Tetrahedral-square tiling honeycombTypeParacompact uniform honeycombSchläfli symbol{(4,4,3,3)} or {(3,3,4,4)}Coxeter diagramsCells{3,3} {4,4} r{4,3} Facestriangle {3}square {4}Vertex figureRhombicuboctahedronCoxeter group[(4,4,3,3)]PropertiesVertex-transitive, edge-transitive In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
Cyclotruncated tetrahedral-square tiling honeycomb Cyclotruncated tetrahedral-square tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt0,1{(4,4,3,3)}Coxeter diagramsCells {4,3} t{4,3} {3,3} t{4,3}Facestriangle {3}square {4}octagon {8}Vertex figureTriangular antiprismCoxeter group[(4,4,3,3)]PropertiesVertex-transitive The cyclotruncated tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cube, truncated cube and truncated square tiling cells, in a triangular antiprism vertex figure. It has a Coxeter diagram, .
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