Order-6-4 triangular honeycomb

<h2 id="geometry">Geometry</h2>
<p>It has four <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a> {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an <a href="/facts/Order-4_hexagonal_tiling/KS6qS8IU">order-4 hexagonal 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></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> {3,61,1}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a> the half symmetry is [3,6,4,1+] = [3,61,1].
</p>
<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/Triangular_tiling/Jx8f8Krh">triangular tiling</a> <a href="/facts/Cell_(geometry)/T2E6BTU4">cells</a>: {3,6,<i>p</i>}
</p>
<table><tbody><tr><th colspan="12">{3,6,p} polytopes</th></tr><tr><th>Space</th><th colspan="5"><a href="/facts/Hyperbolic_space/kI7tnwdk">H3</a></th></tr><tr><th>Form</th><th>Paracompact</th><th colspan="4">Noncompact</th></tr><tr><th>Name</th><th><a href="/facts/Order-6-3_triangular_honeycomb/EjnBjVWg">{3,6,3}</a> </th><th>{3,6,4}</th><th><a href="/facts/Order-6-5_triangular_honeycomb/rkrJCGIM">{3,6,5}</a></th><th><a href="/facts/Order-6-6_triangular_honeycomb/rkrJCGIM">{3,6,6}</a></th><th>... <a href="/facts/Order-6-infinite_triangular_honeycomb/rkrJCGIM">{3,6,∞}</a></th></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Vertexfigure</th><td><a href="/facts/Order-3_hexagonal_tiling/HwG5F0YH">{6,3}</a> </td><td><a href="/facts/Order-4_hexagonal_tiling/KS6qS8IU">{6,4}</a></td><td><a href="/facts/Order-5_hexagonal_tiling/mE0WjMx4">{6,5}</a></td><td><a href="/facts/Order-6_hexagonal_tiling/ydIgQbIS">{6,6}</a></td><td><a href="/facts/Infinite-order_hexagonal_tiling/yM3Lza2w">{6,∞}</a></td></tr></tbody></table>

<h3>Order-6-5 triangular honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-6-5 triangular 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 symbol</a></td><td>{3,6,5}</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">{3,6}</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">{5}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-5_hexagonal_tiling/mE0WjMx4">{6,5}</a> </td></tr><tr><td>Dual</td><td><a href="/facts/Order-6-3_pentagonal_honeycomb/9hsKdU5I">{5,6,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,6,5]</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-6-3 triangular 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> {3,6,5}. It has five <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an <a href="/facts/Order-5_hexagonal_tiling/mE0WjMx4">order-5 hexagonal 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></td><td>Ideal surface</td></tr></tbody></table>

<h3>Order-6-6 triangular honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-6-6 triangular 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>{3,6,6}{3,(6,3,6)}</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/Triangular_tiling/Jx8f8Krh">{3,6}</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_hexagonal_tiling/ydIgQbIS">{6,6}</a> {(6,3,6)} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-6-3_hexagonal_honeycomb/9hsKdU5I">{6,6,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,6,6][3,((6,3,6))]</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-6-6 triangular 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> {3,6,6}. It has infinitely many <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an <a href="/facts/Order-6_triangular_tiling/Jx8f8Krh">order-6 triangular 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></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,(6,3,6)}, Coxeter diagram,  = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,6,1+] = [3,((6,3,6))].
</p>

<h3>Order-6-infinite triangular honeycomb</h3>
<table><tbody><tr><th colspan="2">Order-6-infinite triangular 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>{3,6,∞}{3,(6,∞,6)}</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/Triangular_tiling/Jx8f8Krh">{3,6}</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_hexagonal_tiling/yM3Lza2w">{6,∞}</a> {(6,∞,6)} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-6-3_apeirogonal_honeycomb/9hsKdU5I">{∞,6,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[∞,6,3][3,((6,∞,6))]</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-infinite triangular 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> {3,6,∞}. It has infinitely many <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular 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%25C3%25A9_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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,(6,∞,6)}, Coxeter diagram,  = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,∞,1+] = [3,((6,∞,6))].
</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="https://www.youtube.com/watch?v=wjeeMlHsYjE&">Spherical Video: {3,6,∞} honeycomb with parabolic Möbius transform</a> <a href="/facts/YouTube/db276jst">YouTube</a>, Roice Nelson</li>
<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-6-4 triangular honeycomb open-in-new

Order-6-4 triangular honeycomb