In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space.
Its vertex arrangement is the E6 lattice, and the root system of the E6 Lie group so it can also be called the E6 honeycomb.
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space.
[3] It is constructed by 3 copies of the E6 lattice vertices, one from each of the three branches of the Coxeter diagram.
There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 222 and birectified 222 are isotopic, with only one type of facet: 221, and rectified 122 polytopes respectively.
The birectified 222 honeycomb , has rectified 1 22 polytope facets, , and a proprism {3}×{3}×{3} vertex figure.
Removing a node on the end of one of the 3-node branches leaves the rectified 122, its only facet type, .
Removing a fourth end node defines 2 types of cells: octahedron, 011, and tetrahedron, 020.
Each progressive uniform polytope is constructed from the previous as its vertex figure.