5 21 honeycomb

<table><tbody><tr><th colspan="2">521 honeycomb</th></tr><tr><td>Type</td><td><a href="/facts/9-polytope/UdFL9ns2">Uniform honeycomb</a></td></tr><tr><td>Family</td><td><a href="/facts/Semiregular_k_21_polytope/vv3lWMYd">k21 polytope</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>{3,3,3,3,3,32,1}</td></tr><tr><td><a href="/facts/Coxeter_symbol/PfHvVPBg">Coxeter symbol</a></td><td>521</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a></td><td></td></tr><tr><td>8-faces</td><td><a href="/facts/Octacross/dXFkXtPn">511</a> <a href="/facts/8-simplex/QUhcnmk2">{37}</a> </td></tr><tr><td>7-faces</td><td><a href="/facts/7-simplex/6DwtPplH">{36}</a> Note that there are two distinct orbits of this <a href="/facts/7-simplex/6DwtPplH">7-simplex</a> under the honeycomb's full E ~ 8 {\displaystyle {\tilde {E}}_{8}} <a href="/facts/Coxeter_group/bgzHFm1Q">automorphism group</a>.</td></tr><tr><td>6-faces</td><td><a href="/facts/6-simplex/mInfSIeH">{35}</a> </td></tr><tr><td>5-faces</td><td><a href="/facts/5-simplex/KIfBFfhl">{34}</a> </td></tr><tr><td>4-faces</td><td><a href="/facts/Pentachoron/Rh6lwDcc">{33}</a> </td></tr><tr><td>Cells</td><td><a href="/facts/Tetrahedron/7mkSrl6j">{32}</a> </td></tr><tr><td>Faces</td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a> </td></tr><tr><td>Cell figure</td><td><a href="/facts/Demipenteract/zvCeNbU2">121</a> </td></tr><tr><td>Face figure</td><td><a href="/facts/Gosset_2_21_polytope/nF3nYaQk">221</a> </td></tr><tr><td><a href="/facts/Edge_figure/3u2ySK0C">Edge figure</a></td><td><a href="/facts/Gosset_3_21_polytope/AEUzwA1A">321</a> </td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Gosset_4_21_polytope/QFGglCkk">421</a> </td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Symmetry group</a></td><td> E ~ 8 {\displaystyle {\tilde {E}}_{8}} , [35,2,1]</td></tr></tbody></table>
In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the 521 honeycomb is a <a href="/facts/Uniform_honeycomb/fGXtrgGF">uniform tessellation</a> of 8-dimensional Euclidean space. The symbol 521 is from <a href="/facts/Coxeter/D2l5jWGG">Coxeter</a>, named for the length of the 3 branches of its Coxeter-Dynkin diagram.
By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by <a href="/facts/Maryna_Viazovska/tKNHfw3j">Maryna Viazovska</a> in 2016 using the theory of <a href="/facts/Modular_form/PcnbYR1f">modular forms</a>. Viazovska was awarded the <a href="/facts/Fields_Medal/khFmkEnm">Fields Medal</a> for this work in 2022.
This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).
Each vertex of the 521 honeycomb is surrounded by 2160 <a href="/facts/8-orthoplex/dXFkXtPn">8-orthoplexes</a> and 17280 <a href="/facts/8-simplex/QUhcnmk2">8-simplicies</a>.
The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of Gosset's honeycomb is the semiregular <a href="/facts/4_21_polytope/QFGglCkk">421 polytope</a>. It is the final figure in the <a href="/facts/Semiregular_k_21_polytope/vv3lWMYd">k21</a> family.
This honeycomb is highly regular in the sense that its symmetry group (the affine 
 
 
 
 
 
 
 
 E
 ~
 
 
 
 
 8
 
 
 
 
 {\displaystyle {\tilde {E}}_{8}}
 
 Weyl group) acts transitively on the <a href="/facts/Face_(geometry)/T2E6BTU4">k-faces</a> for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.

5 21 honeycomb open-in-new

5 21 honeycomb