Great rhombated cubic honeycomb
Great rhombated cubic honeycomb | |
---|---|
Rank | 4 |
Type | Uniform |
Space | Euclidean |
Notation | |
Bowers style acronym | Grich |
Coxeter diagram | x4x3x4o () |
Elements | |
Cells | 3N cubes, N truncated octahedra, N great rhombicuboctahedra |
Faces | 6N+12N squares, 8N hexagons, 3N octagons |
Edges | 12N+12N+24N |
Vertices | 24N |
Vertex figure | Sphenoid, edge lengths √2 (3), √3 (2), and √2+√2 (1) |
Measures (edge length 1) | |
Vertex density | |
Dual cell volume | |
Related polytopes | |
Army | Grich |
Regiment | Grich |
Dual | Sphenoidal honeycomb |
Conjugate | Great quasirhombated cubic honeycomb |
Topological properties | |
Orientable | Yes |
Properties | |
Symmetry | R_{4} |
Convex | Yes |
The great rhombated cubic honeycomb, or grich, also known as the cantitruncated cubic honeycomb, is a convex uniform honeycomb. 1 truncated octahedron, 1 cube, and 2 great rhombicuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the cantitruncation of the cubic honeycomb.
This honeycomb can be alternated into a snub rectified cubic honeycomb, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a cantic snub cubic honeycomb, which is also nonuniform.
Vertex coordinates[edit | edit source]
The vertices of a great rhombated 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 great rhombated cubic honeycomb has the following Coxeter diagrams:
- x4x3x4o (regular)
- x3x3x *b4x (S4 symmetry)
- s4x3x4x (as alternated faceting)
Gallery[edit | edit source]
Orthogonal projection into Euclidean plane with V3 symmetry
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. "grich".
- Wikipedia Contributors. "Cantitruncated cubic honeycomb".
- Binnendyk, Eric. "Category 5: Greater Truncates" (#92).