Testing the new site
All posts by jgmoxness
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:
Fun with combining virtual and real visualizations (photography)
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”:
Nested polytopes with non-crystallographic symmetry
I’ve been following an interesting paper titled “Nested polytopes with non-crystallographic symmetry as projected orbits of extended Coxeter groups” which has used my E8 to H4 folding matrix as a basis for not only understanding Lie Algebras/Groups and hyper-dimensional geometry, but also the genetic protein / viral structures of life (very cool)!
The Nested Polytopes paper was first put into Arxiv in Nov of 2014 at about the same time I submitted my related paper on E8 to H4 folding to Vixra. I created this paper in response to discovering that my Rhombic Triacontahedron / QuasiCrystal (D6 projected to 3D using the E8 to H4 Folding Matrix to E8 to H4 folding) work on Wikipedia and this website was being used by Pierre-Philippe Dechant, John Baez and Greg Egan.
The Nested Polytopes paper has since undergone 4 revisions. The first three seemed to be typical (even minor) tweaks, but with a different author list/order in each. Yet, the latest (V4) seems to have a massive change, different author list and a completely different title “Orbits of crystallographic embedding of non-crystallographic groups and applications to virology”.
While I KNOW they were aware of my work, I wasn’t really surprised they never referenced it – as it isn’t published in academic press. I AM a bit surprised by the extensive changes to a single paper on arxiv. They have removed some of the E8 /H4 references (ref: Koca) and added completely different sources. Makes you wonder what’s up with that?
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…