Hexagonal tiling honeycombPerspective projection viewwithin Poincaré disk modelTypeHyperbolic regular honeycombParacompact uniform honeycombSchläfli symbols{6,3,3}t{3,6,3}2t{6,3,6}2t{6,3}t{3[3,3]}Coxeter diagrams ↔ ↔ ↔ ↔ Cells{6,3} Faceshexagon {6}Edge figuretriangle {3}Vertex figuretetrahedron {3,3}DualOrder-6 tetrahedral honeycombCoxeter groups, [3,3,6], [3,6,3], [6,3,6], [6,3], [3[3,3]]PropertiesRegular In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.Coxeter, The Beauty of Geometry, 1999, Chapter 10, Table III
Images
Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces. {6,3,3}{∞,3}One hexagonal tiling cell of the hexagonal tiling honeycombAn order-3 apeirogonal tiling with a green apeirogon and its horocycle
Symmetry constructions
It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3], [3,6,3], [6,3,6], [6,3] and [3[3,3]] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.
Related polytopes and honeycombs The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:
Rectified hexagonal tiling honeycomb Rectified hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolsr{6,3,3} or t1{6,3,3}Coxeter diagrams ↔ Cells{3,3} r{6,3} or Facestriangle {3}hexagon {6}Vertex figuretriangular prismCoxeter groups, [3,3,6], [3,3]PropertiesVertex-transitive, edge-transitive The rectified hexagonal tiling honeycomb, t1{6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The half-symmetry construction alternates two types of tetrahedra.
Hexagonal tiling honeycombRectified hexagonal tiling honeycomb or Related H2 tilingsOrder-3 apeirogonal tilingTriapeirogonal tiling or
Truncated hexagonal tiling honeycomb Truncated hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt{6,3,3} or t0,1{6,3,3}Coxeter diagramCells{3,3} t{6,3} Facestriangle {3}dodecagon {12}Vertex figuretriangular pyramidCoxeter groups, [3,3,6]PropertiesVertex-transitive The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.
It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:
Bitruncated hexagonal tiling honeycomb Bitruncated hexagonal tiling honeycombBitruncated order-6 tetrahedral honeycombTypeParacompact uniform honeycombSchläfli symbol2t{6,3,3} or t1,2{6,3,3}Coxeter diagram ↔ Cellst{3,3} t{3,6} Facestriangle {3}hexagon {6}Vertex figuredigonal disphenoidCoxeter groups, [3,3,6], [3,3]PropertiesVertex-transitive The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.
Cantellated hexagonal tiling honeycomb Cantellated hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolrr{6,3,3} or t0,2{6,3,3}Coxeter diagramCellsr{3,3} rr{6,3} {}×{3} Facestriangle {3}square {4}hexagon {6}Vertex figurewedgeCoxeter groups, [3,3,6]PropertiesVertex-transitive The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.
Cantitruncated hexagonal tiling honeycomb Cantitruncated hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symboltr{6,3,3} or t0,1,2{6,3,3}Coxeter diagramCellst{3,3} tr{6,3} {}×{3} Facestriangle {3}square {4}hexagon {6}dodecagon {12}Vertex figuremirrored sphenoidCoxeter groups, [3,3,6]PropertiesVertex-transitive The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.
Runcinated hexagonal tiling honeycomb Runcinated hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt0,3{6,3,3}Coxeter diagramCells{3,3} {6,3} {}×{6}{}×{3} Facestriangle {3}square {4}hexagon {6}Vertex figureirregular triangular antiprismCoxeter groups, [3,3,6]PropertiesVertex-transitive The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.
Runcitruncated hexagonal tiling honeycomb Runcitruncated hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt0,1,3{6,3,3}Coxeter diagramCellsrr{3,3} {}x{3} {}x{12} t{6,3} Facestriangle {3}square {4}dodecagon {12}Vertex figureisosceles-trapezoidal pyramidCoxeter groups, [3,3,6]PropertiesVertex-transitive The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.
Runcicantellated hexagonal tiling honeycomb Runcicantellated hexagonal tiling honeycombruncitruncated order-6 tetrahedral honeycombTypeParacompact uniform honeycombSchläfli symbolt0,2,3{6,3,3}Coxeter diagramCellst{3,3} {}x{6} rr{6,3} Facestriangle {3}square {4}hexagon {6}Vertex figureisosceles-trapezoidal pyramidCoxeter groups, [3,3,6]PropertiesVertex-transitive The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.
Omnitruncated hexagonal tiling honeycomb Omnitruncated hexagonal tiling honeycombOmnitruncated order-6 tetrahedral honeycombTypeParacompact uniform honeycombSchläfli symbolt0,1,2,3{6,3,3}Coxeter diagramCellstr{3,3} {}x{6} {}x{12} tr{6,3} Facessquare {4}hexagon {6}dodecagon {12}Vertex figureirregular tetrahedron Coxeter groups, [3,3,6]PropertiesVertex-transitive The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.
See also Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3-space Paracompact uniform honeycombs Alternated hexagonal tiling honeycomb
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, , (Chapters 16–17: Geometries on Three-manifolds I,II) N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups, (1999), Volume 4, Issue 4, pp. 329–353, Springer.com, N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002), H3: p. 130, ScienceDirect.com John Baez, Visual Insight: {6,3,3} Honeycomb (2014/03/15) John Baez, Visual Insight: {6,3,3} Honeycomb in Upper Half Space (2013/09/15) John Baez, Visual Insight: Truncated {6,3,3} Honeycomb (2016/12/01)