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Hexagonal tiling honeycomb
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<table><tbody><tr><th colspan="2">Hexagonal tiling honeycomb</th></tr><tr><td colspan="2"><a href="/facts/Perspective_projection/zkNx7Wpf">Perspective projection</a> viewwithin <a href="/facts/Poincar%C3%A9_disk_model/6nCWjbRL">Poincaré disk model</a></td></tr><tr><td>Type</td><td><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Hyperbolic regular honeycomb</a><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{6,3,3}t{3,6,3}2t{6,3,6}2t{6,3[3]}t{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/Hexagonal_tiling/HwG5F0YH">{6,3}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Hexagon/DPzuCzWE">hexagon</a> {6}</td></tr><tr><td><a href="/facts/Edge_figure/3u2ySK0C">Edge figure</a></td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a> {3}</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> {3,3}</td></tr><tr><td><a href="/facts/Dual_polytope/rkIHbmAf">Dual</a></td><td><a href="/facts/Order-6_tetrahedral_honeycomb/stpDuyNA">Order-6 tetrahedral honeycomb</a></td></tr><tr><td><a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [3,3,6] Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3] Z ¯ 3 {\displaystyle {\overline {Z}}_{3}} , [6,3,6] V P ¯ 3 {\displaystyle {\overline {VP}}_{3}} , [6,3[3]] P P ¯ 3 {\displaystyle {\overline {PP}}_{3}} , [3[3,3]]</td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table> <p>In the field of <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic geometry</a>, the hexagonal tiling honeycomb is one of 11 <a href="/facts/Paracompact_uniform_honeycombs/hnmKAVu0">regular paracompact honeycombs</a> in 3-dimensional <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic space</a>. It is <i>paracompact</i> because it has <a href="/facts/Cell_(geometry)/T2E6BTU4">cells</a> composed of an infinite number of faces. Each cell is a <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> whose vertices lie on a <a href="/facts/Horosphere/Mv52WknL">horosphere</a>, a surface in hyperbolic space that approaches a single <a href="/facts/Ideal_point/Td55W94F">ideal point</a> at infinity. </p><p>The <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> of the hexagonal tiling honeycomb is {6,3,3}. Since that of the <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> is {3,3}, the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex. </p>