Order-5 octahedral honeycomb

<h2 id="images">Images</h2>
<table><tbody><tr><td><a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a>(cell centered)</td><td>Ideal surface</td></tr></tbody></table>
<h2 id="related-polytopes-and-honeycombs">Related polytopes and honeycombs</h2>
<p>It a part of a sequence of <a href="/facts/Regular_polychora/lFf8IH1C">regular polychora</a> and honeycombs with <a href="/facts/Octahedron/G38q92xW">octahedral</a> <a href="/facts/Cell_(geometry)/T2E6BTU4">cells</a>: {3,4,<i>p</i>}
</p>
<table><tbody><tr><th colspan="12">{3,4,p} 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>Paracompact</th><th colspan="7">Noncompact</th></tr><tr><th>Name</th><th><a href="/facts/24-cell/tfoGnlAD">{3,4,3}</a> </th><th><a href="/facts/Order-4_octahedral_honeycomb/vvpQj6Mz">{3,4,4}</a></th><th>{3,4,5}</th><th><a href="/facts/Order-6_octahedral_honeycomb/axcmB1c7">{3,4,6}</a></th><th><a href="/facts/Order-7_octahedral_honeycomb/axcmB1c7">{3,4,7}</a></th><th><a href="/facts/Order-8_octahedral_honeycomb/axcmB1c7">{3,4,8}</a></th><th>... <a href="/facts/Infinite-order_octahedral_honeycomb/axcmB1c7">{3,4,∞}</a></th></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Vertexfigure</th><td><a href="/facts/Cube/65lUaAqN">{4,3}</a> </td><td><a href="/facts/Square_tiling/3ym86mvv">{4,4}</a></td><td><a href="/facts/Order-5_square_tiling/YW3Ia52n">{4,5}</a></td><td><a href="/facts/Order-6_square_tiling/3deERRR3">{4,6}</a></td><td><a href="/facts/Order-7_square_tiling/f1nKRk7y">{4,7}</a></td><td><a href="/facts/Order-8_square_tiling/31PjvkLS">{4,8}</a></td><td><a href="/facts/Infinite-order_square_tiling/EjSUkNc8">{4,∞}</a></td></tr></tbody></table>

<h3>Order-6 octahedral honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-6 octahedral honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular honeycomb</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{3,4,6}{3,(3,4,3)}</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/Octahedron/G38q92xW">{3,4}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Hexagon/DPzuCzWE">{6}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-6_square_tiling/3deERRR3">{4,6}</a> {(4,3,4)} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-4-3_hexagonal_honeycomb/RtodEeE2">{6,4,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,4,6][3,((4,3,4))]</td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table>
<p>In the <a href="/facts/Geometry/wTQf5ffQ">geometry</a> of <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic 3-space</a>, the order-6 octahedral honeycomb is a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>) with <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,4,6}. It has six <a href="/facts/Octahedra/G38q92xW">octahedra</a>, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an <a href="/facts/Order-6_square_tiling/3deERRR3">order-6 square tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a>(cell centered)</td><td>Ideal surface</td></tr></tbody></table>
<p>It has a second construction as a uniform honeycomb, <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,(4,3,4)}, Coxeter diagram, , with alternating types or colors of octahedral cells. In <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a> the half symmetry is [3,4,6,1+] = [3,((4,3,4))].
</p>

<h3>Order-7 octahedral honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-7 octahedral honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular honeycomb</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{3,4,7}</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/Octahedron/G38q92xW">{3,4}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Heptagon/b0qIVNhY">{7}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-7_square_tiling/f1nKRk7y">{4,7}</a> </td></tr><tr><td>Dual</td><td><a href="/facts/Order-4-3_heptagonal_honeycomb/RtodEeE2">{7,4,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,4,7]</td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table>
<p>In the <a href="/facts/Geometry/wTQf5ffQ">geometry</a> of <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic 3-space</a>, the order-7 octahedral honeycomb is a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>) with <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,4,7}. It has seven <a href="/facts/Octahedra/G38q92xW">octahedra</a>, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an <a href="/facts/Order-7_square_tiling/f1nKRk7y">order-7 square tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a>(cell centered)</td><td>Ideal surface</td></tr></tbody></table>

<h3>Order-8 octahedral honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-8 octahedral honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular honeycomb</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{3,4,8}</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/Octahedron/G38q92xW">{3,4}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Octagon/H5PuP3wP">{8}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-8_square_tiling/31PjvkLS">{4,8}</a> </td></tr><tr><td>Dual</td><td><a href="/facts/Order-4-3_octagonal_honeycomb/RtodEeE2">{8,4,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,4,8]</td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table>
<p>In the <a href="/facts/Geometry/wTQf5ffQ">geometry</a> of <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic 3-space</a>, the order-8 octahedral honeycomb is a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>) with <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,4,8}. It has eight <a href="/facts/Octahedra/G38q92xW">octahedra</a>, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an <a href="/facts/Order-8_square_tiling/31PjvkLS">order-8 square tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a>(cell centered)</td></tr></tbody></table>

<h3>Infinite-order octahedral honeycomb</h3>
<table><tbody><tr><th colspan="2">Infinite-order octahedral honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular honeycomb</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{3,4,∞}{3,(4,∞,4)}</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/Octahedron/G38q92xW">{3,4}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Apeirogon/r0V9hleb">{∞}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Infinite-order_square_tiling/EjSUkNc8">{4,∞}</a> {(4,∞,4)} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-4_apeirogonal_tiling_honeycomb/JXbJHahy">{∞,4,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[∞,4,3][3,((4,∞,4))]</td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table>
<p>In the <a href="/facts/Geometry/wTQf5ffQ">geometry</a> of <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic 3-space</a>, the infinite-order octahedral honeycomb is a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>) with <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,4,∞}. It has infinitely many <a href="/facts/Octahedra/G38q92xW">octahedra</a>, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an <a href="/facts/Infinite-order_square_tiling/EjSUkNc8">infinite-order square tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a>(cell centered)</td><td>Ideal surface</td></tr></tbody></table>
<p>It has a second construction as a uniform honeycomb, <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,(4,∞,4)}, Coxeter diagram,  = , with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,∞,1+] = [3,((4,∞,4))].
</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">List of regular polytopes</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 (Chapters 16–17: Geometries on Three-manifolds I, II)</li>
<li>George Maxwell, <i>Sphere Packings and Hyperbolic Reflection Groups</i>, JOURNAL OF ALGEBRA 79,78-97 (1982) <a href="http://www.sciencedirect.com/science/article/pii/0021869382903180">[1]</a></li>
<li>Hao Chen, Jean-Philippe Labbé, <i>Lorentzian Coxeter groups and Boyd-Maxwell ball packings</i>, (2013)<a href="https://arxiv.org/abs/1310.8608">[2]</a></li>
<li><a href="https://arxiv.org/abs/1511.02851">Visualizing Hyperbolic Honeycombs arXiv:1511.02851</a> Roice Nelson, <a href="/facts/Henry_Segerman/vymrsJc6">Henry Segerman</a> (2015)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/John_Baez/pjL45P57">John Baez</a>, <i>Visual insights</i>: <a href="http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/">{7,3,3} Honeycomb</a> (2014/08/01) <a href="http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/">{7,3,3} Honeycomb Meets Plane at Infinity</a> (2014/08/14)</li>
<li><a href="/facts/Danny_Calegari/j4YLfXqQ">Danny Calegari</a>, <a href="http://lamington.wordpress.com/2014/03/04/kleinian-a-tool-for-visualizing-kleinian-groups/Kleinian">Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination</a> 4 March 2014. <a href="https://web.archive.org/web/20161109004910/http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf">[3]</a></li></ul>

Order-5 octahedral honeycomb open-in-new

Order-5 octahedral honeycomb