# A4 Group Orbits & Their Polytope Hulls Using Quaternions

Updated: 05/11/2023

This post is a Mathematica evaluation of A4 Group Orbits & Their Polytope Hulls Using Quaternions and nD Weyl Orbits. This is a work-in-process, but the current results are encouraging for evaluating ToE’s.

Please see the this PDF or this Mathematica Notebook (.nb) for the details.

The paper being referenced in this analysis is here.

Of course, as in most cases, this capability has been incorporated into the VisibLie_E8 Demonstration codebase. See below for example screen shots:

This is an example .svg file output from the same interactive demonstration: SVG file output from the VisibLie_E8 Coxeter-Dynkin Interactive Demonstration Pane showing the OmniTruncated A4 Group Vertex Hulls

More code and output images below:

# The (diminished) 120-Cell and it’s relationships to the 5-Cell (A4), the 600-Cell (H4), the two 24-Cells (D4), the Dual Snub 24-Cell, and of course E8!

Updated: 05/03/2023

This post is a Mathematica evaluation of important H4 polytopes involved in the Quaternion construction of nD Weyl Orbits.

Please see the this PDF or this Mathematica Notebook (.nb) for the details.

The paper being referenced in this analysis is here.

I’ve added this PDF or this Mathematica Notebook (.nb) which is having a bit of fun visualizing various (3D) orbits of diminished 120-cell convex hulls. The #5 subset of 408 vertices (diminishing 192) is completely internal to the normal 3D projection of the 120-cell to the chamfered dodecahedron. The #3 has 12 sets of 2 (out of 3) dual snub 24-cell kite cells (that is 24 out of the 96 total cells).

# 3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin ​Geometric Group Theory

It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8->H4 folding matrix) from quaternions given their Weyl Orbits.

If you don’t want to use the interactive Powerpoint, here is the PDF version.

See these Mathematica notebooks here and here for a more detailed look at the analysis of the Koca papers used to produce the results :

Or scroll down to see the .svg images here and here.

# Visualizing the Quaternion Generated Dual to the Snub 24 Cell

I did a Mathematica (MTM) analysis of several important papers here and here from Mehmet Koca, et. al. The resulting MTM output in PDF format is here and the .NB notebook is here. 3D Visualization of the outer hull of the 144 vertex Dual Snub 24 Cell, with vertices colored by overlap count:* The (42) yellow have no overlaps.* The (51) orange have 2 overlaps.* The (18) tetrahedral hull surfaces are uniquely colored.

What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions with judicious selection of the first triad={123}). This includes both the 600 Cell and the 120 Cell and its group theoretic orbits. The 144 vertex Dual Snub 24 Cell is a combination of those 120 Cell orbits, namely T'(24) & S’ (96), along with the D4 24 Cell T(24).

# Visualizing the concentric hulls of E8 with the 24-cells of H4 & H4Φ

I’ve had a number of related posts on this topic, but I wanted to present a few new pictures and PDF documents that combine the 7 concentric hulls forming Platonic solid related sets of E8 vertices projected to 3D with vertices of five pairs of 24-cell objects.

These break down the E8 structure into familiar 3D objects, such as the icosahedron with its dual the dodecahedron, the icosidodecahdron, and the 16-cell with its dual the 8-cell (aka. Tesseract) combined to create the self-dual 24-cell. The yellow spheres are those of the outer hull of singular 30 vertices of an icosidodecahedron. The orange spheres are pairs of icosahedron or dodecahedron vertices. The edges with numbered vertices form one of five 24-cells in the 120 vertex H4Φ scaled 600-cell within E8. Specifically, it is the 3rd rotation (of 4 π/5 rotations) within the 96 vertices of the snub-24-cell. You can see that 6 of the yellow vertices connect to the 24-cell. Rotating that 24-cell four times in 3-space by π/5 gives the connections to the rest of the vertices in H4Φ and completes the 30 vertex icosidodecahedron . The same is true for the 120 vertices of H4 using the corresponding 24-cell in H4. a) 24-cell highlighting the 16-cell (red on-axis vertices) and 8-cell (blue off-axis vertices)b) Snub 24-cell highlighting four π/5 rotations of the 24-cell (black) in red, green, blue, yellow.

The following paper hulls-24cells-combined.pdf (13MB) & interactive Mathematica Notebook hulls-24cells-combined.nb (34MB) contains a comprehensive set of images that show the contents of the 7 concentric hulls of Platonic solid related shapes as well as their integration with one of the rotations of 24-cells in H4 and H4Φ in E8 (the same one used in the image above). Snapshot from the paper showing the 3rd rotation of the snub 24-cell and the icosidodecahedron of H4Φ

In the E8 Petrie projection, every hull (each with 2 or 4 overlapping vertices) pair into left/right patterns. See the set of four icosahedrons that occupy the 3rd hull in both 2D and 3D (the yellow edge sets belongs to the H4Φ and the blue sets belong to H4) :

The pair of dodecahedrons that occupy the 5th hull in both 2D and 3D (the yellow edge set belongs to the H4Φ and the blue belong to H4) :

Below is an animation that cycles through the sequence of 2D Petrie Projections of the pairs of hulls. One cycle shows each frame individually and the other builds the E8 Petrie from the previous 2D hull.

Below are images of the left (H4) and right (H4Φ) 2D Petrie projections of the hulls (which are defined by the Norm’s of 3D projected vertices using the E8->H4 folding matrix rows 2-4 as basis vectors). The two 30 vertex 4 & 7 Icosidodecahedron hulls and four 0th hull (points) are omitted leaving the gaps in the diagram. When these gap vertices are combined with the 48 3rd hull Icosahedrons (above), they make up the 112 integer D8 group assigned to the Bosons (48) and 2nd generation Fermions (64) in the physics model described below.

The combined set of hulls projected using the rows of the folding matrix as basis vectors is shown below from previous work:

This paper hull-list-3b.pdf (4MB) & interactive Mathematica Notebook hull-list-3b.nb (20MB) contains the visualizations and detail E8 vertices associated with each Platonic solid related concentric hulls. This paper 24cell-list-3b.pdf (6MB) & interactive Mathematica Notebook 24cell-list-3b.nb (20MB) contains the visualizations and detail for the five 24-cells in each H4 & H4Φ. Both of these papers also have detail information about its assigned physics particle based on a modified A.G. Lisi model (shown below). This paper describes these particle assignment symmetries in more detail.

# G2, F4, E6, E7, and E8 triality relations in 3D concentric hulls of E8

Here are the G2 gluons connected by their trialities shown in a 3D concentric hull projection of E8 using the E8 to H4 folding matrix basis vectors. This is the 4th hull, which is the outer hull of the inner H4 600 cell (an icosadodecahedron). For more on E8 hulls, see this post. The full F4 group with 10 T2 and 12 T4 trialities affecting the bosons is contained in the outer icosadodecahedron (1st hull) combined with the 3rd (quad icosahedral) and 4th icosadodecahedron hulls.

It is interesting to note that the two icosadodecahedron hulls comprise the 60 vertices of D6. Below is the full E8 with all trialities shown in 3D concentric hull projection. #### Wow – that is nice! Remember, you heard it here first!

BTW – if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at: jgmoxness@theoryofeverthing.org.

# Star of David Projection Basis for E8, E7 and E6

For a full discussion of the particle assignment symmetries involved, see http://vixra.org/pdf/1503.0190v1.pdf

E8 as 240 split real even vertices with  vertex color, size, shape, and labels from a modified A.G. Lisi particle assignment algorithm. The vertex number is the position in the Pascal Triangle (Clifford Algebra) representation of E8. The blue lines forming equilateral triangles represent the  10 Bosonic (or Color) Triality (T2) and 76 Fermionic (T4) rotations (using the 8×8 matrices shown below) that transform each member of the triangle into the next member. E7 as derived from E8 above by taking the vertices of E8 with the last two columns being equal. E6 as derived from E8 above by taking the vertices of E8 with the last three columns being equal. The 48 vertices of F4 are made up of the 112 (D8) +/-1 integer pair vertices that are assigned to bosonic particles, which includes 22 trialities (10 from T2 and 12 from T4). The other 64 of 112  D8  vertices are assigned to the 2nd generation of Fermions. F4 is shown below with its sub-group G2 assigned to the gluons in the larger triality triangles. Projection Basis for X and Y are H (first row) and V (second row) in the matrix below respectively The Triality Matrices are: The 10 Bosonic (Color) Triality Rotation Matrix (T2) equilateral triangles exclusively affect the particles that are derived from F4.

T2 rotates through the bosonic colors indicated by the last 3 columns of the E8 vertices appropriately labeled (r,g,b). The 76 Fermionic (Generation) Triality Rotation Matrix (T4) equilateral triangles. There are 64 fermion particle trialities that are derived from the 128 (BC8) half integer vertex assignments plus the 64 fermions assigned from the 112 (D8) +/-1 integer pair vertices. There is always one integer (2nd Generation) and two 1/2 integer vertices (1st and 3rd Generation) in every fermionic triality.

That is, T4 rotates through the fermionic generations. There are 12 trialities from these 76 that are associated with the inner E6 & F4 Lie Group vertices assigned to bosons, shown below with only the 112 (D8) +/-1 integer pair unlabeled vertices: Of course, there are a number of overlapping vertices in this projection stemming from the 112 (D8) +/-1 Bosonic sector. The following image colors the vertices by overlap count, with 120 Yellow with no overlaps, 24 Cyan with 2 overlaps (48), and 24 Cyan with 3 overlaps (72).

Notice the 8 projection basis vectors with dark gray circles. Of course, these projection basis vectors are the 8 “generator vertices” of E8 vertices with permutations of {+1,0,0,0,0,0,0,0} giving it the full 240+8=248 dimension count. Add to that the -1 “anti-generators” being the opposite end of the projection basis vectors and you get the full 256 vertices representing the full Clifford Algebra on the 9th row of the Pascal Triangle. There are 6720 edges in E8 with an 8D length Sqrt. Just for fun, I introduce to you the 6720 rectified E8 vertices taken from the midpoint of each edge and using the same projection with coloring of the spheres based on overlap count. Enjoy! Here are the G2 gluons connected by their trialities shown in a 3D concentric hull projection of E8 using the E8 to H4 folding matrix basis vectors. This is the 4th hull, which is the outer hull of the inner H4 600 cell (an icosadodecahedron). For more on E8 hulls, see this post. The full F4 group with 10 T2 and 12 T4 trialities affecting the bosons is contained in the outer icosadodecahedron (1st hull) combined with the 3rd (quad icosahedral) and 4th icosadodecahedron hulls. Below is the full E8 with all trialities shown in 3D concentric hull projection. #### Wow – that is nice! Remember, you heard it here first!

BTW – if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at:  jgmoxness@theoryofeverthing.org.

# E8 Projected to the Concentric Hulls of H4+H4 Phi

The 8 concentric projected 3D hulls from the vertices of the 4_21 polytope (using the split real even E8 roots) and a projection basis from 3 of the 4 rows of the E8 to the H4+H4 Phi folding matrix produce from inner to outer (sorted by vertex Norm):

• 4 Points at the origin
• 2 Icosahedron
• 2 Dodecaheron
• 4 Icosahedron
• 2 Dodecaheron
• 2 Icosahedron

Projection Basis:  Looking at it as an orthonormal 3D projection of 2 600-cells (from the fully folded E8 to H4+H4 Phi), which is the same (as it should be), but rotated. Here is just one 600-cell (interior). Here is E7 doing the same procedure: And again for E6 #### Wow – that is nice! Remember, you heard it here first!

BTW – if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at:  jgmoxness@theoryofeverthing.org.

# Rectified D6 to H3 Icosidodecahedrons

#### Icosidodecahedron (Rectified 6-Orthoplex) While the interior vertices are shown, only the exterior 120 hull edges were used in the rectifications.

3D StereoLithography Model for 3D Printing

#### Rectified Icosidodecahedron (Cantellated 6-Orthoplex) 3D StereoLithography Model for 3D Printing

#### Twice Rectified (Cantellated) Icosidodecahedron # Rectified Rhombic Triacontahedrons

#### Rhombic Triacontahedron While the interior vertices are shown, only the exterior 60 hull edges were used in the rectifications.

3D StereoLithography Model for 3D Printing

#### Rectified Rhombic Triacontahedron 3D StereoLithography Model for 3D Printing

#### Twice Rectified (Cantellated) Rhombic Triacontahedron 3D StereoLithography Model for 3D Printing