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!

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!!