Order-3-7 heptagonal honeycomb

<h2 id="geometry">Geometry</h2>
<p>All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an <a href="/facts/Order-7_triangular_tiling/bF5pkIRd">order-7 triangular tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%C3%A9_disk_model/6nCWjbRL">Poincaré disk model</a></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 {<i>p</i>,3,<i>p</i>}:
</p>
<table><tbody><tr><th colspan="12">{p,3,p} regular honeycombs</th></tr><tr><th>Space</th><th><a href="/facts/3-sphere/u67nImpY">S3</a></th><th>Euclidean E3</th><th colspan="5"><a href="/facts/Hyperbolic_space/kI7tnwdk">H3</a></th></tr><tr><th>Form</th><th>Finite</th><th>Affine</th><th>Compact</th><th>Paracompact</th><th colspan="3">Noncompact</th></tr><tr><th>Name</th><th><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></th><th><a href="/facts/Cubic_honeycomb/eMxySFur">{4,3,4}</a></th><th><a href="/facts/Order-5_dodecahedral_honeycomb/fc9FroBY">{5,3,5}</a></th><th><a href="/facts/Order-6_hexagonal_tiling_honeycomb/wATwxBgJ">{6,3,6}</a></th><th><a href="/facts/Order-7_heptagonal_tiling_honeycomb/grCADNuo">{7,3,7}</a></th><th><a href="/facts/Order-8_octagonal_tiling_honeycomb/grCADNuo">{8,3,8}</a></th><th>...<a href="/facts/Infinite-order_apeirogonal_tiling_honeycomb/grCADNuo">{∞,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/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><tr><th>Vertexfigure</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>Order-3-8 octagonal honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-3-8 octagonal 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%C3%A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{8,3,8}{8,(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/Octagonal_tiling/Fjj2z6ED">{8,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Octagon/H5PuP3wP">{8}</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_triangular_tiling/feag8eNn">{3,8}</a> {(3,8,3)} </td></tr><tr><td>Dual</td><td>self-dual</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[8,3,8][8,((3,4,3))]</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-3-8 octagonal 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%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {8,3,8}. It has eight <a href="/facts/Octagonal_tiling/Fjj2z6ED">octagonal tilings</a>, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an <a href="/facts/Order-8_triangular_tiling/feag8eNn">order-8 triangular tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%C3%A9_disk_model/6nCWjbRL">Poincaré disk model</a></td></tr></tbody></table>
<p>It has a second construction as a uniform honeycomb, <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].
</p>

<h3>Order-3-infinite apeirogonal honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-3-infinite apeirogonal 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%C3%A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{∞,3,∞}{∞,(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/Apeirogonal_tiling/r5LH30qt">{∞,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Apeirogon/r0V9hleb">{∞}</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_triangular_tiling/9WnmYawK">{3,∞}</a> {(3,∞,3)}</td></tr><tr><td>Dual</td><td>self-dual</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[∞,3,∞][∞,((3,∞,3))]</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-3-infinite apeirogonal 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%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {∞,3,∞}. It has infinitely many <a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">order-3 apeirogonal tiling</a> {∞,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal tilings existing around each vertex in an <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">infinite-order triangular tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>.
</p>
<table><tbody><tr><td><a href="/facts/Poincar%C3%A9_disk_model/6nCWjbRL">Poincaré disk model</a></td><td>Ideal surface</td></tr></tbody></table>
<p>It has a second construction as a uniform honeycomb, <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {∞,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling 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">List of regular polytopes</a></li>
<li><a href="/facts/Infinite-order_dodecahedral_honeycomb/HCARypC5">Infinite-order dodecahedral 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>) 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-3-7 heptagonal honeycomb open-in-new

Order-3-7 heptagonal honeycomb