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Triangular tiling honeycomb
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<h2 id="symmetry">Symmetry</h2> <p>It has two lower reflective symmetry constructions, as an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternated</a> <a href="/facts/Order-6_hexagonal_tiling_honeycomb/wATwxBgJ">order-6 hexagonal tiling honeycomb</a>, ↔ , and as from , which alternates 3 types (colors) of triangular tilings around every edge. In <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a>, the removal of the 3rd and 4th mirrors, [3,6,3*] creates a new <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a> [3[3,3]], , subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: ↔ . </p> <h2 id="related-tilings">Related Tilings</h2> <p>It is similar to the 2D hyperbolic <a href="/facts/Infinite-order_apeirogonal_tiling/WVaudE59">infinite-order apeirogonal tiling</a>, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface. </p> <h2 id="related-honeycombs">Related honeycombs</h2> <p>The triangular tiling honeycomb is a <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular hyperbolic honeycomb</a> in 3-space, and one of eleven paracompact honeycombs. </p> <table><tbody><tr><th colspan="12"><a href="/facts/List_of_regular_polytopes/Kng25Ymq">11 paracompact regular honeycombs</a></th></tr><tr><td><a href="/facts/Hexagonal_tiling_honeycomb/DD00vEWL">{6,3,3}</a></td><td><a href="/facts/Order-4_hexagonal_tiling_honeycomb/3pSIAwsX">{6,3,4}</a></td><td><a href="/facts/Order-5_hexagonal_tiling_honeycomb/AVt6DFhA">{6,3,5}</a></td><td><a href="/facts/Order-6_hexagonal_tiling_honeycomb/wATwxBgJ">{6,3,6}</a></td><td><a href="/facts/Square_tiling_honeycomb/zlCkJfRo">{4,4,3}</a></td><td><a href="/facts/Order-4_square_tiling_honeycomb/FbysA3Aa">{4,4,4}</a></td></tr><tr><td><a href="/facts/Order-6_tetrahedral_honeycomb/stpDuyNA">{3,3,6}</a></td><td><a href="/facts/Order-6_cubic_honeycomb/mE1rd3oM">{4,3,6}</a></td><td><a href="/facts/Order-6_dodecahedral_honeycomb/nX10JuCf">{5,3,6}</a></td><td>{3,6,3}</td><td><a href="/facts/Order-4_octahedral_honeycomb/vvpQj6Mz">{3,4,4}</a></td></tr></tbody></table> <p>There are <a href="/facts/Paracompact_uniform_honeycombs/hnmKAVu0">nine uniform honeycombs</a> in the [3,6,3] <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a> family, including this regular form as well as the <a href="/facts/Bitruncation_(geometry)/t0xarbyJ">bitruncated</a> form, t1,2{3,6,3}, with all <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</a> facets. </p> [3,6,3] family honeycombs<table><tbody><tr><th>{3,6,3}</th><th><a href="/facts/Rectified_triangular_tiling_honeycomb/EjnBjVWg">r{3,6,3}</a></th><th><a href="/facts/Truncated_triangular_tiling_honeycomb/DD00vEWL">t{3,6,3}</a></th><th><a href="/facts/Cantellated_triangular_tiling_honeycomb/EjnBjVWg">rr{3,6,3}</a></th><th><a href="/facts/Runcinated_triangular_tiling_honeycomb/EjnBjVWg">t0,3{3,6,3}</a></th><th><a href="/facts/Bitruncated_triangular_tiling_honeycomb/EjnBjVWg">2t{3,6,3}</a></th><th><a href="/facts/Cantitruncated_triangular_tiling_honeycomb/EjnBjVWg">tr{3,6,3}</a></th><th><a href="/facts/Runcitruncated_triangular_tiling_honeycomb/EjnBjVWg">t0,1,3{3,6,3}</a></th><th><a href="/facts/Omnitruncated_triangular_tiling_honeycomb/EjnBjVWg">t0,1,2,3{3,6,3}</a></th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <p>The honeycomb is also part of a series of <a href="/facts/Regular_4-polytope/lFf8IH1C">polychora</a> and honeycombs with triangular <a href="/facts/Edge_figure/3u2ySK0C">edge figures</a>. </p> <table><tbody><tr><th colspan="12">{3,<i>p</i>,3} polytopes</th></tr><tr><th>Space</th><th colspan="2"><a href="/facts/3-sphere/u67nImpY">S3</a></th><th colspan="5"><a href="/facts/Hyperbolic_space/kI7tnwdk">H3</a></th></tr><tr><th>Form</th><th colspan="2">Finite</th><th>Compact</th><th>Paracompact</th><th colspan="3">Noncompact</th></tr><tr><th>{3,<i>p</i>,3}</th><th><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></th><th><a href="/facts/24-cell/tfoGnlAD">{3,4,3}</a></th><th><a href="/facts/Icosahedral_honeycomb/4P0s3eNd">{3,5,3}</a></th><th>{3,6,3}</th><th><a href="/facts/Order-7-3_triangular_honeycomb/7hpLNEIR">{3,7,3}</a></th><th><a href="/facts/Order-8-3_triangular_honeycomb/0I4xjWtr">{3,8,3}</a></th><th>... <a href="/facts/Order-infinite-3_triangular_honeycomb/c3KqPvfa">{3,∞,3}</a></th></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Cells</th><td><a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a></td><td><a href="/facts/Octahedron/G38q92xW">{3,4}</a></td><td><a href="/facts/Icosahedron/wWbpnDYp">{3,5}</a></td><td><a href="/facts/Triangular_tiling/Jx8f8Krh">{3,6}</a></td><td><a href="/facts/Order-7_triangular_tiling/bF5pkIRd">{3,7}</a></td><td><a href="/facts/Order-8_triangular_tiling/feag8eNn">{3,8}</a></td><td><a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">{3,∞}</a></td></tr><tr><th>Vertexfigure</th><td><a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a></td><td><a href="/facts/Cube/65lUaAqN">{4,3}</a></td><td><a href="/facts/Dodecahedron/7r519eCl">{5,3}</a></td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">{6,3}</a></td><td><a href="/facts/Heptagonal_tiling/aKhMKTMx">{7,3}</a></td><td><a href="/facts/Octagonal_tiling/Fjj2z6ED">{8,3}</a></td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">{∞,3}</a></td></tr></tbody></table> <h3>Rectified triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Rectified triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>r{3,6,3}h2{6,3,6}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td> ↔ ↔ ↔ </td></tr><tr><td>Cells</td><td><a href="/facts/Trihexagonal_tiling/spAKhDvd">r{3,6}</a> <a href="/facts/Hexagonal_tiling/HwG5F0YH">{6,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3] V P ¯ 3 {\displaystyle {\overline {VP}}_{3}} , [6,3[3]] P P ¯ 3 {\displaystyle {\overline {PP}}_{3}} , [3[3,3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive</td></tr></tbody></table> <p>The rectified triangular tiling honeycomb, , has <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a> and <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> cells, with a <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> vertex figure. </p> <h4>Symmetry</h4> <p>A lower symmetry of this honeycomb can be constructed as a <i>cantic order-6 hexagonal tiling honeycomb</i>, ↔ . A second lower-index construction is ↔ . </p> <h3>Truncated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Truncated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">t{3,6}</a> <a href="/facts/Hexagonal_tiling/HwG5F0YH">{6,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3] V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table> <p>The truncated triangular tiling honeycomb, , is a lower-symmetry form of the <a href="/facts/Hexagonal_tiling_honeycomb/DD00vEWL">hexagonal tiling honeycomb</a>, . It contains <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> facets with a <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedral</a> vertex figure. </p> <h3>Bitruncated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Bitruncated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>2t{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">t{6,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Dodecagon/lKDvfUsm">dodecagon</a> {12}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Tetragonal_disphenoid/7rU7jc7U">tetragonal disphenoid</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> 2 × Y ¯ 3 {\displaystyle 2\times {\overline {Y}}_{3}} , [[3,6,3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive, cell-transitive</td></tr></tbody></table> <p>The bitruncated triangular tiling honeycomb, , has <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</a> cells, with a <a href="/facts/Tetragonal_disphenoid/7rU7jc7U">tetragonal disphenoid</a> vertex figure. </p> <h3>Cantellated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantellated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>rr{3,6,3} or t0,2{3,6,3}s2{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rr{6,3}</a> <a href="/facts/Trihexagonal_tiling/spAKhDvd">r{6,3}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}×{3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Wedge_(geometry)/S9xb50Fs">wedge</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantellated triangular tiling honeycomb, , has <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rhombitrihexagonal tiling</a>, <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> cells, with a <a href="/facts/Wedge_(geometry)/S9xb50Fs">wedge</a> vertex figure. </p> <h4>Symmetry</h4> <p>It can also be constructed as a cantic snub triangular tiling honeycomb, , a half-symmetry form with symmetry [3+,6,3]. </p> <h3>Cantitruncated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantitruncated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>tr{3,6,3} or t0,1,2{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">tr{6,3}</a> <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">t{6,3}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}×{3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}<a href="/facts/Dodecagon/lKDvfUsm">dodecagon</a> {12}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Mirrored_sphenoid/7mkSrl6j">mirrored sphenoid</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantitruncated triangular tiling honeycomb, , has <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">truncated trihexagonal tiling</a>, <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> cells, with a <a href="/facts/Mirrored_sphenoid/7mkSrl6j">mirrored sphenoid</a> vertex figure. </p> <h3>Runcinated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcinated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,3{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Triangular_tiling/Jx8f8Krh">{3,6}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}×{3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Square/JHJeMvmL">square</a> {4}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Hexagonal_antiprism/bt1W4cs6">hexagonal antiprism</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> 2 × Y ¯ 3 {\displaystyle 2\times {\overline {Y}}_{3}} , [[3,6,3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive</td></tr></tbody></table> <p>The runcinated triangular tiling honeycomb, , has <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a> and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> cells, with a <a href="/facts/Hexagonal_antiprism/bt1W4cs6">hexagonal antiprism</a> vertex figure. </p> <h3>Runcitruncated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcitruncated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>t0,1,3{3,6,3}s2,3{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagrams</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">t{3,6}</a> <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rr{3,6}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}×{3}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{}×{6}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Isosceles_trapezoid/Izn5AA4p">isosceles-trapezoidal</a> <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramid</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcitruncated triangular tiling honeycomb, , has <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a>, <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rhombitrihexagonal tiling</a>, <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a>, and <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a> cells, with an <a href="/facts/Isosceles_trapezoid/Izn5AA4p">isosceles-trapezoidal</a> <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h4>Symmetry</h4> <p>It can also be constructed as a runcicantic snub triangular tiling honeycomb, , a half-symmetry form with symmetry [3+,6,3]. </p> <h3>Omnitruncated triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Omnitruncated triangular tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,2,3{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">tr{3,6}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{}×{6}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}<a href="/facts/Dodecagon/lKDvfUsm">dodecagon</a> {12}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Phyllic_disphenoid/7mkSrl6j">phyllic disphenoid</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> 2 × Y ¯ 3 {\displaystyle 2\times {\overline {Y}}_{3}} , [[3,6,3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive</td></tr></tbody></table> <p>The omnitruncated triangular tiling honeycomb, , has <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">truncated trihexagonal tiling</a> and <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a> cells, with a <a href="/facts/Phyllic_disphenoid/7mkSrl6j">phyllic disphenoid</a> vertex figure. </p> <h3>Runcisnub triangular tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcisnub triangular tiling honeycomb</th></tr><tr><td>Type</td><td>Paracompact scaliform honeycomb</td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>s3{3,6,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td><a href="/facts/Trihexagonal_tiling/spAKhDvd">r{6,3}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}x{3}</a> <a href="/facts/Triangular_tiling/Jx8f8Krh">{3,6}</a> <a href="/facts/Triangular_cupola/Pyocot8L">tricup</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3+,6,3]</td></tr><tr><td>Properties</td><td>Vertex-transitive, non-uniform</td></tr></tbody></table> <p>The runcisnub triangular tiling honeycomb, , has <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a>, <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>, <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a>, and <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a> cells. It is <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>, but not uniform, since it contains <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solid</a> <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a> cells. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Convex_uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">Convex uniform honeycombs in hyperbolic space</a></li> <li><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular tessellations of hyperbolic 3-space</a></li> <li><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycombs</a></li></ul> <ul><li><a href="/facts/H.S.M._Coxeter/D2l5jWGG">Coxeter</a>, <i><a href="/facts/Regular_Polytopes_(book)/jomuRNha">Regular Polytopes</a></i>, 3rd. ed., Dover Publications, 1973. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)</li> <li><i>The Beauty of Geometry: Twelve Essays</i> (1999), Dover Publications, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/99035678">99-35678</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-40919-8 (Chapter 10, <a href="http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf">Regular Honeycombs in Hyperbolic Space</a>) Table III</li> <li><a href="/facts/Jeffrey_Weeks_(mathematician)/KjnMwGpQ">Jeffrey R. Weeks</a> <i>The Shape of Space, 2nd edition</i> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I, II)</li> <li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> <i>Uniform Polytopes</i>, Manuscript <ul><li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">N.W. Johnson</a>: <i>The Theory of Uniform Polytopes and Honeycombs</i>, Ph.D. Dissertation, University of Toronto, 1966</li> <li>N.W. Johnson: <i>Geometries and Transformations</i>, (2018) Chapter 13: Hyperbolic Coxeter groups</li></ul></li></ul>