title: Omnitruncated tesseractic honeycomb
text: # Omnitruncated tesseractic honeycomb
  
  omnitruncated tesseractic honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform honeycomb |
  | Schläfli symbol | $t_{0,1,2,3,4}{4,3,3,4}$ |
  | Coxeter diagram | |
  | 4-face type | Omnitruncated tesseract Truncated cuboctahedral prism 8-8 duoprism |
  | Cell type | Truncated cuboctahedron Truncated octahedron Octagonal prism Hexagonal prism Cube |
  | Face type | {4}, {6}, {8} |
  | Vertex figure | irr. 5-cell |
  | Coxeter groups | ${\tilde {C}}_{4}$×2, [[4,3,3,4]] |
  | Properties | Vertex transitive |
  
  In four-dimensional Euclidean geometry, the omnitruncated tesseractic honeycomb is a uniform space-filling honeycomb. It has omnitruncated tesseract, truncated cuboctahedral prism, and 8-8 duoprism facets in an irregular 5-cell vertex figure.
  
  ## Related honeycombs
  
  The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
  
  C4 honeycombs :
  
  | Extended symmetry | Extended diagram | Order | Honeycombs |
  | - | - | - | - |
  | [4,3,3,4]: | | ×1 | $_1,$ $_2,$ $_3,$ $_4,$ $_5,$ $_6,$ $_7,$ $_8,$ $_9,$ $_{10},$ $_{11},$ $_{12},$ $_{13}$ |
  | [[4,3,3,4]] | | ×2 | $_{(1)},$ $_{(2)},$ $_{(13)},$ $_{18}$ $_{(6)},$ $_{19},$ $_{20}$ |
  | $[(3,3)[1^+,4,3,3,4,1^+]]$ ↔ $[(3,3)[3^{1,1,1,1}]]$ ↔ [3,4,3,3] | ↔ ↔ | ×6 | $_{14},$ $_{15},$ $_{16},$ $_{17}$ |
  