paracompact, uniform, honeycombs, tessellation, convex, uniform, polyhedron, cells, example, paracompact, regular, honeycombs, geometry, uniform, honeycombs, hyperbolic, space, tessellations, convex, uniform, polyhedron, cells, dimensional, hyperbolic, space, . Tessellation of convex uniform polyhedron cells Example paracompact regular honeycombs 3 3 6 6 3 3 4 3 6 6 3 4 5 3 6 6 3 5 6 3 6 3 6 3 4 4 3 3 4 4 4 4 4 In geometry uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells In 3 dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs generated as Wythoff constructions and represented by ring permutations of the Coxeter diagrams for each family These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure including ideal vertices at infinity similar to the hyperbolic uniform tilings in 2 dimensions Contents 1 Regular paracompact honeycombs 2 Coxeter groups of paracompact uniform honeycombs 3 Linear graphs 3 1 6 3 3 family 3 2 6 3 4 family 3 3 6 3 5 family 3 4 6 3 6 family 3 5 3 6 3 family 3 6 4 4 3 family 3 7 4 4 4 family 4 Tridental graphs 4 1 3 41 1 family 4 2 4 41 1 family 4 3 6 31 1 family 5 Cyclic graphs 5 1 4 4 3 3 family 5 2 4 4 4 3 family 5 3 4 4 4 4 family 5 4 6 3 3 3 family 5 5 6 3 4 3 family 5 6 6 3 5 3 family 5 7 6 3 6 3 family 6 Loop n tail graphs 6 1 3 3 3 family 6 2 4 3 3 family 6 3 5 3 3 family 6 4 6 3 3 family 7 Multicyclic graphs 7 1 3 family 7 2 3 3 3 family 8 Summary enumerations by family 8 1 Linear graphs 8 2 Tridental graphs 8 3 Cyclic graphs 8 4 Loop n tail graphs 9 See also 10 Notes 11 References Regular paracompact honeycombs Edit Of the uniform paracompact H3 honeycombs 11 are regular meaning that their group of symmetries acts transitively on their flags These have Schlafli symbol 3 3 6 6 3 3 3 4 4 4 4 3 3 6 3 4 3 6 6 3 4 4 4 4 5 3 6 6 3 5 and 6 3 6 and are shown below Four have finite Ideal polyhedral cells 3 3 6 4 3 6 3 4 4 and 5 3 6 11 paracompact regular honeycombs 6 3 3 6 3 4 6 3 5 6 3 6 4 4 3 4 4 4 3 3 6 4 3 6 5 3 6 3 6 3 3 4 4 Name SchlafliSymbol p q r Coxeter Celltype p q Facetype p Edgefigure r Vertexfigure q r Dual Coxetergroup Order 6 tetrahedral honeycomb 3 3 6 3 3 3 6 3 6 6 3 3 6 3 3 Hexagonal tiling honeycomb 6 3 3 6 3 6 3 3 3 3 3 6 Order 4 octahedral honeycomb 3 4 4 3 4 3 4 4 4 4 4 3 4 4 3 Square tiling honeycomb 4 4 3 4 4 4 3 4 3 3 4 4 Triangular tiling honeycomb 3 6 3 3 6 3 3 6 3 Self dual 3 6 3 Order 6 cubic honeycomb 4 3 6 4 3 4 4 3 6 6 3 4 6 3 4 Order 4 hexagonal tiling honeycomb 6 3 4 6 3 6 4 3 4 4 3 6 Order 4 square tiling honeycomb 4 4 4 4 4 4 4 4 4 Self dual 4 4 4 Order 6 dodecahedral honeycomb 5 3 6 5 3 5 5 3 6 6 3 5 6 3 5 Order 5 hexagonal tiling honeycomb 6 3 5 6 3 6 5 3 5 5 3 6 Order 6 hexagonal tiling honeycomb 6 3 6 6 3 6 6 3 6 Self dual 6 3 6 Coxeter groups of paracompact uniform honeycombs Edit These graphs show subgroup relations of paracompact hyperbolic Coxeter groups Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry This is a complete enumeration of the 151 unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains rank 4 paracompact coxeter groups The honeycombs are indexed here for cross referencing duplicate forms with brackets around the nonprimary constructions The alternations are listed but are either repeats or don t generate uniform solutions Single hole alternations represent a mirror removal operation If an end node is removed another simplex tetrahedral family is generated If a hole has two branches a Vinberg polytope is generated although only Vinberg polytope with mirror symmetry are related to the simplex groups and their uniform honeycombs have not been systematically explored These nonsimplectic pyramidal Coxeter groups are not enumerated on this page except as special cases of half groups of the tetrahedral ones Six uniform honeycombs that arise here as alternations have been numbered 152 to 157 after the 151 Wythoffian forms not requiring alternation for their construction Tetrahedral hyperbolic paracompact group summary Coxeter group Simplexvolume Commutator subgroup Unique honeycomb count 6 3 3 0 0422892336 1 6 3 3 3 3 3 15 4 4 3 0 0763304662 1 4 1 4 3 15 3 3 3 0 0845784672 3 3 3 4 6 3 4 0 1057230840 1 6 3 4 1 3 x 15 3 41 1 0 1526609324 3 41 1 4 3 6 3 0 1691569344 3 6 3 8 6 3 5 0 1715016613 1 6 3 5 5 3 3 15 6 31 1 0 2114461680 1 6 31 1 3 x 4 4 3 3 0 2114461680 1 4 3 3 3 x 4 4 4 4 0 2289913985 4 4 4 6 6 3 6 0 2537354016 1 6 3 6 1 3 3 3 8 4 4 3 3 0 3053218647 4 1 4 3 3 4 5 3 3 0 3430033226 5 3 3 4 6 3 3 3 0 3641071004 6 3 3 3 9 3 x 0 4228923360 3 x 1 41 1 1 0 4579827971 1 41 1 1 0 6 3 3 0 5074708032 1 6 3 3 3 3 3 2 6 3 4 3 0 5258402692 6 3 4 3 9 4 4 4 3 0 5562821156 4 1 4 1 4 3 9 6 3 5 3 0 6729858045 6 3 5 3 9 6 3 6 3 0 8457846720 6 3 6 3 5 4 4 4 4 0 9159655942 4 4 4 4 1 3 3 3 1 014916064 3 3 3 0 The complete list of nonsimplectic non tetrahedral paracompact Coxeter groups was published by P Tumarkin in 2003 91 1 93 The smallest paracompact form in H3 can be represented by or or 8734 3 3 8734 which can be constructed by a mirror removal of paracompact hyperbolic group 3 4 4 as 3 4 1 4 160 The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid Another pyramid is or constructed as 4 4 1 4 8734 4 4 8734 160 Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow tie graphs 3 3 4 1 4 3 8734 3 3 8734 3 or 3 4 4 1 4 4 8734 3 3 8734 4 or 4 4 4 1 4 4 8734 4 4 8734 4 or Another nonsimplectic half groups is A radical nonsimplectic subgroup is which can be doubled into a triangular prism domain as Pyramidal hyperbolic paracompact group summary Dimension Rank Graphs H3 5 Linear graphs Edit 6 3 3 family Edit Honeycomb name Coxeter diagram Schlafli symbol Cells by location and count around each vertex Vertex figure Picture 1 2 3 4 1 hexagonal 6 3 3 4 6 6 6 Tetrahedron 2 rectified hexagonal t1 6 3 3 or r 6 3 3 2 3 3 3 3 3 6 3 6 Triangular prism 3 rectified order 6 tetrahedral t1 3 3 6 or r 3 3 6 6 3 3 3 3 2 3 3 3 3 3 3 Hexagonal prism 4 order 6 tetrahedral 3 3 6 3 3 3 Triangular tiling 5 truncated hexagonal t0 1 6 3 3 or t 6 3 3 1 3 3 3 3 3 12 12 Triangular pyramid 6 cantellated hexagonal t0 2 6 3 3 or rr 6 3 3 1 3 3 3 3 2 4 4 3 2 3 4 6 4 7 runcinated hexagonal t0 3 6 3 3 1 3 3 3 3 4 4 3 3 4 4 6 1 6 6 6 8 cantellated order 6 tetrahedral t0 2 3 3 6 or rr 3 3 6 1 3 4 3 4 2 4 4 6 2 3 6 3 6 9 bitruncated hexagonal t1 2 6 3 3 or 2t 6 3 3 2 3 6 6 2 6 6 6 10 truncated order 6 tetrahedral t0 1 3 3 6 or t 3 3 6 6 3 6 6 1 3 3 3 3 3 3 11 cantitruncated hexagonal t0 1 2 6 3 3 or tr 6 3 3 1 3 6 6 1 4 4 3 2 4 6 12 12 runcitruncated hexagonal t0 1 3 6 3 3 1 3 4 3 4 2 4 4 3 1 4 4 12 1 3 12 12 13 runcitruncated order 6 tetrahedral t0 1 3 3 3 6 1 3 6 6 1 4 4 6 2 4 4 6 1 3 4 6 4 14 cantitruncated order 6 tetrahedral t0 1 2 3 3 6 or tr 3 3 6 2 4 6 6 1 4 4 6 1 6 6 6 15 omnitruncated hexagonal t0 1 2 3 6 3 3 1 4 6 6 1 4 4 6 1 4 4 12 1 4 6 12 Alternated forms Honeycomb name Coxeter diagram Schlafli symbol Cells by location and count around each vertex Vertex figure Picture 1 2 3 4 Alt 137 alternated hexagonal 4 3 3 3 3 3 3 4 3 3 3 3 6 6 138 cantic hexagonal 1 3 3 3 3 2 3 6 3 6 2 3 6 6 139 runcic hexagonal 1 4 4 4 1 4 4 3 1 3 3 3 3 3 3 3 3 4 3 4 140 runcicantic hexagonal 1 3 6 6 1 4 4 3 1 3 6 3 6 2 4 6 6 Nonuniform snub rectified order 6 tetrahedral sr 3 3 6 Irr 3 3 3 Nonuniform cantic snub order 6 tetrahedral sr3 3 3 6 Nonuniform omnisnub order 6 tetrahedral ht0 1 2 3 6 3 3 Irr 3 3 3 6 3 4 family Edit There are 15 forms generated by ring permutations of the Coxeter group 6 3 4 or Name of honeycombCoxeter diagramSchlafli symbol Cells by location and count per vertex Vertex figure Picture 0 1 2 3 16 Regular order 4 hexagonal 6 3 4 8 6 6 6 3 3 3 3 17 rectified order 4 hexagonal t1 6 3 4 or r 6 3 4 2 3 3 3 3 4 3 6 3 6 4 4 4 18 rectified order 6 cubic t1 4 3 6 or r 4 3 6 6 3 4 3 4 2 3 3 3 3 3 3 6 4 4 19 order 6 cubic 4 3 6 20 4 4 4 3 3 3 3 3 3 20 truncated order 4 hexagonal t0 1 6 3 4 or t 6 3 4 1 3 3 3 3 4 3 12 12 21 bitruncated order 6 cubic t1 2 6 3 4 or 2t 6 3 4 2 4 6 6 2 6 6 6 22 truncated order 6 cubic t0 1 4 3 6 or t 4 3 6 6 3 8 8 1 3 3 3 3 3 3 23 cantellated order 4 hexagonal t0 2 6 3 4 or rr 6 3 4 1 3 4 3 4 2 4 4 4 2 3 4 6 4 24 cantellated order 6 cubic t0 2 4 3 6 or rr 4 3 6 2 3 4 4 4 2 4 4 6 1 3 6 3 6 25 runcinated order 6 cubic t0 3 6 3 4 1 4 4 4 3 4 4 4 3 4 4 6 1 6 6 6 26 cantitruncated order 4 hexagonal t0 1 2 6 3 4 or tr 6 3 4 1 4 6 6 1 4 4 4 2 4 6 12 27 cantitruncated order 6 cubic t0 1 2 4 3 6 or tr 4 3 6 2 4 6 8 1 4 4 6 1 6 6 6 28 runcitruncated order 4 hexagonal t0 1 3 6 3 4 1 3 4 4 4 1 4 4 4 2 4 4 12 1 3 12 12 29 runcitruncated order 6 cubic t0 1 3 4 3 6 1 3 8 8 2 4 4 8 1 4 4 6 1 3 4 6 4 30 omnitruncated order 6 cubic t0 1 2 3 6 3 4 1 4 6 8 1 4 4 8 1 4 4 12 1 4 6 12 Alternated forms Name of honeycombCoxeter diagramSchlafli symbol Cells by location and count per vertex Vertex figure Picture 0 1 2 3 Alt 87 alternated order 6 cubic h 4 3 6 3 3 3 160 160 3 3 3 3 3 3 3 6 3 6 88 cantic order 6 cubic h2 4 3 6 2 3 6 6 1 3 6 3 6 2 6 6 6 89 runcic order 6 cubic h3 4 3 6 1 3 3 3 1 6 6 6 3 3 4 6 4 90 runcicantic order 6 cubic h2 3 4 3 6 1 3 6 6 1 3 12 12 2 4 6 12 141 alternated order 4 hexagonal h 6 3 4 3 3 3 3 3 3 3 3 3 3 4 6 6 142 cantic order 4 hexagonal h1 6 3 4 1 3 4 3 4 2 3 6 3 6 2 4 6 6 143 runcic order 4 hexagonal h3 6 3 4 1 4 4 4 1 4 4 3 1 3 3 3 3 3 3 3 3 4 4 4 144 runcicantic order 4 hexagonal h2 3 6 3 4 1 3 8 8 1 4 4 3 1 3 6 3 6 2 4 6 8 151 quarter order 4 hexagonal q 6 3 4 3 1 1 3 Nonuniform bisnub order 6 cubic 2s 4 3 6 3 3 3 3 3 3 3 3 3 3 6 3 3 3 Nonuniform runcic bisnub order 6 cubic Nonuniform snub rectified order 6 cubic sr 4 3 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 6 3 3 3 Nonuniform runcic snub rectified order 6 cubic sr3 4 3 6 Nonuniform snub rectified order 4 hexagonal sr 6 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 6 3 3 3 Nonuniform runcisnub rectified order 4 hexagonal sr3 6 3 4 Nonuniform omnisnub rectified order 6 cubic ht0 1 2 3 6 3 4 3 3 3 3 4 3 3 3 4 3 3 3 6 3 3 3 3 6 3 3 3 6 3 5 family Edit Honeycomb nameCoxeter diagramSchlafli symbol Cells by location and count around each vertex Vertex figure Picture 0 1 2 3 31 order 5 hexagonal 6 3 5 20 6 3 Icosahedron 32 rectified order 5 hexagonal t1 6 3 5 or r 6 3 5 2 3 3 3 3 3 5 3 6 2 5 4 4 33 rectified order 6 dodecahedral t1 5 3 6 or r 5 3 6 5 3 5 3 5 2 3 6 6 4 4 34 order 6 dodecahedral 5 3 6 5 5 5 8734 3 6 35 truncated order 5 hexagonal t0 1 6 3 5 or t 6 3 5 1 3 3 3 3 3 5 3 12 12 36 cantellated order 5 hexagonal t0 2 6 3 5 or rr 6 3 5 1 3 5 3 5 2 5 4 4 2 3 4 6 4 37 runcinated order 6 dodecahedral t0 3 6 3 5 1 5 5 5 6 6 4 4 1 6 3 38 cantellated order 6 dodecahedral t0 2 5 3 6 or rr 5 3 6 2 4 3 4 5 2 6 4 4 1 3 6 2 39 bitruncated order 6 dodecahedral t1 2 6 3 5 or 2t 6 3 5 2 5 6 6 2 6 3 40 truncated order 6 dodecahedral t0 1 5 3 6 or t 5 3 6 6 3 10 10 1 3 6 41 cantitruncated order 5 hexagonal t0 1 2 6 3 5 or tr 6 3 5 1 5 6 6 1 5 4 4 2 4 6 10 42 runcitruncated order 5 hexagonal t0 1 3 6 3 5 1 4 3 4 5 1 5 4 4 2 12 4 4 1 3 12 12 43 runcitruncated order 6 dodecahedral t0 1 3 5 3 6 1 3 10 10 1 10 4 4 2 6 4 4 1 3 4 6 4 44 cantitruncated order 6 dodecahedral t0 1 2 5 3 6 or tr 5 3 6 1 4 6 10 2 6 4 4 1 6 3 45 omnitruncated order 6 dodecahedral t0 1 2 3 6 3 5 1 4 6 10 1 10 4 4 1 12 4 4 1 4 6 12 Alternated forms Honeycomb nameCoxeter diagramSchlafli symbol Cells by location and count around each vertex Vertex figure Picture 0 1 2 3 Alt 145 alternated order 5 hexagonal h 6 3 5 20 3 6 12 3 5 5 6 6 146 cantic order 5 hexagonal h2 6 3 5 1 3 5 3 5 2 3 6 3 6 2 5 6 6 147 runcic order 5 hexagonal h3 6 3 5 1 5 5 5 1 4 4 3 1 3 3 3 3 3 3 3 3 4 5 4 148 runcicantic order 5 hexagonal h2 3 6 3 5 1 3 10 10 1 4 4 3 1 3 6 3 6 2 4 6 10 a, wikipedia, wiki, book, books, library,

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