I am working on validating some theoretical work on the triality automorphisms of the split octonions. This is a post with preliminary work on that… for those who are interested 🙂
BTW – It requires the here.
An addendum to the original post (below):
[WolframCDF source=”http://theoryofeverything.org/TOE/JGM/octonion triality checks.cdf” width=”900″ height=”8000″ altimage=”http://theoryofeverything.org/TOE/JGM/octonion triality checks.png” altimagewidth=”900″ altimageheight=”8000″]
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).
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!
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”:
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.
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…
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):
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)
Notice that the binary can be associated with Clifford algebras
I decided to take a quick dive into an interesting idea from Peter Williams related to how the modulus of binomial numbers are related to the Pascal Triangle. I’ve added this to the VisibLie_E8 viewer as well.
Here is a screenshot:
Here is the code I wrote to produce the output below:
Mod 9:
Mod 8:
Mod 7:
Mod 6:
Mod 5:
Mod 4:
Mod 3:
Mod 2:
