Is the 9-space coordinatizion of the roots of $E_6$ "nicely" related to the 8-space coordinatization of these roots as 72 roots of $E_8$?

Question

Context for question:

I am asking this question because my team can now:

i) CERTAINLY show that biomolecular codon space (the space of "genes") instantiates two opposed instances of 4$_2$$_1$, while biomolecular amino acid space (the space of proteins encoded by "genes") instantiates two opposed instances of 1$_2$$_2$

ii) PROBABLY show that biomolecular codon space instantiates two opposed copies of $E_8$ (AS WELL AS their associated Coxter groups), while biomolecular amino acid space also instantiates two opposed copies of $E_6$ (AS WELL AS their associated Coxeter groups).

Background for question:

This link explicitly shows the roots of $E_6$ coordinatized in 8-space as 72 of the roots of $E_8$

This link explicitly shows the roots of $E_6$ symmetrically coordinatized in 9-space

Question:

Are these two coordinatizations "nicely" related in any particular way?

Thanks as always for whatever time you can afford to spend considering this matter.

Answer 1

Yes they are the same figure - 1_22. This diagram gives a projection of 4_21, the root of E8, in an A2 'lace city' with E6 orthogonal to the plane. The lines like p<>p or p> or >x< are 2_31 (126 vert).

           p       p
               >                      p = point
           <       <                  < = 2_21 = /4B  
       p       x       p              > = 2_21 = 4/B  (inverted)
           >       >                  x = 1_22 = 4B/
               <
           p       p

In terms of the nine-dimensional coordinate system, the coordinate system for An is n+1 coordinates that add to zero, in effect $\sum x_i=0$. This is the face-plane of an orthotope, which is a simplex.

The complete /6B or 4_21 is comprised of three polytopes having simplex symmetry.

 /6/   3,-3,0,0,0,0,0,0,0,0
 2/4   2,2,2,-1,-1,-1,-1,-1,-1
 4/2   1,1,1,1,1,1,-2,-2,-2.

The projection of the 1_22 and 2_21, in terms of a tri-triangular coordinate, is to break these nine coordinates into three sets of three, with the same rule, that is, each triplet must add to zero.

   /6/   (3,-3,0) (0,0,0) (0,0,0)          three perpendicular hexagons
   2/4   (2,-1,-1) (2,-1,-1), (2,-1,-1)    tri-triangle prism
   4/2   (1,1,-2)  (1,1,-2)  (1,1,-2)      tri-triangle prism 

In the lace-city that is in the first code-box, the reduction of rules amounts to reducing the whole city to just the central coordinate, marked 'x' in that figure.

Held by a pair of opposite vertices, 4_21 would give a middle section of 2_31, the middle column >,x,<. Held by any of the girthing hexagons (here p), the centre of the hexagon is 1_22. It is similar to an octahedron, by its diameter gives a square, and by its girthing square, just the polar axis.

Answer 2

Wendy's diagram is nothing but the first provided lace city of that page: https://bendwavy.org/klitzing/incmats/fy.htm

There too can be seen, what is meant by her early version of Dynkin diagram linearisations:

/4B = x3o3o3o3o *c3o = 2_21
4/B = o3o3o3o3x *c3o = alternate 2_21
4B/ = o3o3o3o3o *c3x = 1_22

---rk