Tetrahedral-triangular tiling honeycomb

Tetrahedral-triangular tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	{(3,6,3,3)} or {(3,3,6,3)}
Coxeter diagram	File:CDel label6.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch 10l.png or File:CDel label6.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch 01l.png or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2-63.pngFile:CDel node.png
Cells	{3,3} File:Uniform polyhedron-33-t0.svg {3,6} File:Uniform tiling 63-t2-red.svg r{3,3} File:Uniform polyhedron-33-t1.svg
Faces	triangular {3} hexagon {6}
Vertex figure	File:Uniform tiling 63-t02.svg rhombitrihexagonal tiling
Coxeter group	[(6,3,3,3)]
Properties	Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2-63.pngFile:CDel 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.

It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified tetrahedral r{3,3}, becomes the regular octahedron {3,4}.

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

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, (2018) Chapter 13: Hyperbolic Coxeter groups