Icosahedral honeycomb
Icosahedral honeycomb | |
---|---|
Poincaré disk model | |
Type | regular hyperbolic honeycomb |
Schläfli symbol | {3,5,3} |
Coxeter diagram | |
Cells | {3,5} |
Faces | triangle {3} |
Vertex figure | dodecahedron |
Dual | Self-dual |
Coxeter group | J3, [3,5,3] |
Properties | Regular |
The icosahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra, {3,5}, around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral, {5,3}, vertex figure.
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.
Description
The dihedral angle of a Euclidean icosahedron is 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles exactly 120 degrees, so three of these fit around an edge.
Related regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
{5,3,4} |
{4,3,5} |
{3,5,3} |
{5,3,5} |
Related regular polytopes and honeycombs
{3,p,3} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | H3 | |||||||||
Form | Finite | Compact | Paracompact | Noncompact | |||||||
{3,p,3} | {3,3,3} | {3,4,3} | {3,5,3} | {3,6,3} | {3,7,3} | {3,8,3} | ... {3,∞,3} | ||||
Image | |||||||||||
Cells | {3,3} |
{4,3} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} | ||||
Vertex figure |
{3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |
Uniform honeycombs
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.
{3,5,3} |
t1{3,5,3} |
t0,1{3,5,3} |
t0,2{3,5,3} |
t0,3{3,5,3} |
---|---|---|---|---|
t1,2{3,5,3} |
t0,1,2{3,5,3} |
t0,1,3{3,5,3} |
t0,1,2,3{3,5,3} | |
Rectified icosahedral honeycomb
Rectified icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | r{3,5,3} or t1{3,5,3} |
Coxeter diagram | |
Cells | r{3,5} {5,3} |
Faces | triangle {3} Pentagon {5} |
Vertex figure | Triangular prism |
Coxeter group | J3, [3,5,3] |
Properties | Vertex-transitive, edge-transitive |
The rectified icosahedral honeycomb, t1{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:
-
Perspective projections from center of Poincaré disk model
Related honeycomb
There are four rectified compact regular honeycombs:
Image | ||||
---|---|---|---|---|
Symbols | r{5,3,4} |
r{4,3,5} |
r{3,5,3} |
r{5,3,5} |
Vertex figure |
Truncated icosahedral honeycomb
Truncated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | t{3,5,3} or t0,1{3,5,3} |
Coxeter diagram | |
Cells | t{3,5} {5,3} |
Faces | triangle {3} Pentagon {5} |
Vertex figure | triangular pyramid |
Coxeter group | J3, [3,5,3] |
Properties | Vertex-transitive |
The truncated icosahedral honeycomb, t0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.
Related honeycombs
Image | ||||
---|---|---|---|---|
Symbols | t{5,3,4} |
t{4,3,5} |
t{3,5,3} |
t{5,3,5} |
Vertex figure |
Bitruncated icosahedral honeycomb
Bitruncated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | 2t{3,5,3} or t1,2{3,5,3} |
Coxeter diagram | |
Cells | t{5,3} |
Faces | Triangle {3} Dodecagon {10} |
Vertex figure | disphenoid |
Coxeter group | J3×2, [[3,5,3]] |
Properties | Vertex-transitive, edge-transitive, cell-transitive |
The bitruncated icosahedral honeycomb, t1,2{3,5,3}, , has truncated dodecahedron cells with a disphenoid vertex figure.
Related honeycombs
Image | |||
---|---|---|---|
Symbols | 2t{4,3,5} |
2t{3,5,3} |
2t{5,3,5} |
Vertex figure |
Cantellated icosahedral honeycomb
Cantellated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | rr{3,5,3} or t0,2{3,5,3} |
Coxeter diagram | |
Cells | rr{3,5} r{5,3} |
Faces | triangle {3} Square {4} Pentagon {5} |
Vertex figure | triangular prism |
Coxeter group | J3, [3,5,3] |
Properties | Vertex-transitive |
The cantellated icosahedral honeycomb, t0,2{3,5,3}, , has rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure.
Related honeycombs
Four cantellated regular compact honeycombs in H3 | |||||||||||||||
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|
Cantitruncated icosahedral honeycomb
Cantitruncated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | tr{3,5,3} or t0,1,2{3,5,3} |
Coxeter diagram | |
Cells | tr{3,5} r{5,3} {}x{3} {}x{6} |
Faces | Triangle {3} Square {4} Pentagon {5} Hexagon {6} |
Vertex figure | Mirrored sphenoid |
Coxeter group | J3, [3,5,3] |
Properties | Vertex-transitive |
The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, icosidodecahedron, triangular prism and hexagonal prism cells, with a mirrored sphenoid vertex figure.
Related honeycombs
Image | ||||
---|---|---|---|---|
Symbols | tr{5,3,4} |
tr{4,3,5} |
tr{3,5,3} |
tr{5,3,5} |
Vertex figure |
Runcinated icosahedral honeycomb
Runcinated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | t0,3{3,5,3} |
Coxeter diagram | |
Cells | {3,5} {}×{3} |
Faces | Triangle {3} Square {4} |
Vertex figure | pentagonal antiprism |
Coxeter group | J3×2, [[3,5,3]] |
Properties | Vertex-transitive |
The runcinated icosahedral honeycomb, t0,3{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.
- Viewed from center of triangular prism
Related honeycombs
Image | |||
---|---|---|---|
Symbols | t0,3{4,3,5} |
t0,3{3,5,3} |
t0,3{5,3,5} |
Vertex figure |
Runcitruncated icosahedral honeycomb
Runcitruncated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | t0,1,3{3,5,3} |
Coxeter diagram | |
Cells | r{3,5} rr{3,5} {}×{3} {}×{6} |
Faces | Triangle {3} Square {4} Pentagon {5} Hexagon {6} |
Vertex figure | square pyramid |
Coxeter group | J3, [3,5,3] |
Properties | Vertex-transitive |
The runcitruncated icosahedral honeycomb, t0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism and triangular prism cells, with a square pyramid vertex figure.
- Viewed from center of triangular prism
Related honeycombs
Four runcitruncated regular compact honeycombs in H3 | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Omnitruncated icosahedral honeycomb
Omnitruncated icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | t0,1,2,3{3,5,3} |
Coxeter diagram | |
Cells | tr{3,5} {}×{6} |
Faces | Square {4} Hexagon {6} Dodecagon {10} |
Vertex figure | Phyllic disphenoid |
Coxeter group | J3×2, [[3,5,3]] |
Properties | Vertex-transitive |
The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3}, , has truncated icosidodecahedron and pentagonal prism cells, with a tetrahedral vertex figure.
- Centered on hexagonal prism
Related honeycombs
Three omnitruncated regular compact honeycombs in H3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Omnisnub icosahedral honeycomb
Omnisnub icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs in hyperbolic space |
Schläfli symbol | h(t0,1,2,3{3,5,3}) |
Coxeter diagram | |
Cells | sr{3,5} s{2,3} irr. {3,3} |
Faces | Square {4} Pentagon {5} |
Vertex figure | |
Coxeter group | J3×2, [[3,5,3]]+ |
Properties | Vertex-transitive |
The omnisnub icosahedral honeycomb, h(t0,1,2,3{3,5,3}), , has snub dodecahedron, octahedron, and tetrahedron cells, with an irregular vertex figure. It is vertex-transitive, but can't be made with uniform cells.
Partially diminished icosahedral honeycomb
Partially diminished icosahedral honeycomb Parabidiminished icosahedral honeycomb | |
---|---|
Type | Uniform honeycombs |
Schläfli symbol | pd{3,5,3} |
Coxeter diagram | - |
Cells | {5,3} s{2,10} |
Faces | Triangle {3} Pentagon {5} |
Vertex figure | tetrahedrally diminished dodecahedron |
Coxeter group | 1/5[3,5,3]+ |
Properties | Vertex-transitive |
The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a nonwythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the {3,5,3} are diminished at opposite vertices (parabidiminished), leaving a pentagonal antiprism (parabidiminished icosahedron) core, and creating new dodecahedron cells above and below.[1][2]
See also
- Seifert–Weber space
- List of regular polytopes
- Convex uniform honeycombs in hyperbolic space
- 11-cell - An abstract regular polychoron which shares the {3,5,3} Schläfli symbol.
References
- ↑ Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005)
- ↑ http://www.bendwavy.org/klitzing/incmats/pt353.htm
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
- Norman Johnson Uniform Polytopes, Manuscript
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups
- Klitzing, Richard. "Hyperbolic H3 honeycombs hyperbolic order 3 icosahedral tesselation".