Tetrahedral-icosahedral honeycomb

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Tetrahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Semiregular honeycomb
Schläfli symbol {(3,3,5,3)}
Coxeter diagram CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.png or CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 01l.png or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2-53.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t0.png
{3,5} Uniform polyhedron-53-t2.png
r{3,3} Uniform polyhedron-33-t1.png
Faces triangular {3}
pentagon {5}
Vertex figure Uniform t2 5333 honeycomb verf.png
Coxeter group [(5,3,3,3)]
Properties Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in a icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2-53.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.

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}.


Wide-angle perspective views
H3 5333-0010 center ultrawide.png
Centered on octahedron

See also[edit]


  • 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, (2015) Chapter 13: Hyperbolic Coxeter groups