Hyperbolic tetrahedral-octahedral honeycomb explained

bgcolor=#e7dcc3 colspan=2	Tetrahedron-octahedron honeycomb
Type	Compact uniform honeycomb Semiregular honeycomb
Schläfli symbol	or
Coxeter diagram	or or
Cells
Faces
Vertex figure	rhombicuboctahedron
Coxeter group	[(4,3,3,3)]
Properties	Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in a rhombicuboctahedron vertex figure.

It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified tetrahedral r, becomes the regular octahedron .

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• Tetrahedral-octahedral honeycomb - similar Euclidean honeycomb,
• Tetrahedral-cubic honeycomb

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition (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, (2015) Chapter 13: Hyperbolic Coxeter groups