Edited as per Jim Humphreys 9/16/2018 to make it clearer that the 192 are 192 of the 240 total roots of E8, and also to add this link for information on Gosset’s polytope 4_21: https://en.wikipedia.org/wiki/4_21_polytope

Edited 9/18/2018 to add new information from Wendy Krieger (see bottom of this post.)

Edited 9/24/2018 to add new information provided by Dr David Richter (see bottom of this post)

Edited 10/1/2018 to add new information provided by Dr Derek Smith (see bottom of this post.)

Edited 10/21/2018 to provide new information from Roger Bagula re “Krieger-tetrahedra” in relation to 2x2x matrices of SL(2,C) type. (Roger Bagula is the depositor of OEIS A152459.)

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 neighborsâ 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 weâre 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 2âs.

As weâve discussed previously, there are geometric symmetries of the lattice taking any one of the tetrahedra to any other, so thereâs 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 arenât 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 -âs
+ + + + 3 x y z x, y, and z are + and -, with an odd number of -âs
0 0 0 0 2 p q r p, q, and r are -2, 0, and 2, with exactly two 0âs
```

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

There may be other ways to describe ânearest neighborsâ that might yield different answers, e.g. take vectors close to the 4 centers of the tetrahedronâs faces, or close to the 6 centers of the tetrahedronâs 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.
```