Dodecahedral-icosahedral honeycomb

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Dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol {(3,5,3,5)} or {(5,3,5,3)}
Coxeter diagram CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label5.png or CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch.pngCDel label5.png
Cells {5,3} Uniform polyhedron-53-t0.png
{3,5} Uniform polyhedron-53-t2.png
r{5,3} Uniform polyhedron-53-t1.png
Faces triangular {3}
pentagon {5}
Vertex figure Uniform t0 5353 honeycomb verf.png
rhombicosidodecahedron
Coxeter group [(5,3)[2]]
Properties Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron 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.

Images

Wide-angle perspective views:

Related honeycombs

There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png: CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png, CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png, CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label5.png, CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png, CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png.

Rectified dodecahedral-icosahedral honeycomb

Rectified dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol r{(5,3,5,3)}
Coxeter diagrams CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png or CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 01l.pngCDel label5.png
Cells r{5,3} Uniform polyhedron-53-t1.png
rr{3,5} Uniform polyhedron-53-t02.png
Faces triangular {3}
decagon {10}
Vertex figure Uniform t02 5353 honeycomb verf.png
cuboid
Coxeter group [[(5,3)[2]]], CDel label5.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel label5.png
Properties Vertex-transitive, edge-transitive

The rectified dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosidodecahedron, and rhombicosidodecahedron cells, in a cuboid vertex figure. It has a Coxeter diagram CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png.

H3 5353-1010 center ultrawide.png

Perspective view from center of rhombicosidodecahedron

Cyclotruncated dodecahedral-icosahedral honeycomb

Cyclotruncated dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol ct{(5,3,5,3)}
Coxeter diagrams CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label5.png or CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png
Cells t{5,3} Uniform polyhedron-53-t01.png
{3,5} Uniform polyhedron-53-t2.png
Faces triangular {3}
decagon {10}
Vertex figure Uniform t01 5353 honeycomb verf.png
square antiprism
Coxeter group [[(5,3)[2]]], CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label5.png
Properties Vertex-transitive, edge-transitive

The cyclotruncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated dodecahedron, icosahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label5.png.

H3 5353-1100 center ultrawide.png

Perspective view from center of icosahedron

Cyclotruncated icosahedral-dodecahedral honeycomb

Cyclotruncated icosahedral-dodecahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol ct{(3,5,3,5)}
Coxeter diagrams CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png or CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 01l.pngCDel label5.png
Cells {5,3} Uniform polyhedron-53-t0.png
t{3,5} Uniform polyhedron-53-t12.png
Faces triangular {3}
square {4}
hexagon {6}
Vertex figure Uniform t12 5353 honeycomb verf.png
triangular antiprism
Coxeter group [[(5,3)[2]]], CDel label5.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.pngCDel label5.png
Properties Vertex-transitive, edge-transitive

The cyclotruncated icosahedral-dodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron, truncated icosahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png.

H3 5353-0110 center ultrawide.png

Perspective view from center of dodecahedron

It can be see as somewhat analogous to the pentahexagonal tiling with pentagonal and hexagonal faces:

H2 tiling 355-5.png

Truncated dodecahedral-icosahedral honeycomb

Truncated dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol t{(5,3,5,3)}
Coxeter diagrams CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png or CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png or
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 01l.pngCDel label5.png or CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png
Cells t{3,5} Truncated icosahedron.png
t{5,3} Truncated dodecahedron.png
rr{3,5} Small rhombicosidodecahedron.png
tr{5,3} Great rhombicosidodecahedron.png
Faces triangular {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure Uniform t012 5353 honeycomb verf.png
trapezoidal pyramid
Coxeter group [(5,3)[2]]
Properties Vertex-transitive

The truncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosahedron, truncated dodecahedron, rhombicosidodecahedron, truncated icosidodecahedron cells, in a trapezoidal pyramid vertex figure. It has a Coxeter diagram CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png.

H3 5353-1101 center ultrawide.png

Perspective view from center of truncated icosahedron

Omnitruncated dodecahedral-icosahedral honeycomb

Omnitruncated dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol tr{(5,3,5,3)}
Coxeter diagrams CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png
Cells tr{3,5} Great rhombicosidodecahedron.png
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure Uniform t0123 5353 honeycomb verf.png
Rhombic disphenoid
Coxeter group [(2,2)+[(5,3)[2]]], CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label5.png
Properties Vertex-transitive, edge-transitive, cell-transitive

The omnitruncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosidodecahedron cells, in a rhombic disphenoid vertex figure. It has a Coxeter diagram CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png.

H3 5353-1111 center ultrawide.png

Perspective view from center of truncated icosidodecahedron

See also

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)
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
  • 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
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