Octahedral honeycomb
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Octahedral honeycomb | |
---|---|
Rank | 4 |
Type | Regular, paracompact |
Space | Hyperbolic |
Notation | |
Bowers style acronym | Octh |
Coxeter diagram | o4o4o3x () |
Schläfli symbol | {3,4,4} |
Elements | |
Cells | NM octahedra |
Faces | 4NM triangles |
Edges | 3NM |
Vertices | 6N |
Vertex figure | Square tiling, edge length 1 |
Measures (edge length 1) | |
Circumradius | 0 |
Related polytopes | |
Army | Octh |
Regiment | Octh |
Dual | Square tiling honeycomb |
Abstract & topological properties | |
Orientable | Yes |
Properties | |
Symmetry | [4,4,3] |
Convex | Yes |
The order-4 octahedral honeycomb, or just octahedral honeycomb, is a paracompact regular tiling of 3D hyperbolic space. 4 ideal octahedra meet at each edge. All vertices are ideal points at infinity, with infinitely many octahedra meeting at each vertex in a square tiling arrangement.
Representations[edit | edit source]
The octahedral honeycomb has the following Coxeter diagrams:
- o4o4o3x (full symmetry)
- o4o4o *b3x (octahedra of two types)
- o3x3o4o4*a (octahedra of three types, x4o4x verf)
Related polytopes[edit | edit source]
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Square tiling honeycomb | squah | {4,4,3} | ||
Truncated square tiling honeycomb | tisquah | t{4,4,3} | ||
Rectified square tiling honeycomb | risquah | r{4,4,3} | ||
Octahedral-square tiling honeycomb | osquah | 2t{4,4,3} | ||
Rectified octahedral honeycomb | rocth | r{3,4,4} | ||
Truncated octahedral honeycomb | tocth | t{3,4,4} | ||
Octahedral honeycomb | octh | {3,4,4} | ||
Small rhombated square tiling honeycomb | srisquah | rr{4,4,3} | ||
Great rhombated square tiling honeycomb | grisquah | tr{4,4,3} | ||
Small rhombated octahedral honeycomb | srocth | rr{3,4,4} | ||
Great rhombated octahedral honeycomb | grocth | tr{3,4,4} | ||
Small prismated square tiling honeycomb | sidposquah | t_{0,3}{4,4,3} | ||
Prismatorhombated octahedral honeycomb | procth | t_{0,1,3}{4,4,3} | ||
Prismatorhombated square tiling honeycomb | prisquah | t_{0,1,3}{3,4,4} | ||
Great prismated square tiling honeycomb | gidposquah | t_{0,1,2,3}{4,4,3} |
External links[edit | edit source]
- Klitzing, Richard. "octh".
- Wikipedia Contributors. "Order-4 octahedral honeycomb".
- lllllllllwith10ls. "Category 1: Regulars" (#14).