex | Yes |
The rectified cubic honeycomb, or rich, is a convex uniform honeycomb. 2 octahedra and 4 cuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the rectification of the cubic honeycomb. It is also the rectification of the tetrahedral-octahedral honeycomb.
Vertex coordinates[edit | edit source]
The vertices of a rectified cubic honeycomb of edge length 1 are given by all permutations of:
where i, j, and k range over the integers.
Representations[edit | edit source]
A rectified cubic honeycomb has the following Coxeter diagrams:
- (full symmetry)
- (S_{4} symmetry, as rectified tetrahedral-octahedral honeycomb)
- (S_{4} symmetry)
- (P_{4} symmetry, as rectified cyclotetrahedral honeycomb)
- (as alternated faceting)
- (as alternated faceting)
- qo4ox3xo4oq&#zx
Gallery[edit | edit source]
Related polytopes[edit | edit source]
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 links[edit | edit source]
- Klitzing, Richard. "rich".
- Wikipedia Contributors. "Rectified cubic honeycomb".
- Binnendyk, Eric. "Category 3: Rich regiment" (#22).