title: 2 41 polytope
text: # 2 41 polytope
  
  | $4_{21}$ | $1_{42}$ | $2_{41}$ |
  | - | - | - |
  | Rectified $4_{21}$ | Rectified $1_{42}$ | Rectified $2_{41}$ |
  | Birectified $4_{21}$ | Trirectified $4_{21}$ |
  | Orthogonal projections in $E_6$ Coxeter plane |||
  
  In 8-dimensional geometry, the $2_{41}$ is a uniform 8-polytope, constructed within the symmetry of the $E_8$ group.
  
  Its Coxeter symbol is $2_{41},$ describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.
  
  The rectified $2_{41}$ is constructed by points at the mid-edges of the $2_{41}.$ The birectified $2_{41}$ is constructed by points at the triangle face centers of the $2_{41},$ and is the same as the rectified $1_{42}.$
  
  These polytopes are part of a family of 255 $(2^8$ − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
  
  ## $2_{41}$ polytope
  
  $2_{41}$ polytope :
  
  | Type | Uniform 8-polytope |
  | - | - |
  | Family | $2_{k1}$ polytope |
  | Schläfli symbol | ${3,3,3^{4,1}}$ |
  | Coxeter symbol | $2_{41}$ |
  | Coxeter diagram | |
  | 7-faces | 17520: 240 $2_{31}$ 17280 ${3^6}$ |
  | 6-faces | 144960: 6720 $2_{21}$ 138240 ${3^5}$ |
  | 5-faces | 544320: 60480 $2_{11}$ 483840 ${3^4}$ |
  | 4-faces | 1209600: 241920 ${2_{01}$ 967680 ${3^3}$ |
  | Cells | 1209600 ${3^2}$ |
  | Faces | 483840 {3} |
  | Edges | 69120 |
  | Vertices | 2160 |
  | Vertex figure | $1_{41}$ |
  | Petrie polygon | 30-gon |
  | Coxeter group | $E_8,$ $[3^{4,2,1}]$ |
  | Properties | convex |
  
  The $2_{41}$ is composed of 17,520 facets (240 $2_{31}$ polytopes and 17,280 7-simplices), 144,960 6-faces (6,720 $2_{21}$ polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 $2_{11}$ and 483,840 5-simplices), 1,209,600 4-faces (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.
  
  This polytope is a facet in the uniform tessellation, $2_{51}$ with Coxeter-Dynkin diagram:
  
  ### Alternate names
  
  * E. L. Elte named it $V_{2160}$ (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
  * It is named $2_{41}$ by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
  * Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)
  
  ### Coordinates
  
  The 2160 vertices can be defined as follows:
  
      16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)
      1120 permutations of (±2,±2,±2,±2,0,0,0,0) of (trirectified 8-orthoplex)
      1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an odd number of minus-signs
  
  ### Construction
  
  It is created by a Wythoff construction upon a set of 8 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 7-simplex: . There are 17280 of these facets
  
  Removing the node on the end of the 4-length branch leaves the $2_{31},$ . There are 240 of these facets. They are centered at the positions of the 240 vertices in the $4_{21}$ polytope.
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, $1_{41},$ .
  
  Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
  
  | | Configuration matrix |
  | - | - |
  | $E_8$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_3$ | $f_4$ | $f_4$ | $f_5$ | $f_5$ | $f_6$ | $f_6$ | $f_7$ | $f_7$ | k-figure | notes |
  | $D_7$ | ( ) | $f_0$ | 2160 | 64 | 672 | 2240 | 560 | 2240 | 280 | 1344 | 84 | 448 | 14 | 64 | h{4,3,3,3,3,3} | $E_8/D_7$ = 192*10!/64/7! = 2160 |
  | $A_6A_1$ | { } | $f_1$ | 2 | 69120 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | r{3,3,3,3,3} | $E_8/A_6A_1$ = 192*10!/7!/2 = 69120 |
  | $A_4A_2A_1$ | {3} | $f_2$ | 3 | 3 | 483840 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 |x{3,3,3} | $E_8/A_4A_2A_1$ = 192*10!/5!/3!/2 = 483840 |
  | $A_3A_3$ | {3,3} | $f_3$ | 4 | 6 | 4 | 1209600 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}V( ) | $E_8/A_3A_3$ = 192*10!/4!/4! = 1209600 |
  | $A_4A_3$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 5 | 241920 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | $E_8/A_4A_3$ = 192*10!/5!/4! = 241920 |
  | $A_4A_2$ | {3,3,3} | $f_4$ | 5 | 10 | 10 | 5 | * | 967680 | 1 | 3 | 3 | 3 | 3 | 1 | {3}V( ) | $E_8/A_4A_2$ = 192*10!/5!/3! = 967680 |
  | $D_5A_2$ | ${3,3,3^{1,1}}$ | $f_5$ | 10 | 40 | 80 | 80 | 16 | 16 | 60480 | * | 3 | 0 | 3 | 0 | {3} | $E_8/D_5A_2$ = 192*10!/16/5!/2 = 40480 |
  | $A_5A_1$ | {3,3,3,3} | $f_5$ | 6 | 15 | 20 | 15 | 0 | 6 | * | 483840 | 1 | 2 | 2 | 1 | { }V( ) | $E_8/A_5A_1$ = 192*10!/6!/2 = 483840 |
  | $E_6A_1$ | ${3,3,3^{2,1}}$ | $f_6$ | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 6720 | * | 2 | 0 | { } | $E_8/E_6A_1$ = 192*10!/72/6! = 6720 |
  | $A_6$ | {3,3,3,3,3} | $f_6$ | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 138240 | 1 | 1 | { } | $E_8/A_6$ = 192*10!/7! = 138240 |
  | $E_7$ | ${3,3,3^{3,1}}$ | $f_7$ | 126 | 2016 | 10080 | 20160 | 4032 | 12096 | 756 | 4032 | 56 | 576 | 240 | * | ( ) | $E_8/E_7$ = 192*10!/72!/8! = 240 |
  | $A_7$ | {3,3,3,3,3,3} | $f_7$ | 8 | 28 | 56 | 70 | 0 | 56 | 0 | 28 | 0 | 8 | * | 17280 | ( ) | $E_8/A_7$ = 192*10!/8! = 17280 |
  
  ### Visualizations
  
  | E8 [30] | [20] | [24] |
  | - | - | - |
  | (1) | | |
  | E7 [18] | E6 [12] | [6] |
  | | (1,8,24,32) | |
  
  Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.
  
  | D3 / B2 / A3 [4] | D4 / B3 / A2 [6] | D5 / B4 [8] |
  | - | - | - |
  | D6 / B5 / A4 [10] | D7 / B6 [12] | D8 / B7 / A6 [14] |
  | | (1,3,9,12,18,21,36) | |
  | B8 [16/2] | A5 [6] | A7 [8] |
  
  ### 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 | $2_{−1,1}$ | $2_{01}$ | $2_{11}$ | $2_{21}$ | $2_{31}$ | $2_{41}$ | $2_{51}$ | $2_{61}$ |
  
  ## Rectified 2_41 polytope
  
  Rectified $2_{41}$ polytope :
  
  | Type | Uniform 8-polytope |
  | - | - |
  | Schläfli symbol | $t_1{3,3,3^{4,1}}$ |
  | Coxeter symbol | $t_1(2_{41})$ |
  | Coxeter diagram | |
  | 7-faces | 19680 total: 240 $t_1(2_{21})$ 17280 $t_1{3^6}$ 2160 $1_{41}$ |
  | 6-faces | 313440 |
  | 5-faces | 1693440 |
  | 4-faces | 4717440 |
  | Cells | 7257600 |
  | Faces | 5322240 |
  | Edges | 19680 |
  | Vertices | 69120 |
  | Vertex figure | rectified 6-simplex prism |
  | Petrie polygon | 30-gon |
  | Coxeter group | $E_8,$ $[3^{4,2,1}]$ |
  | Properties | convex |
  
  The rectified $2_{41}$ is a rectification of the $2_{41}$ polytope, with vertices positioned at the mid-edges of the $2_{41}.$
  
  ### Alternate names
  
  * Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)
  
  ### Construction
  
  It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the $E_8$ Coxeter group.
  
  The facet information can be extracted from its Coxeter-Dynkin diagram: .
  
  Removing the node on the short branch leaves the rectified 7-simplex: .
  
  Removing the node on the end of the 4-length branch leaves the rectified $2_{31},$ .
  
  Removing the node on the end of the 2-length branch leaves the 7-demicube, $1_{41}.$
  
  The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .
  
  ### Visualizations
  
  Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.
  
  | E8 [30] | [20] | [24] |
  | - | - | - |
  | (1) | | |
  | E7 [18] | E6 [12] | [6] |
  | | (1,8,24,32) | |
  | D3 / B2 / A3 [4] | D4 / B3 / A2 [6] | D5 / B4 [8] |
  | D6 / B5 / A4 [10] | D7 / B6 [12] | D8 / B7 / A6 [14] |
  | | (1,3,9,12,18,21,36) | |
  | B8 [16/2] | A5 [6] | A7 [8] |
  