The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with {\tilde{E}}_6 symmetry. 24 of them have doubled symmetry
3,3,32,2 with 2 equally ringed branches, and 7 have sextupled (3
!) symmetry
3,32,2,2 with identical rings on all 3 branches. 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.
Birectified 222 honeycomb The
birectified 222 honeycomb , has
rectified 1 22 polytope facets, , and a
proprism {3}×{3}×{3}
vertex figure. Its facets are centered on the
vertex arrangement of
E6* lattice, as: : ∪ ∪
Construction 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}×{3}×{3}, . Removing a node on the end of one of the 3-node branches leaves the
rectified 122, its only
facet type, . Removing a second end node defines 2 types of 5-faces:
birectified 5-simplex, 022 and
birectified 5-orthoplex, 0211. Removing a third end node defines 2 types of 4-faces:
rectified 5-cell, 021, and
24-cell, 0111. Removing a fourth end node defines 2 types of cells:
octahedron, 011, and
tetrahedron, 020.
k22 polytopes The 222 honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by
Coxeter as k22 series. The final is a paracompact hyperbolic honeycomb,
322. Each progressive
uniform polytope is constructed from the previous as its
vertex figure. The 222 honeycomb is third in another dimensional series 22k. == Notes ==