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Alternated hexagonal tiling honeycomb
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<table><tbody><tr><th colspan="2">Alternated hexagonal tiling honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/Paracompact_uniform_honeycomb/hnmKAVu0">Paracompact uniform honeycomb</a><a href="/facts/Semiregular_honeycomb/Xw9TURVR">Semiregular honeycomb</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>h{6,3,3}s{3,6,3}2s{6,3,6}2s{6,3[3]}s{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/Tetrahedron/7mkSrl6j">{3,3}</a> <a href="/facts/Triangular_tiling/Jx8f8Krh">{3[3]}</a> </td></tr><tr><td>Faces</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/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a></td></tr><tr><td><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeter groups</a></td><td> P ¯ 3 {\displaystyle {\overline {P}}_{3}} , [3,3[3]]1/2 V ¯ 3 {\displaystyle {\overline {V}}_{3}} , [6,3,3]1/2 Y ¯ 3 {\displaystyle {\overline {Y}}_{3}} , [3,6,3]1/2 Z ¯ 3 {\displaystyle {\overline {Z}}_{3}} , [6,3,6]1/2 V P ¯ 3 {\displaystyle {\overline {VP}}_{3}} , [6,3[3]]1/2 P P ¯ 3 {\displaystyle {\overline {PP}}_{3}} , [3[3,3]]</td></tr><tr><td>Properties</td><td>Vertex-transitive, edge-transitive, <a href="/facts/Quasiregular_honeycomb/1ukrFhec">quasiregular</a></td></tr></tbody></table> <p>In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a <a href="/facts/Semiregular_polytope/Xw9TURVR">semiregular</a> tessellation with <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> and <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a> cells arranged in an <a href="/facts/Octahedron/G38q92xW">octahedron</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>. It is named after its construction, as an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alteration</a> of a <a href="/facts/Hexagonal_tiling_honeycomb/DD00vEWL">hexagonal tiling honeycomb</a>. </p><p>A <a href="/facts/Honeycomb_(geometry)/seyRBcel">geometric honeycomb</a> is a <i>space-filling</i> of <a href="/facts/Polyhedron/UC0p9FTF">polyhedral</a> or higher-dimensional <i>cells</i>, so that there are no gaps. It is an example of the more general mathematical <i>tiling</i> or <i><a href="/facts/Tessellation/7BWDywXC">tessellation</a></i> in any number of dimensions. </p><p>Honeycombs are usually constructed in ordinary <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean</a> ("flat") space, like the <a href="/facts/Convex_uniform_honeycomb/78IkTl7A">convex uniform honeycombs</a>. They may also be constructed in <a href="/facts/Non-Euclidean_geometry/efKa7De2">non-Euclidean spaces</a>, such as <a href="/facts/Uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">hyperbolic uniform honeycombs</a>. Any finite <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytope</a> can be projected to its <a href="/facts/Circumsphere/m6EPvP2E">circumsphere</a> to form a uniform honeycomb in spherical space. </p>