Testing the new site
Category Archives: Physics
Visualizing E6 as a subgroup of E8 for a Leptoquark SUSY GUT?
MyToE was mentioned in a Luboš Motl blog post comment, so I thought I would throw out a few ideas and offer to help Luboš visualize the models/thinking around developing an E6 Leptoquark SUSY GUT.
Here are the E6 Dynkin related constructs, including the detail Hasse visualization.
For this post, I use Split Real Even (SRE) E8 vertices created by dot product of the 120 positive (and 120 negative roots) with the following Simple Roots Matrix (SRM), which generates the Cartan as well:
This is a list of E6 vertices as a subset of SRE E8 by taking off the orthogonal right-side 2 dimensions (or Dynkin nodes in red) vertices with identical entries).
This is an interesting E6 projection with basis vectors of:
X={2, -1, 1, 1/2, -1, -(1/2), 0, 0}
Y= {0, 0, Sqrt[3], Sqrt[3]/2, Sqrt[3], Sqrt[3]/2, 0, 0}
Adding a third Z={0,0,Sqrt[3],Sqrt[3]/2,Sqrt[3],Sqrt[3]/2,0,0}, we get a 3D projection:
Luboš> “the maximum subgroup we may embed to E6 is actually SO(10)×U(1).”
Since D5 (which is the same as E5 in terms of the Dynkin diagram topology) relates to SO(10), it is interesting to look at the complement of E6 vertices and the 40 D5 vertices contained in E6 (leaving 32):
Here is the D5 Dynkin and its Cartan algebra:
Higgs Boson Decay Modes added to VisibLie_E8 demonstration on MTM 10.3
A code snippet…
Updated VisibLie_E8 demonstrations to the latest Mathematica 10.2
I also added some new capability, such as hyperbolic edge projections and 2D Fourier Transforms through E8.
Unfortunately, I had to disable 3D graphic clipping and slicing planes due to a bug introduced in this version (it has been submitted to Wolfram.com).
2D Fourier Transforms on E8 Vertex Projections
Interesting stuff as it relates to QuasiCrystals and non-Crystallographic analysis.
Original source demonstration from John Holland modified and integrated into my VisibLie_E8 viewer…Cool!
QuasiCrystals and Geometry
I found a 1995 book (PDF online) “QuasiCrystals and Geometry” by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below:
12-gon Rectified E8
Diffraction
Overlay
18-gon Rectified E8
Diffraction
Overlay
30-gon Rectified E8
Diffraction
Overlay
and another 12-gon rectified E8 pattern overlay from M. & N. Koca’s in “12-fold Symmetric Quasicrystallography from affine E6, B6, and F4”:
Fun with Tutte-Coxeter, Beordijk-Coxeter, E8 and H4
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.
3D Hyperbolic Projection of the Rhombic Triacontahedron QuasiCrystal
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…
Opinion- Pentaquarks
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:
My E8 F4 projection and so called "Sacred Geometry"
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:
Copyright J Gregory Moxness 2015
Rectified E8 and F4 Triality (Original Work)
WikiMedia Commons source images:
Metatron’s Cube (User:Barfly2001)
Flower of Life (User:Life_of_Riley)
Tree of Life (User:AnonMoos)