Tag Archives: 2D

Personal, Sculpture and Art

Info about my creative work: The Universe, Math, Physics, & Art

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I think the most well-known (2D) image I’ve created so far (above) is found on Wikipedia’s (WP) math/geometry pages. It is what has been called the Public Relations (PR) or publicity photo for E8, which is an 8D (or 248 dimensional as the math guys count it) geometric object. It has been used in books, papers, math/science conference promotional materials, etc. That WP page also has what many used to consider “uninteresting” – my 3D versions of the same object. E8 is used in various theories to understand how the Universe operates way below the atomic level (i.e. the quantum stuff). I think it is responsible for what I call the shape of the Universe. As others have said, “who knew the Universe had a shape?”

In (very) layman’s terms, that 2D image of E8 looks sort-of-like how the lowly 4D Tesseract cube would look if it grew up into an 8D object, but a bit more interesting. The Tesseract was made (more) popular in the modern Avengers movies. In 2015 before I had ever seen any of those movies, I created a laser etched 3D projection of E8 within a jewel cut optical crystal cube and put a photo of that object lit from below by a blue LED on my website’s main homepage. Someone mentioned the resemblance and I was surprised at the likeness, motivating me to go see the movies.

Another of my more notable math/physics images has to do with the discovery of QuasiCrystals made by a guy, Dan Shechtman, who was ostracized from academia for the “impossible” idea of crystals having rotational symmetries beyond 2,3,4 and 6. He is now a Nobel Laureate. The WP image above is on that QuasiCrystal page as an overlay of part of E8 projected over a picture made from shooting a beam of x-rays at an icosahedral Ho-Mg-Zn quasicrystal.  I didn’t actually do that experiment with the x-ray beams, but I did similar ones in my college days on a particle accelerator called the Cockroft-Walton Kevatron – see below for a picture of me (the one with the beard ;–)  in the physics lab assembling a moon dust experiment. One of the professors had gotten the accelerator out of Germany after having worked on the Manhattan project.

E8 and its 4D children, the 600-cell and 120-cell (pages on which I have some work, amongst others) and its grandkids (2 of the 3D 5 Platonic Solids, one of which is the 3D version of the 2D Pentagon) are all related to the Fibonacci numbers and the Golden Ratio. So that kind of explains why most of my 2D art, 3D objects and sculptures (e.g. furniture like the dodecahedron table below), and 4D youtube animations all use the Golden Ratio theme.

Greg

Physics

Cloud Based VisibLie_E8 Demonstration

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The cloud deployments don’t have all the needed features as the fully licensed Mathematica notebooks, so I included a few of the panes that seem to work for the most part. Some 3D and animation features won’t work, but it is a start. Bear in mind that the response time is slow.

Link to the demonstration.

A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:

1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein

2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle

3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize

4) Math: Group Theory: Dynkin Diagram Algebra Create

5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize

6) Physics: Quantum Elements: Fundamental Quantum Element Select

7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity

8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays

9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation

10) Chemistry: Atomic Elements: 4D Periodic Table Element Select

11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select

12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding

13) Biology: Human Neurology: OrchOR Quantum Consciousness

14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori

15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria

16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier

17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life

18) CompSci: Human/Machine Interfaces: nD Human Machine Interface

Physics

The free Wolfram CDF Player v. 13 works with my VisibLie E8 ToE demonstration on Win10

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In case you’re interested, I just verified the demo works on the free Mathematica CDF player v.13 for Win10.

Just go to https://www.wolfram.com/player/ install, download and open the app:

https://theoryofeverything.org/TOE/JGM/ToE_Demonstration.nb

There is a ton of other cool interactive stuff in there. FYI – Some features don’t work without a full Mathematica license.

Enjoy.

Physics

Working with E8+++

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I wanted to confirm some work being done with E11 (or more specifically the Extended E8+++), so I used my “VisibLie” notebook (which includes the “SuperLie” package for analyzing Lie Algebras) to get the following information:

I first created the Dynkin diagram which produces the Cartan Matrix:

This allows the evaluation of the EigenSystem of the Cartan matrix as follows:

As well as using the SuperLie package to evaluate the Positive Roots, Weights and Heights of the E8+++ (The first 80 are shown. For the full list in .pdf click here or click on the image below):

The 4870 positive roots up to height 47 generate a Hasse diagram as follows (with 3 sections zoomed in):

I am working on replication of the Petrie projection by A.G. Lisi based on his prescription using his basis vectors for 2D projection and a Simple Roots Matrix:

Adjusting this E11 Simple Roots Matrix (srmE11) to match the Dynkin diagram above and then verifying it creates the same Cartan matrix, we have:

To get the 9740 E8+++ vertices, we simply use the dot product of the transpose of srmE11 against the positive and negative root vectors (first 50 shown):

A first cut at plotting these involves projecting them to a 2D using the dot product of the 2 basis vectors provided, which gives the following ListPlot:

The limited results from SuperLie are consistent with the image given by Lisi on the Bee’s BackReaction blog, after eliminating the vertical “levels” in my raw plot.

Email me if you want more detail, see errors, or would like to help.

Greg
JGMoxness@theoryofeverything.org

Physics

Star of David Projection Basis for E8, E7 and E6

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For a full discussion of the particle assignment symmetries involved, see http://vixra.org/pdf/1503.0190v1.pdf

E8 as 240 split real even vertices with  vertex color, size, shape, and labels from a modified A.G. Lisi particle assignment algorithm. The vertex number is the position in the Pascal Triangle (Clifford Algebra) representation of E8. The blue lines forming equilateral triangles represent the  10 Bosonic (or Color) Triality (T2) and 76 Fermionic (T4) rotations (using the 8×8 matrices shown below) that transform each member of the triangle into the next member.

E7 as derived from E8 above by taking the vertices of E8 with the last two columns being equal.

E6 as derived from E8 above by taking the vertices of E8 with the last three columns being equal.

The 48 vertices of F4 are made up of the 112 (D8) +/-1 integer pair vertices that are assigned to bosonic particles, which includes 22 trialities (10 from T2 and 12 from T4). The other 64 of 112  D8  vertices are assigned to the 2nd generation of Fermions. F4 is shown below with its sub-group G2 assigned to the gluons in the larger triality triangles. Projection Basis for X and Y are H (first row) and V (second row) in the matrix below respectively
The Triality Matrices are:

The 10 Bosonic (Color) Triality Rotation Matrix (T2) equilateral triangles exclusively affect the particles that are derived from F4.

T2 rotates through the bosonic colors indicated by the last 3 columns of the E8 vertices appropriately labeled (r,g,b).

The 76 Fermionic (Generation) Triality Rotation Matrix (T4) equilateral triangles. There are 64 fermion particle trialities that are derived from the 128 (BC8) half integer vertex assignments plus the 64 fermions assigned from the 112 (D8) +/-1 integer pair vertices. There is always one integer (2nd Generation) and two 1/2 integer vertices (1st and 3rd Generation) in every fermionic triality.

That is, T4 rotates through the fermionic generations.

There are 12 trialities from these 76 that are associated with the inner E6 & F4 Lie Group vertices assigned to bosons, shown below with only the 112 (D8) +/-1 integer pair unlabeled vertices:

Of course, there are a number of overlapping vertices in this projection stemming from the 112 (D8) +/-1 Bosonic sector. The following image colors the vertices by overlap count, with 120 Yellow with no overlaps, 24 Cyan with 2 overlaps (48), and 24 Cyan with 3 overlaps (72).

Notice the 8 projection basis vectors with dark gray circles. Of course, these projection basis vectors are the 8 “generator vertices” of E8 vertices with permutations of {+1,0,0,0,0,0,0,0} giving it the full 240+8=248 dimension count. Add to that the -1 “anti-generators” being the opposite end of the projection basis vectors and you get the full 256 vertices representing the full Clifford Algebra on the 9th row of the Pascal Triangle.

There are 6720 edges in E8 with an 8D length Sqrt[2].

Just for fun, I introduce to you the 6720 rectified E8 vertices taken from the midpoint of each edge and using the same projection with coloring of the spheres based on overlap count. Enjoy!

Here are the G2 gluons connected by their trialities shown in a 3D concentric hull projection of E8 using the E8 to H4 folding matrix basis vectors. This is the 4th hull, which is the outer hull of the inner H4 600 cell (an icosadodecahedron). For more on E8 hulls, see this post.

The full F4 group with 10 T2 and 12 T4 trialities affecting the bosons is contained in the outer icosadodecahedron (1st hull) combined with the 3rd (quad icosahedral) and 4th icosadodecahedron hulls.

Below is the full E8 with all trialities shown in 3D concentric hull projection.

Wow – that is nice! Remember, you heard it here first!

BTW – if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at:  jgmoxness@theoryofeverthing.org.