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Order-5 hexagonal tiling honeycomb
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<h2 id="symmetry">Symmetry</h2> <p>A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular <a href="/facts/Regular_dodecahedron/xClYqhcp">dodecahedral</a> fundamental domains, and an <a href="/facts/Regular_icosahedron/1U93vmXV">icosahedral</a> <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a> with 6 axial infinite-order (ultraparallel) branches. </p> <h2 id="images">Images</h2> <p>The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact <a href="/facts/Order-5_apeirogonal_tiling/wF0evtKd">order-5 apeirogonal tiling</a>, {∞,5}, with five <a href="/facts/Apeirogon/r0V9hleb">apeirogonal</a> faces meeting around every vertex. </p> <h2 id="related-polytopes-and-honeycombs">Related polytopes and honeycombs</h2> <p>The order-5 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><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>{6,3,5}</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>There are <a href="/facts/Convex_uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">15 uniform honeycombs</a> in the [6,3,5] <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a> family, including this regular form, and its regular dual, the <a href="/facts/Order-6_dodecahedral_honeycomb/nX10JuCf">order-6 dodecahedral honeycomb</a>. </p> [6,3,5] family honeycombs<table><tbody><tr><th>{6,3,5}</th><th><a href="/facts/Rectified_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">r{6,3,5}</a></th><th><a href="/facts/Truncated_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">t{6,3,5}</a></th><th><a href="/facts/Cantellated_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">rr{6,3,5}</a></th><th><a href="/facts/Runcinated_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">t0,3{6,3,5}</a></th><th><a href="/facts/Cantitruncated_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">tr{6,3,5}</a></th><th><a href="/facts/Runcitruncated_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">t0,1,3{6,3,5}</a></th><th><a href="/facts/Omnitruncated_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">t0,1,2,3{6,3,5}</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_dodecahedral_honeycomb/nX10JuCf">{5,3,6}</a></th><th><a href="/facts/Rectified_order-6_dodecahedral_honeycomb/nX10JuCf">r{5,3,6}</a></th><th><a href="/facts/Truncated_order-6_dodecahedral_honeycomb/nX10JuCf">t{5,3,6}</a></th><th><a href="/facts/Cantellated_order-6_dodecahedral_honeycomb/nX10JuCf">rr{5,3,6}</a></th><th><a href="/facts/Bitruncated_order-6_dodecahedral_honeycomb/nX10JuCf">2t{5,3,6}</a></th><th><a href="/facts/Cantitruncated_order-6_dodecahedral_honeycomb/nX10JuCf">tr{5,3,6}</a></th><th><a href="/facts/Runcitruncated_order-6_dodecahedral_honeycomb/nX10JuCf">t0,1,3{5,3,6}</a></th><th><a href="/facts/Omnitruncated_order-6_dodecahedral_honeycomb/nX10JuCf">t0,1,2,3{5,3,6}</a></th></tr></tbody></table> <p>The order-5 hexagonal tiling honeycomb has a related <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternation</a> honeycomb, represented by ↔ , with <a href="/facts/Regular_icosahedron/1U93vmXV">icosahedron</a> and <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a> cells. </p><p>It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> facets: </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><a href="/facts/Hexagonal_tiling_honeycomb/DD00vEWL">{6,3,3}</a></th><th><a href="/facts/Order-4_hexagonal_tiling_honeycomb/3pSIAwsX">{6,3,4}</a></th><th>{6,3,5}</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> <p>It is also part of a sequence of <a href="/facts/Regular_polychora/lFf8IH1C">regular polychora</a> and honeycombs with <a href="/facts/Regular_icosahedron/1U93vmXV">icosahedral</a> vertex figures: </p> <table><tbody><tr><th colspan="8">{p,3,5} polytopes</th></tr><tr><th>Space</th><th><a href="/facts/3-sphere/u67nImpY">S3</a></th><th colspan="6"><a href="/facts/Hyperbolic_space/kI7tnwdk">H3</a></th></tr><tr><th>Form</th><th>Finite</th><th colspan="2">Compact</th><th>Paracompact</th><th colspan="3">Noncompact</th></tr><tr><th>Name</th><th><a href="/facts/600-cell/tNBtDUe9">{3,3,5}</a></th><th><a href="/facts/Order-5_cubic_honeycomb/nPw9L9Ak">{4,3,5}</a></th><th><a href="/facts/Order-5_dodecahedral_honeycomb/fc9FroBY">{5,3,5}</a></th><th>{6,3,5}</th><th><a href="/facts/Order-5_heptagonal_tiling_honeycomb/z4sSkLD3">{7,3,5}</a></th><th><a href="/facts/Order-5_octagonal_tiling_honeycomb/z4sSkLD3">{8,3,5}</a></th><th>... <a href="/facts/Order-5_apeirogonal_tiling_honeycomb/z4sSkLD3">{∞,3,5}</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/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 order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Rectified order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>r{6,3,5} or t1{6,3,5}</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/Regular_icosahedron/1U93vmXV">{3,5}</a> <a href="/facts/Trihexagonal_tiling/spAKhDvd">r{6,3}</a> or h2{6,3}</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/Pentagonal_prism/CtvKLxcy">pentagonal prism</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {{\overline {HV}}_{3}}} , [5,3,6] H P ¯ 3 {\displaystyle {{\overline {HP}}_{3}}} , [5,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive</td></tr></tbody></table> <p>The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, has <a href="/facts/Regular_icosahedron/1U93vmXV">icosahedron</a> and <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a> facets, with a <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prism</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p><p>It is similar to the 2D hyperbolic <a href="/facts/Infinite-order_square_tiling/EjSUkNc8">infinite-order square tiling</a>, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface. </p> r{p,3,5}<table><tbody><tr><th>Space</th><th><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>Finite</th><th colspan="2">Compact</th><th>Paracompact</th><th colspan="2">Noncompact</th></tr><tr><th>Name</th><th><a href="/facts/Rectified_600-cell/S2DeX7LG">r{3,3,5}</a></th><th><a href="/facts/Rectified_order-5_cubic_honeycomb/nPw9L9Ak">r{4,3,5}</a></th><th><a href="/facts/Rectified_order-5_dodecahedral_honeycomb/fc9FroBY">r{5,3,5}</a></th><th><a href="/facts/Rectified_order-5_hexagonal_tiling_honeycomb/AVt6DFhA">r{6,3,5}</a></th><th>r{7,3,5}</th><th>... r{∞,3,5}</th></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Cells<a href="/facts/Icosahedron/wWbpnDYp">{3,5}</a></th><td><a href="/facts/Octahedron/G38q92xW">r{3,3}</a></td><td><a href="/facts/Cuboctahedron/hD7p0Mem">r{4,3}</a></td><td><a href="/facts/Icosidodecahedron/P0JluaVA">r{5,3}</a></td><td><a href="/facts/Trihexagonal_tiling/spAKhDvd">r{6,3}</a></td><td><a href="/facts/Triheptagonal_tiling/Is89Gv5q">r{7,3}</a></td><td>r{∞,3}</td></tr></tbody></table> <h3>Truncated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Truncated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t{6,3,5} or t0,1{6,3,5}</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/Regular_icosahedron/1U93vmXV">{3,5}</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/Pentagonal_pyramid/tN5CBPjK">pentagonal pyramid</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has <a href="/facts/Regular_icosahedron/1U93vmXV">icosahedron</a> and <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</a> facets, with a <a href="/facts/Pentagonal_pyramid/tN5CBPjK">pentagonal pyramid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Bitruncated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Bitruncated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>2t{6,3,5} or t1,2{6,3,5}</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/Truncated_icosahedron/LR8ugaY9">t{3,5}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Pentagon/xoq5oJeQ">pentagon</a> {5}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6] H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} , [5,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> and <a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a> facets, with a <a href="/facts/Digonal_disphenoid/7mkSrl6j">digonal disphenoid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Cantellated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantellated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>rr{6,3,5} or t0,2{6,3,5}</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/Icosidodecahedron/P0JluaVA">r{3,5}</a> <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rr{6,3}</a> <a href="/facts/Pentagonal_prism/CtvKLxcy">{}x{5}</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/Pentagon/xoq5oJeQ">pentagon</a> {5}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has <a href="/facts/Icosidodecahedron/P0JluaVA">icosidodecahedron</a>, <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rhombitrihexagonal tiling</a>, and <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prism</a> facets, with a <a href="/facts/Wedge_(geometry)/S9xb50Fs">wedge</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Cantitruncated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantitruncated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>tr{6,3,5} or t0,1,2{6,3,5}</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_icosahedron/LR8ugaY9">t{3,5}</a> <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">tr{6,3}</a> <a href="/facts/Pentagonal_prism/CtvKLxcy">{}x{5}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Pentagon/xoq5oJeQ">pentagon</a> {5}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has <a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a>, <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">truncated trihexagonal tiling</a>, and <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prism</a> facets, with a <a href="/facts/Mirrored_sphenoid/7mkSrl6j">mirrored sphenoid</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Runcinated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcinated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,3{6,3,5}</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">{6,3}</a> <a href="/facts/Regular_dodecahedron/xClYqhcp">{5,3}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{}x{6}</a> <a href="/facts/Pentagonal_prism/CtvKLxcy">{}x{5}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Square/JHJeMvmL">square</a> {4}<a href="/facts/Pentagon/xoq5oJeQ">pentagon</a> {5}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has <a href="/facts/Regular_dodecahedron/xClYqhcp">dodecahedron</a>, <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a>, <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prism</a>, and <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a> facets, with an irregular <a href="/facts/Triangular_antiprism/G38q92xW">triangular antiprism</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Runcitruncated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcitruncated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,3{6,3,5}</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> <a href="/facts/Rhombicosidodecahedron/AaDckLGP">rr{5,3}</a> <a href="/facts/Pentagonal_prism/CtvKLxcy">{}x{5}</a> <a href="/facts/Dodecagonal_prism/50AfF1sz">{}x{12}</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/Pentagon/xoq5oJeQ">pentagon</a> {5}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">truncated hexagonal tiling</a>, <a href="/facts/Rhombicosidodecahedron/AaDckLGP">rhombicosidodecahedron</a>, <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prism</a>, and <a href="/facts/Dodecagonal_prism/50AfF1sz">dodecagonal 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> <h3>Runcicantellated order-5 hexagonal tiling honeycomb</h3> <p>The runcicantellated order-5 hexagonal tiling honeycomb is the same as the <a href="/facts/Runcitruncated_order-6_dodecahedral_honeycomb/nX10JuCf">runcitruncated order-6 dodecahedral honeycomb</a>. </p> <h3>Omnitruncated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Omnitruncated order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,2,3{6,3,5}</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_icosidodecahedron/d5v9VpFY">tr{5,3}</a> <a href="/facts/Decagonal_prism/dIjtR9zW">{}x{10}</a> <a href="/facts/Dodecagonal_prism/50AfF1sz">{}x{12}</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/Decagon/GdHnJjMK">decagon</a> {10}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} , [5,3,6]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">truncated trihexagonal tiling</a>, <a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">truncated icosidodecahedron</a>, <a href="/facts/Decagonal_prism/dIjtR9zW">decagonal prism</a>, and <a href="/facts/Dodecagonal_prism/50AfF1sz">dodecagonal prism</a> facets, with an irregular <a href="/facts/Tetrahedral/7mkSrl6j">tetrahedral</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Alternated order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Alternated order-5 hexagonal tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a><a href="/facts/Semiregular_honeycomb/Xw9TURVR">Semiregular honeycomb</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>h{6,3,5}</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[3]}</a> <a href="/facts/Icosahedron/wWbpnDYp">{3,5}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} , [5,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive, <a href="/facts/Quasiregular_honeycomb/1ukrFhec">quasiregular</a></td></tr></tbody></table> <p>The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a> and <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a> facets, with a <a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. It is a <a href="/facts/Quasiregular_honeycomb/1ukrFhec">quasiregular honeycomb</a>. </p> <h3>Cantic order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Cantic order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>h2{6,3,5}</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">h2{6,3}</a> <a href="/facts/Truncated_icosahedron/LR8ugaY9">t{3,5}</a> <a href="/facts/Icosidodecahedron/P0JluaVA">r{5,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}<a href="/facts/Pentagon/xoq5oJeQ">pentagon</a> {5}<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%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} , [5,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔ , has <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a>, <a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a>, and <a href="/facts/Icosidodecahedron/P0JluaVA">icosidodecahedron</a> facets, with a <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Runcic order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcic order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>h3{6,3,5}</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[3]}</a> <a href="/facts/Rhombicosidodecahedron/AaDckLGP">rr{5,3}</a> <a href="/facts/Dodecahedron/7r519eCl">{5,3}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}x{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/Pentagon/xoq5oJeQ">pentagon</a> {5}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} , [5,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔ , has <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>, <a href="/facts/Rhombicosidodecahedron/AaDckLGP">rhombicosidodecahedron</a>, <a href="/facts/Dodecahedron/7r519eCl">dodecahedron</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> facets, with a <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <h3>Runcicantic order-5 hexagonal tiling honeycomb</h3> <table><tbody><tr><th colspan="2">Runcicantic order-5 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>h2,3{6,3,5}</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">h2{6,3}</a> <a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">tr{5,3}</a> <a href="/facts/Truncated_dodecahedron/PVTAsUWQ">t{5,3}</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">{}x{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/Decagon/GdHnJjMK">decagon</a> {10}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Rectangle/mdLfIGKA">rectangular</a> <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramid</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} , [5,3[3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive</td></tr></tbody></table> <p>The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔ , has <a href="/facts/Trihexagonal_tiling/spAKhDvd">trihexagonal tiling</a>, <a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">truncated icosidodecahedron</a>, <a href="/facts/Truncated_dodecahedron/PVTAsUWQ">truncated dodecahedron</a>, and <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> facets, with a <a href="/facts/Rectangle/mdLfIGKA">rectangular</a> <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramid</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></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> <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>