This is a cloud based .ppt version. These show 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 :
Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). “Catalan Solids Derived From 3D-Root Systems and Quaternions”. Journal of Mathematical Physics. 51 (4). arXiv:0908.3272. doi:10.1063/1.3356985.
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).
3D Visualization of the outer hull of the alternate 96 vertex Snub 24 Cell (S’)Visualization of the concentric hulls of the Alternate Snub 24 CellVarious 2D Coxeter Plane Projections with vertex overlap color coding.3D Visualization of the outer hull of M(192) as one of the W(D4) C3 orbits of the 120-Cell (600)3D Visualization of the outer hull of N(288) that are the 120-Cell (600) Complement of the W(D4) C3 orbits T'(24)+S'(96)+M (192)3D Visualization of the outer hull of the 120-Cell (600) generated using T’3D Visualization of the outer hull of the 120-Cell (600) generated using T3D Animation of the 5 quaternion generated 24-cell outer hulls consecutively adding to make the 600-Cell.3D Animation of the 5 quaternion generated 600-cell outer hulls consecutively adding to make the 120-Cell.
I’ve published a version of these demonstrations in the cloud, so no Mathematica installation is needed.
Please click here for the interactive Hindu Sri Yantra demonstration shown below.
Image of the User Interface (UI) of the 3D Sri Yantra demonstration
Please click here for the interactive Buddhist (Tao/Confucian) I-Ching demonstration shown below.
Image of the User Interface (UI) of the 2D I-Ching demonstration.
The Mathematica Cloud version doesn’t work in 3D as well as using a local .NB version with the free Mathematica CDF player. You can File->Download the local demonstration and use it after installing the CDF player found here.
Of course, the full licensed Mathematica Notebook works the best for this. See this file here for the Hindu 3D Sri Yantra and here for the Buddhist 2D I-Ching notebooks.
There is even a full pane 15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria here.
Here is a list of other Interactive Demonstrations within my VisibLie-E8 package:
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
This is a link to the free cloud Mathematica demonstration. (Note: You need to enable “Dynamic Behavior” aka. interactivity in the upper left corner).
Please bear in mind that this demonstration is written for a full Mathematica licensed viewer. The cloud deployments are limited in interactivity, especially those that involve 3D and significant computation. Also, be patient – it takes a minute to load and more than a few seconds to respond to any mouse click interactions.
The utility of the cloud demo of this 4D (3D+color) Periodic Table is in visualizing it in 2D or 3D (from the left side menu) and building up n=1 to 8. Select the Stowe vs. Scerri display for different 3D models. The explode view slider helps distribute the lattices in the model.
The 2D/3D electron density representations for each atom’s orbitals are too slow for the cloud, so they don’t show anything. The isotope and list-picker of internet curated element data also does not function.
For an explanation of this pane #10 in the suite of 18 VisibLie-E8 demonstrations, please see this link.
High resolution 4D (3D+color) images of the demonstration.High resolution 2D images from the demonstration.
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
The cloud deployments don’t have all the needed features as the fully licensed Mathematica notebooks, so I included a few of the panes that seem to work for the most part. Some 3D and animation features won’t work, but it is a start. Bear in mind that the response time is slow.
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.
Dedicated to the pursuit of beauty and Truth in Nature!
