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Order-5 octahedral honeycomb
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<table><tbody><tr><th colspan="2">Order-5 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>{3,4,5}</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/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/Pentagon/xoq5oJeQ">{5}</a></td></tr><tr><td>Vertex figure</td><td><a href="/facts/Order-5_square_tiling/YW3Ia52n">{4,5}</a> </td></tr><tr><td>Dual</td><td><a href="/facts/Order-4-3_pentagonal_honeycomb/RtodEeE2">{5,4,3}</a></td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter group</a></td><td>[3,4,5]</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-5 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> {3,4,5}. It has five <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-5_square_tiling/YW3Ia52n">order-5 square tiling</a> <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>. </p>