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Tesseractic honeycomb
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<table><tbody><tr><th colspan="2">Tesseractic honeycomb</th></tr><tr><td colspan="2"><a href="/facts/Perspective_projection/zkNx7Wpf">Perspective projection</a> of a 3x3x3x3 red-blue chessboard.</td></tr><tr><td>Type</td><td><a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular 4-space honeycomb</a><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td>Family</td><td><a href="/facts/Hypercubic_honeycomb/MQxLbDeB">Hypercubic honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbols</a></td><td>{4,3,3,4}t0,4{4,3,3,4}{4,3,31,1}{4,4}(2){4,3,4}×{∞}{4,4}×{∞}(2){∞}(4)</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a></td><td></td></tr><tr><td>4-face type</td><td><a href="/facts/Tesseract/tlqouBoU">{4,3,3}</a> </td></tr><tr><td>Cell type</td><td><a href="/facts/Cube/65lUaAqN">{4,3}</a> </td></tr><tr><td>Face type</td><td><a href="/facts/Square_(geometry)/JHJeMvmL">{4}</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Octahedron/G38q92xW">{3,4}</a>(<a href="/facts/Octahedron/G38q92xW">octahedron</a>)</td></tr><tr><td>Vertex figure</td><td><a href="/facts/16-cell/X0YAfobw">{3,3,4}</a>(<a href="/facts/16-cell/X0YAfobw">16-cell</a>)</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td> C ~ 4 {\displaystyle {\tilde {C}}_{4}} , [4,3,3,4] B ~ 4 {\displaystyle {\tilde {B}}_{4}} , [4,3,31,1]</td></tr><tr><td>Dual</td><td><a href="/facts/Self-dual_polytope/rkIHbmAf">self-dual</a></td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>, <a href="/facts/Edge-transitive/p5j7GhwX">edge-transitive</a>, <a href="/facts/Face-transitive/ukLVQlK2">face-transitive</a>, <a href="/facts/Cell-transitive/ukLVQlK2">cell-transitive</a>, 4-face-transitive</td></tr></tbody></table> <p>In <a href="/facts/Four-dimensional_space/vFQd0M7T">four-dimensional</a> <a href="/facts/Euclidean_geometry/V6cTcoCy">euclidean geometry</a>, the tesseractic honeycomb is one of the three <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular</a> space-filling <a href="/facts/Tessellation/7BWDywXC">tessellations</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycombs</a>), represented by <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {4,3,3,4}, and consisting of a packing of <a href="/facts/Tesseract/tlqouBoU">tesseracts</a> (4-<a href="/facts/Hypercube/q2mRPbIz">hypercubes</a>). </p><p>Its <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> is a <a href="/facts/16-cell/X0YAfobw">16-cell</a>. Two tesseracts meet at each cubic <a href="/facts/Cell_(geometry)/T2E6BTU4">cell</a>, four meet at each <a href="/facts/Square_(geometry)/JHJeMvmL">square</a> <a href="/facts/Face_(geometry)/T2E6BTU4">face</a>, eight meet on each <a href="/facts/Edge_(geometry)/3lFN6sEz">edge</a>, and sixteen meet at each <a href="/facts/Vertex_(geometry)/ID3cea9z">vertex</a>. </p><p>It is an analog of the <a href="/facts/Square_tiling/3ym86mvv">square tiling</a>, {4,4}, of the plane and the <a href="/facts/Cubic_honeycomb/eMxySFur">cubic honeycomb</a>, {4,3,4}, of 3-space. These are all part of the <a href="/facts/Hypercubic_honeycomb/MQxLbDeB">hypercubic honeycomb</a> family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are <a href="/facts/Self-dual_polytope/rkIHbmAf">self-dual</a>. </p>