E9 honeycomb

DISPLAYTITLE:E9 honeycomb In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded. E10 is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, but there are three simplest ones, with a single ring at the end of its 3 branches: 621, 261, 162. The 621 honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E10 Coxeter group. This honeycomb is highly regular in the sense that its symmetry group (the affine E9 Weyl group) acts transitively on the k-faces for k ≤ 7. All of the k-faces for k ≤ 8 are simplices. This honeycomb is last in the series of k21 polytopes, enumerated by Thorold Gosset in 1900, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 521. It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 9-orthoplex, 711. Removing the node on the end of the 1-length branch leaves the 9-simplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 521 honeycomb. The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 421 polytope. The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 321 polytope. The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 221 polytope.