page_title: Convex uniform honeycomb
section_title: Hyperbolic forms
hierarchical_section_title: Convex uniform honeycomb / Hyperbolic forms
context_page_description: In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
  Twenty-eight such honeycombs are known:
  the familiar cubic honeycomb and 7 truncations thereof;
  the alternated cubic honeycomb and 4 truncations thereof;
  10 prismatic forms based on the uniform plane tilings;
  5 modifications of some of the above by elongation and/or gyration.
  They can be considered the three-dimensional analogue to the uniform tilings of the plane.
  The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra.
context_section_description: There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family.
  From these 9 families, there are a total of 76 unique honeycombs generated:
  [3,5,3] :  - 9 forms
  [5,3,4] :  - 15 forms
  [5,3,5] :  - 9 forms
  [5,31,1] :  - 11 forms (7 overlap with [5,3,4] family, 4 are unique)
  [(4,3,3,3)] :  - 9 forms
  [(4,3,4,3)] :  - 6 forms
  [(5,3,3,3)] :  - 9 forms
  [(5,3,4,3)] :  - 9 forms
  [(5,3,5,3)] :  - 6 forms
  The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is in the {3,5,3} family.
caption: The order-4 dodecahedral honeycomb, {5,3,4} in perspective
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