Tag Archives: 3D
E8 to H4 Folding Amplituhedron Surface Visualization
…nuff said, enjoy!
or in “physics mode”
Hopf Fibration and Chaotic Attractors, etc.
I’ve added some new features to my VisibLie_E8 ToE Demonstration. Some of it comes from Richard Hennigan’s Rotating The Hopf Fibration and Enrique Zeleny’s A Collection Of Chaotic Attractors . These are excellent demonstrations that I’ve now included with the features of my integrated ToE demonstration, since they are not only great visualizations, but relate to the high-dimensional physics of E8, octonions and their projections. This gives the opportunity to change the background and color schemes, as well as output 3D models or stereoscopic L/R and red-cyan anaglyph images.
I’ve also used David Madore’s help to calculate the symbolic value of the E7 18-gon and 20-gon symmetries of E8. It uses the nth roots of unity (18 and 20, in this case) and applies a recursive dot product matrix based on the Weyl group centralizer elements of a given conjugacy class of E8. I ended up using a combination of Mathematica Group Theory built-in functions, SuperLie and also LieART packages. These symbolic projection values are:
My latest paper on E8, the H4 folding matrix, and integration with octonions and GraviGUT physics models.
The full paper with appendix can be found on this site, or w/o appendix on viXra. (13 pages, 14 figures, 20Mb). The 22 page appendix contains the E8 algebra roots, Hasse diagram, and a complete integrated E8-particle-octonion list.
Abstract:
This paper will present various techniques for visualizing a split real even E8 representation in 2 and 3 dimensions using an E8 to H4 folding matrix. This matrix is shown to be useful in providing direct relationships between E8 and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects. A direct linkage between E8, the folding matrix, fundamental physics particles in an extended Standard Model GraviGUT, quaternions, and octonions is introduced, and its importance is investigated and described.
If you would like to cite this, you can use this BibTex format if you like (remove the LaTex href tag structure if you don’t use the hyperref package):
Interactive Hasse Visualizations and StereoScopic viewing
Along with added options for Left-Right and Red-Cyan Anaglyph stereoscopic outputs on the left side control panel…
I’ve added Interactive Hasse Visualizations to the #2 Dynkin Demonstration Pane. There is now a checkbox for showing the detail root vector data and Hasse visualizations (instead of the default interactive Dynkin pane). This is done from an integration of SuperLie 2.07 by P. Grozman.
If you have a full licensed Mathematica, use ToE_Demonstration.nb. For use with the free CDF Player, use ToE_Demonstration.cdf or as an interactive web page.
New TheoryOfEverything Visualizer Features
I cleaned up some of the N-Body physics screens, and created a few animation sequences showing the simulation runs available (if you have Mathematica and source code (available upon request and appropriate use-case)).
This is a video of a preliminary Galaxy formation in N-Body gravitational physics.
This is a video of the solar system (not yet using the OpenCL N-Body code for GPU parallelism).
This is a video of the Compton Effect in 3D, which I plan on using to show how Big Bang Inflationary Quantum effects are explained.
I’ve also improved the capabilities of the other demonstrations.
Avengers Tesseract Cosmic Cube
I finally got around to watching the Avengers movie and noticed that the Tesseract Cosmic Cube looked much like the hyper-dimensional projections that I make with laser etched optical crystal.
The blue light is projected from a multi-colored LED base.
While I do have tesseract projections, the E8 projection on my home page seems most similar. Here are a few blue crystal photos I’ve taken that are also similar.
A few new 3D Virtual World pics
New Laser Etched Crystal Pics
The 3rd (Z) basis vector for Bathsheba & Wizzy's 600 Cell
While the 3D model I used to create the 2D Van Oss projection isomorphic to E8 Petrie (and a beautiful pentagonal view), it was not the same as what was being used by Richter in his 3D “pre-Van-Oss” construction. Given my H (or x) and V (or y), the 3rd basis vector for this projections is most likely:
Z={0, -0.0801064, 0, 0.236818, 0, 0.0801064, 0, -0.236818}
which reproduces the Richter, Bathsheba and Wizzy’s 3D models. Interestingly, it produces one face (shown above) that is the same as all the orthonormal faces of 2 concentric 600 Cells (at the Golden Ratio). The 3rd unique face is:
I replaced the 3D spin movie of this on my main page with this new projection.
