Category Archives: Physics

Fun with Tutte-Coxeter, Beordijk-Coxeter, E8 and H4

August 17, 2015 jgmoxness Leave a comment

In reference to a G+ post by Baez (w/Greg Egan), it’s interesting to note the link to E8’s outer ring of the Petrie projection of a split real even E8, which creates a Beordijk-Coxeter helix.

Beordijk-Coxeter helix in 2D

Beordijk-Coxeter helix in 3D

The Beordijk-Coxeter helix connects the nearest 6 vertices on the outer ring. The Tutte-Coxeter graph is created in 3 (blk,grn,red) sets of edges by taking the (outer) ring and skipping (6,8,12) or counting (7,9,13) vertices. It shows there are 2 perfect pentagons and 1 pentagram (with different radii due to the difference in distance between the sets of vertices used).

Of course, the crystallographic E8 is manifestly related to the 5 fold symmetry of the pentagon, with its integral relationship to the non-crystallographic H4 group (and its Coxeter-Dynkin diagram) through E8 to H4 folding using the Golden ratio Phi.

It is interesting to note that the skipping of 5+(1,3,7) vertices is similar to the creation of the 120 (240) vertex positions of H4 (E8) Petrie projection by adding to the 24 vertices of the 8-cell and 16-cell (which make up the self-dual 24-cell) the 96 vertices of the Snub 24-cell. This is done through 4 rotations skipping 5 vertices.

Also notice the (1,3,7) are the number of the imaginary parts of Complex, Quaternion, and Octonion numbers, also integrally related to E8.

Physics

3D Hyperbolic Projection of the Rhombic Triacontahedron QuasiCrystal

August 4, 2015 jgmoxness Leave a comment

Projected from E8 to 6-Cube and then to Rhombic Triacontahedron using 3 of 4 rows of the E8-to-H4 folding matrix.

and a Tesseract for good measure…

Personal, Physics

Opinion- Pentaquarks

July 20, 2015 jgmoxness Leave a comment

The recent announcement for the potential LHC realization of composite particle resonance of the pentaquark (or meson-baryon molecule) is interesting. This, along with a prior confirmation of the dimeson tetraquark Z(4430), and prognostications for hexaquarks (or dibaryons), expand the “zoo” of composite particles now numbering in the thousands.

If by chance we can use these new discoveries to discern the mathematical pattern (symmetry) well enough to predict the particle masses, lifetimes, branching ratios and CKM (quark) PMNS (neutrino) mixing matrix ratios – that would be COOL!

Yet, given the current size of this zoo of particles, it seems that adding a few more bags of quarks to the mix may not do the trick, but I am always hopeful.

IMNSHO (In My Not So Humble Opinion ;-)… it is more a task of “naming the animals”.

or in Ernest Rutherford’s words:
“All science is either physics or stamp collecting.”

“The pentaquark search may be physics’ version of ‘stamp collecting'”. JGM

Hexaqark with “Physics” shapes:

Decays:

My attempt to organize the stamp collection of chemistry into a 4D periodic table:

Personal, Philosophy, Physics

My E8 F4 projection and so called "Sacred Geometry"

July 7, 2015 jgmoxness Leave a comment

Please note, this is about the GEOMETRY used in what has come to be called “Sacred Geometry”. Upon further investigation, along with the Kabbalah’s “Tree of Life”, it seems even the Jewish “Star of David” (Solomon) is medieval and may not be sourced in authentic ancient Abrahamic theology.

As it stands, the so-called “sacred” origin of these, including the pentagon that shows up in the icosahedral/dodecahedral Platonic solids due to the Sqrt(5) Golden Ratio (Phi) relationship within my E8 to H4 folding, is likely medieval and mostly pagan. I am merely pointing out that E8 contains the geometry that many find “interesting” for whatever reason, and am not promoting the underlying cults that use them.

Video (Best in HD):

My E8 F4 projection w/blue triality edges and physics particle assignments:

My E8 F4 projection with 6720 edge lines:

My E8 F4 projection rectified:

Star Of David Overlay:

Metatron’s Cube Overlay:

Flower of Life Overlay:

Kabbalah Tree of Life Overlay:

WikiMedia Commons source images:
Metatron’s Cube (User:Barfly2001)
Flower of Life (User:Life_of_Riley)
Tree of Life (User:AnonMoos)

Physics

A new Lisi Arxiv paper along with a new E8 projection

June 29, 2015 jgmoxness Leave a comment

For Lisi’s latest ideas on a “Lie Group Cosmology (LGC)” ToE, see http://arxiv.org/abs/1506.08073.
Be warned, it is over 40 heavy pages.

He also posted a new projection on FB.

He uses the (H)orizontal and (V)ertical projection vectors of:
H = {.12, -.02, .02, .08, .33, -.49, -.49, .63};
V = {-.04, .03, .13, .05, .20, .83, -.45, .19};

Here it is rendered with H and V reordered to match in VisibLie_E8 tool:
H/V order is {2, 3, 4, 7, 1, 6, 5, 8};
Note: the particle assignments in my VisibLie_E8 tool (as in the Interactive visualizations on this website) are not the same and have different triality rotation matrices.

The pic below has a slightly altered projection that creates a bit more hexagonal symmetry and integer projections (without deviating more than .05 off his coordinates):
Note: vertex colors indicate the number of overlapping vertices.
This uses H/V projection vectors (reverse ordered back to his coordinate system).
Symbolic:
H={3 Sqrt[3]/40, 0, 0, 3 Sqrt[3]/40, 1/3, 1/2, 1/2, -2/3};
V={-3/40, 0, 3/20, 3/40, 3/20, -4/5, 1/2, -3/20}};
Numeric:
H={.13, 0., 0.,.13,.33,.5, .5,-.66`};
V={-.075, 0., .15, .075, .15,-.8, .5,-.15};

Here is Lisi’s version with my coordinates:

Here is a version of the full hexagonal (Star of David) triality representation using coordinates I found with the same process used above. Particle color, shape and size are based on the assigned particle quantum numbers (spin, color, generation). There are 86 trialities indicated by blue triangles.
Projection vectors (in my “physics” coordinate system) are:
H={2-4/Sqrt[3],0,0,-Sqrt[(2/3)]+Sqrt[2],0,0,Sqrt[2],0};
V={0,-2+4/Sqrt[3],Sqrt[2/3]-Sqrt[2],0,0,0,0,-Sqrt[2]};

See also a rectified version of this, where the particle overlaps show the inner beauty of E8!!

The underlying quantum symmetry patterns within this coordinate system are:

Physics

Hyperbolic E8 Visualizations

June 24, 2015 jgmoxness Leave a comment

Beginning to play with visualizing E8 and subgroups using hyperbolic Poincare projections and tiling – in 2D/3D/nD

Philosophy, Physics

The latest VisibLie_E8

May 17, 2015 jgmoxness Leave a comment

It now has 3D clipping (and slicing) plane control sliders on the the left side controls within a new multiplexed slider control. I also added the 2D I-Ching and 2D/3D Sanskrit Sri Yantra visualizations.

Physics

Pascal Triangle and Mod 2-9 Sierpinski Maps

April 22, 2015 jgmoxness Leave a comment

I decided to take a quick dive into an interesting idea from Peter Williams related to how the modulus of binomial numbers are related to the Pascal Triangle. I’ve added this to the VisibLie_E8 viewer as well.

Here is a screenshot:

Here is the code I wrote to produce the output below:

Mod 9:

Mod 8:

Mod 7:

Mod 6:

Mod 5:

Mod 4:

Mod 3:

Mod 2:

Philosophy, Physics

New features and options – Lite version, MIDI Music, Hebrew/English Nearest Word Gematria

April 15, 2015 jgmoxness Leave a comment

I created a “Lite” version (4-10 Mb) of the VisibLie_E8 CDF/NB/Intereactive application, so now you can interact with the main E8 and particle physics panes w/o loading the full 100 Mb version (which contains multiple versions of parametric spherical harmonics for atoms, Molecule/DNA protein data, 4 gematria word value indexed Hebrew/Greek Bibles, solar system and Universe simulation dynamics, etc.)

I added some demonstration capability to the Chords (Beauty of Math/Music) pane. It now has full instrumentation for creating MIDI tones along with the various tones in available chords.

I added a cool Nearest Word Graph to the Gematria pane. This uses the Wolfram curated LanguageData / DictionaryLookup to find Hebrew and English words that are close in spelling (no Greek available yet)-: