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

# 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.

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

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