If you are working with topological field theoretic Hopf fibrations, chaotic attractors, and/or E8 and any of its extensive set of maximal subgroups, polytopes, lattices and codes or its folded (rotated) H4 family of polytopes (e.g. 600 and 120 cells in their vertex, cell, face, and edge orientations), VisibLie_E8 can group theoretically integrate these concepts visually!
Click on the PNG image to get the SVG version.
A few more geometries:
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:
I am playing around with the Wolfram Tweet-a-Program, and the Wolfram Language (i.e. Mathematica) on the Wolfram Cloud.
What’s really cool is that you can now interact with advanced math and HPC on your phone/tablet.
Here are a few results…
@Wolframtap
BTW – you will need a WolframID (and be logged into WolframCloud.com) to interact with these pages.
Octonions: The Fano Plane & Cubic
Navier-Stokes Chaos Theory, 6D Calabi-Yau and 3D/4D Surface visualizations
Solar System (from NBody Universe Simulator)
I’ve added an interactive (Chaos) pane (#7). It is taken from Zeleny’s Five Mode Truncation Of The Navier-Stokes Equations. It now includes 2D and 3D visualizations of these equations. This will be a basis for study of a superfluid space-time theory to merge GR with QM.
