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5-cell honeycomb
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<h2 id="structure">Structure</h2> <p>Cells of the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> are ten <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedrons</a> and 20 <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prisms</a>, corresponding to the ten <a href="/facts/5-cell/Rh6lwDcc">5-cells</a> and 20 <a href="/facts/Rectified_5-cell/2UAMuo2P">rectified 5-cells</a> that meet at each vertex. All the vertices lie in parallel realms in which they form <a href="/facts/Alternated_cubic_honeycomb/mX4CxSvp">alternated cubic honeycombs</a>, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="alternate-names">Alternate names</h2> <ul><li>Cyclopentachoric tetracomb</li> <li>Pentachoric-dispentachoric tetracomb</li></ul> <h2 id="projection-by-folding">Projection by folding</h2> <p>The <i>5-cell honeycomb</i> can be projected into the 2-dimensional <a href="/facts/Square_tiling/3ym86mvv">square tiling</a> by a <a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">geometric folding</a> operation that maps two pairs of mirrors into each other, sharing the same <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>: </p> <table><tbody><tr><th> A ~ 3 {\displaystyle {\tilde {A}}_{3}} </th><td></td></tr><tr><th> C ~ 2 {\displaystyle {\tilde {C}}_{2}} </th><td></td></tr></tbody></table> <p>Two different <a href="/facts/Aperiodic_tiling/WFJMWDg4">aperiodic tilings</a> with 5-fold symmetry can be obtained by projecting two-dimensional slices of the honeycomb: the <a href="/facts/Penrose_tiling/usLcMvNN">Penrose tiling</a> composed of rhombi, and the <a href="/facts/T%C3%BCbingen_triangle/u6NBB5LD">Tübingen triangle</a> tiling composed of isosceles triangles.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="a4-lattice">A4 lattice</h2> <p>The <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> of the <i>5-cell honeycomb</i> is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>, the <a href="/facts/Runcinated_5-cell/7A0II7du">runcinated 5-cell</a> represent the 20 roots of the A ~ 4 {\displaystyle {\tilde {A}}_{4}} Coxeter group.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> It is the 4-dimensional case of a <a href="/facts/Simplectic_honeycomb/hCmbARWt">simplectic honeycomb</a>. </p><p>The A*4 lattice<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> is the union of five A4 lattices, and is the dual to the <a href="/facts/Omnitruncated_5-simplex_honeycomb/qpE5F5jg">omnitruncated 5-simplex honeycomb</a>, and therefore the <a href="/facts/Voronoi_cell/AhQY5LD8">Voronoi cell</a> of this lattice is an <a href="/facts/Omnitruncated_5-cell/7A0II7du">omnitruncated 5-cell</a> </p> ∪ ∪ ∪ ∪ = dual of <h2 id="related-polytopes-and-honeycombs">Related polytopes and honeycombs</h2> <p>The <i>tops</i> of the 5-cells in this honeycomb adjoin the <i>bases</i> of the 5-cells, and vice versa, in adjacent <a href="/facts/Lamination_(geology)/kbBpdCQM">laminae</a> (or layers); but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb. <a href="/facts/Octahedral_prism/8UVHkxvl">Octahedral prisms</a> and <a href="/facts/Tetrahedral_prism/oQ720tUe">tetrahedral prisms</a> may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>This honeycomb is one of <a href="/facts/Uniform_5-polytope/mqxsoqwY">seven unique uniform honeycombs</a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> constructed by the A ~ 4 {\displaystyle {\tilde {A}}_{4}} <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a>. The symmetry can be multiplied by the symmetry of rings in the <a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter–Dynkin diagrams</a>: </p> <table><tbody><tr><th colspan="5">A4 honeycombs</th></tr><tr><th>Pentagonsymmetry</th><th><a href="/facts/Coxeter_notation/ArUw6goe">Extendedsymmetry</a></th><th>Extendeddiagram</th><th>Extendedgroup</th><th>Honeycomb diagrams</th></tr><tr><th>a1</th><th>[3[5]]</th><th></th><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} </td><td>(None)</td></tr><tr><th>i2</th><th>[[3[5]]]</th><th></th><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2</td><td> 1, <a href="/facts/Rectified_5-cell_honeycomb/YgY6eSTr">2</a>, <a href="/facts/Cyclotruncated_5-cell_honeycomb/YgY6eSTr">3</a>,<p> <a href="/facts/Truncated_5-cell_honeycomb/YgY6eSTr">4</a>, <a href="/facts/Cantellated_5-cell_honeycomb/YgY6eSTr">5</a>, <a href="/facts/Bitruncated_5-cell_honeycomb/YgY6eSTr">6</a></p></td></tr><tr><th>r10</th><th>[5[3[5]]]</th><th></th><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×10</td><td> <a href="/facts/Omnitruncated_5-cell_honeycomb/YgY6eSTr">7</a></td></tr></tbody></table> <h3>Rectified 5-cell honeycomb</h3> <table><tbody><tr><th colspan="2">Rectified 5-cell honeycomb</th></tr><tr><td colspan="2">(No image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,2{3[5]} or r{3[5]}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>4-face types</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">t1{33}</a> <a href="/facts/Cantellated_5-cell/pm158bSU">t0,2{33}</a> <a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{33}</a> </td></tr><tr><td>Cell types</td><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a> <a href="/facts/Octahedron/G38q92xW">Octahedron</a> <a href="/facts/Cuboctahedron/hD7p0Mem">Cuboctahedron</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">Triangular prism</a> </td></tr><tr><td>Vertex figure</td><td>triangular elongated-antiprismatic prism</td></tr><tr><td><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></td><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2[3[5]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a></td></tr></tbody></table> <p>The rectified 4-simplex honeycomb or rectified 5-cell honeycomb is a space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>. </p> <h4>Alternate names</h4> <ul><li>small cyclorhombated pentachoric tetracomb</li> <li>small prismatodispentachoric tetracomb</li></ul> <h3>Cyclotruncated 5-cell honeycomb</h3> <table><tbody><tr><th colspan="2">Cyclotruncated 5-cell honeycomb</th></tr><tr><td colspan="2">(No image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td>Family</td><td><a href="/facts/Truncated_simplectic_honeycomb/Au7wa2P4">Truncated simplectic honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1{3[5]}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>4-face types</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a> <a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a> <a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a> </td></tr><tr><td>Cell types</td><td><a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a> <a href="/facts/Truncated_tetrahedron/K9ItvmUK">t{3,3}</a> </td></tr><tr><td>Face types</td><td><a href="/facts/Triangle/cRXUwsMn">Triangle</a> {3}<a href="/facts/Hexagon/DPzuCzWE">Hexagon</a> {6}</td></tr><tr><td>Vertex figure</td><td><a href="/facts/Tetrahedral_antiprism/X0YAfobw">Tetrahedral antiprism</a>[3,4,2+], order 48</td></tr><tr><td><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></td><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2[3[5]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a></td></tr></tbody></table> <p>The cyclotruncated 4-simplex honeycomb or cyclotruncated 5-cell honeycomb is a space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>. It can also be seen as a birectified 5-cell honeycomb. </p><p>It is composed of <a href="/facts/5-cell/Rh6lwDcc">5-cells</a>, <a href="/facts/Truncated_5-cell/oqF6jkb7">truncated 5-cells</a>, and <a href="/facts/Bitruncated_5-cell/oqF6jkb7">bitruncated 5-cells</a> facets in a ratio of 2:2:1. Its <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> is a <a href="/facts/Tetrahedral_antiprism/X0YAfobw">tetrahedral antiprism</a>, with 2 <a href="/facts/Regular_tetrahedron/7mkSrl6j">regular tetrahedron</a>, 8 <a href="/facts/Triangular_pyramid/7mkSrl6j">triangular pyramid</a>, and 6 <a href="/facts/Tetragonal_disphenoid/7rU7jc7U">tetragonal disphenoid</a> cells, defining 2 <a href="/facts/5-cell/Rh6lwDcc">5-cell</a>, 8 <a href="/facts/Truncated_5-cell/oqF6jkb7">truncated 5-cell</a>, and 6 <a href="/facts/Bitruncated_5-cell/oqF6jkb7">bitruncated 5-cell</a> facets around a vertex. </p><p>It can be constructed as five sets of parallel <a href="/facts/Hyperplane/2s2ycYYd">hyperplanes</a> that divide space into two half-spaces. The 3-space hyperplanes contain <a href="/facts/Quarter_cubic_honeycomb/22qQazbw">quarter cubic honeycombs</a> as a collection facets.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h4>Alternate names</h4> <ul><li>Cyclotruncated pentachoric tetracomb</li> <li>Small truncated-pentachoric tetracomb</li></ul> <h3>Truncated 5-cell honeycomb</h3> <table><tbody><tr><th colspan="2">Truncated 4-simplex honeycomb</th></tr><tr><td colspan="2">(No image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,2{3[5]} or t{3[5]}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>4-face types</td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t0,1{33}</a> <a href="/facts/Cantitruncated_5-cell/pm158bSU">t0,1,2{33}</a> <a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{33}</a> </td></tr><tr><td>Cell types</td><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a> <a href="/facts/Truncated_tetrahedron/K9ItvmUK">Truncated tetrahedron</a> <a href="/facts/Truncated_octahedron/GA98b4TH">Truncated octahedron</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">Triangular prism</a> </td></tr><tr><td>Vertex figure</td><td>triangular elongated-antiprismatic pyramid</td></tr><tr><td><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></td><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2[3[5]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a></td></tr></tbody></table> <p>The truncated 4-simplex honeycomb or truncated 5-cell honeycomb is a space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>. It can also be called a cyclocantitruncated 5-cell honeycomb. </p> <h4>Alaternate names</h4> <ul><li>Great cyclorhombated pentachoric tetracomb</li> <li>Great truncated-pentachoric tetracomb</li></ul> <h3>Cantellated 5-cell honeycomb</h3> <table><tbody><tr><th colspan="2">Cantellated 5-cell honeycomb</th></tr><tr><td colspan="2">(No image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,3{3[5]} or rr{3[5]}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>4-face types</td><td><a href="/facts/Cantellated_5-cell/pm158bSU">t0,2{33}</a> <a href="/facts/Bitruncated_5-cell/oqF6jkb7">t1,2{33}</a> <a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{33}</a> </td></tr><tr><td>Cell types</td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">Truncated tetrahedron</a> <a href="/facts/Octahedron/G38q92xW">Octahedron</a> <a href="/facts/Cuboctahedron/hD7p0Mem">Cuboctahedron</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">Triangular prism</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">Hexagonal prism</a> </td></tr><tr><td>Vertex figure</td><td>Bidiminished rectified pentachoron</td></tr><tr><td><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></td><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2[3[5]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a></td></tr></tbody></table> <p>The cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>. It can also be called a cycloruncitruncated 5-cell honeycomb. </p> <h4>Alternate names</h4> <ul><li>Cycloprismatorhombated pentachoric tetracomb</li> <li>Great prismatodispentachoric tetracomb</li></ul> <h3>Bitruncated 5-cell honeycomb</h3> <table><tbody><tr><th colspan="2">Bitruncated 5-cell honeycomb</th></tr><tr><td colspan="2">(No image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,2,3{3[5]} or 2t{3[5]}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>4-face types</td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{33}</a> <a href="/facts/Cantitruncated_5-cell/pm158bSU">t0,1,2{33}</a> <a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{33}</a> </td></tr><tr><td>Cell types</td><td><a href="/facts/Cuboctahedron/hD7p0Mem">Cuboctahedron</a> <p><a href="/facts/Truncated_octahedron/GA98b4TH">Truncated octahedron</a> <a href="/facts/Truncated_tetrahedron/K9ItvmUK">Truncated tetrahedron</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">Hexagonal prism</a> <a href="/facts/Triangular_prism/Yjp4Kw9M">Triangular prism</a> </p></td></tr><tr><td>Vertex figure</td><td>tilted rectangular <a href="/facts/Duopyramid/B400kvjL">duopyramid</a></td></tr><tr><td><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></td><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×2[3[5]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a></td></tr></tbody></table> <p>The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>. It can also be called a cycloruncicantitruncated 5-cell honeycomb. </p> <h4>Alternate names</h4> <ul><li>Great cycloprismated pentachoric tetracomb</li> <li>Grand prismatodispentachoric tetracomb</li></ul> <h3>Omnitruncated 5-cell honeycomb</h3> <table><tbody><tr><th colspan="2">Omnitruncated 4-simplex honeycomb</th></tr><tr><td colspan="2">(No image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></td></tr><tr><td>Family</td><td><a href="/facts/Omnitruncated_simplectic_honeycomb/URt9M5Ms">Omnitruncated simplectic honeycomb</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,2,3,4{3[5]} or tr{3[5]}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>4-face types</td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a> </td></tr><tr><td>Cell types</td><td><a href="/facts/Truncated_octahedron/GA98b4TH">t0,1,2{3,3}</a> <a href="/facts/Hexagonal_prism/oxYc0nF2">{6}x{}</a> </td></tr><tr><td>Face types</td><td><a href="/facts/Square_(geometry)/JHJeMvmL">{4}</a><a href="/facts/Hexagon/DPzuCzWE">{6}</a></td></tr><tr><td>Vertex figure</td><td>Irr. <a href="/facts/5-cell/Rh6lwDcc">5-cell</a></td></tr><tr><td><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></td><td> A ~ 4 {\displaystyle {\tilde {A}}_{4}} ×10, [5[3[5]]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>, <a href="/facts/Cell-transitive/ukLVQlK2">cell-transitive</a></td></tr></tbody></table> <p>The omnitruncated 4-simplex honeycomb or omnitruncated 5-cell honeycomb is a space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>. It can also be seen as a cyclosteriruncicantitruncated 5-cell honeycomb. . </p><p>It is composed entirely of <a href="/facts/Omnitruncated_5-cell/7A0II7du">omnitruncated 5-cell</a> (omnitruncated 4-simplex) facets. </p><p><a href="/facts/Coxeter/D2l5jWGG">Coxeter</a> calls this Hinton's honeycomb after <a href="/facts/Charles_Howard_Hinton/15oS42Pf">C. H. Hinton</a>, who described it in his book <i>The Fourth Dimension</i> in 1906.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>The facets of all <a href="/facts/Omnitruncated_simplectic_honeycomb/URt9M5Ms">omnitruncated simplectic honeycombs</a> are called <a href="/facts/Permutohedron/No1KH0rE">permutohedra</a> and can be positioned in <i>n+1</i> space with integral coordinates, permutations of the whole numbers (0,1,..,n). </p> <h4>Alternate names</h4> <ul><li>Omnitruncated cyclopentachoric tetracomb</li> <li>Great-prismatodecachoric tetracomb</li></ul> <h4>A4* lattice</h4> <p>The A*4 lattice is the union of five A4 lattices, and is the dual to the omnitruncated 5-cell honeycomb, and therefore the <a href="/facts/Voronoi_cell/AhQY5LD8">Voronoi cell</a> of this lattice is an <a href="/facts/Omnitruncated_5-cell/7A0II7du">omnitruncated 5-cell</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> ∪ ∪ ∪ ∪ = dual of <h2 id="alternated-form">Alternated form</h2> <p>This honeycomb can be <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternated</a>, creating <a href="/facts/Runcinated_5-cell/7A0II7du">omnisnub 5-cells</a> with irregular <a href="/facts/5-cell/Rh6lwDcc">5-cells</a> created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10. </p> <h2 id="see-also">See also</h2> <p>Regular and uniform honeycombs in 4-space: </p> <ul><li><a href="/facts/Tesseractic_honeycomb/fpucRJJn">Tesseractic honeycomb</a></li> <li><a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a></li> <li><a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a></li> <li><a href="/facts/Truncated_24-cell_honeycomb/uQN2kXg1">Truncated 24-cell honeycomb</a></li> <li><a href="/facts/Snub_24-cell_honeycomb/bzFwdh7X">Snub 24-cell honeycomb</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> <i>Uniform Polytopes</i>, Manuscript (1991)</li> <li><i>Kaleidoscopes: Selected Writings of H.S.M. Coxeter</i>, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6 <a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[1]</a> <ul><li>(Paper 22) H.S.M. Coxeter, <i>Regular and Semi Regular Polytopes I</i>, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)</li> <li>(Paper 24) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes III</i>, [Math. Zeit. 200 (1988) 3-45]</li></ul></li> <li>George Olshevsky, <i>Uniform Panoploid Tetracombs</i>, Manuscript (2006) <i>(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)</i> Model 134</li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/flat.htm">"4D Euclidean tesselations"</a>., x3o3o3o3o3*a - cypit - O134, x3x3x3x3x3*a - otcypit - 135, x3x3x3o3o3*a - gocyropit - O137, x3x3o3x3o3*a - cypropit - O138, x3x3x3x3o3*a - gocypapit - O139, x3x3x3x3x3*a - otcypit - 140</li> <li>Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals, Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1209.1878">1209.1878</a></li></ul> <table><tbody><tr><th colspan="7"><ul><li>v</li><li>t</li><li>e</li></ul>Fundamental convex <a href="/facts/Regular_honeycomb/Kng25Ymq">regular</a> and <a href="/facts/Uniform_honeycomb/fGXtrgGF">uniform honeycombs</a> in dimensions 2–9</th></tr><tr><th>Space</th><th><a href="/facts/Coxeter_group/bgzHFm1Q">Family</a></th><td> A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} </td><td> C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} </td><td> B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} </td><td> D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} </td><td> G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}} </td></tr><tr><th>E2</th><th><a href="/facts/Uniform_tiling/QVK02ksI">Uniform tiling</a></th><td><a href="/facts/Triangular_tiling/Jx8f8Krh">0[3]</a></td><td><a href="/facts/Square_tiling/3ym86mvv">δ3</a></td><td><a href="/facts/Square_tiling/3ym86mvv">hδ3</a></td><td><a href="/facts/Square_tiling/3ym86mvv">qδ3</a></td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal</a></td></tr><tr><th>E3</th><th><a href="/facts/Uniform_convex_honeycomb/78IkTl7A">Uniform convex honeycomb</a></th><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">0[4]</a></td><td><a href="/facts/Cubic_honeycomb/eMxySFur">δ4</a></td><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">hδ4</a></td><td><a href="/facts/Quarter_cubic_honeycomb/22qQazbw">qδ4</a></td><td></td></tr><tr><th>E4</th><th><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></th><td>0[5]</td><td><a href="/facts/Tesseractic_honeycomb/fpucRJJn">δ5</a></td><td><a href="/facts/16-cell_honeycomb/lbktV8Nr">hδ5</a></td><td><a href="/facts/Rectified_tesseractic_honeycomb/P266Jc7y">qδ5</a></td><td><a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a></td></tr><tr><th>E5</th><th><a href="/facts/Uniform_5-honeycomb/pxqFTw2F">Uniform 5-honeycomb</a></th><td><a href="/facts/5-simplex_honeycomb/XSLrRsLz">0[6]</a></td><td><a href="/facts/5-cubic_honeycomb/bqAJvIBI">δ6</a></td><td><a href="/facts/5-demicubic_honeycomb/eYp2lMB8">hδ6</a></td><td><a href="/facts/Quarter_5-cubic_honeycomb/r3AAQHbg">qδ6</a></td><td></td></tr><tr><th>E6</th><th><a href="/facts/Uniform_6-honeycomb/suCJSvTN">Uniform 6-honeycomb</a></th><td><a href="/facts/6-simplex_honeycomb/H7LWND7E">0[7]</a></td><td><a href="/facts/6-cubic_honeycomb/2FdezLvD">δ7</a></td><td><a href="/facts/6-demicubic_honeycomb/oxttB0n0">hδ7</a></td><td><a href="/facts/Quarter_6-cubic_honeycomb/vDRQk9kS">qδ7</a></td><td><a href="/facts/2_22_honeycomb/cvXxbk6L">222</a></td></tr><tr><th>E7</th><th><a href="/facts/Uniform_7-honeycomb/M93v3h3c">Uniform 7-honeycomb</a></th><td><a href="/facts/7-simplex_honeycomb/dq8hy2lI">0[8]</a></td><td><a href="/facts/7-cubic_honeycomb/Im2bVBf7">δ8</a></td><td><a href="/facts/7-demicubic_honeycomb/QYm4RCGK">hδ8</a></td><td><a href="/facts/Quarter_7-cubic_honeycomb/GjVHMBLq">qδ8</a></td><td><a href="/facts/1_33_honeycomb/tYL93yF1">133</a> • <a href="/facts/3_31_honeycomb/VlInrcea">331</a></td></tr><tr><th>E8</th><th><a href="/facts/Uniform_8-honeycomb/UdFL9ns2">Uniform 8-honeycomb</a></th><td><a href="/facts/8-simplex_honeycomb/eoAGHg7J">0[9]</a></td><td><a href="/facts/8-cubic_honeycomb/fx3hM5DD">δ9</a></td><td><a href="/facts/8-demicubic_honeycomb/baWsKzQc">hδ9</a></td><td>qδ9</td><td><a href="/facts/1_52_honeycomb/LAJXFvnV">152</a> • <a href="/facts/2_51_honeycomb/XthKHy0m">251</a> • <a href="/facts/5_21_honeycomb/f971VRma">521</a></td></tr><tr><th>E9</th><th><a href="/facts/Uniform_9-honeycomb/Qc5NglV8">Uniform 9-honeycomb</a></th><td>0[10]</td><td>δ10</td><td>hδ10</td><td>qδ10</td><td></td></tr><tr><th>E10</th><th>Uniform 10-honeycomb</th><td>0[11]</td><td>δ11</td><td>hδ11</td><td>qδ11</td><td></td></tr><tr><th>E<i>n</i>-1</th><th>Uniform (<i>n</i>-1)-<a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a></th><td><a href="/facts/Simplectic_honeycomb/hCmbARWt">0[<i>n</i>]</a></td><td><a href="/facts/Hypercubic_honeycomb/MQxLbDeB">δ<i>n</i></a></td><td><a href="/facts/Alternated_hypercubic_honeycomb/zrU9sG2e">hδ<i>n</i></a></td><td><a href="/facts/Quarter_hypercubic_honeycomb/7Je1f2st">qδ<i>n</i></a></td><td><a href="/facts/Uniform_1_k2_polytope/eqTstjgl">1k2</a> • <a href="/facts/Uniform_2_k1_polytope/pQv1I7AC">2k1</a> • <a href="/facts/Uniform_k_21_polytope/vv3lWMYd">k21</a></td></tr></tbody></table> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Olshevsky (2006), Model 134 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Baake, M.; Kramer, P.; Schlottmann, M.; Zeidler, D. (December 1990). "Planar Patterns with Fivefold Symmetry as Sections of Periodic Structures in 4-Space". International Journal of Modern Physics B. 04 (15n16): 2217–2268. doi:10.1142/S0217979290001054. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"The Lattice A4". <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html" target="_blank">http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"A4 root lattice - Wolfram|Alpha". <a href="https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3" target="_blank">https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"The Lattice A4". <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html" target="_blank">http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>mathworld: Necklace, OEIS sequence A000029 8-1 cases, skipping one with zero marks <a href="http://mathworld.wolfram.com/Necklace.html" target="_blank">http://mathworld.wolfram.com/Necklace.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Olshevsky, (2006) Model 135 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. (The classification of Zonohededra, page 73) <a href="0-486-40919-8" target="_blank">0-486-40919-8</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>The Lattice A4* <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html" target="_blank">http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>