Category Archives: Physics

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).

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 Cell
Various 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 T
3D 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.

Interactive Cloud VisibLie-E8 4D Periodic Table

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:

1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein

2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle

3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize

4) Math: Group Theory: Dynkin Diagram Algebra Create

5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize

6) Physics: Quantum Elements: Fundamental Quantum Element Select

7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity

8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays

9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation

10) Chemistry: Atomic Elements: 4D Periodic Table Element Select

11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select

12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding

13) Biology: Human Neurology: OrchOR Quantum Consciousness

14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori

15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria

16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier

17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life

18) CompSci: Human/Machine Interfaces: nD Human Machine Interface

Cloud Based VisibLie_E8 Demonstration

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.

Link to the demonstration.

A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:

1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein

2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle

3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize

4) Math: Group Theory: Dynkin Diagram Algebra Create

5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize

6) Physics: Quantum Elements: Fundamental Quantum Element Select

7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity

8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays

9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation

10) Chemistry: Atomic Elements: 4D Periodic Table Element Select

11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select

12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding

13) Biology: Human Neurology: OrchOR Quantum Consciousness

14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori

15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria

16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier

17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life

18) CompSci: Human/Machine Interfaces: nD Human Machine Interface

My VisbLie E8 demonstration system for Mathematica v13

The newer version of the VisibLieE8-NewDemo-v13.nb (130 Mb) will work with those who have a full Mathematica v13 license. It is backward compatible to earlier versions. There are a few bug fixes from the older version of ToE_Demonstration.nb (130 Mb), which should work on v13 and older versions as well.

For more detail on the modules, see this blog post.

Please be patient, it is very large and can take 10 minutes to load, depending on your Internet connection, memory and CPU speed.

The free Wolfram CDF Player v. 13 works with my VisibLie E8 ToE demonstration on Win10

In case you’re interested, I just verified the demo works on the free Mathematica CDF player v.13 for Win10.

Just go to https://www.wolfram.com/player/ install, download and open the app:

https://theoryofeverything.org/TOE/JGM/ToE_Demonstration.nb

There is a ton of other cool interactive stuff in there. FYI – Some features don’t work without a full Mathematica license.

Enjoy.

12-fold Symmetric Quasicrystallography from affine E6, B6, and F4

This post is an analysis of a June 2013 paper by Mehmet Koca, Nazife Koca, and Ramazan Koc. That paper contains various well-known Coxeter plane projections of hyper-dimensional polytopes as well as a new direct point distribution of the quasicrystallographic weight lattice for E6 (their Figure 3), as well as the quasicrystal lattices of B6 and F4.

Koca / Koc Figure 3 E6 Quasicrystallographic Weight Lattice

Wha‍t is interesting about this projection is that it precisely matches the point distribution (to within a small number of vertices) from a rectified E8 projection using a set of basis vectors I discovered in December of 2009, published in Wikipedia (WP) in February of 2010 here.

Rectified E8 in my “Triality” projection basis:
x=(2-4/√3   ,   0         ,   1-1/√3   ,      1-1/√3   ,   0   ,   -1   ,   1   ,   0   )
y=(   0   ,   -2+4/√3   ,   -1+1/√3   ,   1-1/√3   ,   0   ,   1/√3   ,   1/√3   ,   -2/√3   )

Rectification of E8 is a process of replacing the 240 vertices of E8 with points that represent the midpoint of each of the 6720 edges. In this projection, there are overlaps which are indicated by different colors in the color-coded WP image linked above.

The image below is an overlay of the above images highlighting the 12*(9+3+26+7)=540 points that are not overlapping:

Annotated overlap comparison showing missing 432 overlaps.

It is interesting to note that with a 30° rotation of my projection, the missing overlaps are reduced to 12*(15+2)=204.

Annotated overlap comparison showing missing 204 overlaps.

Given the paper’s explanation for the methods using E6 (720) with 6480 edges as a projection through a 4D 3-sphere window defined by q1 and q6, it may be insightful to study my projection basis for E8’s triality relationships with the Koca/Koc paper’s defined 4D 3-sphere.

For more information on why my projection basis is called the E8 Triality projection, see this post.