**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

regarding limitations on his computer-graphic capabilities.

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

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.

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.

PLEASE STOP. $\endgroup$ – Andy Putman Jan 21 '18 at 21:233more comments