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Regular tiling of hyperbolic 3-space From Wikipedia, the free encyclopedia
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
| Order-4 dodecahedral honeycomb | |
|---|---|
| |
| Type | Hyperbolic regular honeycomb Uniform hyperbolic honeycomb |
| Schläfli symbol | {5,3,4} {5,31,1} |
| Coxeter diagram |       ↔   |
| Cells | {5,3} (dodecahedron) |
| Faces | {5} (pentagon) |
| Edge figure | {4} (square) |
| Vertex figure |  octahedron |
| Dual | Order-5 cubic honeycomb |
| Coxeter group | BH3, [4,3,5] DH3, [5,31,1] |
| Properties | Regular, Quasiregular honeycomb |
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.
The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.
It has a half symmetry construction, {5,31,1}, with two types (colors) of dodecahedra in the Wythoff construction. ↔ .
A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model
There are four regular compact honeycombs in 3D hyperbolic space:
{5,3,4} |
 {4,3,5} |
 {3,5,3} |
 {5,3,5} |
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form.
| {5,3,4} |
r{5,3,4} |
t{5,3,4} |
rr{5,3,4} |
t0,3{5,3,4} |
tr{5,3,4} |
t0,1,3{5,3,4} |
t0,1,2,3{5,3,4} |
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| {4,3,5} |
r{4,3,5} |
t{4,3,5} |
rr{4,3,5} |
2t{4,3,5} |
tr{4,3,5} |
t0,1,3{4,3,5} |
t0,1,2,3{4,3,5} |
There are eleven uniform honeycombs in the bifurcating [5,31,1] Coxeter group family, including this honeycomb in its alternated form. This construction can be represented by alternation (checkerboard) with two colors of dodecahedral cells.
This honeycomb is also related to the 16-cell, cubic honeycomb, and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures:
This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:
, ...| Rectified order-4 dodecahedral honeycomb | |
|---|---|
| Type | Uniform honeycombs in hyperbolic space |
| Schläfli symbol | r{5,3,4} r{5,31,1} |
| Coxeter diagram | ↔ |
| Cells | r{5,3}  {3,4}  |
| Faces | triangle {3} pentagon {5} |
| Vertex figure |  square prism |
| Coxeter group | , [4,3,5] , [5,31,1] |
| Properties | Vertex-transitive, edge-transitive |
The rectified order-4 dodecahedral honeycomb, , has alternating octahedron and icosidodecahedron cells, with a square prism vertex figure.
There are four rectified compact regular honeycombs:
| Image | |
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|---|---|---|---|---|
| Symbols | r{5,3,4} |
r{4,3,5} |
r{3,5,3} |
r{5,3,5} |
| Vertex figure |
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, ...| Truncated order-4 dodecahedral honeycomb | |
|---|---|
| Type | Uniform honeycombs in hyperbolic space |
| Schläfli symbol | t{5,3,4} t{5,31,1} |
| Coxeter diagram | ↔ |
| Cells | t{5,3}  {3,4} |
| Faces | triangle {3} decagon {10} |
| Vertex figure |  square pyramid |
| Coxeter group | , [4,3,5] , [5,31,1] |
| Properties | Vertex-transitive |
The truncated order-4 dodecahedral honeycomb, , has octahedron and truncated dodecahedron cells, with a square pyramid vertex figure.
It can be seen as analogous to the 2D hyperbolic truncated order-4 pentagonal tiling, t{5,4} with truncated pentagon and square faces:
| Image | |
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|---|---|---|---|---|
| Symbols | t{5,3,4} |
t{4,3,5} |
t{3,5,3} |
t{5,3,5} |
| Vertex figure |
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, ...| Bitruncated order-4 dodecahedral honeycomb Bitruncated order-5 cubic honeycomb | |
|---|---|
| Type | Uniform honeycombs in hyperbolic space |
| Schläfli symbol | 2t{5,3,4} 2t{5,31,1} |
| Coxeter diagram | ↔  |
| Cells | t{3,5}  t{3,4}  |
| Faces | square {4} pentagon {5} hexagon {6} |
| Vertex figure |  digonal disphenoid |
| Coxeter group | , [4,3,5] , [5,31,1] |
| Properties | Vertex-transitive |
The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, , has truncated octahedron and truncated icosahedron cells, with a digonal disphenoid vertex figure.
| Image | |
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|---|---|---|---|
| Symbols | 2t{4,3,5} |
2t{3,5,3} |
2t{5,3,5} |
| Vertex figure |
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, ...| Cantellated order-4 dodecahedral honeycomb | |
|---|---|
| Type | Uniform honeycombs in hyperbolic space |
| Schläfli symbol | rr{5,3,4} rr{5,31,1} |
| Coxeter diagram | ↔ |
| Cells | rr{3,5}  r{3,4}  {}x{4}  |
| Faces | triangle {3} square {4} pentagon {5} |
| Vertex figure |  wedge |
| Coxeter group | , [4,3,5] , [5,31,1] |
| Properties | Vertex-transitive |
The cantellated order-4 dodecahedral honeycomb, , has rhombicosidodecahedron, cuboctahedron, and cube cells, with a wedge vertex figure.
| Four cantellated regular compact honeycombs in H3 | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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, ...| Cantitruncated order-4 dodecahedral honeycomb | |
|---|---|
| Type | Uniform honeycombs in hyperbolic space |
| Schläfli symbol | tr{5,3,4} tr{5,31,1} |
| Coxeter diagram | ↔ |
| Cells | tr{3,5}  t{3,4} {}x{4} |
| Faces | square {4} hexagon {6} decagon {10} |
| Vertex figure |  mirrored sphenoid |
| Coxeter group | , [4,3,5] , [5,31,1] |
| Properties | Vertex-transitive |
The cantitruncated order-4 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated octahedron, and cube cells, with a mirrored sphenoid vertex figure.
| Image | |
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|---|---|---|---|---|
| Symbols | tr{5,3,4} |
tr{4,3,5} |
tr{3,5,3} |
tr{5,3,5} |
| Vertex figure |
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The runcinated order-4 dodecahedral honeycomb is the same as the runcinated order-5 cubic honeycomb.
...| Runcitruncated order-4 dodecahedral honeycomb | |
|---|---|
| Type | Uniform honeycombs in hyperbolic space |
| Schläfli symbol | t0,1,3{5,3,4} |
| Coxeter diagram | |
| Cells | t{5,3} rr{3,4}  {}x{10}  {}x{4} |
| Faces | triangle {3} square {4} decagon {10} |
| Vertex figure |  isosceles-trapezoidal pyramid |
| Coxeter group | , [4,3,5] |
| Properties | Vertex-transitive |
The runcitruncated order-4 dodecahedral honeycomb, , has truncated dodecahedron, rhombicuboctahedron, decagonal prism, and cube cells, with an isosceles-trapezoidal pyramid vertex figure.
| Four runcitruncated regular compact honeycombs in H3 | |||||||||||||||
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The runcicantellated order-4 dodecahedral honeycomb is the same as the runcitruncated order-5 cubic honeycomb.
The omnitruncated order-4 dodecahedral honeycomb is the same as the omnitruncated order-5 cubic honeycomb.
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