Note #1: title of this post was modified as per a great suggestion from Todd Trimble

Note #2 ( to site moderators): I truly consider this question to be a legit "meta" question about the MathOverflow community - in particular, the general attitude of the MathOverflow community toward a particular tool which is available to them courtesy of HSM Coxeter. But Todd Trimble thinks not, so in deference to the opinion of an experienced MO member, I've reworded the title of the question as he suggested - see discussion below in comments.)

Here is the question I'd like to post to MO, if appropriate.

Background:

Please see my recent question at MathOverflow:

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

Question:

Assume for the sake of discussion that in the examples in the question above:

(a) projection of the vertices of the 8-space cross-polytope onto the vertices of the 3-space octagonal prism DOES preserve all important properties of the roots of $E_8$ (as expressed in their usual {128,112} coodinatization in 8-space)

(b) projection of the vertices of the 9-space cross-polytope onto the vertices of the 3-space nonagonal antiprism DOES preserve all important properties of the roots of $E_6$ (as expressed in 9-space in their usual {18,27,27} coordinatization)

Then under these assumptions, is there still some reason why you personally would NOT think it worthwhile to investigate the roots of $E_6$ and $E_8$ and their relations to one another using the visualizations of these roots in 3-space that can be obtained via Coxeter's projections?

If so, what would your reason be?

For example, would your reason simply arise from your own "personal mathematical aesthetic", e.g a distaste for working from "diagrams" rather than linear symbolic expressions? (I mention this possibility because Dr. (Sam) Eilenberg once chewed me out for doing a 140-page proof about certain properties of the "derivation trees" (ordered finite rooted directed trees with labelled nodes) of derivations in context-free grammars - he told me that in his opinion as an automata-theorist, nothing worthwhile could ever be learned from the properties of "diagrams".)

Or is there some other reason you personally wouldn't use such projections?

fullyappropriate for MO meta, and you might get some good ideas on how to turn your poll into a good MO question. $\endgroup$ – Todd Trimble♦ Dec 10 '17 at 14:33