Octahedral-dodecahedral honeycomb

In the geometry of hyperbolic 3-space, the octahedron-dodecahedron honeycomb is a compact uniform honeycomb, constructed from dodecahedron, octahedron, and icosidodecahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

Octahedron-dodecahedron honeycomb
TypeCompact uniform honeycomb
Schläfli symbol{(5,3,4,3)} or {(3,4,3,5)}
Coxeter diagram or
Cells{3,4}
{5,3}
r{5,3}
Facestriangular {3}
pentagon {5}
Vertex figure
rhombicuboctahedron
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 dodecahedron
gollark: Oh no.
gollark: Anyway, concrete, glass, urbanism and high-performance computing good; fields, isolated cottages and manually farming bad.
gollark: If you don't have an infographic for that I can't possibly believe it.
gollark: This would displease me. I dislike "cottagecore".
gollark: First aid is valid, but "helping friends with mental and emotional problems" sounds extremely hard to teach. Although I guess that also applies to independent learning and stuff, and the solution is probably to structure stuff such that it arises easily instead of trying to manually teach it.

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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