title: 2 51 honeycomb
text: # 2 51 honeycomb
  
  $2_{51}$ honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform tessellation |
  | Family | $2_{k1}$ polytope |
  | Schläfli symbol | ${3,3,3^{5,1}}$ |
  | Coxeter symbol | $2_{51}$ |
  | Coxeter-Dynkin diagram | |
  | 8-face types | $2_{41}$ ${3^7}$ |
  | 7-face types | $2_{31}$ ${3^6}$ |
  | 6-face types | $2_{21}$ ${3^5}$ |
  | 5-face types | $2_{11}$ ${3^4}$ |
  | 4-face type | ${3^3}$ |
  | Cells | ${3^2}$ |
  | Faces | {3} |
  | Edge figure | $0_{51}$ |
  | Vertex figure | $1_{51}$ |
  | Edge figure | $0_{51}$ |
  | Coxeter group | ${\tilde {E}}_{8}$, $[3^{5,2,1}]$ |
  
  In 8-dimensional geometry, the $2_{51}$ honeycomb is a space-filling uniform tessellation. It is composed of $2_{41}$ polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the $2_{k1}$ family.
  
  ## Construction
  
  It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.
  
  The facet information can be extracted from its Coxeter-Dynkin diagram.
  
  Removing the node on the short branch leaves the 8-simplex.
  
  Removing the node on the end of the 5-length branch leaves the $2_{41}.$
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, $1_{51}.$
  
  The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, $0_{51}.$
  
  ## Related polytopes and honeycombs
  
  $2_{k1}$ figures in n dimensions :
  
  | Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
  | - | - | - | - | - | - | - | - | - |
  | n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
  | Coxeter group | $E_3=A_2A_1$ | $E_4=A_4$ | $E_5=D_5$ | $E_6$ | $E_7$ | $E_8$ | $E_9$ = ${\tilde {E}}_{8}$ = $E_8^+$ | $E_{10}$ = ${\bar {T}}_{8}$ = $E_8^{++}$ |
  | Coxeter diagram | | | | | | | | |
  | Symmetry | $[3^{−1,2,1}]$ | $[3^{0,2,1}]$ | $[[3^{1,2,1}]]$ | $[3^{2,2,1}]$ | $[3^{3,2,1}]$ | $[3^{4,2,1}]$ | $[3^{5,2,1}]$ | $[3^{6,2,1}]$ |
  | Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ∞ |
  | Graph | | | | | | | - | - |
  | Name | $2_{−1,1}$ | $2_{01}$ | $2_{11}$ | $2_{21}$ | $2_{31}$ | $2_{41}$ | $2_{51}$ | $2_{61}$ |
  