In geometry, the 2<sub>22</sub> honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,3<sup>2,2</sup>}. It is constructed from 2<sub>21</sub> facets and has a 1<sub>22</sub> vertex figure, with 54 2<sub>21</sub> polytopes around every vertex.
Its vertex arrangement is the E<sub>6</sub> lattice, and the root system of the E<sub>6</sub> Lie group so it can also be called the E<sub>6</sub> honeycomb.
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space.
The facet information can be extracted from its Coxeter–Dynkin diagram, .
Removing a node on the end of one of the 2-node branches leaves the 2<sub>21</sub>, its only facet type,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1<sub>22</sub>, .
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t<sub>2</sub>{3<sup>4</sup>}, .
The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, .
Each vertex of this tessellation is the center of a 5-sphere in the densest known packing in 6 dimensions, with kissing number 72, represented by the vertices of its vertex figure 1<sub>22</sub>.
The 2<sub>22</sub> honeycomb's vertex arrangement is called the E<sub>6</sub> lattice.
The E<sub>6</sub><sup>2</sup> lattice, with 3,3,3<sup>2,2</sup> symmetry, can be constructed by the union of two E<sub>6</sub> lattices:
The E<sub>6</sub><sup>*</sup> lattice (or E<sub>6</sub><sup>3</sup>) with 3,3<sup>2,2,2</sup> symmetry. The Voronoi cell of the E<sub>6</sub><sup>*</sup> lattice is the rectified 1<sub>22</sub> polytope, and the Voronoi tessellation is a bitruncated 2<sub>22</sub> honeycomb. It is constructed by 3 copies of the E<sub>6</sub> lattice vertices, one from each of the three branches of the Coxeter diagram.
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
The 2<sub>22</sub> honeycomb is one of 127 uniform honeycombs (39 unique) with symmetry. 24 of them have doubled symmetry 3,3,3<sup>2,2</sup> with 2 equally ringed branches, and 7 have sextupled (3!) symmetry 3,3<sup>2,2,2</sup> with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 2<sub>22</sub> and birectified 2<sub>22</sub> are isotopic, with only one type of facet: 2<sub>21</sub>, and rectified 1<sub>22</sub> polytopes respectively.
The birectified 2<sub>22</sub> honeycomb , has rectified 1 22 polytope facets, , and a proprism {3}ÃÂ{3}ÃÂ{3} vertex figure.
Its facets are centered on the vertex arrangement of E<sub>6</sub><sup>*</sup> lattice, as:
The facet information can be extracted from its Coxeter–Dynkin diagram, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}ÃÂ{3}ÃÂ{3}, .
Removing a node on the end of one of the 3-node branches leaves the rectified 1<sub>22</sub>, its only facet type, .
Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 0<sub>22</sub> and birectified 5-orthoplex, 0<sub>211</sub>.
Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 0<sub>21</sub>, and 24-cell, 0<sub>111</sub>.
Removing a fourth end node defines 2 types of cells: octahedron, 0<sub>11</sub>, and tetrahedron, 0<sub>20</sub>.
The 2<sub>22</sub> honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k<sub>22</sub> series. The final is a paracompact hyperbolic honeycomb, 3<sub>22</sub>. Each progressive uniform polytope is constructed from the previous as its vertex figure.
The 2<sub>22</sub> honeycomb is third in another dimensional series 2<sub>2k</sub>.