$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Question

Edited 1/21/2018 to add the following:

Here is a DropBox link

https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0

to a PDF showing how my team used biomolecular first principles to arrive at a set of 240 biomolecular objects (which we believe to be an instantiation of the roots of $E_8$), and more generally, how we arrived at related sets of biomolecular objects with the cardinalities of the Zumkeller numbers (176,240,336) and the correponding edge-magic injection label numbers (11,15,21).

Original question:

In Regular Polytopes, Coxeter shows that the vertices of every n-dimensional cross-polytope (hyperoctahedron) project onto the vertices of an n-gonal (anti-)prism.

Question 1:

Has this projection ever been used to visualize properties of $E_8$ in 3-space via the octagonal prism (i.e. by expressing roots in terms of the basis defined by the vectors from the center of the prism to its vertices)

Question 2:

Has this projection ever been used to visualize properties of $E_6$ in 3-space via:

i) the nonagonal antiprism (when the roots of $E_6$ are coordinatized in 9-space)

ii) the octagonal prism (when the roots of $E_6$ are coordinatized in 8-space as a subset of the roots of $E_8$.)

Question 3:

Have these projections ever been used to visualize relationships between $E_6$ amd $E_8$ in 3-space?

Or are there important properties of $E_6$ and $E_8$ that would not be preserved by such projections?

Please note that this question is related to a comment by Tobias Kildetoft in this question

https://math.stackexchange.com/questions/2556791/when-an-e-6-and-e-8-lattice-are-co-located-by-root-co-location-how-do-mat

regarding limitations on his computer-graphic capabilities.

Thanks as always for any time anyone can afford to spend considering this matter.

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01/04/2018 10pm US EDST: "PS" added at the suggestion of Todd Trimble.

Since $E_6$ is a subgroup of $E_8$ (with roots occurring as a subset of the roots of $E_8$), there will, in general, be patterns of spatial relationships between the points of the $E_6$ lattice and the points of the $E_8$ lattice.

My team is very interested in the nature of these spatial relationships (for reasons which I won't go into here), but it is difficult for us to visualize these relationships as they truly exist in n > 3 -spaces.

So my question was actually posted in order to find out whether the projections mentioned in the above question would faithfully preserve the spatial relationships in question, because if so, then the projected lattices (or portions thereof) would be very helpful to us.

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01/05/2018: 1pm US EDST:

This second "PS" addresses two concerns which had appeared in comments:

1) with respect to points in the $E_6$ lattice and points in the $E_8$ lattice, my first "PS" made reference to "patterns of spatial relationships" between these points, and this reference was too vague by usual and customary MO standards;

2) I didn't explain WHY my team was interested in "patterns of spatial relationships" between these two sets of points.

I. What "patterns of spatial relationships" in particular (between points in the $E_6$ lattice and points in the $E_8$ lattice) ?

My team is interested in whether any points in the $E_8$ lattice tend to "cluster" around any points in the $E_6$ lattice and if so, where, how, and why.

II. Why is my team interested in the question of whether such "clustering" exists?

I think I can best answer this question as follows - hope this answer is satisfactory.

My team is working at two biomolecular levels simultaneously:

1) the level of DNA and RNA polynucleotides and their associated energetics

2) the level of protein polypeptides (amino acid chains) and two of their associated properties (amino acid hydroaffinity and associated tRNA synthetase class)

In addition, because these two levels are interrelated by what is commonly called the "genetic code", my team is working at the junction of these two levels, i.e. the interface at which DNA genes are transcribed into RNA messages which are then translated into the polypeptide chains of protein "primary structures."

At the polynucleotide level, we have empirically determined a set of 240 special nonanucleotides ("tricodons") over the DNA alphabet {t,c,a,g} (or equivalent RNA alphabet {u,c,a,g}, and we have several good reasons to suspect that these 240 special nonanucleotides are an instantiation of the roots of $E_8$.

At the polypeptide level, these 240 special nonanucleotides translate (via the "genetic code" into a set of 72 tripeptides (ordered 3-tuples of amino acids) and again, we have several good reasons to suspect that these 72 tripeptides are an instantiation of the roots of $E_6$.

And what we suspect is that:

1) the "full-precision" genetic code as we know it TODAY (in all its minor variations across the different kingdoms of species or organisms) may have originally arisen as a set of less precise relationships between nonanuclotides and tripeptides;

2) it MAY be possible to characterize this early set of less precise relationships in terms of the way points of the $E_8$ lattice cluster around points of the $E_6$ lattice, IF such clustering does in fact exist.

Notes:

1) the above is somewhat of an over-simplification, but I think it will suffice to convey the general idea;

2) by "full-precision" genetic code, I simply mean that the present-day genetic code is constructed such that every codon encodes exactly one amino acid - though the reverse is of course not true, inasmuch as the "standard" genetic code has 61 "non-STOP" codons encoding only 20 amino acids.

Answer 1

In this more recent thread:

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

I have reported the details of two constructions obtained by Wendy Krieger and Richard Klitzing.

Taken together, these two constructions define various relationships between the roots of one chosen E8 (4_21) and the roots of various E6's (1_22's) defined by two additional copies of E8 determined by our chosen E8.

So, what we are seeing in our biomolecular data is NOT one E6 within one E8, but rather a set of E6's in two E8's defined by an initially selected E8.

Answer 2

I've been waiting for a good image of A(m+n-1) in terms of A(m-1) A(n-1) to appear in my mind, and now i have it.

The 72 in 84,72,84 is A8. The largest A group in E6 is A5, the relevant section is point, 2/2, /4/, 2/2, pt. 2/2 is the birectified simplex, /4/ is the runcinated or expanded simplex.

A8 has a decomposition into A5.A2, where A2 is the hexagon. The decomposition gives a /4/, and /1/ orthogonal, the remaining cross-section is /4 * /1 + 4/ * 1/. Since A5 A2 makes only seven dimensions, the eighth is formed by three layers /4 . /1, then /4/ & /1/, then 4/ . 1/. This is the way that it is with all of the Am.An decompositions of A(m+n-1).

If we return to the A8, this projects as a hexagon, with inscribed hexagram and point. The centre becomes A5, the ring is the hexagon /1/. and the six vertices of the inner hexagon, are alternately /4./1 (the points are /4, the product of this with the triangle is /4 /1), these are 'above' the paper, (which is A5A2 = 7 dimensions), and 4/ * 1/ below the paper.

Since six dimensions is represented by a line in this diagram, the line that runs through A5 and a /4 and 4/ (any of the product bases), represents an embedded A6.

The largest shared symmety between A8 and E6, is the 32-vertex eutactic star formed by the centre A5 and a pair of opposite vertices, make an A5A1. The remaining forty vertices form two 2/2 opposite each other. This 2/2 is 1:9:9:1, in the same lattitudes that a cube forms, and in fact contains six diametric cubes. (actually, 20*6/8 = 15 diametric cubes).

However, because these forty points fall on the same line in the projection in A5A2, you can't include both the polar points and these points.

Answer 3

Wendy Krieger and Richard Klitzing have kindly explained to me how the 72 roots of $E_6$ co-locate with 72 of the 240 roots of $E_8$ in 8-space - it's subtle, but really quite simple, as follows:

1) as is well-known, the 240 roots of $E_8$ are instantiated by the 240 vertices of Coxeter's polytope $4$$_2$$_1$ and the 72 roots of $E_6$ are instantiated by the 72 vertices of Coxeter's polytope $1$$_2$$_2$;

2) as is less well-known, the $4$$_2$$_1$ can be decomposed into three objects derived from 8-simplices, where these objects have 84, 72, and 84 vertices each;

3) similarly, the $1$$_2$$_2$ can be decomposed into three objects derived from 5-simplices, where these objects have 21, 30, and 21 vertices each (and where each of the objects with 21 vertices further decomposes into a point plus a 5-simplex)

4) these 21, 30, and 21 co-locate with 21, 30, and 21 of the 84, 72, and 84 respectively.

BUT . . . although Wendy and Richard have thus kindly provided a simple way to think about the "co-location" of $E_8$ and $E_6$ roots in 8-space, this nice visualization does NOT really help us to understand whether points in the $E_8$ lattice do or do not "cluster" around points of the $E_6$ lattice in any interesting way, and if so, how in particular.

So, my personal opinion is STILL that Coxeter's prismatic projection of the usual orthogonal basis in 8-space onto the vectors from the center of an octagonal prism to its vertices (in familiar 3-space) would in fact provide a useful tool for seeing whether or not any interesting clustering does in fact occur.

Answer 4

In my previous answer and Wendy's previous answer, these two alternative decompositions of $1$$_2$$_2$ were discussed:

{21,30,21}

{20,32,20}

Here is Wendy's explanation of why BOTH of these are valid. (For my team's empirical purposes, {20,32,20} is probably better than {21,30,21}. although I am not absolutely certain at this point.)

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They're both valid. Here is A5-A2 projection of A8

     p      p                 p point
        T                     T 5-simplex, above plane  /4
     t      t                 t 5-simplex, below plane, 4/ (inverted
 p      a      p             a runcinated 6-simplex /4/
     T      T
        t
      p     p

This is the lace-city for A8, the 72 in 84/72/84, as projected against A5.

The decomposition into 20,32,20 is given by one of the lines through p-a-p. This is the largest shared symmetry, and the eutactic star of A5A1.

The line through taT corresponds to a runcinated 7-simplex /6/, the other sections are pTtp, and p,

pT is a 7-simplex, and pTtp, is a 7-simplex antiprism, xo3oo3oo3oo3oo3ox&#q, this means it's a pair of 7-simplexes, in dual position, laced together by edges of sqrt(2). Like a 7-orthotope, stretched by keeping a pair of opposite faces the same size.

Given that these thirteen points of the lace city are the same 13 points of the E8, the two sets of 84 must lie in the same points as T, a, t.