Icosahedral honeycomb

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Icosahedral honeycomb
H3 353 CC center.png
Poincaré disk model
Type regular hyperbolic honeycomb
Schläfli symbol {3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
Vertex figure Order-3 icosahedral honeycomb verf.png
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[edit]

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[edit]

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

Four regular compact honeycombs in H3
H3 534 CC center.png
{5,3,4}
H3 435 CC center.png
{4,3,5}
H3 353 CC center.png
{3,5,3}
H3 535 CC center.png
{5,3,5}

Related regular polytopes and honeycombs[edit]

Uniform honeycombs[edit]

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}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png, also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.

[3,5,3] family honeycombs
{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t1{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t0,1{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t0,2{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,3{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
H3 353 CC center.png H3 353 CC center 0100.png H3 353-0011 center ultrawide.png H3 353-1010 center ultrawide.png H3 353-1001 center ultrawide.png
t1,2{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,3{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
H3 353-0110 center ultrawide.png H3 353-1110 center ultrawide.png H3 353-1101 center ultrawide.png H3 353-1111 center ultrawide.png

Rectified icosahedral honeycomb[edit]

Rectified icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{3,5,3} or t1{3,5,3}
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells r{3,5} Uniform polyhedron-53-t1.png
{5,3} Uniform polyhedron-53-t0.png
Faces triangle {3}
Pentagon {5}
Vertex figure Rectified icosahedral honeycomb verf.png
Triangular prism
Coxeter group J3, [3,5,3]
Properties Vertex-transitive, edge-transitive

The rectified icosahedral honeycomb, t1{3,5,3}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:

H3 353 CC center 0100.pngRectified icosahedral honeycomb.png
Perspective projections from center of Poincaré disk model

Related honeycomb[edit]

There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H3
Image H3 534 CC center 0100.png H3 435 CC center 0100.png H3 353 CC center 0100.png H3 535 CC center 0100.png
Symbols r{5,3,4}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
r{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
r{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
r{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Vertex
figure
Rectified order-4 dodecahedral honeycomb verf.png Rectified order-5 cubic honeycomb verf.png Rectified icosahedral honeycomb verf.png Rectified order-5 dodecahedral honeycomb verf.png

Truncated icosahedral honeycomb[edit]

Truncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{3,5,3} or t0,1{3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells t{3,5} Uniform polyhedron-53-t12.png
{5,3} Uniform polyhedron-53-t0.png
Faces triangle {3}
Pentagon {5}
Vertex figure Truncated icosahedral honeycomb verf.png
triangular pyramid
Coxeter group J3, [3,5,3]
Properties Vertex-transitive

The truncated icosahedral honeycomb, t0,1{3,5,3}, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.

H3 353-0011 center ultrawide.png

Related honeycombs[edit]

Four truncated regular compact honeycombs in H3
Image H3 435-0011 center ultrawide.png H3 534-0011 center ultrawide.png H3 353-0011 center ultrawide.png H3 535-0011 center ultrawide.png
Symbols t{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t{5,3,5}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Vertex
figure
Truncated order-4 dodecahedral honeycomb verf.png Truncated order-5 cubic honeycomb verf.png Truncated icosahedral honeycomb verf.png Truncated order-5 dodecahedral honeycomb verf.png

Bitruncated icosahedral honeycomb[edit]

Bitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol 2t{3,5,3} or t1,2{3,5,3}
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells t{5,3} Uniform polyhedron-53-t01.png
Faces Triangle {3}
Decagon {10}
Vertex figure Bitruncated icosahedral honeycomb verf.png
disphenoid
Coxeter group J3×2, [[3,5,3]]
Properties Vertex-transitive, edge-transitive, cell-transitive

The bitruncated icosahedral honeycomb, t1,2{3,5,3}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png, has truncated dodecahedron cells with a disphenoid vertex figure.

H3 353-0110 center ultrawide.png

Related honeycombs[edit]

Three bitruncated regular compact honeycombs in H3
Image H3 534-0110 center ultrawide.png H3 353-0110 center ultrawide.png H3 535-0110 center ultrawide.png
Symbols 2t{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
2t{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Vertex
figure
Bitruncated order-5 cubic honeycomb verf.png Bitruncated icosahedral honeycomb verf.png Bitruncated order-5 dodecahedral honeycomb verf.png

Cantellated icosahedral honeycomb[edit]

Cantellated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{3,5,3} or t0,2{3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells rr{3,5} Uniform polyhedron-53-t02.png
r{5,3} Uniform polyhedron-53-t1.png
Faces triangle {3}
Square {4}
Pentagon {5}
Vertex figure Cantellated icosahedral honeycomb verf.png
triangular prism
Coxeter group J3, [3,5,3]
Properties Vertex-transitive

The cantellated icosahedral honeycomb, t0,2{3,5,3}, CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png, has rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure.

H3 353-1010 center ultrawide.png

Related honeycombs[edit]

Cantitruncated icosahedral honeycomb[edit]

Cantitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{3,5,3} or t0,1,2{3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells tr{3,5} Uniform polyhedron-53-t02.png
r{5,3} Uniform polyhedron-53-t1.png
{}x{3} Triangular prism.png
{}x{6} Hexagonal prism.png
Faces Triangle {3}
Square {4}
Pentagon {5}
Hexagon {6}
Vertex figure Cantitruncated icosahedral honeycomb verf.png
Mirrored sphenoid
Coxeter group J3, [3,5,3]
Properties Vertex-transitive

The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png, has truncated icosidodecahedron, icosidodecahedron, triangular prism and hexagonal prism cells, with a mirrored sphenoid vertex figure.

H3 353-1110 center ultrawide.png

Related honeycombs[edit]

Four cantitruncated regular compact honeycombs in H3
Image H3 534-1110 center ultrawide.png H3 534-0111 center ultrawide.png H3 353-1110 center ultrawide.png H3 535-1110 center ultrawide.png
Symbols tr{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
tr{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
tr{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{5,3,5}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Vertex
figure
Cantitruncated order-4 dodecahedral honeycomb verf.png Cantitruncated order-5 cubic honeycomb verf.png Cantitruncated icosahedral honeycomb verf.png Cantitruncated order-5 dodecahedral honeycomb verf.png

Runcinated icosahedral honeycomb[edit]

Runcinated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,3{3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells {3,5} Uniform polyhedron-53-t2.png
{}×{3} Triangular prism.png
Faces Triangle {3}
Square {4}
Vertex figure Runcinated icosahedral honeycomb verf.png
pentagonal antiprism
Coxeter group J3×2, [[3,5,3]]
Properties Vertex-transitive

The runcinated icosahedral honeycomb, t0,3{3,5,3}, CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png, has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.

H3 353-1001 center ultrawide.png

Viewed from center of triangular prism

Related honeycombs[edit]

Three runcinated regular compact honeycombs in H3
Image H3 534-1001 center ultrawide.png H3 353-1001 center ultrawide.png H3 535-1001 center ultrawide.png
Symbols t0,3{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,3{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Vertex
figure
Runcinated order-5 cubic honeycomb verf.png Runcinated icosahedral honeycomb verf.png Runcinated order-5 dodecahedral honeycomb verf.png

Runcitruncated icosahedral honeycomb[edit]

Runcitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells r{3,5} Uniform polyhedron-53-t12.png
rr{3,5} Uniform polyhedron-53-t02.png
{}×{3} Triangular prism.png
{}×{6} Hexagonal prism.png
Faces Triangle {3}
Square {4}
Pentagon {5}
Hexagon {6}
Vertex figure Runcitruncated icosahedral honeycomb verf.png
square pyramid
Coxeter group J3, [3,5,3]
Properties Vertex-transitive

The runcitruncated icosahedral honeycomb, t0,1,3{3,5,3}, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png, has truncated icosahedron, rhombicosidodecahedron, hexagonal prism and triangular prism cells, with a square pyramid vertex figure.

H3 353-1101 center ultrawide.png

Viewed from center of triangular prism

Related honeycombs[edit]

Omnitruncated icosahedral honeycomb[edit]

Omnitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,2,3{3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells tr{3,5} Uniform polyhedron-53-t012.png
{}×{6} Hexagonal prism.png
Faces Square {4}
Hexagon {6}
Dodecagon {10}
Vertex figure Omnitruncated icosahedral honeycomb verf.png
Phyllic disphenoid
Coxeter group J3×2, [[3,5,3]]
Properties Vertex-transitive

The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3}, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png, has truncated icosidodecahedron and pentagonal prism cells, with a tetrahedral vertex figure.

H3 353-1111 center ultrawide.png

Centered on hexagonal prism

Related honeycombs[edit]

Omnisnub icosahedral honeycomb[edit]

Omnisnub icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h(t0,1,2,3{3,5,3})
Coxeter diagram CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Cells sr{3,5} Uniform polyhedron-53-s012.png
s{2,3} Trigonal antiprism.png
irr. {3,3} Tetrahedron.png
Faces Square {4}
Pentagon {5}
Vertex figure Snub icosahedral honeycomb verf.png
Coxeter group J3×2, [[3,5,3]]+
Properties Vertex-transitive

The omnisnub icosahedral honeycomb, h(t0,1,2,3{3,5,3}), CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png, 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[edit]

Partially diminished icosahedral honeycomb
Parabidiminished icosahedral honeycomb
Type Uniform honeycombs
Schläfli symbol pd{3,5,3}
Coxeter diagram -
Cells {5,3} Uniform polyhedron-53-t0.png
s{2,10} Pentagonal antiprism.png
Faces Triangle {3}
Pentagon {5}
Vertex figure Partial truncation order-3 icosahedral honeycomb verf.png
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]

H3 353-pd center ultrawide.png

H3 353-pd center ultrawide2.png

See also[edit]

References[edit]

  1. ^ Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [1]
  2. ^ http://www.bendwavy.org/klitzing/incmats/pt353.htm