The 3D vertex shape and size represent theoretically assigned extended SM particle types based on a modified A.G. Lisi model.
Tag Archives: H4
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…
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!
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.
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).
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.
E8 vertices projected to 2D pentagonal projection
5-cube in 3D
6-cube edges projected to the Rhombic-Triacontahedron using 3 of 4 rows of my E8-H4 folding matrix.
Rhombic-Triacontahedron with inner edges removed
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.
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.
Same as above also showing the inner E8 ring Boerdijk-Coxeter helix.
Platonic solids
Analyzing individual Fermi 4 particle "cell interactions" in 3D on Boerdijk–Coxeter helix rings
The Boerdijk–Coxeter helix is a 4D helix (of 3D tetrahedral cells) that makes up the vertices on 4 of the concentric rings of E8 Petrie projection (or the H4 and H4φ rings of the 2 600 cells in E8).
Outer (Ring 4) of H4 in 2D with non-physics vertices of all 8 rings of E8 in the background
Outer (Ring 4) of H4 in 3D with physics vertices
Ring 3 of H4 in 3D with physics vertices
Ring 2 of H4 in 3D with physics vertices
Inner (Ring 1) of H4φ in 3D with physics vertices
Combined 4 rings of H4 in 3D with physics vertices
Outer (Ring 4) of H4φ in 3D with physics vertices
Ring 3 of H4φ in 3D with physics vertices
Ring 2 of H4φ in 3D with physics vertices
Inner (Ring 1) of H4φ in 3D with physics vertices
Combined 4 rings of H4φ in 3D with physics vertices
Combined 8 rings in 3D with physics vertices
Boerdijk–Coxeter helix
The Boerdijk–Coxeter helix is a 4D helix (of 3D tetrahedral cells) that makes up the vertices on 4 of the concentric rings of E8 Petrie projection (or the H4 and H4φ rings of the 2 600 cells in E8).
Outer (Ring 4) of H4 in 2D with non-physics vertices of all 8 rings of E8 in the background
Outer (Ring 4) of H4 in 3D with physics vertices
Ring 3 of H4 in 3D with physics vertices
Ring 2 of H4 in 3D with physics vertices
Inner (Ring 1) of H4φ in 3D with physics vertices
Combined 4 rings of H4 in 3D with physics vertices
Outer (Ring 4) of H4φ in 3D with physics vertices
Ring 3 of H4φ in 3D with physics vertices
Ring 2 of H4φ in 3D with physics vertices
Inner (Ring 1) of H4φ in 3D with physics vertices
Combined 4 rings of H4φ in 3D with physics vertices
Combined 8 rings in 3D with physics vertices