The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.[1]
Order-4 octahedral honeycomb |
Perspective projection view within Poincaré disk model |
Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {3,4,4} {3,41,1} |
Coxeter diagrams | ↔ ↔ ↔ |
Cells | {3,4} |
Faces | triangle {3} |
Edge figure | square {4} |
Vertex figure | square tiling, {4,4} |
Dual | Square tiling honeycomb, {4,4,3} |
Coxeter groups | , [3,4,4] , [3,41,1] |
Properties | Regular |
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
A half symmetry construction, [3,4,4,1+], exists as {3,41,1}, with two alternating types (colors) of octahedral cells: ↔ .
A second half symmetry is [3,4,1+,4]: ↔ .
A higher index sub-symmetry, [3,4,4*], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: .
This honeycomb contains and that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings and , respectively:
The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs.
More information 11 paracompact regular honeycombs ...
11 paracompact regular honeycombs |
{6,3,3} |
{6,3,4} |
{6,3,5} |
{6,3,6} |
{4,4,3} |
{4,4,4} |
{3,3,6} |
{4,3,6} |
{5,3,6} |
{3,6,3} |
{3,4,4} |
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There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form.
More information {4,4,3}, r{4,4,3} ...
[4,4,3] family honeycombs
{4,4,3}
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r{4,4,3}
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t{4,4,3}
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rr{4,4,3}
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t0,3{4,4,3}
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tr{4,4,3}
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t0,1,3{4,4,3}
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t0,1,2,3{4,4,3}
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{3,4,4}
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r{3,4,4}
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t{3,4,4}
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rr{3,4,4}
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2t{3,4,4}
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tr{3,4,4}
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t0,1,3{3,4,4}
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t0,1,2,3{3,4,4}
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It is a part of a sequence of honeycombs with a square tiling vertex figure:
More information Space, E3 ...
{p,4,4} honeycombs |
Space |
E3 |
H3 |
Form |
Affine |
Paracompact |
Noncompact |
Name |
{2,4,4} |
{3,4,4} |
{4,4,4} |
{5,4,4} |
{6,4,4} |
..{∞,4,4} |
Coxeter
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Cells |
{2,4}
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{3,4}
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{4,4}
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{5,4}
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{6,4}
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{∞,4}
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It a part of a sequence of regular polychora and honeycombs with octahedral cells:
More information {3,4,p} polytopes, Space ...
{3,4,p} polytopes |
Space |
S3 |
H3 |
Form |
Finite |
Paracompact |
Noncompact |
Name |
{3,4,3}
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{3,4,4}
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{3,4,5}
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{3,4,6}
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{3,4,7}
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{3,4,8}
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... {3,4,∞}
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Image |
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Vertex figure |
{4,3}
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{4,4}
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{4,5}
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{4,6}
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{4,7}
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{4,8}
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{4,∞}
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Rectified order-4 octahedral honeycomb
More information , ...
Rectified order-4 octahedral honeycomb |
Type | Paracompact uniform honeycomb |
Schläfli symbols | r{3,4,4} or t1{3,4,4} |
Coxeter diagrams | ↔ ↔ ↔ |
Cells | r{4,3} {4,4} |
Faces | triangle {3} square {4} |
Vertex figure | square prism |
Coxeter groups | , [3,4,4] , [3,41,1] |
Properties | Vertex-transitive, edge-transitive |
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The rectified order-4 octahedral honeycomb, t1{3,4,4}, has cuboctahedron and square tiling facets, with a square prism vertex figure.
Truncated order-4 octahedral honeycomb
More information , ...
Truncated order-4 octahedral honeycomb |
Type | Paracompact uniform honeycomb |
Schläfli symbols | t{3,4,4} or t0,1{3,4,4} |
Coxeter diagrams | ↔ ↔ ↔ |
Cells | t{3,4} {4,4} |
Faces | square {4} hexagon {6} |
Vertex figure | square pyramid |
Coxeter groups | , [3,4,4] , [3,41,1] |
Properties | Vertex-transitive |
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The truncated order-4 octahedral honeycomb, t0,1{3,4,4}, has truncated octahedron and square tiling facets, with a square pyramid vertex figure.
Cantellated order-4 octahedral honeycomb
More information , ...
Cantellated order-4 octahedral honeycomb |
Type | Paracompact uniform honeycomb |
Schläfli symbols | rr{3,4,4} or t0,2{3,4,4} s2{3,4,4} |
Coxeter diagrams |
↔ |
Cells | rr{3,4} {}x4 r{4,4} |
Faces | triangle {3} square {4} |
Vertex figure | wedge |
Coxeter groups | , [3,4,4] , [3,41,1] |
Properties | Vertex-transitive |
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The cantellated order-4 octahedral honeycomb, t0,2{3,4,4}, has rhombicuboctahedron, cube, and square tiling facets, with a wedge vertex figure.
Cantitruncated order-4 octahedral honeycomb
More information , ...
Cantitruncated order-4 octahedral honeycomb |
Type | Paracompact uniform honeycomb |
Schläfli symbols | tr{3,4,4} or t0,1,2{3,4,4} |
Coxeter diagrams | ↔ |
Cells | tr{3,4} {}x{4} t{4,4} |
Faces | square {4} hexagon {6} octagon {8} |
Vertex figure | mirrored sphenoid |
Coxeter groups | , [3,4,4] , [3,41,1] |
Properties | Vertex-transitive |
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The cantitruncated order-4 octahedral honeycomb, t0,1,2{3,4,4}, has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.