All posts by jgmoxness

Physics

12-fold Symmetric Quasicrystallography from affine E6, B6, and F4

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This post is an analysis of a June 2013 paper by Mehmet Koca, Nazife Koca, and Ramazan Koc. That paper contains various well-known Coxeter plane projections of hyper-dimensional polytopes as well as a new direct point distribution of the quasicrystallographic weight lattice for E6 (their Figure 3), as well as the quasicrystal lattices of B6 and F4.

Koca / Koc Figure 3 E6 Quasicrystallographic Weight Lattice

Wha‍t is interesting about this projection is that it precisely matches the point distribution (to within a small number of vertices) from a rectified E8 projection using a set of basis vectors I discovered in December of 2009, published in Wikipedia (WP) in February of 2010 here.

Rectified E8 in my “Triality” projection basis:
x=(2-4/√3   ,   0         ,   1-1/√3   ,      1-1/√3   ,   0   ,   -1   ,   1   ,   0   )
y=(   0   ,   -2+4/√3   ,   -1+1/√3   ,   1-1/√3   ,   0   ,   1/√3   ,   1/√3   ,   -2/√3   )

Rectification of E8 is a process of replacing the 240 vertices of E8 with points that represent the midpoint of each of the 6720 edges. In this projection, there are overlaps which are indicated by different colors in the color-coded WP image linked above.

The image below is an overlay of the above images highlighting the 12*(9+3+26+7)=540 points that are not overlapping:

Annotated overlap comparison showing missing 432 overlaps.

It is interesting to note that with a 30° rotation of my projection, the missing overlaps are reduced to 12*(15+2)=204.

Annotated overlap comparison showing missing 204 overlaps.

Given the paper’s explanation for the methods using E6 (720) with 6480 edges as a projection through a 4D 3-sphere window defined by q1 and q6, it may be insightful to study my projection basis for E8’s triality relationships with the Koca/Koc paper’s defined 4D 3-sphere.

For more information on why my projection basis is called the E8 Triality projection, see this post.

Physics

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

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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.

I welcome any help or advice or additional examples.

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).

Physics

Messier Marathon 2021

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The idea is to observe the 110 astronomical Messier objects which are visible to some of the Northern Hemisphere in March/April. The best result is to do it all in one night, which is difficult unless everything works out seamlessly (110 over 11 dark hours leaves only 6 minutes per image, so it isn’t so much about quality but speed).

So I decided to try and set up my astro gear to automate the process of taking 110 Messier objects in one night. 

In testing it the evening of March 2, it was going really well so I decided to pull an unplanned all-night’er and got 100 of the 110 (with the missing 10 not being visible to my scope before sunrise due to the obstruction of my house, as well as it being a bit early in the Messier season). 

Click here for a video compilation of the shots I took (sorted by their Messier number each using a 2 minute monochrome exposure):

Physics

5-color graph solutions to the Hadwiger–Nelson problem

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These are a simple animated visualizations I created while studying the problem. They contain a sequence of 2s frames starting at 130 vertices and incrementing 62. The red numbered text are the first few starting vertex numbers showing the hexagonal structure of the construction. On the left is a list of sorted and color coded vertices by the number of edges on a vertex for that frame’s shown vertices (it also lists in the right column the number of vertices with that number of edges & color). The solution data was obtained from: https://github.com/marijnheule/CNP-SAT

The 510 vertex / 2504 edge 5-color graph solution to the Hadwiger–Nelson problem
The 874 vertex / 4461 edge 5-color graph solution to the Hadwiger–Nelson problem

Here is a link to a Mathematica Notebook (50Mb) that has interactive ListAnimated graphs with vertex data (vertex number, symbolic and numeric coordinates, number of edges on the vertex in the frame) in Tooltip on mouseover.

Physics

3D Visualization of the rays of E6 & E7 in Kochen-specker theory by Ruuge & Waegell/Aravind

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In several papers on BKS proofs, Arthur Ruuge’s “Exceptional and Non-Crystallograpic Root Systems and the Kochen-Specker Theorem” https://arxiv.org/abs/0906.2696v1 and Mordecai Waegell & P.K. Aravind’s “Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8” https://arxiv.org/abs/1502.04350v2, in addition to E8, E6 and E7 is studied. Using the visualization developed for my recent paper and prior papers, I present here the related visualizations for E6 and E7 as discussed in those papers.

The 72 E6 vertices derived from E8 and projected to 3D using the “E8->H4” basis vectors. The 15*72=1080 edges are shown in the upper left, the 36 anti-podal rays are shown in the upper right along with the hull group vertex counts and norm distance. The bottom image shows the 4 hulls – yellow Icosahedron (12), cyan dodecahedron (20) and the orange/pink pentagonal prisms (40)).
The 126 E7 vertices derived from E8 and projected to 3D using the “E8->H4” basis vectors. The 16*126=2016 edges are shown in the upper left, the 63 anti-podal rays are shown in the upper right along with the hull group vertex counts and norm distance. The bottom image shows the 4 hulls – orange 2 overlapping Icosahedrons (24), pink2 overlapping dodecahedron (40), and cyan & gray icosidodecahedrons(60) with 2 vertices at the origin.

Physics

My Latest paper published on Vixra – 3D Polytope Hulls of E8 4_21, 2_41, and 1_42

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https://vixra.org/pdf/2005.0200v1.pdf

or also available directly from this website:

https://theoryofeverything.org/TOE/JGM/3D_Polytope_Hulls_of_E8-421-241-142.pdf

Using rows 2 through 4 of a unimodular 8x8 rotation matrix, the vertices of E8 421, 241, and 142 are projected to 3D and then gathered & tallied into groups by the norm of their projected locations. The resulting Platonic and Archimedean solid 3D structures are then used to study E8’s relationship to other research areas, such as sphere packings in Grassmannian spaces, using E8 Eisenstein Theta Series in recent proofs for optimal 8D and 24D sphere packings, nested lattices, and quantum basis critical parity proofs of the Bell-Kochen-Specker (BKS) theorem.

A few new Figures from the paper.

FIG. 6: Pair of overlapping rhombicosidodecahedrons from
3rd largest hull of the 74 hulls in 142
FIG. 13: 421 & Polytope projected to various 3D spaces
Each 3D projection shown lists the projection name, the nu-
meric basis vectors used, and the 421 & 142 overlap color coded
vertex groups, and the projection with vertices (larger) &
6720 edges and the 142 vertices (smaller)
FIG. 14: Concentric hulls of 241 in Platonic 3D projection
with vertex count in each hull and increasing opacity and
varied surface colors.
a) 24 individual concentric hulls
b) In groups of 8 hulls
FIG. 15: Concentric hulls of 142 in Platonic 3D projection
with vertex count in each hull and increasing opacity and
varied surface colors.
a) 74 individual concentric hulls
b) In groups of 8 hulls
FIG. 18: E8’s outer two hulls scaled to unit norms in Platonic
3D projection with vertex counts color coded by overlaps
a) 54 vertex (42 unique) 421=241 icosidodecahedron (30 yel-
low) & two overlapping icosahedrons (12 red) scaled 1.051
b) 100 vertex (80 unique) 142 non-uniform rhombicosidodeca-
hedron (60 yellow) & two overlapping dodecahedrons (20 red)
scaled 1.0092
c) 154 vertex (122 unique) combination of a & b
d) 208 vertex (122 unique) combination same as c with color
coded vertex counts for both 421 & 241
Note: The internal numbers of the image are the 8 axis (pro-
jection basis vectors).

Now for a few new visualizations that are not in the paper…

Various 3D projections of 2_41
Various 3D projections of 4_21

Physics

3D visualization of E8 1_42 polytope

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This is what I expect to be the first ever 3D visualization of the E8 1_42 polytope with 17280 vertices showing concentric hulls of Platonic solid related structures!

E8 1_42 polytope with 17280 vertices showing concentric hulls of Platonic solid related structures! Vertex colors represent the overlap counts.
Each of 74 concentric hulls based on 3D Norm’d vertex positions (with varying opacity in sets of 8). Vertex counts in each hull listed above.
9 sets of 8 concentric hulls plus the last 2 outer hulls, Vertex counts in each
of 8 hull listed above. Notice this is a combination of two overlapped dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).

For the sake of completeness in visualization, see below for various projections to 2D. Click these links for a higher resolution PNG or the SVG version.

Physics

Nested Lattices of E8 in Complex Projective 4-Space

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I read an interesting article about a pattern discovered by Warren D. Smith (discussed at length here):

“The sum of the first three terms in the Eisenstein E_4(q)  Series Integers of the Theta series of the E8 lattice is a perfect fourth power: 1 + 240 + 2160 = 2401 = 7^4”

So I decided to visualize the 2401=1+240+2160 vertex patterns of E8 using my Mathematica codebased toolset based on some previous work I put on my Wikipedia talk page.

The image below represents various projections showing 6720 edges of the 240 E8 vertices, plus a black vertex at the origin, and the 2160 Witting Polytope E8 2 _ 41 vertices using the same projection basis (listed at the top of each image along with the color coded vertex overlaps). Click these links for a higher resolution PNG or the SVG version.

Some of the particular projections of the Witting Polytope may need 8D rotations applied to the basis vectors to find better symmetries with the Gosset, but this is a start using my standard set of projections.

The 240 vertices of the Gosset Polytope are generated using various permutations:

(* E8 4_21 vertices *)
e8421 = Union@Join[
Eperms8@{1, 1, 1, 1, 1, 1, 1, 1}/2,
perms8@{1, 1, 0, 0, 0, 0, 0, 0}];

The 2160 vertices of the Witting Polytope are generated using various permutations:

(* E8 2_41 vertices *)
e8241=Union@Join[
perms8[{1,0,0,0,0,0,0,0}4],
perms8[{1,1,1,1,0,0,0,0}2],
Eperms8[({2,0,0,0,0,0,0,0}+1)]]/4;

Another view shows just the 2160 Witting Polytope vertices. Click these links for a higher resolution PNG or the SVG version.

2160 Witting Polytope (vertices only)

Another great source of visualizations on E8 and this Witting Polytope is here.

Now visualizing in 3D the structure in 3D using rows 2-4 of the E8->H4 folding matrix, we get:

Witting Polytope with 2160 E8 2 _ 41 vertices sorted by Norm distance into 3D Platonic solid related concentric hull structures using rows 2-4 of the E8->H4 folding matrix. The last image shows overlap counts using colored vertices.
Sets of 8 concentric hulls with hull vertex counts above each.
Full set of concentric hulls with color coded vertices showing overlap counts. Norm’d vertex groups in red.