title: 1 52 honeycomb
text: # 1 52 honeycomb
  
  $1_{52}$ honeycomb :
  
  | (No image) ||
  | - | - |
  | Type | Uniform tessellation |
  | Family | $1_{k2}$ polytope |
  | Schläfli symbol | ${3,3^{5,2}}$ |
  | Coxeter symbol | $1_{52}$ |
  | Coxeter-Dynkin diagram | |
  | 8-face types | $1_{42}$ $1_{51}$ |
  | 7-face types | $1_{32}$ $1_{41}$ |
  | 6-face types | $1_{22}$ ${3^{1,3,1}}$ ${3^5}$ |
  | 5-face types | $1_{21}$ ${3^4}$ |
  | 4-face type | $1_{11}$ ${3^3}$ |
  | Cells | ${3^2}$ |
  | Faces | {3} |
  | Vertex figure | birectified 8-simplex: $t_2{3^7}$ |
  | Coxeter group | ${\tilde {E}}_{8}$, $[3^{5,2,1}]$ |
  
  In geometry, the $1_{52}$ honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains $1_{42}$ and $1_{51}$ facets, in a birectified 8-simplex vertex figure. It is the final figure in the $1_{k2}$ polytope 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 end of the 2-length branch leaves the 8-demicube, $1_{51}.$
  
  Removing the node on the end of the 5-length branch leaves the $1_{42}.$
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 8-simplex, $0_{52}.$
  
  ## Related polytopes and honeycombs
  
  $1_{k2}$ 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 (order) | $[3^{−1,2,1}]$ | $[3^{0,2,1}]$ | $[3^{1,2,1}]$ | $[[3^{2,2,1}]]$ | $[3^{3,2,1}]$ | $[3^{4,2,1}]$ | $[3^{5,2,1}]$ | $[3^{6,2,1}]$ |
  | Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ∞ |
  | Graph | | | | | | | - | - |
  | Name | $1_{−1,2}$ | $1_{02}$ | $1_{12}$ | $1_{22}$ | $1_{32}$ | $1_{42}$ | $1_{52}$ | $1_{62}$ |
  