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 surround each edge, and 12 icosahedra surround each vertex, in a regular dodecahedral 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.

Honeycomb seen in perspective outside Poincare's model disk

Related regular honeycombs

There are four regular compact honeycombs in 3D hyperbolic space:

Four regular compact honeycombs in H³

{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, t_1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.

[3,5,3] family honeycombs

{3,5,3}	t1{3,5,3}	t0,1{3,5,3}	t0,2{3,5,3}	t0,3{3,5,3}
	120px	120px	120px	120px
	t1,2{3,5,3}	t0,1,2{3,5,3}	t0,1,3{3,5,3}	t0,1,2,3{3,5,3}
	120px	120px	120px	120px

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	80px 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:

240px180px
Perspective projections from center of Poincaré disk model

Related honeycomb

There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	r{5,3,4}	r{4,3,5}	r{3,5,3}	r{5,3,5}
Vertex figure	100px	100px	100px	100px

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	80px triangular pyramid
Coxeter group	J3, [3,5,3]
Properties	Vertex-transitive

The truncated icosahedral honeycomb, t_0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.

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Related honeycombs

Four truncated regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	t{5,3,4}	t{4,3,5}	t{3,5,3}	t{5,3,5}
Vertex figure	100px	100px	100px	100px

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, t_1,2{3,5,3}, , has truncated dodecahedron cells with a disphenoid vertex figure.

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Related honeycombs

Three bitruncated regular compact honeycombs in H³

Image	100px	100px	100px
Symbols	2t{4,3,5}	2t{3,5,3}	2t{5,3,5}
Vertex figure	100px

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	80px triangular prism
Coxeter group	J3, [3,5,3]
Properties	Vertex-transitive

The cantellated icosahedral honeycomb, t_0,2{3,5,3}, , has rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure.

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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	80px Mirrored sphenoid
Coxeter group	J3, [3,5,3]
Properties	Vertex-transitive

The cantitruncated icosahedral honeycomb, t_0,1,2{3,5,3}, , has truncated icosidodecahedron, icosidodecahedron, triangular prism and hexagonal prism cells, with a mirrored sphenoid vertex figure.

480px

Related honeycombs

Four cantitruncated regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	tr{5,3,4}	tr{4,3,5}	tr{3,5,3}	tr{5,3,5}
Vertex figure	100px	100px	100px	100px

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, t_0,3{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.

480px

Viewed from center of triangular prism

Related honeycombs

Three runcinated regular compact honeycombs in H³

Image	100px	100px	100px
Symbols	t0,3{4,3,5}	t0,3{3,5,3}	t0,3{5,3,5}
Vertex figure	100px

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	80px square pyramid
Coxeter group	J3, [3,5,3]
Properties	Vertex-transitive

The runcitruncated icosahedral honeycomb, t_0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism and triangular prism cells, with a square pyramid vertex figure.

480px

Viewed from center of triangular prism

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	80px Phyllic disphenoid
Coxeter group	J3×2, [[3,5,3]]
Properties	Vertex-transitive

The omnitruncated icosahedral honeycomb, t_0,1,2,3{3,5,3}, , has truncated icosidodecahedron and pentagonal prism cells, with a tetrahedral vertex figure.

480px

Centered on hexagonal prism

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	80px
Coxeter group	J3×2, [[3,5,3]]+
Properties	Vertex-transitive

The omnisnub icosahedral honeycomb, h(t_0,1,2,3{3,5,3}), , has snub dodecahedron, octahedron, and tetrahedron cells, with an irregular vertex figure. It is vertex-uniform, 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	80px 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.

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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

• 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
• Richard Klitzing, Hyperbolic H3 honeycombs, hyperbolic order 3 icosahedral tesselation