title: 1 33 honeycomb
text: # 1 33 honeycomb
  
  $1_{33}$ honeycomb :
  
  | (no image) ||
  | - | - |
  | Type | Uniform tessellation |
  | Schläfli symbol | ${3,3^{3,3}}$ |
  | Coxeter symbol | $1_{33}$ |
  | Coxeter-Dynkin diagram | or |
  | 7-face type | $1_{32}$ |
  | 6-face types | $1_{22}$ $1_{31}$ |
  | 5-face types | $1_{21}$ ${3^4}$ |
  | 4-face type | $1_{11}$ ${3^3}$ |
  | Cell type | $1_{01}$ |
  | Face type | {3} |
  | Cell figure | Square |
  | Face figure | Triangular duoprism |
  | Edge figure | Tetrahedral duoprism |
  | Vertex figure | Trirectified 7-simplex |
  | Coxeter group | ${\tilde {E}}_{7}$, $[[3,3^{3,3}]]$ |
  | Properties | vertex-transitive, facet-transitive |
  
  In 7-dimensional geometry, $1_{33}$ is a uniform honeycomb, also given by Schläfli symbol ${3,3^{3,3}},$ and is composed of $1_{32}$ facets.
  
  ## Construction
  
  It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.
  
  The facet information can be extracted from its Coxeter-Dynkin diagram.
  
  Removing a node on the end of one of the 3-length branch leaves the $1_{32},$ its only facet type.
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, $0_{33}.$
  
  The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.
  
  ## Kissing number
  
  Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.
  
  ## Geometric folding
  
  The ${\tilde {E}}_{7}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
  
  | ${\tilde {E}}_{7}$ | ${\tilde {F}}_{4}$ |
  | - | - |
  | ${3,3^{3,3}}$ | {3,3,4,3} |
  
  ## $E_7^*$ lattice
  
  ${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144. Both ${\tilde {E}}_{7}$ and ${\tilde {A}}_{7}$ can be seen as affine extension from $A_{7}$ from different nodes:
  
  The $E_7^*$ lattice (also called $E_7^2)$ has double the symmetry, represented by $[[3,3^{3,3}]].$ The Voronoi cell of the $E_7^*$ lattice is the $1_{32}$ polytope, and voronoi tessellation the $1_{33}$ honeycomb. The $E_7^*$ lattice is constructed by 2 copies of the $E_7$ lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four $A_7^*$ lattices, also called $A_7^4:$
  
      ∪ = ∪ ∪ ∪ = dual of .
  
  ### Related polytopes and honeycombs
  
  The $1_{33}$ is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as $1_{3k}$ series. The final is a noncompact hyperbolic honeycomb, $1_{34}.$
  
  $1_{3k}$ dimensional figures :
  
  | Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
  | - | - | - | - | - | - | - |
  | n | 4 | 5 | 6 | 7 | 8 | 9 |
  | Coxeter group | $A_3A_1$ | $A_5$ | $D_6$ | $E_7$ | ${\tilde {E}}_{7}$$=E_7^+$ | ${\bar {T}}_{8}$$=E_7^{++}$ |
  | Coxeter diagram | | | | | | |
  | Symmetry | $[3^{−1,3,1}]$ | $[3^{0,3,1}]$ | $[3^{1,3,1}]$ | $[3^{2,3,1}]$ | $[[3^{3,3,1}]]$ | $[3^{4,3,1}]$ |
  | Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | ∞ |
  | Graph | | | | | - | - |
  | Name | $1_{3,-1}$ | $1_{30}$ | $1_{31}$ | $1_{32}$ | $1_{33}$ | $1_{34}$ |
  
  **Rectified $1_{33}$ honeycomb**
  
  Rectified $1_{33}$ honeycomb :
  
  | (no image) ||
  | - | - |
  | Type | Uniform tessellation |
  | Schläfli symbol | ${3^{3,3,1}}$ |
  | Coxeter symbol | $0_{331}$ |
  | Coxeter-Dynkin diagram | or |
  | 7-face type | Trirectified 7-simplex Rectified 1_32 |
  | 6-face types | Birectified 6-simplex Birectified 6-cube Rectified 1_22 |
  | 5-face types | Rectified 5-simplex Birectified 5-simplex Birectified 5-orthoplex |
  | 4-face type | 5-cell Rectified 5-cell 24-cell |
  | Cell type | {3,3} {3,4} |
  | Face type | {3} |
  | Vertex figure |×{3,3}×{3,3} |
  | Coxeter group | ${\tilde {E}}_{7}$, $[[3,3^{3,3}]]$ |
  | Properties | vertex-transitive, facet-transitive |
  
  The rectified $1_{33}$ or $0_{331},$ Coxeter diagram has facets and , and vertex figure .
  
  