Tetrahedral-square tiling honeycomb

In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

Tetrahedral-square tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbol{(4,4,3,3)} or {(3,3,4,4)}
Coxeter diagrams
Cells{3,3}
{4,4}
r{4,3}
Facestriangle {3}
square {4}
Vertex figure
Rhombicuboctahedron
Coxeter group[(4,4,3,3)]
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.

Cyclotruncated tetrahedral-square tiling honeycomb
Cyclotruncated tetrahedral-square tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1{(4,4,3,3)}
Coxeter diagrams
Cells {4,3}
t{4,3}
{3,3}
t{4,3}
Facestriangle {3}
square {4}
octagon {8}
Vertex figure
Triangular antiprism
Coxeter group[(4,4,3,3)]
PropertiesVertex-transitive

The cyclotruncated tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cube, truncated cube and truncated square tiling cells, in a triangular antiprism vertex figure. It has a Coxeter diagram, .

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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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