Icosahedral honeycomb

Poincaré disk model
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol{3,5,3}
Coxeter diagram
Cells{5,3} (regular icosahedron)
Faces{3} (triangle)
Edge figure{3} (triangle)
Vertex figure
dodecahedron
DualSelf-dual
Coxeter groupJ3, [3,5,3]
PropertiesRegular

In geometry, 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 around each edge, and 12 icosahedra around 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 regular icosahedron is around 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 of exactly 120 degrees, so three of those can fit around an edge.

Honeycomb seen in perspective outside Poincare's model disk

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 is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:

{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}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
Vertex
figure
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}

It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:

{p,5,p} regular honeycombs
Space H3
Form Compact Noncompact
Name {3,5,3} {4,5,4} {5,5,5} {6,5,6} {7,5,7} {8,5,8} ...{∞,5,∞}
Image
Cells
{p,5}
{3,5}
{4,5}
{5,5}
{6,5}
{7,5}
{8,5}
{∞,5}
Vertex
figure
{5,p}
{5,3}
{5,4}
{5,5}
{5,6}
{5,7}
{5,8}
{5,∞}

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 group, [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}
Facespentagon {5}
hexagon {6}
Vertex figure
triangular pyramid
Coxeter group, [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
tetragonal disphenoid
Coxeter group, [[3,5,3]]
PropertiesVertex-transitive, edge-transitive, cell-transitive

The bitruncated icosahedral honeycomb, t1,2{3,5,3}, , has truncated dodecahedron cells with a tetragonal 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}
{}x{3}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group, [3,5,3]
PropertiesVertex-transitive

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

Four cantellated regular compact honeycombs in H3
Image
Symbols rr{5,3,4}
 rr{4,3,5}
 rr{3,5,3}
 rr{5,3,5}
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}
t{5,3}
{}x{3}
Facestriangle {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure
mirrored sphenoid
Coxeter group, [3,5,3]
PropertiesVertex-transitive

The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular 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 group, [[3,5,3]]
PropertiesVertex-transitive, edge-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
Cellst{3,5}
rr{3,5}
{}×{3}
{}×{6}
Facestriangle {3}
square {4}
pentagon {5}
hexagon {6}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group, [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 an isosceles-trapezoidal pyramid vertex figure.

The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb.

Viewed from center of triangular prism
Four runcitruncated regular compact honeycombs in H3
Image
Symbols t0,1,3{5,3,4}
 t0,1,3{4,3,5}
 t0,1,3{3,5,3}
 t0,1,3{5,3,5}
Vertex
figure

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 group, [[3,5,3]]
PropertiesVertex-transitive

The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3}, , has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.

Centered on hexagonal prism
Three omnitruncated regular compact honeycombs in H3
Image
Symbols t0,1,2,3{4,3,5}
 t0,1,2,3{3,5,3}
 t0,1,2,3{5,3,5}
Vertex
figure

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}
Facestriangle {3}
pentagon {5}
Vertex figure
Coxeter group[[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 cannot 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{2,5}
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 non-Wythoffian 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) Archived 2013-10-07 at the Wayback Machine
  2. ↑ "Pd{3,5,3".}
  • 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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