Random Article Summary #147

The 222 honeycomb is a geometric pattern that fills six-dimensional space in a uniform way. It is described by the Schläfli symbol {3,3,32,2}. This honeycomb is built from 221-shaped facets, and at each vertex there are 54 of those 221 polytopes meeting. The arrangement of its vertices follows the E6 lattice. Because of this connection to the E6 Lie group’s root system, it is also called the E6 honeycomb. The honeycomb is built using a Wythoff construction based on seven hyperplane mirrors in 6-dimensional space, and its facet types can be read from its Coxeter–Dynkin diagram. If you remove a node at the end of one 2-node branch, the only facet type you get is 221, and the vertex figure is found by modifying the diagram to produce 122. Continuing this process, the edge figure becomes the vertex figure of 122 (a birectified 5-simplex, t2{34}), and the face figure becomes the vertex figure of that (a triangular duoprism, {3}×{3}). In this 6-dimensional tessellation, every vertex acts as the center of a 5-sphere from the densest known packing. Each such sphere is “kissing” 72 other spheres, and this count is shown by the vertices of its vertex figure, 122. The 222 honeycomb’s vertex pattern is known as the E6 lattice. By combining two related E6 lattices, you can build the E6’2 lattice with 3,3,3,2,2 symmetry. This uses the E6* (E63) lattice, whose Voronoi cell is the rectified 1-2-2 polytope and whose Voronoi tiling forms a bitruncated 222 honeycomb. The group can be connected to another structure through a geometric folding. This folding allows the honeycomb to be projected into the 4-dimensional 16-cell honeycomb.