title: E6 polytope
text: # E6 polytope
  
  Orthographic projections in the $E_6$ Coxeter plane :
  
  | $2_{21}$ | $1_{22}$ |
  | - | - |
  
  In 6-dimensional geometry, there are 39 uniform polytopes with $E_6$ symmetry. The two simplest forms are the $2_{21}$ and $1_{22}$ polytopes, composed of 27 and 72 vertices respectively.
  
  They can be visualized as symmetric orthographic projections in Coxeter planes of the $E_6$ Coxeter group, and other subgroups.
  
  ## Graphs
  
  Symmetric orthographic projections of these 39 polytopes can be made in the $E_6,$ $D_5,$ $D_4,$ $D_2,$ $A_5,$ $A_4,$ $A_3$ Coxeter planes. $A_k$ has k+1 symmetry, $D_k$ has 2(k-1) symmetry, and $E_6$ has 12 symmetry.
  
  Six symmetry planes graphs are shown for 9 of the 39 polytopes in the $E_6$ symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.
  
  | # | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter diagram Names |
  | - | - | - | - | - | - | - | - |
  | # | $Aut(E_6)$ [18/2] | $E_6$ [12] | $D_5$ [8] | $D_4$ / $A_2$ [6] | $A_5$ [6] | $D_3$ / $A_3$ [4] | Coxeter diagram Names |
  | 1 | | | | | | | $2_{21}$ Icosihepta-heptacontidipeton (jak) |
  | 2 | | | | | | | Rectified $2_{21}$ Rectified icosihepta-heptacontidipeton (rojak) |
  | 3 | | | | | | | Trirectified $2_{21}$ Trirectified icosihepta-heptacontidipeton (harjak) |
  | 4 | | | | | | | Truncated $2_{21}$ Truncated icosihepta-heptacontidipeton (tojak) |
  | 5 | | | | | | | Cantellated $2_{21}$ Cantellated icosihepta-heptacontidipeton |
  | # | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter diagram Names |
  | # | $Aut(E_6)$ [18] | $E_6$ [12] | $D_5$ [8] | $D_4$ / $A_2$ [6] | $A_5$ [6] | $D_6$ / $A_4$ [10] | $D_3$ / $A_3$ [4] | Coxeter diagram Names |
  | 6 | | | | | | | | $1_{22}$ Pentacontatetrapeton (mo) |
  | 7 | | | | | | | | Rectified $1_{22}$ / Birectified $2_{21}$ Rectified pentacontatetrapeton (ram) |
  | 8 | | | | | | | | Birectified $1_{22}$ Birectified pentacontatetrapeton (barm) |
  | 9 | | | | | | | | Truncated $1_{22}$ Truncated pentacontatetrapeton (tim) |
  