Tag Archives: Octonion

Mathematica MyToE on the Wolfram Cloud

I am playing around with the Wolfram Tweet-a-Program, and the Wolfram Language (i.e. Mathematica) on the Wolfram Cloud.

What’s really cool is that you can now interact with advanced math and HPC on your phone/tablet.

Here are a few results…


ByzPbCqIQAAzUJq.png large

BTW – you will need a WolframID (and be logged into WolframCloud.com) to interact with these pages.

Octonions: The Fano Plane & Cubic


Dynkin Diagrams


E8 and Subgroup Projections


Particle Selector


Hadron Builder


Navier-Stokes Chaos Theory, 6D Calabi-Yau and 3D/4D Surface visualizations


Solar System (from NBody Universe Simulator)

4D Periodic Table


The Comprehensive Split Octonions and their Fano Planes

I am pleased to announce the availability of splitFano.pdf, a 321 page pdf file with the 3840=480*8 split octonion permutations (with Fano planes and multiplication tables). These are organized into “flipped” and “non-flipped” pairs associated with the 240 assigned particles to E8 vertices (sorted by Fano plane index or fPi).

There are 7 sets of split octonions for each of the 480 “parent” octonions (each of which is defined by 30 sets of 7 triads and 16 7 bit “sign masks” which reverse the direction of the triad multiplication). The 7 split octonions are identified by selecting a triad. The complement of {1,2,3,4,5,6,7} and the triad list leaves 4 elements which are the rows/colums corresponding to the negated elements in the multiplication table (highlighted with yellow background). The red arrows in the Fano Plane indicate the potential reversal due to this negation that defines the split octonions. The selected triad nodes are yellow, and the other 4 are cyan (25MB).

These allow for the simplification of Maxwell’s four equations which define electromagnetism (aka.light) into a single equation.

I believe this is the only comprehensive presentation of all 3840 Split Fano Planes with their multiplication tables available.

Below is the first page of the comprehensive split octonion list.


Connecting the Octonion Fano Plane to the Atomic Elements

There are 28 octonion Fano plane triangles that correspond directly to the 28 Trott quartic curve bitangents. These bitangents are directly related to the Legendre functions used in the Shroedinger spherical harmonic electron orbital probability densities. Shown below is a graphic of these overlaid onto the n=5, l=2, m=1 element, which is assigned to gold (Au). When using an algorithm based on the E8 positive algebra root assignments, the “flipped” Fano plane has E8 algebra root number 79 (the atomic number of Au) and split real even group number of 228 (in Clifford/Pascal triangle order). For more information, please see the complete list of integrated Fano indices here with full explanation of the layout here.


Integration of the Atomic Elements to E8 and Octonions

While it still needs some work – I’ve integrated the atomic elements to the Octonions, E8 and theoretical Lisi eSM model particle assignments (along with Wolfram’s NKS Cellular Automata linked to the Clifford Algebra/Pascal Triangle binary assignments). I have combined all these visualizations with the 2D/3D electron orbitals (based on the symmetry of the {n,l,m,s} quantum numbers (from the Stowe-Janet-Scerri Periodic Table. I totally understand this is not easy to dig into w/o some effort, but … it looks cool 😉

It is shown in an updated version of Fano.pdf. This is a very large and complex 30Mb file – with 241 pages. It shows the Lisi particle assignments, the E8 roots, split real even (SRE) E8 vertex and the Lisi “physics rotation”. It also shows two Fano plane and cubic derived from the symmetries of the E8 particle assignments (and all the relevant construction of it). See the interactive demo or the Mathematica Notebook for a more “navigational look” at the integration.