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Order-infinite-3 triangular honeycomb
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<h2 id="geometry">Geometry</h2> <p>It has three <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">Infinite-order triangular tiling</a> {3,∞} 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-3_apeirogonal_tiling/UEkWcKPh">order-3 apeirogonal tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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> <h2 id="related-polytopes-and-honeycombs">Related polytopes and honeycombs</h2> <p>It is a part of a sequence of regular honeycombs with <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">Infinite-order triangular tiling</a> <a href="/facts/Cell_(geometry)/T2E6BTU4">cells</a>: {3,∞,<i>p</i>}. </p><p>It is a part of a sequence of regular honeycombs with <a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">order-3 apeirogonal tiling</a> <a href="/facts/Vertex_figures/3u2ySK0C">vertex figures</a>: {<i>p</i>,∞,3}. </p><p>It is a part of a sequence of self-dual regular honeycombs: {<i>p</i>,∞,<i>p</i>}. </p> <h3>Order-infinite-4 triangular honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-4 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,∞,4}{3,∞1,1}</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/Infinite-order_triangular_tiling/9WnmYawK">{3,∞}</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/Square/JHJeMvmL">{4}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">{∞,4}</a> r{∞,∞} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-3_square_honeycomb/c3KqPvfa">{4,∞,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,∞,4][3,∞1,1]</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-infinite-4 triangular honeycomb (or 3,∞,4 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}. </p><p>It has four <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">infinite-order triangular tilings</a>, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an <a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">order-4 apeirogonal tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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,∞1,1}, Coxeter diagram, , with alternating types or colors of infinite-order triangular tiling cells. In <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a> the half symmetry is [3,∞,4,1+] = [3,∞1,1]. </p> <h3>Order-infinite-5 triangular honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-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 symbols</a></td><td>{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/Infinite-order_triangular_tiling/9WnmYawK">{3,∞}</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/Pentagon/xoq5oJeQ">{5}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-5_apeirogonal_tiling/wF0evtKd">{∞,5}</a> </td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-3_pentagonal_honeycomb/c3KqPvfa">{5,∞,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,∞,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-infinite-3 triangular honeycomb (or 3,∞,5 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,∞,5}. It has five <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">infinite-order triangular tiling</a>, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an <i>order-5 apeirogonal tiling</i> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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-infinite-6 triangular honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-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}{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/Infinite-order_triangular_tiling/9WnmYawK">{3,∞}</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_apeirogonal_tiling/3jXwGWfp">{∞,6}</a> {(∞,3,∞)} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-3_hexagonal_honeycomb/c3KqPvfa">{6,∞,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[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-infinite-6 triangular honeycomb (or 3,∞,6 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/Infinite-order_triangular_tiling/9WnmYawK">infinite-order triangular tiling</a>, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an <i>order-6 apeirogonal tiling</i>, {∞,6}, <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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-infinite-7 triangular honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-7 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,∞,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/Infinite-order_triangular_tiling/9WnmYawK">{3,∞}</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>{∞,7} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-3_heptagonal_honeycomb/c3KqPvfa">{7,∞,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,∞,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-infinite-7 triangular honeycomb (or 3,∞,6 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,∞,7}. It has infinitely many <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">infinite-order triangular tiling</a>, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an <i>order-7 apeirogonal tiling</i>, {∞,7}, <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <table><tbody><tr><td>Ideal surface</td></tr></tbody></table> <h3>Order-infinite-infinite triangular honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-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,∞,∞}{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/Infinite-order_triangular_tiling/9WnmYawK">{3,∞}</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_apeirogonal_tiling/WVaudE59">{∞,∞}</a> {(∞,∞,∞)} </td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-3_apeirogonal_honeycomb/c3KqPvfa">{∞,∞,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[∞,∞,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-infinite-infinite triangular honeycomb (or 3,∞,∞ 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,∞,∞}. It has infinitely many <a href="/facts/Infinite-order_triangular_tiling/9WnmYawK">infinite-order triangular tiling</a>, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an <i>infinite-order apeirogonal tiling</i>, {∞,∞}, <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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,(∞,∞,∞)}, Coxeter diagram, = , with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is [3,∞,∞,1+] = [3,((∞,∞,∞))]. </p> <h3>Order-infinite-3 square honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-3 square 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>{4,∞,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/Infinite-order_square_tiling/EjSUkNc8">{4,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Square/JHJeMvmL">{4}</a></td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">{∞,3}</a></td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-4_triangular_honeycomb/c3KqPvfa">{3,∞,4}</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[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-infinite-3 square honeycomb (or 4,∞,3 honeycomb) a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>). Each infinite cell consists of a <a href="/facts/Heptagonal_tiling/aKhMKTMx">heptagonal tiling</a> whose vertices lie on a <a href="/facts/Hypercycle_(geometry)/5w8YYW78">2-hypercycle</a>, each of which has a limiting circle on the ideal sphere. </p><p>The <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> of the <i>order-infinite-3 square honeycomb</i> is {4,∞,3}, with three infinite-order square tilings meeting at each edge. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of this honeycomb is an order-3 apeirogonal tiling, {∞,3}. </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-infinite-3 pentagonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-3 pentagonal 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>{5,∞,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/Infinite-order_pentagonal_tiling/s3D0FjrM">{5,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Pentagon/xoq5oJeQ">{5}</a></td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Heptagonal_tiling/aKhMKTMx">{∞,3}</a></td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-5_triangular_honeycomb/c3KqPvfa">{3,∞,5}</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[5,∞,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-infinite-3 pentagonal honeycomb (or 5,∞,3 honeycomb) a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>). Each infinite cell consists of an <a href="/facts/Infinite-order_pentagonal_tiling/s3D0FjrM">infinite-order pentagonal tiling</a> whose vertices lie on a <a href="/facts/Hypercycle_(geometry)/5w8YYW78">2-hypercycle</a>, each of which has a limiting circle on the ideal sphere. </p><p>The <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> of the <i>order-6-3 pentagonal honeycomb</i> is {5,∞,3}, with three <i>infinite-order pentagonal tilings</i> meeting at each edge. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of this honeycomb is a heptagonal tiling, {∞,3}. </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-infinite-3 hexagonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-3 hexagonal 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>{6,∞,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/Infinite-order_hexagonal_tiling/yM3Lza2w">{6,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Hexagon/DPzuCzWE">{6}</a></td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">{∞,3}</a></td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-6_triangular_honeycomb/c3KqPvfa">{3,∞,6}</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[6,∞,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-infinite-3 hexagonal honeycomb (or 6,∞,3 honeycomb) a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>). Each infinite cell consists of an <a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">order-3 apeirogonal tiling</a> whose vertices lie on a <a href="/facts/Hypercycle_(geometry)/5w8YYW78">2-hypercycle</a>, each of which has a limiting circle on the ideal sphere. </p><p>The <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> of the <i>order-infinite-3 hexagonal honeycomb</i> is {6,∞,3}, with three infinite-order hexagonal tilings meeting at each edge. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of this honeycomb is an order-3 apeirogonal tiling, {∞,3}. </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-infinite-3 heptagonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-3 heptagonal 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>{7,∞,3}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>Cells</td><td>{7,∞} </td></tr><tr><td>Faces</td><td><a href="/facts/Heptagon/b0qIVNhY">{7}</a></td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">{∞,3}</a></td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-7_triangular_honeycomb/c3KqPvfa">{3,∞,7}</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[7,∞,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-infinite-3 heptagonal honeycomb (or 7,∞,3 honeycomb) a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>). Each infinite cell consists of an infinite-order heptagonal tiling whose vertices lie on a <a href="/facts/Hypercycle_(geometry)/5w8YYW78">2-hypercycle</a>, each of which has a limiting circle on the ideal sphere. </p><p>The <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> of the <i>order-infinite-3 heptagonal honeycomb</i> is {7,∞,3}, with three infinite-order heptagonal tilings meeting at each edge. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of this honeycomb is an order-3 apeirogonal tiling, {∞,3}. </p> <table><tbody><tr><td>Ideal surface</td></tr></tbody></table> <h3>Order-infinite-3 apeirogonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-3 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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>{∞,∞,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/Infinite-order_apeirogonal_tiling/WVaudE59">{∞,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Apeirogon/r0V9hleb">Apeirogon</a> {∞}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Infinite-order_apeirogonal_tiling/WVaudE59">{∞,3}</a></td></tr><tr><td>Dual</td><td><a href="/facts/Order-infinite-infinite_triangular_honeycomb/c3KqPvfa">{3,∞,∞}</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[∞,∞,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-infinite-3 apeirogonal honeycomb (or ∞,∞,3 honeycomb) a regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>). Each infinite cell consists of an <a href="/facts/Infinite-order_apeirogonal_tiling/WVaudE59">infinite-order apeirogonal tiling</a> whose vertices lie on a <a href="/facts/Hypercycle_(geometry)/5w8YYW78">2-hypercycle</a>, each of which has a limiting circle on the ideal sphere. </p><p>The <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> of the apeirogonal tiling honeycomb is {∞,∞,3}, with three <i>infinite-order apeirogonal tilings</i> meeting at each edge. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of this honeycomb is an infinite-order apeirogonal tiling, {∞,3}. </p><p>The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an <a href="/facts/Apollonian_gasket/4lrnp6zV">Apollonian gasket</a> pattern of circles inside a largest circle. </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-infinite-4 square honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-4 square 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>{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/Infinite-order_square_tiling/EjSUkNc8">{4,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Square/JHJeMvmL">{4}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Square/JHJeMvmL">{4}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">{∞,4}</a>{∞,∞}</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>[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 order-infinite-4 square honeycomb (or 4,∞,4 honeycomb) 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> {4,∞,4}. </p><p>All vertices are ultra-ideal (existing beyond the ideal boundary) with four <a href="/facts/Infinite-order_square_tiling/EjSUkNc8">infinite-order square tilings</a> existing around each edge and with an <a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">order-4 apeirogonal tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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> {4,∞1,1}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [4,∞,4,1+] = [4,∞1,1]. </p> <h3>Order-infinite-5 pentagonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-5 pentagonal 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>{5,∞,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/Infinite-order_pentagonal_tiling/s3D0FjrM">{5,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Pentagon/xoq5oJeQ">{5}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Pentagon/xoq5oJeQ">{5}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-5_apeirogonal_tiling/wF0evtKd">{∞,5}</a></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>[5,∞,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-infinite-5 pentagonal honeycomb (or 5,∞,5 honeycomb) 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> {5,∞,5}. </p><p>All vertices are ultra-ideal (existing beyond the ideal boundary) with five infinite-order pentagonal tilings existing around each edge and with an <a href="/facts/Order-5_apeirogonal_tiling/wF0evtKd">order-5 apeirogonal tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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-infinite-6 hexagonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-6 hexagonal 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>{6,∞,6}{6,(∞,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/Infinite-order_hexagonal_tiling/yM3Lza2w">{6,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Hexagon/DPzuCzWE">{6}</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}</a> {(5,3,5)} </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>[6,∞,6][6,((∞,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-infinite-6 hexagonal honeycomb (or 6,∞,6 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> {6,∞,6}. It has six <a href="/facts/Infinite-order_hexagonal_tiling/yM3Lza2w">infinite-order hexagonal tilings</a>, {6,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an <a href="/facts/Order-6_apeirogonal_tiling/3jXwGWfp">order-6 apeirogonal tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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> {6,(∞,3,∞)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,∞,6,1+] = [6,((∞,3,∞))]. </p> <h3>Order-infinite-7 heptagonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-7 heptagonal 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>{7,∞,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/Order-5_heptagonal_tiling/cU67vNad">{7,∞}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Heptagon/b0qIVNhY">{7}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Heptagon/b0qIVNhY">{7}</a></td></tr><tr><td>Vertex figure</td><td>{∞,7} </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>[7,∞,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-infinite-7 heptagonal honeycomb (or 7,∞,7 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> {7,∞,7}. It has seven infinite-order heptagonal tilings, {7,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 apeirogonal tiling <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. </p> <table><tbody><tr><td>Ideal surface</td></tr></tbody></table> <h3>Order-infinite-infinite apeirogonal honeycomb</h3> <table><tbody><tr><th colspan="2">Order-infinite-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%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{∞,∞,∞}{∞,(∞,∞,∞)}</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">{∞,∞}</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_apeirogonal_tiling/WVaudE59">{∞,∞}</a> {(∞,∞,∞)}</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>[∞,∞,∞][∞,((∞,∞,∞))]</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-infinite-infinite apeirogonal honeycomb (or ∞,∞,∞ 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> {∞,∞,∞}. It has infinitely many <a href="/facts/Infinite-order_apeirogonal_tiling/WVaudE59">infinite-order apeirogonal tiling</a> {∞,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order apeirogonal tilings existing around each vertex in an <a href="/facts/Infinite-order_apeirogonal_tiling/WVaudE59">infinite-order apeirogonal tiling</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</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> {∞,(∞,∞,∞)}, Coxeter diagram, , with alternating types or colors of 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></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, Henry Segerman (2015)</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.youtube.com/watch?v=GRo_FQm2KRc">Hyperbolic Catacombs Carousel: {3,∞,3} honeycomb</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>