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


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

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

  • $\begingroup$ @JyrkiLahtonen - ???? If you have any thoughts on this, I'd of course be grateful if you'd take a moment to share them. $\endgroup$ – David Halitsky Dec 5 '17 at 2:36
  • 1
    $\begingroup$ What does it mean to instantiate something? $\endgroup$ – Tobias Kildetoft Dec 5 '17 at 8:26
  • $\begingroup$ @TobiasKildetoft - thanks for stopping by. When I say that our data instantiate certain groups, I simply mean that: i) there are certain observable patterns in our real-wold data; ii) these patterns are related to one another in ways that can certainly be described by the structure of certain Coxeter groups, and probably be described by the structure of the root-systems of the algebraic-groups associated with these Coxeter groups. The idea is no different from saying that crystals of garnet (the mineral) "instantiate" the structure of the rhombic dodecahedron. $\endgroup$ – David Halitsky Dec 5 '17 at 10:59
  • $\begingroup$ @TobiasKildetoft - I distinguish here between Coxeter groups and correponding root-systems here because in my opinion, it is LOGICALLY possible for a real-world object to have the structure of a Coxeter group, even when there is no reason to invoke the associated algebraic-group in order to describe the structure or behavior of the object. But maybe this isn't true - it may be the case that when you find a Coxeter group in nature, there is ALWAYS an algebraic-group at work "behind the scenes". $\endgroup$ – David Halitsky Dec 5 '17 at 11:06
  • $\begingroup$ Why do you expect the Coxeter group to show up at all? That requires the objects you have (which correspond to the roots) to somehow define reflections in some space, which does not seem to be a feature here. $\endgroup$ – Tobias Kildetoft Dec 5 '17 at 11:23

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


  • $\begingroup$ thank you very much for providing the link and the additional explanatory observation . . . $\endgroup$ – David Halitsky Jan 5 '18 at 0:06
  • $\begingroup$ Could you please provide some explanation of where that link goes and what it says? $\endgroup$ – Xander Henderson Jan 5 '18 at 0:33
  • $\begingroup$ The link points to R Klitzing's site, which is hosted on some other private site. "fy" is the short name allocated by Jonathan Bowers for 4_21. I use a different term: /6B. The site mostly is a listing of surtopes (surface polytopes), and what kinds are incident (parts of or contains) other surtopes. "Lace cities" is an invention of mine, which shows the sections against a 2d axis, which allows a plan view. $\endgroup$ – wendy.krieger Jan 5 '18 at 9:58
  • $\begingroup$ Garret Lisi uses this same diagram as the 'magic star'. He is involved in the physics of E8, and using missing holes in fy to find missing particles. $\endgroup$ – wendy.krieger Jan 5 '18 at 10:17

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.

  • $\begingroup$ @wemdykrieger - thank you - I want to be sure that I understand what you're saying: speaking purely geometrically, there actually IS a 1_22 in a 4_21? Correct? $\endgroup$ – David Halitsky Dec 27 '17 at 13:00
  • $\begingroup$ @WendyKrieger - for the benefit of myself and others, could you give just two or three examples of vertices (roots of $E_6$) with both their 9-space and 8-space coordinates? This would help us (or at least me) further understand how your beautiful diagram plays into the these two sets of coordinates. Also, could you explain the notations /4B, 4/B, and4B/ for 2_21's. I thought I knew 2_21 literature very well, but have never seen those symbols. $\endgroup$ – David Halitsky Dec 27 '17 at 13:11

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