This blogs.ams.org/visualinsight article is an American Mathematical Society post based on some of my work.
This animated Gif from the article is from Greg Egan
This blogs.ams.org/visualinsight article is an American Mathematical Society post based on some of my work.
This animated Gif from the article is from Greg Egan
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):
This is a little clip containing three different 8 dimensional rotations of E8. While it may seem like a simple rotation of an object in 3-space, if you look carefully (especially the first rotation), you can see sets of vertices moving in different directions.
The object (the 240 vertices of E8 along with 1220 of 6720 edges of 8D length Sqrt[2]) is not “moving”. It is the camera that is moving in an 8 dimensional space.
Best when viewed in HD!
The previous 3D animations here and here are created by projecting the 8 dimensional object of E8 into 3 and simply rotating (spinning) that object around in 3-space. I like to call those animations 4D, since animation visualizes changes over time – a 4D space-time object if you will…
Best viewed in HD mode…
Last year I showed that the Dynkin diagram of the Lie Algebra, Group and Lattice of E8 is related to the Coxeter-Dynkin of the 600 Cell of H4 through a folding of their diagrams:
I had found the E8 to H4 folding matrix several years ago after reading several papers by Koca and another paper by Dechant, on the topic. The matrix I found is:
Notice that C600=Transpose[C600] and the Quaternion–Octonion like structure (ala. Cayley-Dickson) within the folding matrix. Only the first 4 rows are needed for folding, but I use the 8×8 matrix to rotate 8D vectors easily.
The following x,y,z vectors project E8 to its Petrie projection on one face (or 2 of 6 cubic faces, which are the same).
On another face are the orthonormal H4 600 cell and the concentric H4/Phi. There are 6720 edges with 8D Norm’d length of Sqrt[2], but I only show 1220 in order to prevent obscuring the nodes.
These values were calculated using the dot product of the Inverse[4*C600] with the following H4 Petrie projection vertices (aka. the Van Oss projection), where I’ve added the z vector for the proper 3D projection. Notice I am still using 8D basis vectors with the last 4 zero, as this maintains scaling due to the left-right symmetries in C600).
The Split Real Even (SRE) E8 vertices are generated from the root system for the E8 Dynkin diagram and its resulting Cartan matrix:
This is input from my Mathematica “VisibLie” application. It is shown with the assigned physics particles that make up the simple roots matrix entries:
.
Please note, the Cartan matrix can also be generated by srE8.Transpose[srE8]
It takes Transpose[srE8] and applies the dot product against the 120 positive and 120 negative algebra roots generated by the Mathematica “SuperLie” package (shown along with its Hasse diagram, which I generate in the full version of my Mathematica notebook):
Interestingly, E8 in addition to containing the 8D structures D8 and BC8 and the 4D Polychora, contains the 7D E7, as well as the 6D structures of the Hexeract or 6 Cube. I also showed several years ago that this projects down to the 3D as the Rhombic Triacontahedron. The Rhombic Triacontahedron (of Quasicrystal fame) contains the Platonic Solids including the Icosahedron and Dodecahedron! This is shown to be done through folding of the 6-Cube using rows 2 through 4 of the E8 to H4 folding matrix!!
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