Tag Archives: Math

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

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

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:
StarOfDavid1

My E8 F4 projection with 6720 edge lines:
E8-F4mox1-black

My E8 F4 projection rectified:
F4Rect-Sacred-1a-StarOfDavid

Star Of David Overlay:
F4Rect-Sacred-1b-StarOfDavid

Metatron’s Cube Overlay:
F4Rect-Sacred-1a-Metatron

Flower of Life Overlay:
F4Rect-Sacred-1a-Flower

Kabbalah Tree of Life Overlay:
F4Rect-Sacred-1a-Tree

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)

Adding sociology and psychology panes to VisibLie_E8 MyToE Demonstration

These are more personal interest than hard math/science as the other panes – but they are, IMO, interesting asides.

The sociology pane deals with the idea of Old Testament Hebrew and New Testament Koine Greek Gematria. The histogram shows the distribution of words used in the Bible according to their gematria value in Hebrew or Greek. Each word in each verse is selectable and shown in the distribution. Each word in the verse is colored by their value in Hebrew (if Old Testament), Greek and English.
Note: Hebrew is properly presented from right to left.

Gematria-snap

outGematria3

outGematria2

The psychology pane deals with the idea of the mathematical beauty of music.

The list of 18 panes is now:
1) Math:____ Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein
2) Math:____ Number Theory:_________ Mod 2-9 Pascal and Sierpinski Triangle
3) Math:____ Geometric Calculus:______ Octonion Fano Plane-Cubic Visualize
4) Math:____ Group Theory:__________ Dynkin Diagram Algebra Create
5) Math:____ Representation Theory:___ E8 Lie Algebra Subgroups Visualize
6) Physics:__ Quantum Elements:______ Fundamental Quantum Element Select
7) Physics:__ Particle Theory:_________ CKM(q)-PMNS(ν) Mix CPT Unitarity
8) Physics:__ Hadronic Elements:______ Composite Quark-Gluon Select Decays
9) Physics:__ Relativistic Cosmology:___ N-Body Bohmian GR-QM Simulation
10) Chemistry: Atomic Elements:_______ 4D Periodic Table Element Select
11) Chemistry:Molecular Crystallography: 4D Molecule Visualization Select
12) Biology:_ Genetic Crystallography:___ 4D Protein/DNA/RNA E8-H4 Folding
13) Biology:_ Human Neurology:________ OrchOR Quantum Consciousness
14) Psychology:Music Theory & Cognition: Chords, Lambdoma, H4, and Tori
15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria
16) CompSci: Quantum Computing:______ Poincare-Bloch Sphere /Qubit Fourier
17) CompSci: Artificial Intelligence:_______ nD Conway’s Game Of Life
18) CompSci: Human/Machine Interfaces:_ nD Human Machine Interface