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Hexagonal tiling honeycomb
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<h2 id="images">Images</h2> <p>Viewed in perspective outside of a <a href="/facts/Poincar%C3%A9_disk_model/6nCWjbRL">Poincaré disk model</a>, the image above shows one <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> cell within the honeycomb, and its mid-radius <a href="/facts/Horosphere/Mv52WknL">horosphere</a> (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, <a href="/facts/Asymptote/nvLPuxKa">asymptoting</a> towards a single ideal point. It can be seen as similar to the <a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">order-3 apeirogonal tiling</a>, {∞,3} of H2, with <a href="/facts/Horocycle/sBR9wnuu">horocycles</a> circumscribing vertices of <a href="/facts/Apeirogon/r0V9hleb">apeirogonal</a> faces. </p> <table><tbody><tr><th>{6,3,3}</th><th>{∞,3}</th></tr><tr><td></td><td></td></tr><tr><td>One hexagonal tiling cell of the hexagonal tiling honeycomb</td><td>An <a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">order-3 apeirogonal tiling</a> with a green apeirogon and its horocycle</td></tr></tbody></table> <h2 id="symmetry-constructions">Symmetry constructions</h2> <p>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[3]] and [3[3,3]] , having 1, 4, 6, 12 and 24 times <a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">larger fundamental domains respectively</a>. In <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a> 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 <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff construction</a>. </p> <h2 id="related-polytopes-and-honeycombs">Related polytopes and honeycombs</h2> <p>The hexagonal tiling honeycomb is a <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular hyperbolic honeycomb</a> in 3-space, and one of 11 which are paracompact. </p> <table><tbody><tr><th colspan="12"><a href="/facts/List_of_regular_polytopes/Kng25Ymq">11 paracompact regular honeycombs</a></th></tr><tr><td>{6,3,3}</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><a href="/facts/Triangular_tiling_honeycomb/EjnBjVWg">{3,6,3}</a></td><td><a href="/facts/Order-4_octahedral_honeycomb/vvpQj6Mz">{3,4,4}</a></td></tr></tbody></table> <p>It is <a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">one of 15 uniform paracompact honeycombs</a> in the [6,3,3] Coxeter group, along with its dual, the <a href="/facts/Order-6_tetrahedral_honeycomb/stpDuyNA">order-6 tetrahedral honeycomb</a>. </p> <table><tbody><tr><th colspan="12">[6,3,3] family honeycombs</th></tr><tr><th>{6,3,3}</th><th><a href="/facts/Rectified_hexagonal_tiling_honeycomb/DD00vEWL">r{6,3,3}</a></th><th><a href="/facts/Truncated_hexagonal_tiling_honeycomb/DD00vEWL">t{6,3,3}</a></th><th><a href="/facts/Cantellated_hexagonal_tiling_honeycomb/DD00vEWL">rr{6,3,3}</a></th><th><a href="/facts/Runcinated_hexagonal_tiling_honeycomb/DD00vEWL">t0,3{6,3,3}</a></th><th><a href="/facts/Cantitruncated_hexagonal_tiling_honeycomb/DD00vEWL">tr{6,3,3}</a></th><th><a href="/facts/Runcitruncated_hexagonal_tiling_honeycomb/DD00vEWL">t0,1,3{6,3,3}</a></th><th><a href="/facts/Omnitruncated_hexagonal_tiling_honeycomb/DD00vEWL">t0,1,2,3{6,3,3}</a></th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan="2"></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Order-6_tetrahedral_honeycomb/stpDuyNA">{3,3,6}</a></th><th><a href="/facts/Rectified_order-6_tetrahedral_honeycomb/stpDuyNA">r{3,3,6}</a></th><th><a href="/facts/Truncated_order-6_tetrahedral_honeycomb/stpDuyNA">t{3,3,6}</a></th><th><a href="/facts/Cantellated_order-6_tetrahedral_honeycomb/stpDuyNA">rr{3,3,6}</a></th><th><a href="/facts/Bitruncated_order-6_tetrahedral_honeycomb/DD00vEWL">2t{3,3,6}</a></th><th><a href="/facts/Cantitruncated_order-6_tetrahedral_honeycomb/stpDuyNA">tr{3,3,6}</a></th><th><a href="/facts/Runcitruncated_order-6_tetrahedral_honeycomb/stpDuyNA">t0,1,3{3,3,6}</a></th><th><a href="/facts/Omnitruncated_order-6_tetrahedral_honeycomb/DD00vEWL">t0,1,2,3{3,3,6}</a></th></tr></tbody></table> <p>It is part of a sequence of <a href="/facts/Regular_polychora/lFf8IH1C">regular polychora</a>, which include the <a href="/facts/5-cell/Rh6lwDcc">5-cell</a> {3,3,3}, <a href="/facts/Tesseract/tlqouBoU">tesseract</a> {4,3,3}, and <a href="/facts/120-cell/W4nroBjJ">120-cell</a> {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedral</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a>. </p> <table><tbody><tr><th colspan="9">{p,3,3} honeycombs</th></tr><tr><th colspan="2">Space</th><th colspan="3"><a href="/facts/3-sphere/u67nImpY">S3</a></th><th colspan="4"><a href="/facts/Hyperbolic_space/kI7tnwdk">H3</a></th></tr><tr><th colspan="2">Form</th><th colspan="3">Finite</th><th>Paracompact</th><th colspan="3">Noncompact</th></tr><tr><th colspan="2">Name</th><th><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></th><th><a href="/facts/Tesseract/tlqouBoU">{4,3,3}</a></th><th><a href="/facts/120-cell/W4nroBjJ">{5,3,3}</a></th><th>{6,3,3}</th><th><a href="/facts/Heptagonal_tiling_honeycomb/UPTjY8ue">{7,3,3}</a></th><th><a href="/facts/Octagonal_tiling_honeycomb/UPTjY8ue">{8,3,3}</a></th><th>... <a href="/facts/Apeirogonal_tiling_honeycomb/UPTjY8ue">{∞,3,3}</a></th></tr><tr><th colspan="2">Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th rowspan="5"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagrams</a></th><th>1</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>4</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>6</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>12</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>24</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th colspan="2">Cells{p,3}</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> <p>It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> cells: </p> <table><tbody><tr><th colspan="12">{6,3,<i>p</i>} honeycombs <ul><li>v</li><li>t</li><li>e</li></ul></th></tr><tr><th>Space</th><th colspan="7"><a href="/facts/Hyperbolic_space/kI7tnwdk">H3</a></th></tr><tr><th>Form</th><th colspan="4"><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact</a></th><th colspan="3">Noncompact</th></tr><tr><th>Name</th><th>{6,3,3}</th><th><a href="/facts/Order-4_hexagonal_tiling_honeycomb/3pSIAwsX">{6,3,4}</a></th><th><a href="/facts/Order-5_hexagonal_tiling_honeycomb/AVt6DFhA">{6,3,5}</a></th><th><a href="/facts/Order-6_hexagonal_tiling_honeycomb/wATwxBgJ">{6,3,6}</a></th><th><a href="/facts/Order-7_hexagonal_tiling_honeycomb/9YlHxNNK">{6,3,7}</a></th><th><a href="/facts/Order-8_hexagonal_tiling_honeycomb/9YlHxNNK">{6,3,8}</a></th><th>... <a href="/facts/Infinite-order_hexagonal_tiling_honeycomb/9YlHxNNK">{6,3,∞}</a></th></tr><tr><th>Coxeter</th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Vertexfigure{3,p}</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></tbody></table> <h3>Rectified hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Rectified hexagonal 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>r{6,3,3} or t1{6,3,3}</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/Tetrahedron/7mkSrl6j">{3,3}</a> <a href="/facts/Trihexagonal_tiling/spAKhDvd">r{6,3}</a> or </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%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6] P ¯ 3 {\displaystyle {\overline {P}}_{3}} , [3,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive</td></tr></tbody></table> <p>The rectified hexagonal tiling honeycomb, t1{6,3,3}, has <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedral</a> and <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a> facets, with a <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. The half-symmetry construction alternates two types of tetrahedra. </p> <table><tbody><tr><th>Hexagonal tiling honeycomb</th><th>Rectified hexagonal tiling honeycomb or </th></tr><tr><td></td><td></td></tr><tr><th colspan="2">Related H2 tilings</th></tr><tr><th>Order-3 apeirogonal tiling</th><th>Triapeirogonal tiling or </th></tr><tr><td></td><td></td></tr></tbody></table> <h3>Truncated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Truncated hexagonal 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{6,3,3} or t0,1{6,3,3}</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/Tetrahedron/7mkSrl6j">{3,3}</a> <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/Tetrahedron/7mkSrl6j">triangular pyramid</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedral</a> and <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</a> facets, with a <a href="/facts/Tetrahedron/7mkSrl6j">triangular pyramid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p><p>It is similar to the 2D hyperbolic <a href="/facts/Truncated_order-3_apeirogonal_tiling/x98K6sIB">truncated order-3 apeirogonal tiling</a>, t{∞,3} with apeirogonal and triangle faces: </p> <h3>Bitruncated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Bitruncated hexagonal tiling honeycombBitruncated order-6 tetrahedral 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{6,3,3} or t1,2{6,3,3}</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_tetrahedron/K9ItvmUK">t{3,3}</a> <a href="/facts/Hexagonal_tiling/HwG5F0YH">t{3,6}</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/Digonal_disphenoid/7mkSrl6j">digonal disphenoid</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6] P ¯ 3 {\displaystyle {\overline {P}}_{3}} , [3,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a> and <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> cells, with a <a href="/facts/Digonal_disphenoid/7mkSrl6j">digonal disphenoid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Cantellated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantellated hexagonal 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{6,3,3} or t0,2{6,3,3}</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/Octahedron/G38q92xW">r{3,3}</a> <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rr{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%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, has <a href="/facts/Octahedron/G38q92xW">octahedron</a>, <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rhombitrihexagonal tiling</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> cells, with a <a href="/facts/Wedge_(geometry)/S9xb50Fs">wedge</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Cantitruncated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantitruncated hexagonal 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{6,3,3} or t0,1,2{6,3,3}</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_tetrahedron/K9ItvmUK">t{3,3}</a> <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">tr{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%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, has <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a>, <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">truncated trihexagonal tiling</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> cells, with a <a href="/facts/Mirrored_sphenoid/7mkSrl6j">mirrored sphenoid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Runcinated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcinated hexagonal 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{6,3,3}</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/Tetrahedron/7mkSrl6j">{3,3}</a> <a href="/facts/Hexagonal_tiling/HwG5F0YH">{6,3}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{}×{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}<a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>irregular <a href="/facts/Triangular_antiprism/G38q92xW">triangular antiprism</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a>, <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a>, <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> cells, with an irregular <a href="/facts/Triangular_antiprism/G38q92xW">triangular antiprism</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Runcitruncated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcitruncated hexagonal 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>t0,1,3{6,3,3}</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/Cuboctahedron/hD7p0Mem">rr{3,3}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}x{3}</a> <a href="/facts/Dodecagonal_prism/50AfF1sz">{}x{12}</a> <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/Square/JHJeMvmL">square</a> {4}<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/Isosceles_trapezoid/Izn5AA4p">isosceles-trapezoidal</a> <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramid</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, has <a href="/facts/Cuboctahedron/hD7p0Mem">cuboctahedron</a>, <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a>, <a href="/facts/Dodecagonal_prism/50AfF1sz">dodecagonal prism</a>, and <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</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> <h3>Runcicantellated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcicantellated hexagonal tiling honeycombruncitruncated order-6 tetrahedral 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>t0,2,3{6,3,3}</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_tetrahedron/K9ItvmUK">t{3,3}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{}x{6}</a> <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rr{6,3}</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/Isosceles_trapezoid/Izn5AA4p">isosceles-trapezoidal</a> <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramid</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, has <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a>, <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a>, and <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rhombitrihexagonal tiling</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> <h3>Omnitruncated hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Omnitruncated hexagonal tiling honeycombOmnitruncated order-6 tetrahedral 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{6,3,3}</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_octahedron/GA98b4TH">tr{3,3}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{}x{6}</a> <a href="/facts/Dodecagonal_prism/50AfF1sz">{}x{12}</a> <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">tr{6,3}</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>irregular <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, has <a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a>, <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a>, <a href="/facts/Dodecagonal_prism/50AfF1sz">dodecagonal prism</a>, and <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">truncated trihexagonal tiling</a> cells, with an irregular <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </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> <li><a href="/facts/Alternated_hexagonal_tiling_honeycomb/1aaatIbv">Alternated hexagonal tiling honeycomb</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> <a href="https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf">Archived</a> 2016-06-10 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</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 (Chapters 16–17: Geometries on Three-manifolds I,II)</li> <li>N. W. Johnson, <a href="/facts/Ruth_Kellerhals/e8tM0VgF">R. Kellerhals</a>, J. G. Ratcliffe, S. T. Tschantz, <i>The size of a hyperbolic Coxeter simplex</i>, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 <a href="https://link.springer.com/article/10.1007%2FBF01238563">[1]</a> <a href="https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf">[2]</a></li> <li>N. W. Johnson, <a href="/facts/Ruth_Kellerhals/e8tM0VgF">R. Kellerhals</a>, J. G. Ratcliffe, S. T. Tschantz, <i>Commensurability classes of hyperbolic Coxeter groups</i>, (2002) H3: p130. <a href="http://www.sciencedirect.com/science/article/pii/S0024379501004773">[3]</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/John_Baez/pjL45P57">John Baez</a>, <i>Visual Insight</i>: <a href="http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/">{6,3,3} Honeycomb</a> (2014/03/15)</li> <li><a href="/facts/John_Baez/pjL45P57">John Baez</a>, <i>Visual Insight</i>: <a href="http://blogs.ams.org/visualinsight/2013/09/15/633-honeycomb-in-upper-half-space/">{6,3,3} Honeycomb in Upper Half Space</a> (2013/09/15)</li> <li><a href="/facts/John_Baez/pjL45P57">John Baez</a>, <i>Visual Insight</i>: <a href="http://blogs.ams.org/visualinsight/2016/12/01/truncated-633-honeycomb/">Truncated {6,3,3} Honeycomb</a> (2016/12/01)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>