title: Truncated tesseractic honeycomb
text: # Truncated tesseractic honeycomb
  
  Truncated tesseractic honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform 4-honeycomb |
  | Schläfli symbol | t{4,3,3,4} $t{4,3,3^{1,1}}$ |
  | Coxeter-Dynkin diagram | |
  | 4-face type | truncated tesseract 16-cell |
  | Cell type | Truncated cube Tetrahedron |
  | Face type | {3}, {8} |
  | Vertex figure | octahedral pyramid |
  | Coxeter group | ${\tilde {C}}_{4}$ = [4,3,3,4] ${\tilde {B}}_{4}$ = $[4,3,3^{1,1}]$ |
  | Dual | |
  | Properties | vertex-transitive |
  
  In four-dimensional Euclidean geometry, the truncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a truncation of a tesseractic honeycomb creating truncated tesseracts, and adding new 16-cell facets at the original vertices.
  
  ## 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 | $_{14},$ $_{15},$ $_{16},$ $_{17}$ |
  