Cubic-icosahedral honeycomb

In the geometry of hyperbolic 3-space, the cubic-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, cube, and cuboctahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

Cubic-icosahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symbol{(4,3,5,3)} or {(3,5,3,4)}
Coxeter diagram or
Cells{4,3}
{3,5}
r{4,3}
Facestriangle {3}
square {4}
Vertex figure
icosidodecahedron
Coxeter group[(5,3,4,3)]
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.

Images
Wide-angle perspective view

Centered on icosahedron
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See also
  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes

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
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