Tetrahedral-icosahedral honeycombTypeCompact uniform honeycombSemiregular honeycombSchläfli symbol{(3,3,5,3)}Coxeter diagram or or Cells{3,3} {3,5} r{3,3} Facestriangle {3}Vertex figurerhombicosidodecahedronCoxeter group[(5,3,3,3)]PropertiesVertex-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 an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram , and is named by its two regular cells.
It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, the octahedron comes from the rectified tetrahedron .
Images + Wide-angle perspective viewsCentered on octahedron
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