Based on the comments from Wendy Krieger below, I accept this observation from მამუკა ჯიბლაძე as an acceptable answer to the question
"product of all coordinates is 1 for E, -1 for O and 0 for A"
Comments from Wendy Krieger 11/06/2018:
There is a geometric description here, which has to deal with
reflections. A reflection in a mirror will produce a reversed image,
that is, convert 1 to -1.
If you place two mirrors at right-angles, the image in the corner is a
double reflection, or -1 * -1 = +1. So if you looked at yourself in
this arrangement, when you lift your right hand, the image lift its
right hand too. In a single reflection, a right hand is reflected to
the image's left hand.
The most likely source for this is the rectangular mirrors (the ones
like x=0, and y=0, and z=0), are flipping the image back and forward.
What it probably means in terms of your experiment is that A-X and A+X
somehow add to the same measure, but the difference from A is being
inverted.
The whole of E8 is a eutactic star. The two 24-cells are eutactic
stars, severally and jointly, and the thing you are looking at is one of
the 16chora inscribed in the 24-cells.
The 24ch is the group of order 192, can be divided into three 4r's
(16ch) or lines at right angles. These three can be labeled say N, E,
O. The vertices of the form (1,1,1,1) etc, can be formed by an 'even'
sum of four N axies, or an odd sum of four O axies. even and odd here
means that the axies are labeled as +1 to -1, and an e/o number of
negatives are used.
Likewise, (2,0,0,0) can be formed by an 'even' number of E or an 'odd'
number of O vectors.
The signs then correspond to reflections in the N star, which inverts
exactly one axis (eg w,x,y,z -> -w,x,y,z) and to get to other points,
you have to reflect in the x, y, and/or z mirror too.
The progression from O to N to E is a linear operation in odd
dimensions, but involves some sort of turn in even ones. This is why
they can be treated symmetrically in 4 and 8 dimensions.
Added 20 Dec 2018:
For those familiar with at least the rudiments of biomolecular translation (the process by which messages transcribed from genes are translated into proteins), the following conceptual cheat-sheet may help explain why an affirmative answer to the "generalized Kronecker delta" question was so important to my research team. In order to be able to deliver on the claims made in this cheat-sheet, we needed to be able to show that given Wendy's construction, we can define a "high" (+1), "low" (-1), and a midpoint between them (0). And the affirmative answer provided to the question by @მამუკაჯიბლაძე tells us that this is possible.
Conceptual Cheat-Sheet
Our analysis deals solely with sets of dicodons and the sets of dipeptides
they encode, NOT with individual codons and encoded amino acids, nor with
individual dicodons and encoded dipeptides.
Our analysis deals solely with the energetic properties of sets of dicodons,
the affinity properties of sets of dipeptides (hydrophobicity), and the
synthetase affiliation properties of sets of dipeptides (Class I or Class
II). No other properties of dicodons or dipeptides are relevant to the
analysis.
Our analysis identifies certain energetic symmetries in the energetic
patterns exhibited by our sets of dicodons.
Our analysis identifies certain affinity symmetries in the affinity patterns
exhibited by our sets of dipeptides.
Our analysis identifies certain affiliation symmetries in the affiliation
patterns exhibited by our sets of dipeptides.
Our analysis identifies certain consistent symmetry relations between
energetic symmetries and affinity symmetries, and also between energetic
symmetries and affiliation symmetries.
These symmetry relations hold for: i) both amino acids of dipeptides (AA1
and AA2 in a dipeptide AA1AA2); or ii) AA1 only; or iii) AA2 only; or iv)
neither. And we interpret (i-iv) as suggesting that sets of dipeptides
assumed functionality in protein structure according to this rough 3-way
chronology:
Early onset of functionality: sets of dipeptides exhibiting symmetry
relations to their dicodon sets for both
AA1And AA2
Late onset of functionality: sets of dipeptides exhibiting symmetry
relations to their dicodon sets for
neither AA1 nor AA2
Onset midway: sets of dipeptides exhibiting symmetry
relations to their dicodon set for AA1
or AA2 but not botj
Our rationale for this chronology is that early onset dipeptide sets assumed
functionality when mRNA energetics were still important in early translation
systems, whereas late onset dipeptides assumed functionality in relatively
mature translation systems in which mRNA energetics were relatively less
important (primarily due to the advent of the water-tight ribosome.)
This hypothetical chronology is fully falsifiable by determining if it makes
correct predictions with respect to “early-late” pairs of SCOP protein
families within SCOP superfamilies (where protein family age is taken as the
ranking assigned by GCA and his team.)
Added 12/30/2018:
Since the 72 vertices of the polytope 1_22 occur as 72 of the 240 vertices of the polytope 4_21 (corresponding to the fact that the roots of E8 contain a copy of the roots of E6), Dr. Richard Klitzing kindly investigated whether there might be an empirically relevant construction inside 1_22 which might be intrinsically related to Wendy Krieger's empirically relevant tetrahedral construction within 4_21. Richard has determined that in 1_22, one can find 40 pairs of opposed pentachora (aka 5-cells, aka hypertetrahedrons in 4-space) and these 40 pairs seem to be empirically relevant to our biomolecular results. However, Richard and Wendy have not yet determined whether Richard's pentachora are related to Wendy's tetrahedra in any meaningful way. Also, note that this question is a new particular version of the more general question asked in this post last year: https://mathoverflow.net/questions/288114/e-6-e-8-and-coxeters-anti-prismatic-projections-of-the-n-dimensional-cr
Here are further details on the pentachora from Richard, in reponse to my question to him as to whether his 40 pairs of pentachora can be divided into three subsets of 10, 20, and 10.
Those 2 x 40 pentachora within that E_6 figure 1_22 (= mo) had been obtained
from its D_5 subsymmetry, when being seen as hin (= 1_21, hemipenteract)
atop rat (rectified 5D crosspolytope) atop alternate hin (i.e. the “other”
hemipenteract) as xoo3ooo3oox *b3oxo3ooo&#xt. In fact, each hemipenteract
layer shows up 40 tetrahedra which are solely connected to hexadecachora
(within that very layer). (In fact, those are the following tetrahedra:
x3o3o *b3.3.) Those tetrahedra furthermore connect to the medial layer
(featuring exactly 40 vertices) via tips, thus becoming pyramids with
tetrahedral bases, aka the mentioned pentachora.
Now you ask whether those 40 pentachora (on either side) would be divisable
into a sum of 10+20+10. Thus your question readily is transfering down one
dimension, yielding the quest to divide those 40 tetrahedra within either hin.
And indeed, when considering hin in turn as an axial stack (aka lace tower)
of a point atop a rectified penteract atop a (relatively inverted) penteract,
i.e. in axial A_4 symmetry, as ooo3oxo3ooo3oox&#xt, then those 40 tetrahedra
show up as
10x oo.3ox.3...3...&#xt
20x ...3.xo3.oo3...&#xt
10x ...3...3.oo3.ox&#xt
which is nothing but a positive answer to your quest.
Thus in total you break the E_6 symmetry down into a lace city (i.e. a 2D
position space) with a 4D perpendicular object space. That perp space then
still features A_4 symmetry. (Btw., the corresponding position space
geometry is readily visible displayed as the 2nd provided lace city of
https://bendwavy.org/klitzing/incmats/mo.htm.)
And here is my response to Richard:
Thank you very much, Richard!
I am very glad that there is an affirmative answer to the question,
because it arose from a consideration of this empirically derived table:
Min
Asym 2 1 0
yr>ry 1 5 3 9
yr=ry 8 9 2 19
yr<ry 1 6 5 12
10 20 10 40
I won't bother at the moment to tell you what the counts in this
matrix are - it's too complicated for a short email.)
But I do want to draw your attention to the fact that the 3x3 matrix here
is our empirical counterpart to Wendy's 3x3 matrix from which her three
sets of 16 tetrahedra can be selected in 6 different ways.
And in my own personal opinion, the fact that we get these 10-20-10
column sums strongly suggests that your pentachora in 1_22 (E6)
can in fact be systematically related to Wendy's tetrahedra in 4_21 (E8),
assuming of couse that we choose to locate the1_22 inside a 4_21
(i.e. to locate a copy of the roots of E6 inside the roots of E8.)
Added 12/31/2018:
Note that in the 3x3 immediately above, these two 2x2's occur
5 3
9 2
9 2
6 5
where:
i) 5*2 = 10, 3*9 = 27, and 10*27 = 270
ii) 9*5 = 45, 2*6 = 12, and 45*12 = 540
iii) 2*270 = 540.
Coincidence? Perhaps. Perhaps not.
Added 1/5/2019 as a more sensible refinement of the above note:
Consider any three 3x3's:
SM:
sm11 sm12 sm13
sm21 sm22 sm23
sm31 sm32 sm33
SD:
sd11 sd12 sd13
sd21 sd22 sd23
sd31 sd32 sd33
MD:
md11 md12 md13
md21 md22 md23
md31 md32 md33
which exhibit the triple cross-ratio:
(sm12*sm13)/(sm32*sm33) = (sd12*sd13)/(sd32*sd33) = (md12*md13)/(md32*md33)
Our empirical data exhibit three such 3x3's:
Min
2 1 0
yr>ry 1 5 3 9
yr=ry 8 9 2 19
yr<ry 1 6 5 12
10 20 10 40
Cross-Diff
1 14 25
yr>ry 3 3 3 9
yr=ry 7 6 6 19
yr<ry 3 6 3 12
13 15 12 40
Cross-Diff
1 14 25
2 5 1 4 10
1 7 6 7 20
0 1 8 1 10
13 15 12 40
and it will therefore be interesting to see if Richard Klitzing can see a geometric realization of this linear-algebraic cross-ratio in his construction involving 40 pairs of pentachora within 1_22 (roots of E6.)
Added 25 January 2019:
I asked Wendy Krieger this question:
> Recall that we have 3 distinct sets with 16 Krieger-tetrahedra each,
> for a total of 48 distinct tetrahedra that consume 192 vertices of the
> 4_21.
>
> How many distinct 16-cells can be found among these 48 tetrahedra (in
> various ways, of course)?
>
> Please clarify when you have a moment - thanks.
Her answer:
The 48 distinct tetrahedra are opposite faces, by pairs, of 24 16-cells:
192 vertices -> 3 * (16ch * 16 ch)-prisms = 3*8*8 = 192.
At each prism, we take each vertex of the first 16ch, * a 16ch, gives 8
16ch per prism.
Each 16ch gives two opposite tetrahedra.
So there are 24 such 16ch.
Added 27 Jan 2018
I asked Wendy Krieger this question:
Wendy has told us that any set of 16 Krieger-tetrahedra is spanned by
eight 16-cells. (Each of these 16-cells pairs-off 2 of the tetrahedra.)
By duality, these eight 16-cells define eight tesseracts with a total
of 128 vertices.
But how many distinct vertices among these 128?
Is it easy for you to answer this question?
If so, then depending on your answer, here is my next question.
How many distinct penteracts can we find in the distinct vertices of
the 128?
Her answer:
The questions are both easy.
If you take your 4_21 as having coordinates at OO, EE, NN, then the dual
tesseracts will form at places like OE+ON, EO+EN, NO+NE. None of these
are in the 4_21, but alternate vertices lie in the other 4_21's that
form in the 3*3 grid.
There are 48 such tesseracts, that form into a chain of six, that is,
you get a skew hexagon formed by a ring of six tesseracts joined at the
opposite cubes, eg OE-ON-EN-EO-NO-NE-OE...
The vertices of these 48*16 = 256, lie in two sets of 192 vertices of
each of the two E8s. Because of this, one can replace the eight cubes
with a chain of eight 16-cells that follow the same pattern.
The alternation of tesseracts can be seen from arranging a hexagon of
matches, and removing every second one. But this still preserves the
384 vertices of two sets of 192, but each of the 24 remaining tesseracts
are un-connected.
There are no distinct tesseracts in this assembly, that i have seen.
Her answer here tells us how any choice of 48 Krieger-tetrahedra in a 4_21 are co-located with various sets of 80 Klitzing-tetrahedra in 1_22.