Order-5 dodecahedral honeycomb

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Order-5 dodecahedral honeycomb
H3 535 CC center.png
Perspective projection view
from center of Poincaré disk model
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {5,3,5}
Coxeter-Dynkin diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces pentagon {5}
Vertex figure Order-5 dodecahedral honeycomb verf.png
{3,5}
Dual Self-dual
Coxeter group K3, [5,3,5]
Properties Regular

The order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

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 regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.

Images[edit]

It is analogous to the 2D hyperbolic order-5 pentagonal tiling, {5,5}

Order 5 dodecahedral honeycomb.png

Related polytopes and 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}

There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells.

There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, of this honeycomb has all truncated icosahedron cells.

The Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb.

This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:

This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

Rectified order-5 dodecahedral honeycomb[edit]

Rectified order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{5,3,5}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Cells r{5,3} Uniform polyhedron-53-t1.png
{3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
pentagon {5}
Vertex figure Rectified order-5 dodecahedral honeycomb verf.png
pentagonal prism
Coxeter group K3, [5,3,5]
Properties Vertex-transitive, edge-transitive

The rectified order-5 dodecahedral honeycomb, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure.

H3 535 CC center 0100.png

Related tilings and honeycomb[edit]

It can be seen as analogous to the 2D hyperbolic order-4 pentagonal tiling, r{5,5}

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
r{p,3,5}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name r{3,3,5}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
r{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
r{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
r{6,3,5}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
r{7,3,5}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
... r{∞,3,5}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
Image Stereographic rectified 600-cell.png H3 435 CC center 0100.png H3 535 CC center 0100.png H3 635 boundary 0100.png
Cells
Icosahedron.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform polyhedron-33-t1.png
r{3,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cuboctahedron.png
r{4,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.png
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 237-2.png
r{7,3}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-2.png
r{∞,3}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png

Truncated order-5 dodecahedral honeycomb[edit]

Truncated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{5,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Cells r{5,3} Uniform polyhedron-53-t01.png
{3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
pentagon {5}
dodecagon {10}
Vertex figure Truncated order-5 dodecahedral honeycomb verf.png
pentagonal pyramid
Coxeter group K3, [5,3,5]
Properties Vertex-transitive

The truncated order-5 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure.

H3 535-0011 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 order-5 dodecahedral honeycomb[edit]

Truncated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol 2t{5,3,5}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Cells t{3,5} Uniform polyhedron-53-t12.png
Faces triangle {3}
pentagon {5}
hexagon {6}
Vertex figure Bitruncated order-5 dodecahedral honeycomb verf.png
disphenoid
Coxeter group K3×2, [[5,3,5]]
Properties Vertex-transitive, edge-transitive, cell-transitive

The bitruncated order-5 dodecahedral honeycomb, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, has truncated icosahedron cells, with a disphenoid vertex figure.

H3 535-0110 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 order-5 dodecahedral honeycomb[edit]

Cantellated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{5,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Cells rr{5,3} Uniform polyhedron-53-t02.png
r{3,5} Uniform polyhedron-53-t1.png
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure Cantellated order-5 dodecahedral honeycomb verf.png
triangular prism
Coxeter group K3, [5,3,5]
Properties Vertex-transitive

The cantellated order-5 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, has alternating rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure.

H3 535-1010 center ultrawide.png

Related honeycombs[edit]

Cantitruncated order-5 dodecahedral honeycomb[edit]

Cantitruncated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{5,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Cells tr{5,3} Uniform polyhedron-53-t012.png
r{3,5} Uniform polyhedron-53-t1.png
{}x{5} Pentagonal prism.png
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure Cantitruncated order-5 dodecahedral honeycomb verf.png
Mirrored sphenoid
Coxeter group K3, [5,3,5]
Properties Vertex-transitive

The cantitruncated order-5 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, has truncated icosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.

H3 535-1110 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 order-5 dodecahedral honeycomb[edit]

Runcinated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,3{5,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Cells {5,3} Uniform polyhedron-53-t0.png
{}x{5} Pentagonal prism.png
Faces square {4}
pentagon {5}
Vertex figure Runcinated order-5 dodecahedral honeycomb verf.png
triangular antiprism
Coxeter group K3×2, [[5,3,5]]
Properties Vertex-transitive, edge-transitive

The runcinated order-5 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png, has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure.

H3 535-1001 center ultrawide.png

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 order-5 dodecahedral honeycomb[edit]

Runcitruncated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{5,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Cells t{5,3} Uniform polyhedron-53-t01.png
rr{5,3} Uniform polyhedron-53-t02.png
{}x{5} Pentagonal prism.png
Faces Triangle {3}
Square {4}
Pentagon {5}
Decagon {10}
Vertex figure Runcitruncated order-5 dodecahedral honeycomb verf.png
quad pyramid
Coxeter group K3, [5,3,5]
Properties Vertex-transitive

The runcitruncated order-5 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png, has truncated dodecahedron, icosidodecahedron and pentagonal prism cells, with a distorted square pyramid vertex figure.

H3 535-1101 center ultrawide.png

Related honeycombs[edit]

Omnitruncated order-5 dodecahedral honeycomb[edit]

omnitruncated order-5 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,2,3{5,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
Cells tr{5,3} Uniform polyhedron-53-t012.png
{}x{10} Dodecagonal prism.png
Faces Square {4}
Hexagon {6}
Decagon {10}
Vertex figure Omnitruncated order-5 dodecahedral honeycomb verf.png
Phyllic disphenoid
Coxeter group K3×2, [[5,3,5]]
Properties Vertex-transitive

The omnitruncated order-5 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png, has truncated icosidodecahedron and decagonal prism cells, with a disphenoid vertex figure.

H3 535-1111 center ultrawide.png

Related honeycombs[edit]

See also[edit]

References[edit]

  • 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