# The (diminished) 120-Cell and it’s relationships to the 5-Cell (A4), the 600-Cell (H4), the two 24-Cells (D4), the Dual Snub 24-Cell, and of course E8!

Updated: 05/03/2023

This post is a Mathematica evaluation of important H4 polytopes involved in the Quaternion construction of nD Weyl Orbits.

Please see the this PDF or this Mathematica Notebook (.nb) for the details.

The paper being referenced in this analysis is here.

I’ve added this PDF or this Mathematica Notebook (.nb) which is having a bit of fun visualizing various (3D) orbits of diminished 120-cell convex hulls. The #5 subset of 408 vertices (diminishing 192) is completely internal to the normal 3D projection of the 120-cell to the chamfered dodecahedron. The #3 has 12 sets of 2 (out of 3) dual snub 24-cell kite cells (that is 24 out of the 96 total cells).

# 3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin ​Geometric Group Theory

It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8->H4 folding matrix) from quaternions given their Weyl Orbits.

If you don’t want to use the interactive Powerpoint, here is the PDF version.

See these Mathematica notebooks here and here for a more detailed look at the analysis of the Koca papers used to produce the results :

Or scroll down to see the .svg images here and here.

Greg Moxness, Tucson AZ

# Visualizing the Quaternion Generated Dual to the Snub 24 Cell

I did a Mathematica (MTM) analysis of several important papers here and here from Mehmet Koca, et. al. The resulting MTM output in PDF format is here and the .NB notebook is here.

What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions with judicious selection of the first triad={123}). This includes both the 600 Cell and the 120 Cell and its group theoretic orbits. The 144 vertex Dual Snub 24 Cell is a combination of those 120 Cell orbits, namely T'(24) & S’ (96), along with the D4 24 Cell T(24).

# Updated Analysis of RCHO Bi-Octonion Standard Model (Cohl Furey Papers)

I’ve updated an analysis I did on the work of Cohl Furey’s papers from several years ago. Since then, she added another paper: https://arxiv.org/abs/1910.08395v1

The short pdf version of my analysis (with some detail cells collapsed) is here (34 pages), and a longer version is here (no collapsed cells and 51 pages). These pdf’s are a direct output from my Mathematica (MTM) Notebook. I will follow up with a LaTex paper on the topic soon.

This notebook has code built in to operate symbolically on native MTM reals, complexes, and quaternionic forms, as well as my custom code to handle the octonions, and now the bi-octonions (which doesn’t assign the octonion e1 to be equivalent to the complex imaginary (I)). That change also applies to the native quaternion assignments where of e1=I, e2=J, and e3=K in order to work with quater-octonions. This was a fairly trivial change to make since it simply involves removing the conversion of complex (and quaternion) operators from being involved in the octonionic multiplication.

Please note that my previous analysis here (from Feb. 2019) made the mistake of not commenting out these operations. As such, it was operating on octonions (not complexified bi-octonions), so some of my concerns were resolved based on correcting that error.

The bottom line is that I did validate much of the work presented in the referenced papers, with the exception of some 3 generation SM charge (Q) assignments in that latest paper (Oct. 2019).

I am very interested here in the suggestion at the very end of that paper [5] in the Addendum Section IX(B/C) on Multi-actions splitting spinor spaces, Lie algebras/groups, and Jordan algebras. I suspect having the ability to create a machine (i.e. a symbolic engine such as MTM) to operate on and visualize these structures as hyper-dimensional physical elements is critical to making progress in understanding our Universe more thoroughly.

While I have had some success in replicating quark color exchange, as well as flavor changes (e.g. green u2 to d3 quark exchange using g13), there doesn’t seem to be a complete description of how to construct each of those color and flavor exchange actions from the examples given. So for reference I present all possible combinations of these actions across the particle/anti-particle definitions (see the image linked in the last paragraph of this post). This comment about limited examples also applies to replicating the 3 generation charge (Q) calculation using the sS constructs mentioned above.

Below is an example image of the 3 generation SM from the 2019 paper built from bi-octonions (with my octonion multiplication table reductions applied. The anti-particles (not shown) are simply the complex-conjugate of these. While I show in string form of Q, I am not showing the commutations based evaluations for them due to the questions / issues I have on how to get it to work.

The image below shows more detail of the 3 generation SM from 2014 with my code implementing the reductions. This leaves off the charge (Q) which was not defined as above in 2014 (AFAIK).

The image below shows a simple construction of the 0-V to 6-V splitting of the Mf Clifford algebraic structures, which I generated using MTM Subsets:

The rather large (long) image here checks all SM particle color and flavor changing actions and includes the anti-particles. The output is extensive and given my open questions on the formalism presented, the accuracy likely deviates from the intent of [5], but it is interesting to show how everything transforms. If no transform is found for a particular action, it outputs an * for that action. If a color or flavor changing transformation action is found, it identifies that action with the list of particles to which the transformation applies. Note: it only identifies a transformed particle if the source particle has a non-zero reduced value and the resulting match is exact (red) or a +/- integer factor of that particle (blue).

# Mathematica Analysis of Cohl Furey’s octonion and Clifford group theoretic ℂ⊗O assignments to standard model particles

I’ve read some recent papers by Cohl Furey and was intrigued by the potential relationship between the octonion and Clifford group theoretic assignments to standard model particles. Since I have developed an extensive Mathematica notebook to perform symbolic analysis using these structures (derived primarily for E8 Lie group work), I decided to follow her suggestion… “The reader is encouraged to check that C⊗O forms the 64-complex-dimensional Clifford algebra Cl(6), generated by the set {i e1,…i e6} acting on f”.

So to that end, I created the this .pdf with some preliminary results of that analysis. I found a few minor issues, but so far the model seems to symbolically compute consistently. Very cool!

# Introducing the Sedenion Fano Tesseract Mnemonic

The Sedenion “Fano Tesseract” Mnemonic is an extension of the “Fano Cube” idea introduced by John Baez in his much cited blog post. The Fano Cube identifies each valid each triad by a hyper-plane which intersects the e_0 node.

Notice in this VisibLie_E8 output for the pane #3 “Fano Visualization Demonstration”, there are 35 sedenion triads, 7 of which are from the octonion used as an upper left quadrant base for a Cayley-Dickson doubling (highlighted in red).

The 16 vertices of the tesseract are sorted by the same “triad flattening” process used to construct a consistent Fano Plane Mnemonic for all 480 unique octonion multiplication tables.

As in the Fano Cube, the edges are highlighted in Cyan if they are selected in the n1-n3 buttons. Unlike the Fano Plane and Cube, the edges represented by the split octonion multiplication table columns/rows are not highlighted in red.

While there are 32 edges in the formal tesseract, each valid sedenion triad is identified by a hyper-plane which intersects the e_0 node, which are not necessarily those of the formal tesseract.

Here is the computation of the same sedenion table given in the Sedenion Wikipedia article as well as from this website: http://www.derivativesinvesting.net/article/307057068/a-few-hypercomplex-numbers/, which uses the harder to read IJKL style notation.

# Incorporating Sedenions into the VisibLie_E8 visualizer

Here is a snapshot of the UI.

It is taken from an example given on this website: http://www.derivativesinvesting.net/article/307057068/a-few-hypercomplex-numbers/, which uses the harder to read IJKL style notation.

Compare the upper left quadrant of the multiplication table. It is the same as the octonion of E8 #164 (non-Flipped) used to generate the sedenion using the Cayley-Dickson doubling. It can be read off using the Fano Plane (or Fano Cube) mnemonic. My next post will show the Sedenion forms of these, namely the Fano Tetrahedron and Tesseract!

Notice the highlighting of the interesting aspect of sedenions, where some non-zero p=a+b and q=c+d give a product pq=0. These are given in a list of the {a,b,c,d} where p={a+b} and q={c+d} and pq=0 (along with the deltas between the indices), always 42 excluding trivial juxtapositions. In this case, it is the row entry with {3,12,5,10}.

Here is the computation of the same table in the format given in the Sedenion Wikipedia article.

# G2 as the automorphism group of the (split) octonion algebra

This post describes the derivation of G2 automorphisms for each of the 480 unique octonion multiplication tables, as well as each of the 7 split octonions (created from negating the 4 row/column entries which are not members of the split #, which is an index to one of the 7 triads that make up the octonion).

The Exceptional Lie Algebra/Group G2 is identified by its Dynkin diagram and/or associated Cartan Matrix (also shown here with its Hasse diagram):

The particular octonion multiplication table selected in the example referenced above is associated in my model as a “Non Flipped E8 #164”. The multiplication table in various formats (IJKL, e_n, and Numeric) is:

The selected multiplication table (fm) is the basis for the following octonion symbolic math:
[wlcode](* Octonion math from the matrix using 8D vector representations for x and y, this has the Subscript[\[ScriptE], 0]=1 first *)
octProduct[a_, b_] := Block[{c3 = Array[0 &, 8]},
Do[c3[[If[i != j, Abs@fm[[i, j]] + 1, 1]]]+=Sign@fm[[i, j]] a[[i]] b[[j]],
{i, 8}, {j, 8}];
Chop@c3];

(* Complex, quaternion and octonian multiplication Subscript[\[ScriptE], a]\[SmallCircle]Subscript[\[ScriptE], b] *)
quat2oct@in_ := Block[{a3},
If[QuaternionQ@in,
a3 = FromQuaternion@in;
Switch[Length@a3,
0, Re@a3 + Im@a3 Subscript[\[ScriptE], 1],
2, If[Length@a[[1]] > 0, a3,
1\[SmallCircle](Re@a3[[1]] +
Im@a3[[1]] Subscript[\[ScriptE], 1]) + a3[[2]]]],
3, Re@a3[[1]] + Im@a3[[1]] Subscript[\[ScriptE], 1] +
a3[[2 ;;]]],
“not Quaternion”] /. Most@Thread[octIJKL -> oct]];

(* Octonion math with conversions, leave final as octonion *)
SmallCircle[a_List, b_List] := octonion@octProduct[a, b];
SmallCircle[a_, b_Quaternion] := a\[SmallCircle]quat2oct@b;
SmallCircle[a_Quaternion, b_] := quat2oct@a\[SmallCircle]b;
SmallCircle[a_, b_Complex] := a\[SmallCircle]ToQuaternion@b;
SmallCircle[a_Complex, b_] := ToQuaternion@a\[SmallCircle]b;
SmallCircle[a_, b_] := oct2List@a\[SmallCircle]oct2List@b;

associator[a_List, b_List,c_List] := (a\[SmallCircle]b)\[SmallCircle]c-a\[SmallCircle](b\[SmallCircle]c);
associator[a_, b_, c_] :=associator[oct2List@a, oct2List@b, oct2List@c];[/wlcode]

Each set of 21 upper triangle pairs of octonion elements (utOct) has 14 “derivations” which are null
(alternatively, the NullSpace of the derivation vectors of utOct has 7 which are NOT null…)

A derivation pair D_{x, y} is defined by:
[wlcode](* Commutator *)
commutator[x_,y_]:=x\[SmallCircle]y-y\[SmallCircle]x;
CircleDot[x_,y_]:=commutator[x, y];

(* Commutator matrix *)
comMat[x_, y_] := x.y – y.x;
comMat[{x_, y_}] := comMat[x, y];

(* Derivation matrix and vector *)
derivation[x_,y_][a_]:=(x\[CircleDot]y)\[CircleDot]a-3 associator[x,y,a];
Square[{x_, y_, a_}] := derivation[x, y][a];
derMatrix[x_,y_]:=Table[Coefficient[\[Square]{x,y,oct[[i]]},oct[[j]]], {j, 7}, {i,7}];
derVector[x_,y_]:=derMatrix[x, y]//Flatten;[/wlcode]
The 14 null pairs define G2 for a SPECIFIC octonion multiplication matrix.

For each of 7 non-null derivations there is a triple of D_{x1, y1}=D_{x2, y2}+D_{x3, y3} related derivations.

There is a 7×21 matrix describing the non-NullSpace for the octonion given in the Wolfram MathSource reference above along with the 7 non-null triples. This is generated by g2Null:
[wlcode]g2Null:=NullSpace[derVector[Subscript[\[ScriptE], #1], Subscript[\[ScriptE],#2]]&@@@utOct//Transpose];

(* Formatting all non-Null upper triangle pairs derivation entries *)
outG2Null := MatrixForm[parallelMap[Grid[{
{Row[
Subscript[D, Row@Flatten@##] & /@
Partition[Flatten@{utOct[[Flatten@Position[#, -1]]]}, 2]]},{“==”},
{Row[
Subscript[D, Row@Flatten@##] & /@
Partition[Flatten@{utOct[[Flatten@Position[#, 1]]]}, 2]]}(*),
{“\[LongDash]\[LongDash]\[LongDash]\[LongDash]”}**)}] &,
g2Null]];[/wlcode]

getG2Null (below) retrieves two octonion multiplication matrices for the flipped and non-flipped index on one of 240 E8 particles.
[wlcode]formG2Null := MatrixForm@{
Row@{Style[Row@{“flip=”, flip}, Blue],
Style[” split # \[DownArrow],”, Red]},
g2NullTbl = Table[{
split,
setFM[#, flip, split];
outG2Null},
{split, 0, 7}];
MatrixForm[Column[#, Center] & /@ noNull@# & /@ Table[
(* Color the common Der in each column *)
cmnD =first@Select[g2NullTbl[[All,2,1,i,1,3,1,1,All]],
Length@# == 1 &];
cmnDclr = Style[cmnD, Magenta];
If[j == 1, {“”, Row@{
(* Highlight the missing index in the triple D_{x,y} for each row *)
Style[Complement[Range@7,Union@Flatten@g2NullTbl[[All,2,1,i,1,3,1,1,All,2,1]]][[1]],Darker@Green],
g2NullTbl[[1, 2, 1, i]] /. cmnD -> cmnDclr}},
(* Display only non-null derivations from the splits if not the same as the base octonion *)
If[g2NullTbl[[1,2,1,i]] =!= g2NullTbl[[j,2,1,i]],
{Style[g2NullTbl[[j, 1]], Red],
g2NullTbl[[j,2,1,i]] /. cmnD -> cmnDclr}]],
{i,7},{j,8}]]} &;

getG2Null := ColumnForm[{
Style[Row@{“E8#=”, #}, Darker@Green],
Row@{flip = False; formG2Null@#, flip = True; formG2Null@#}},
Center] &;[/wlcode]

Here we show the output of getG2Null for the example non-Flipped E8 #164):

Emergent patterns:

1. For each of the 7 non-null derivation triples (the first column of the 7 rows), there are precisely 3 (of the 7) split octonions that don’t share the exact same non-null triple derivation pair as the parent (non-split) octonion (these deviations are shown in column 2 thru 4, along with the associated Fano plane diagrams for the parent and 7 split octonions of E8 #164).
2. Each non-null derivation triple contains 6 of 7 indices, and each row is missing a different index (highlighted in green before the column 1 triple). The sequence of missing element numbers in each row of the given example follows the row number.
3. It is only the equality relationship (signs) of the non-Null triple that change in 3 of the 7 splits, not the D_{x,y} itself.
4. There are always 2 pairs of 2 D_{x,y} which are negative (i.e. with a -1 entry in the NullSpace matrix located above the “==”) that occur across the row.
5. There is always a common positive D_{x,y} (i.e. with a +1 entry in the NullSpace matrix located below the “==”) in each entry of the row (colored magenta). The common positive entries in the 7 rows suggest a “distinguished” non-null indicator for the 14=21-7 G2 automorphism.
6. All octonion multiplication matrices have the first 3 rows of distinguished entries of {6,7},{5,7} and {5,6} in that order (i.e. the last 2 rows of utOct).
7. There are several possible choices for G2 automorphism sets of 14 elements within each of 480 octonions based on the 7 non-Null entries. Interestingly, there are the 4 sets of rows in utOct which sum to 14 elements, specifically rows {{1, 2, 4}, {1, 2, 5, 6}, {1, 2, 4, 6}, {2, 3, 4, 5}}). The distinguished entries of the flipped E8 #164 example below (as in the MathSource post referenced above) suggests a G2 created by rows {1,2,4} of utOct. Although, not all G2 sets must use complete rows as in this example.

This is octonionG2-nonNull, a 480 page (15 MB) which lists each of the 7 non-Null parent octonion derivations (and for each of those, the 3 other split octonion non-null derivations) which are NOT a member of the 14 null derivations for each of 480 octonion multiplication matrices and 7 splits for each (480*8=3840). It includes the Fano plane mnemonics for the parent and 7 split octionions. This is a smaller 240 page (1MB) version of octonionG2-nonNull.pdf without the Fano plane mnemonics.

# Octonion triality testing

I am working on validating some theoretical work on the triality automorphisms of the split octonions. This is a post with preliminary work on that… for those who are interested 🙂

BTW – It requires the here.

An addendum to the original post (below):

[WolframCDF source=”http://theoryofeverything.org/TOE/JGM/octonion triality checks.cdf” width=”900″ height=”8000″ altimage=”http://theoryofeverything.org/TOE/JGM/octonion triality checks.png” altimagewidth=”900″ altimageheight=”8000″]

# My latest paper on E8, the H4 folding matrix, and integration with octonions and GraviGUT physics models.

The full paper with appendix can be found on this site, or w/o appendix on viXra. (13 pages, 14 figures, 20Mb). The 22 page appendix contains the E8 algebra roots, Hasse diagram, and a complete integrated E8-particle-octonion list.

Abstract:
This paper will present various techniques for visualizing a split real even E8 representation in 2 and 3 dimensions using an E8 to H4 folding matrix. This matrix is shown to be useful in providing direct relationships between E8 and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects. A direct linkage between E8, the folding matrix, fundamental physics particles in an extended Standard Model GraviGUT, quaternions, and octonions is introduced, and its importance is investigated and described.

If you would like to cite this, you can use this BibTex format if you like (remove the LaTex href tag structure if you don’t use the hyperref package):

@article{Moxness2014:006,
title={{The 3D Visualization of E8 using an H4 Folding Matrix}},
author={Moxness, J. G.},
journal={href{http://vixra.org/abs/1411.0130}{www.vixra.org/abs/1411.0130}},
year={2014},
month = nov,