Order-6 tetrahedral honeycombPerspective projection viewwithin Poincaré disk modelTypeHyperbolic regular honeycombParacompact uniform honeycombSchläfli symbols{3,3,6}{3,3}Coxeter diagrams ↔ Cells{3,3} Facestriangle {3}Edge figurehexagon {6}Vertex figure triangular tilingDualHexagonal tiling honeycombCoxeter groups, [3,3,6], [3,3]PropertiesRegular, quasiregular In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.Coxeter, The Beauty of Geometry, 1999, Chapter 10, Table III
Symmetry constructions
The order-6 tetrahedral honeycomb has a second construction as a uniform honeycomb, with Schläfli symbol {3,3}. This construction contains alternating types, or colors, of tetrahedral cells. In Coxeter notation, this half symmetry is represented as [3,3,6,1+] ↔ [3,((3,3,3))], or [3,3]: ↔ .
Related polytopes and honeycombs The order-6 tetrahedral honeycomb is analogous to the two-dimensional infinite-order triangular tiling, {3,∞}. Both tessellations are regular, and only contain triangles and ideal vertices.
The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
This honeycomb is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the hexagonal tiling honeycomb.
The order-6 tetrahedral honeycomb is part of a sequence of regular polychora and honeycombs with tetrahedral cells.
It is also part of a sequence of honeycombs with triangular tiling vertex figures.
Rectified order-6 tetrahedral honeycomb Rectified order-6 tetrahedral honeycombTypeParacompact uniform honeycombSemiregular honeycombSchläfli symbolsr{3,3,6} or t1{3,3,6}Coxeter diagrams ↔ Cellsr{3,3} {3,6} Facestriangle {3}Vertex figurehexagonal prismCoxeter groups, [3,3,6], [3,3]PropertiesVertex-transitive, edge-transitive The rectified order-6 tetrahedral honeycomb, t1{3,3,6} has octahedral and triangular tiling cells arranged in a hexagonal prism vertex figure. Perspective projection view within Poincaré disk model
Truncated order-6 tetrahedral honeycomb Truncated order-6 tetrahedral honeycombTypeParacompact uniform honeycombSchläfli symbolst{3,3,6} or t0,1{3,3,6}Coxeter diagrams ↔ Cellst{3,3} {3,6} Facestriangle {3}hexagon {6}Vertex figurehexagonal pyramidCoxeter groups, [3,3,6], [3,3]PropertiesVertex-transitive The truncated order-6 tetrahedral honeycomb, t0,1{3,3,6} has truncated tetrahedron and triangular tiling cells arranged in a hexagonal pyramid vertex figure.
Bitruncated order-6 tetrahedral honeycomb The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb.
Cantellated order-6 tetrahedral honeycomb Cantellated order-6 tetrahedral honeycombTypeParacompact uniform honeycombSchläfli symbolsrr{3,3,6} or t0,2{3,3,6}Coxeter diagrams ↔ Cellsr{3,3} r{3,6} {}x{6} Facestriangle {3}square {4}hexagon {6}Vertex figureisosceles triangular prismCoxeter groups, [3,3,6], [3,3]PropertiesVertex-transitive The cantellated order-6 tetrahedral honeycomb, t0,2{3,3,6} has cuboctahedron, trihexagonal tiling, and hexagonal prism cells arranged in an isosceles triangular prism vertex figure.
Cantitruncated order-6 tetrahedral honeycomb Cantitruncated order-6 tetrahedral honeycombTypeParacompact uniform honeycombSchläfli symbolstr{3,3,6} or t0,1,2{3,3,6}Coxeter diagrams ↔ Cellstr{3,3} t{3,6} {}x{6} Facessquare {4}hexagon {6}Vertex figuremirrored sphenoidCoxeter groups, [3,3,6], [3,3]PropertiesVertex-transitive The cantitruncated order-6 tetrahedral honeycomb, t0,1,2{3,3,6} has truncated octahedron, hexagonal tiling, and hexagonal prism cells connected in a mirrored sphenoid vertex figure.
Runcinated order-6 tetrahedral honeycomb The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb.
Runcitruncated order-6 tetrahedral honeycomb The runcitruncated order-6 tetrahedral honeycomb is equivalent to the runcicantellated hexagonal tiling honeycomb.
Runcicantellated order-6 tetrahedral honeycomb The runcicantellated order-6 tetrahedral honeycomb is equivalent to the runcitruncated hexagonal tiling honeycomb.
Omnitruncated order-6 tetrahedral honeycomb The omnitruncated order-6 tetrahedral honeycomb is equivalent to the omnitruncated hexagonal tiling honeycomb.
See also Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3-space Paracompact uniform honeycombs
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, (1999), Dover Publications, , , (Chapter 10, Regular Honeycombs in Hyperbolic Space ) Table III 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