Original question (without additional information from Wendy):

Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:

Taking the E8 as {128,112}, of radius 2, we get

16 tetrahedra at (1,1,1,1)E, (1,1,1,1)E

16 tetrahedra at (1,1,1,1)O (1,1,1,1)O

16 tetrahedra in (2,0,0,0)A (2,0,0,0)A

In the first two,

E means take an even number of sign-changes in the bracket.

O means take an odd number of sign-changes in each bracket.

A means all permutations, all change of sign in the brackets.

The vertices of the tetrahedron then comes from three coordinates in a given set, so these are the coordinates of a tetrahedron in the first set, using the first three coordinates.

1,1,1,1 1,1,1,1

1,-1,-1,1 1,1,1,1

-1,1,-1,1 1,1,1,1

-1,-1,-1,1 1,1,1,1

Question:

Is it possible to define a generalized Kronecker delta function which takes Wendy's E, O, and A sets to 1, -1, and 0 respectively (or -1, 1, 0)?

See this link for definition of the GENERALIZED Kronecker delta:

https://en.wikipedia.org/wiki/Kronecker_delta

New information from Wendy Krieger (added 9/18):

Of the 240 total roots of E8, 192 are consumed by Wendy's 48 tetrahedra, leaving 48. And she has discovered that these 48 define two 24-cells. So in addition to the generalized Kronecker delta question, there is an additional question as to whether the 48 vertices of the two 24-cells "organize" the 48 tetrahedra in any interesting way.

Wendy has provided an affirmative and interesting answer to this question:

The centres of the 48 octahedral faces of each of the two 24-cells, in rectangular product, produce a 3*3 array of the 48 tetrahedra, of which there are six distinct sets of 24.

See this link for definition of the famous and unique 24-cell, which has no analogs in spaces lower or higher than 4 dimensions:

https://en.wikipedia.org/wiki/24-cell

New information from Dr. David Richter (9/24/2018)

Dr. Richter has kindly suggested that Wendy's construction is not new within the literature of Lie-algebras. He writes:

"It does not seem new to me. Eugene Dynkin studied these things in depth in the 1940's, for example. (In Russia during World War II. He avoided the draft due to poor eyesight.) You should look up his article "Semisimple subalgebras of semisimple Lie algebras". This was published in the Translations of the American Mathematical Society in 1952. Although he does not use the language of regular polytopes, I think you will find the structures that you describe, encoded as rank -8 Lie subalgebras of the E(8) Lie algebra."

New information from Dr. Derek Smith provided 10/1/2018)

There are 16 "nearest neighbors” to any one of Wendy's tetrahedra, in the following sense.

Consider the tetrahedron T that has its four vertices in R^8 given by the following rows (here + is 1, - is -1, and we’re using the even coordinate system, scaled up by a factor of 2 to avoid some fractions, so minimal non-zero vectors have norm 8):

```
+ + + + + + + +
+ - - + + + + +
- + - + + + + +
- - + + + + + +
```

The center c of T is the average of those vectors, namely c = 0 0 0 + + + + +.

The vectors in E8 closest to c are the four vertices of T, each of whose (squared) distance from c is 3. The next possible distance is 5, and the 16 vectors in E8 that achieve this are of the form 0 0 0 a b c d e, where each of the five letters is either 0 or 2, and there are an even number of 2’s.

As we’ve discussed previously, there are geometric symmetries of the lattice taking any one of the tetrahedra to any other, so there’s nothing special about this one. For instance, if we had taken

```
2 0 0 0 2 0 0 0
0 2 0 0 2 0 0 0
0 0 2 0 2 0 0 0
0 0 0 2 2 0 0 0
```

as our tetrahedron instead, with center c = 1/2 1/2 1/2 1/2 2 0 0 0, then the 16 vectors at distance 5 from c aren’t as easily described in one sentence; but you can check that they are

```
0 0 0 0 0 0 0 0
0 0 0 0 4 0 0 0
+ + + + + a b c a, b, and c are + and -, with an even number of -‘s
+ + + + 3 x y z x, y, and z are + and -, with an odd number of -‘s
0 0 0 0 2 p q r p, q, and r are -2, 0, and 2, with exactly two 0’s
```

for a total of 1+1+4+4+6=16 vectors.

There may be other ways to describe “nearest neighbors” that might yield different answers, e.g. take vectors close to the 4 centers of the tetrahedron’s faces, or close to the 6 centers of the tetrahedron’s edges.

Email from Roger Bagula 10/21/2018 (and mine to him):

My email to Roger Bagula (10/20/2018):

```
My research team seems to have uncovered a pretty clear biomolecular
instantiation of OEIS A152459.
Furthermore, this instantiation appears to be connected with a particular
decomposition of the root-system of E8 (via the geometry of 4_21 or
a Dynkin sub-algebra).
```

His email to me (10/21/2018):

```
Thank you for letting me know about your discovery.
That idea / result appears to be an amazing connection of SL(2,C) type
of 2x2 matrices to the higher geometry.
I wish you luck as E_8 symmetry breaking is near my heart, LOL.
```

Email from Jim Humphreys (10/21/2018) re "SL(2,C)"

```
Probably the quoted phrase just refers to the group of complex 2x2
matrices of determinant 1, usually referred to as the (Lie or linear
algebraic) group SL(2.C).
Jim Humphreys
```

Email from Roger Bagula 10/22/2018 with some further thoughts

```
This morning I was thinking about E_8 from the Thurston-Weeks minimal
hyperbolic 3 manifold approach.
The original Poincare dodecahedral universe model is elliptical,
but the Siefert-Weber dodecahedral space is hyperbolic.
Weeks contends that the Cosmic microwave background radiation should be
invertible as an X-ray crystal is, so the the spectrum of anomalous lines
would reflect the geometry of the universe that contains them. (That is
looking at the bare wiggles on the blackbody peak /anisotropy.) The Fourier
transform inversion approach suggests itself.
https://www.physicsmyths.org.uk/imgs/wmap_spec.gif
A_5 as a 3d point group is Ih (Icosahedral group) as 60 elements, so the 240
of your 8 dimensional model (4_21) is 4 orthogonal Icosahedral groups as a
matrix group:
{{A_5,A_5},
{A_5,-A_5}}
In trace-Character terms that looks very much like an Hadamard matrix self-
similar construction, as the Ih :A_5 character table is already an
{{C,C},
{C,-C}}
symmetry.
In spectral terms this situation is a good thing as only a limited number of
vibrations are spectrally active in the Infrared . Weeks in one of his
papers points out the observed spectra is more limited than expected from
the hyperbolic 3 manifold symmetry models available.
The C60 buckyball is an elliptical molecular analog to this sort of Ih
symmetry: (with a spectral list of frequencies)
http://www.public.asu.edu/~cosmen/C60_vibrations/newc60revcorr.pdf
The Raman shift in the 1466/cm to 1474/cm region looks like a backward
cosmic microwave background anisotropy.
There are actually very few symmetry models that would be able to have such
a limited spectral set in the microwave region. E_8 as an Ih ( A_5)
hyperbolic 3 manifold is one of them.
```

Added 10/28/2018 - a possible different way to derive Wendy's sets of 16 tetrahedra within the 4_21 polytope:

```
Recall that inside the 4_21 polytope which instantiates the roots of E8,
Wendy's 3x3 matrix (derived from two 24-cells in the 4_21) generates
48 tetrahedra inside the E8 (in six different ways), and that
each such set of 48 divides naturally into three sets of 16 tetrahedra.
Nature is telling us very loudly and clearly that it must be possible
to take the sets of 16 tetrahedra in sets of five in some reasonable way,
and further, that it must be possible to take four such sets
of eighty tetrahedra in some reasonable way.
So, can anyone think of any reasonable geometric, lattice-theoretic,
or group-theoretic way to get 5*16 = 80 and 4*80 = 320,
using the various sets of 16 tetrahedra generated by Wendy's construction?
```

Added 11/1/2018:

Roger Bagula has just reported that the group SO(27) appears to be occurring within our biomolecular instantiation of the "Krieger-tetrahedra" in 4_21. This may be of possible relevance since 27*26 = 702, where 702 is the number of 4-faces of 1_22 (which realizes the 72 roots of E6 within the 240 roots of E8 realized by 4_21.)

Added 11/2/2018:

An important piece of Mathematica code from Roger Bagula (important because the value 351 and the triple of absolute values |1,14,25| occur in an obviously related way within the energetics our biomolecular instantiation of Wendy's "Krieger-tetrahedra".)

```
A linear relationship between {1,14,25} and {-351,351}:
(* Mathematica*)
(*diff vector*)
v = {1, 14, 25}
{1, 14, 25}
(* plane vectors*)
w1 = {-1, x, y}
{-1, x, y}
w2 = {1, y, x}
{1, y, x}
(*linear solutions*)
NSolve[{v.w1 - 351 == 0, v.w2 + 351 == 0}, {x, y}]
{{x -> -32., y -> 32.}}
(*Check*)
-1*1 - 32*14 + 32*25
351
1 + 32*14 - 32*25
-351
(*end*)
32=16*2
The SO(27) group:
number of elements; n(n-1)/2=27*26/2=351
```

Added 11/4/2018: i) an email from Dr. Richard Klitzing describing 40 pairs of pentachora within 1_22 (the polytope which realizes the roots of E6); and ii) an email from me to Richard and Wendy Krieger conveying a possible reason why our biomolecular energetic data might be showing not only patterns related to Wendy's tetrahedra within 4_21, but also Richard's pentachora within 1_22.

Email 11/4/2018 from Dr. Richard Klitzing:

```
In my prior email, I described a setup of 40 pairs of pentachora
(i.e. tetrahedral pyramids), being pairwise connected at their tips,
which are to be found within 1_22. And that this number of 40 furthermore
is being divisable as 40=1+1+2+12+12+12.
Btw. this can be depicted in the following lace city display
(aka projection of vertex set onto 2D) of mo = 1_22, which projects along o
rhtogonal A_3 x A_1 symmetry:
a B a -- x3o3o *b3o3o (hin)
o O c O o -- o3o3o *b3x3o (rat)
b A b -- o3o3x *b3o3o (alt. hin)
| | | | +-- o3o3o3o3o (point)
| | | +------ o3o3x3o3o (dot)
| | +---------- x3o3o3o3x (scad)
| +-------------- o3o3x3o3o (dot)
+------------------ o3o3o3o3o (point)
where:
o = o3o3o o (point)
a = x3o3o o (tet)
b = o3o3x o (dual tet)
A = x3o3o x (tepe)
B = o3o3x x (inv. tepe)
O = o3x3o x (ope)
c = compound of
x3o3x o (co)
o3o3o u (ortho u-line)
```

Email 11/4/2018 from me to Dr. Richard Klitzing and Wendy Krieger:

```
This is just a note to convey my own personal suspicion as to why
our biomolecular energetic data seem to be showing internal structure
related to pentachora within 1_22 as well as tetrahedra within 4_21
(where 1_22 and 4_21 respectively realize the roots of E6 and E8.)
On the empirical biomolecular energetic side, we calculate energetic
values for certain linear ordered 6-tuples, where these 6-tuples
come in natural sets of 64 that can be placed into correspondence
with the 64 vertices of one of Wendy's sets of 16 tetrahedra within 4_21.
But to calculate an energetic value for any ordered 6-tuple (1,2,3,4,5,6)
in our various sets of 64 6-tuples, our protocol demands that we decompose
the 6-tuple into five overlapping 2-tuples (12,23,34,45,56), in accordance
with the analysis in the seminal paper by Jacques Fresco
(the senior scientific member of our team, now Emeritus Princeton):
https://www.ncbi.nlm.nih.gov/pubmed/6193821
So, if we take eight of our biomolecular 6-tuples that correspond to two
of Wendy's tetrahedra in 4_21, these eight will contain 40 of our
(overlapping) 2-tuples, and this is why I think that our data are
ALSO showing patterns pointing to the pentachora within 1_22, i.e.
patterns involving the triples of integers (1,14,25) in relation to
the integer 351.
For this reason, it would be very nice if some regular relationship
could be shown between the tetrahedra in 4_21 and the pentchora within 1_22.
But as Wendy has pointed out, it is not a trivial task to determine
whether any such relationship exists, and if so, what this relationship is.
In this regard, Roger Bagula and our team's statistical guru Ray Koopman
are now doing further numerical analyses of our (1,14,25) patterns in
relation to our (351, 702, 1404) patterns, and perhaps these analyses
will provide some clues as to how the pentachora in 1_22 might lie
"nicely" with respect to the tetrahedra in 4_21.
But again, all of the above is nothing more than a suspicion, at least
at this point.
```

Added 25 November 2018

For those who find this thread of interest, here is something important to remember. Wendy defines a 3x3 matrix of 16 cells, each of which is the dual of a 4-cube (tesseract). So when we select any set of three 16-cells from this matrix choosing one cell from each row and one cell from each column of the 3x3 (e.g. cells 11, 22, 33), we are implicitly choosing three 4-cubes - namely the duals of the three 16-cells which we have selected. And if we then use Coxeter's standard orthogonal projection of 4-cube to rhombic dodecahedron (in which two vertices of the 4-cube go to the center of the rhombic dodecahedron), some very nice things happen on the scientific side of the house, particularly if we make the slightly unusual choice to use Coxeter's projection so that the vertices 1010 and 0101 becomes the poles of the dodecahedron (instead of 1111 and 0000, which are the usual choices.)

Added 11/29/2018

Here is a correction by Wendy Krieger to my comment just above.

```
From two given 24-cells which are orthogonal, you can make nine bi-16ch
prisms. Any three of these that are not in the same row or column make
a 4_21.
Each of the bi-16chora, make 16 tetrahedra, which (supposing some
four-axis * four-axis), lie by pairs at the vertices of a perpendicular
16-choron. What this means, is for example, an octahedron can be
regarded as a pair of triangles opposite each other. If this octahedron
lies in a prism-product with something else, then each vertex of the
'something else' gives an octahedron, and thus a pair of opposite but
parallel triangles."
```

So, each set of "Krieger-tetrahedra" actually defines eight 16-cells, each of which defines eight dual 4-cubes which can each be projected into a rhombic dodecahedron in three of its four dimensions.

Added 12/9/2018:

See the post by Dr. Richard Klitzing with time stamp "Sun Dec 09, 2018 2:58 pm" in this thread at the "hi.gher.Space" bulletin board:

http://hi.gher.space/forum/viewtopic.php?f=25&t=1881

Richard has succeeded in understanding and documenting the full structure of the scaliform polytope "codify", which is his name for the polytope with 192 vertices derived from 4_21 by Wendy's construction (in which she removes the 48 vertices of two 24-cells from 4_21.)