Tag Archives: H4

Comparing nearest edges between Split Real Even (SRE) E8

I have been intrigued with Richter’s arXiv 0704.3091 “Triacontagonal Coordinates for the E8 root system” paper.

Comparing nearest edges between Split Real Even (SRE) E8 and Richter’s Complex 4D Golden Ratio Petrie Projection Model:

Please note, the vertex numbers are reordered from Richter’s paper in order to be consistent with ArXiv quant-ph 1502.04350 “Parity Proofs of Kochen-Specker theorem based on the Lie Algebra E8” by Aravind and Waegell. This paper uses the beautiful symmetries of E8 as a basis for proof sets related to the Bell inequalities of quantum mechanics. Below is a graphic showing that vertex number ordering:

Richter’s model with 74 nearest edges per vertex (Complex 4D Norm’d length Sqrt[2], with 8880 total):

Richter’s model with only 26 nearest edges per vertex (Complex 4D Norm’d length 1, 3120 total):

The above unity length edge pattern has is more consistent with the SRE E8 below.

The SRE E8 Petrie Projection (with 56 Norm’d edge length Sqrt[2] for each vertex, 6720 total). This is equivalent to folding E8 to H4 with an 8×8 rotation matrix which creates a 4D-left H4+H4*Phi and a 4D-right H4+H4*Phi.

Please note, the vertex number is that of the E8 vertices rotated (or “folded”) to H4 in 2D Petrie projection. The vertex numbers are in canonical binary order from E8’s 1:1 correspondence with the 9th row of the Pascal Triangle (eliminating the 16 generator/anti-generator vertices of E8 found in the 2nd and 8th column of the Pascal Triangle). The graphic below shows the vertex number ordering:

Here we take the 4D-left half and project to two concentric rings of H4 and H4 Phi (Golden Ratio) with 56 nearest edges per vertex (Norm’d length of unity or Phi for each vertex):

The animation frames are sorted by the ArcTan[y/x] of the vertex position in each ring (sorted H4*Phi outer to inner, then H4 outer to inner).

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. 

The work of Frans G. Marcelis

 

https://fgmarcelis.wordpress.com

http://members.home.nl/fg.marcelis/

Really cool analysis of E8 geometry. I would love to collaborate, but can’t seem to find any contact info. If anyone knows him or can contact him, let me know.

I found his ’79-’81 Masters Thesis “Collectieve beweging in atoomkernen” from tue.nl

He co-authored a paper B. J. Verhaar, J. de Kam, A. M. Schulte, F. Marcelis: Simplified description of y vibrations in permanently deformed nuclei. Phys. Rev. C18 (1978) 523.

and see he participated in the HISPARC collaboration.

 

Some of my work that may support his is shown in this post, which shows how there are 10 self dual 24 cells in 2 sets of 5 (one for each H4 600 cell within E8). These are broken down by dual 16 vertex 8-cells and 8 vertex 16 cells. Marcelis creates a set of 14 8-cells (224 vertices) plus a basis of 16 vertices.

snub-pics

I replicate some of his work below using my Mathematica demonstration code as a base. When clicking a given vertex in my tool, it generates the edges to the 56 nearest vertices to that clicked vertex (as measured in all 8 dimensions). These are indeed the same vertices as those that intersect the rings of E8 Petrie Projection after translation to the clicked point.

Ring intersections that complement the 240 E8 vertices from 3{3}3{4}2. This is like clicking every 5th vertex (6) on the inner ring of E8 Petrie Projection.

Vertices from 3{3}3{4}2

Inner 3{4}3

6 Outer 3{3}3

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
quasi-Rect421-12gon0b
Diffraction
quasi-Rect421-12gon1b
Overlay
quasi-Rect421-12gon2b

18-gon Rectified E8
quasi-Rect421-18gon0b
Diffraction
quasi-Rect421-18gon1b
Overlay
quasi-Rect421-18gon2b

30-gon Rectified E8
quasi-Rect421-30gon0b
Diffraction
quasi-Rect421-30gon1b
Overlay
quasi-Rect421-30gon2b

and another 12-gon rectified E8 pattern overlay from M. & N. Koca’s in “12-fold Symmetric Quasicrystallography from affine E6, B6, and F4”:
koca0b

koca1b

koca2b

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?

E8-8D-Polytopes-2

E8-6D-StarPolytopes-2

E8-4D-Polychora-2

E8-3D-Platonic-2

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
helix2Db
Beordijk-Coxeter helix in 3D
cells6004b

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).

e8ring8-1b

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.

snub-pics

Universal Time’s Arrow as a Broken Symmetry of an 8D Crystallographic E8 folding to self dual H4+H4φ 4D QuasiCrystal spacetime

Animations of Wolfram’s Cellular Automaton rule 224 in a 3D version of Conway’s Game of Life.

While this does not yet incorporate the E8 folding to 4D H4+H4φ construct, the notion of applying it, as the title suggests, to a fundamental particle physics simulation of a “Quantum Computational Universe” or an emergent (non)crystallographic genetic DNA (aka. Life) is an interesting thought…

Watch a few notional movies, some in left-right stereo 3D. Best viewed in HD mode.

More to come, soon!
outAI2

These are snapshots of all the different initial conditions on the rule 224. Each set of 25 has a different object style and/or color gradient selection.
AI 1-25

AI 26-50

AI 51-75

AI 76-100

AI 101-125

AI 126-150

AI 151-175

AI 176-200

AI 201-225

AI 226-250

AI 251-253

Playing around visualizing non-crystallographic DNA/RNA

I’ve got interaction between the DNA/RNA protein visualizations (e.g. “1F8V- PARIACOTO VIRUS REVEALS A DODECAHEDRAL CAGE OF DUPLEX RNA) with the VisibLie_E8 projections of crystallographic E8 to non-crystallographic H4 (and to dodecahedral H3 in 3 dimensions, of course).

These pics are a simple (naive) merge of the D6 projected using the E8 to H4 folding matrix and the Protein DB at http://www.rcsb.org/ for 1F8V).

outE80prot

DNAscreen

outDNA

outDNAprot

E8, H4, Quasi-Crystals, Penrose Tiling, Boerdijk-Coxeter Helices and an AMS blog post on the topic

This pic is an overlay of an image from Greg Egan on the AMS blog VisualInsight on top of one I created several years ago for the quasicrystal wikipedia page.

It uses my E8-H4 folding matrix to project E8 vertices to several interesting objects. The 5 dimensional 5-cube (Penteract) and the related 3D the Rhombic-Triacontahedron, as well as this 2D overlay on the Ho-Mg-Zn electron diffraction pattern.

Ho-Mg-Zn_E8-5Cube-baez-egan-overlay

E8 vertices projected to 2D pentagonal projection
E8-5Cube

5-cube in 3D
5-cube-2

6-cube edges projected to the Rhombic-Triacontahedron using 3 of 4 rows of my E8-H4 folding matrix.
6Cube-QuasiCrystal-low

Rhombic-Triacontahedron with inner edges removed
RhombicTricontahedron

The Boerdijk-Coxeter helix is also related to these structures through the Golden Ratio.

Edges on the outer ring of the E8 Petrie projection related to Boerdijk-Coxeter helix.
helix2Db

Same as above in 3D with the tetrahedral cell faces and 3D vertex shape-color-size based on quantum particle parameters from a theoretical physics model.
cells6004b

Same as above also showing the inner E8 ring Boerdijk-Coxeter helix.
inner-outerP
Platonic solids
E8-3D-Platonic-2