page_title: Uniform 9-polytope
section_title: Regular and uniform honeycombs
hierarchical_section_title: Uniform 9-polytope / Regular and uniform honeycombs
caption_attribution_description: en:Coxeter-Dynkin diagram table of correspondences and automorphisms
context_page_description: In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
  A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets.
context_section_description: There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 8-space:
  Regular and uniform tessellations include:
   45 uniquely ringed forms
  8-simplex honeycomb: {3} 
   271 uniquely ringed forms
  Regular 8-cube honeycomb: {4,3⁶,4}, 
  : 383 uniquely ringed forms, 255 shared with , 128 new
  8-demicube honeycomb: h{4,3⁶,4} or {31,1,3⁵,4},  or 
  , [31,1,3⁴,31,1]: 155 unique ring permutations, and 15 are new, the first, , Coxeter called a quarter 8-cubic honeycomb, representing as q{4,3⁶,4}, or qδ₉.
   511 forms
  5₂₁ honeycomb:
  2₅₁ honeycomb: 
  1₅₂ honeycomb:
caption: Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.
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