Tetrahedral-cubic honeycomb

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Tetrahedron-cube honeycomb
Type Compact uniform honeycomb
Schläfli symbol {(4,3,3,3)} or {(3,3,3,4)}
Coxeter diagram CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.png or CDel label4.pngCDel branch 01r.pngCDel 3ab.pngCDel branch.png or CDel node 1.pngCDel split1-43.pngCDel nodes.pngCDel split2.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t0.png
{4,3} 40px
r{4,3} Uniform polyhedron-43-t1.png
Faces triangular {3}
square {4}
Vertex figure Uniform t0 4333 honeycomb verf.png
rhombicuboctahedron
Coxeter group [(4,3,3,3)]
Properties Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the tetrahedron-cube honeycomb is a compact uniform honeycomb, constructed from cube, tetrahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, CDel node 1.pngCDel split1-43.pngCDel nodes.pngCDel split2.pngCDel node.png, and is named by its two regular cells.

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
H3 4333-1000 center ultrawide.png
Centered on cube

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