Top Qs
Timeline
Chat
Perspective

Dodecahedral-icosahedral honeycomb

From Wikipedia, the free encyclopedia

Dodecahedral-icosahedral honeycomb
Remove ads

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.

Dodecahedral-icosahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symbol{(3,5,3,5)} or {(5,3,5,3)}
Coxeter diagram or
Cells{5,3}
{3,5}
r{5,3}
Facestriangle {3}
pentagon {5}
Vertex figureThumb
rhombicosidodecahedron
Coxeter group[(5,3)[2]]
PropertiesVertex-transitive, edge-transitive

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.

Remove ads

Images

Wide-angle perspective views:

Summarize
Perspective

There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group : , , , , .

Rectified dodecahedral-icosahedral honeycomb

More information Rectified dodecahedral-icosahedral honeycomb ...

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 .

Thumb

Perspective view from center of rhombicosidodecahedron

Cyclotruncated dodecahedral-icosahedral honeycomb

More information Cyclotruncated dodecahedral-icosahedral honeycomb ...

The cyclotruncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated dodecahedron and icosahedron cells, in a pentagonal antiprism vertex figure. It has a Coxeter diagram .

Thumb

Perspective view from center of icosahedron

Cyclotruncated icosahedral-dodecahedral honeycomb

More information Cyclotruncated icosahedral-dodecahedral honeycomb ...

The cyclotruncated icosahedral-dodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron and truncated icosahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .

Thumb

Perspective view from center of dodecahedron

It can be seen as somewhat analogous to the pentahexagonal tiling, which has pentagonal and hexagonal faces:

Thumb

Truncated dodecahedral-icosahedral honeycomb

More information Truncated dodecahedral-icosahedral honeycomb ...

The truncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosahedron, truncated dodecahedron, rhombicosidodecahedron, and truncated icosidodecahedron cells, in a trapezoidal pyramid vertex figure. It has a Coxeter diagram .

Thumb

Perspective view from center of truncated icosahedron

Omnitruncated dodecahedral-icosahedral honeycomb

More information Omnitruncated dodecahedral-icosahedral honeycomb ...

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 .

Thumb

Perspective view from center of truncated icosidodecahedron
Remove ads

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, (2018) Chapter 13: Hyperbolic Coxeter groups
Remove ads
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads