Icosahedral honeycomb

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.

Icosahedral honeycomb

Poincaré disk model
Typeregular hyperbolic honeycomb
Schläfli symbol{3,5,3}
Coxeter diagram
Cells{3,5}
Facestriangle {3}
Vertex figure
dodecahedron
DualSelf-dual
Coxeter groupJ3, [3,5,3]
PropertiesRegular

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.

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}

It a part of a sequence of regular polychora and honeycombs {3,p,3}:

It a part of a sequence of regular polychora and honeycombs {p,5,p}:

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] family honeycombs
{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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{3,5,3} or t1{3,5,3}
Coxeter diagram
Cellsr{3,5}
{5,3}
Facestriangle {3}
Pentagon {5}
Vertex figure
Triangular prism
Coxeter groupJ3, [3,5,3]
PropertiesVertex-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

There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H3
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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{3,5,3} or t0,1{3,5,3}
Coxeter diagram
Cellst{3,5}
{5,3}
Facestriangle {3}
Pentagon {5}
Vertex figure
triangular pyramid
Coxeter groupJ3, [3,5,3]
PropertiesVertex-transitive

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

Four truncated regular compact honeycombs in H3
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
TypeUniform honeycombs in hyperbolic space
Schläfli symbol2t{3,5,3} or t1,2{3,5,3}
Coxeter diagram
Cellst{5,3}
FacesTriangle {3}
Decagon {10}
Vertex figure
disphenoid
Coxeter groupJ3×2, [[3,5,3]]
PropertiesVertex-transitive, edge-transitive, cell-transitive

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

Three bitruncated compact honeycombs in H3
Image
Symbols 2t{4,3,5}
2t{3,5,3}
2t{5,3,5}
Vertex
figure

Cantellated icosahedral honeycomb

Cantellated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{3,5,3} or t0,2{3,5,3}
Coxeter diagram
Cellsrr{3,5}
r{5,3}
Facestriangle {3}
Square {4}
Pentagon {5}
Vertex figure
triangular prism
Coxeter groupJ3, [3,5,3]
PropertiesVertex-transitive

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

Cantitruncated icosahedral honeycomb

Cantitruncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{3,5,3} or t0,1,2{3,5,3}
Coxeter diagram
Cellstr{3,5}
r{5,3}
{}x{3}
{}x{6}
FacesTriangle {3}
Square {4}
Pentagon {5}
Hexagon {6}
Vertex figure
Mirrored sphenoid
Coxeter groupJ3, [3,5,3]
PropertiesVertex-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.

Four cantitruncated regular compact honeycombs in H3
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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,3{3,5,3}
Coxeter diagram
Cells{3,5}
{}×{3}
FacesTriangle {3}
Square {4}
Vertex figure
pentagonal antiprism
Coxeter groupJ3×2, [[3,5,3]]
PropertiesVertex-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
Three runcinated regular compact honeycombs in H3
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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,3{3,5,3}
Coxeter diagram
Cellsr{3,5}
rr{3,5}
{}×{3}
{}×{6}
FacesTriangle {3}
Square {4}
Pentagon {5}
Hexagon {6}
Vertex figure
square pyramid
Coxeter groupJ3, [3,5,3]
PropertiesVertex-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

Omnitruncated icosahedral honeycomb

Omnitruncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,2,3{3,5,3}
Coxeter diagram
Cellstr{3,5}
{}×{6}
FacesSquare {4}
Hexagon {6}
Dodecagon {10}
Vertex figure
Phyllic disphenoid
Coxeter groupJ3×2, [[3,5,3]]
PropertiesVertex-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

Omnisnub icosahedral honeycomb

Omnisnub icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh(t0,1,2,3{3,5,3})
Coxeter diagram
Cellssr{3,5}
s{2,3}
irr. {3,3}
FacesSquare {4}
Pentagon {5}
Vertex figure
Coxeter groupJ3×2, [[3,5,3]]+
PropertiesVertex-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
TypeUniform honeycombs
Schläfli symbolpd{3,5,3}
Coxeter diagram-
Cells{5,3}
s{2,10}
FacesTriangle {3}
Pentagon {5}
Vertex figure
tetrahedrally diminished
dodecahedron
Coxeter group1/5[3,5,3]+
PropertiesVertex-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

References

  1. Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005)
  2. 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, (2018) Chapter 13: Hyperbolic Coxeter groups
  • Klitzing, Richard. "Hyperbolic H3 honeycombs hyperbolic order 3 icosahedral tesselation".
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