I decided to visualize my decomposition of the 10 self-dual 24-cells of E8. These are shown in the Petrie projection of E8 that is split into H4 and H4φ using my E8 to H4 folding matrix. The interesting aspect of this is that the selection of the canonical H4 24-cell determines the other 4 24-cells that make up the 96 vertices of the H4 snub-24-cell by rotating 4 times by π/5. Scaling the vertices by φ on the Petrie projection give the other 5 24-cell+snub-24-cell vertices.

It is also interesting to note that (my modified) Lisi physics particle assignments (with particle/anti-particle pairing) create 30 nicely ordered 4-particle/anti-particle sets that (fairly consistently) distribute over type, spin, generation, and color. These are visualized in 3D with shape, size and color defined by the quantum numbers of the assigned particle. The 3D projection is the same as that of the Petrie with a 3rd Z vector added. For more detail, see my latest paper.

The last set of images (using only the “math version” round black vertices and numeric identifier from the position in the Pascal triangle/Clifford Algebra canonical lexicographic ordering) are the 2D rotations of 24-cells around face-3 of this projection (which turns out to be the orthonormal shape of the H4 600 cells).

This blogs.ams.org/visualinsight article is an American Mathematical Society post based on some of my work. I’ve made an animation of the D6 polytope with 60 vertices and 480 edges of 6D length Sqrt(2). It is similar to the one in the article by Greg Egan. This version was made from my VisibLie E8 demonstration software.

Best viewed in HD mode.

The process for making this using the .nb version of VisibLie E8 ToE demonstration:
1) Side menu selections:

• slide “inches” to 4 (lowers the px resolution)
• select “3D”
• set bckGrnd “b” (black)
• uncheck the shwAxes box
• check the physics box (to give it interesting vertex shapes/colors)
• change the fileExt to “.avi” and check the fileOut box (to save the .avi)
• check the artPrint box (to give it cylinder-like edges)

2) Go to the #9) HMI nD Human Interface pane:
2a) Top menu selections:

• set pScale to .08 (to increase the vertex shape sizes)

2b) Inside the pane selections:

• set “eRadius” to .01 (for thicker edge lines)

3) Go to the #2) Dynkin pane and interactively create (or select from the drop down menu) the D6 diagram.
BTW – this actually selects the subset of vertices from the E8 polytope that are in D6.
 

4) Go to the #3) E8 Lie Algebra pane:
4a) Top menu selections:

• E8->H4 projection
• check the edges box
• “spin” animation
• 30 animation “steps”

or simply upload this file or copy / paste the following into a file with extension of “.m” and select it from the “input file” button on the #9 HMI nD Human Interface pane.

(* This is an auto generated list from e8Flyer.nb *)
new :={
artPrint=True;
inches=4;
physics=True;
p3D=” 3D”;
bckGrnd=GrayLevel[0];
fileExt=”.avi”;
shwAxes=False;
scale=0.08;
selPrj=”E8->H4″;
lie=”D6″;
cylR=0.01;
showEdges=True;
steps=30;
pthRot=”Spin”;
selPrj=”E8->H4″;
p3D=” 3D”;
};new;

In the next version I will just make this another “MetaFavorite”, so it will be in the dropdown menu in all versions.
Enjoy!