All posts by jgmoxness

Physics

Hyperbolic Dynkin Diagrams

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I am pleased to announce the availability of the comprehensive set of 238 Hyperbolic Dynkin diagrams ranks 3 to 10. These are enumerated here. They are constructed interactively using the mouse GUI (or manually through keyboard entry of the node / line tables) from this Mathematica notebook or free CDF player demonstration document or interactive CDF player web page.

All of these demonstrations have now been updated to include the new options on Dynkin diagram creation, including left (vs. right) affine topology recognition, black&white vs. color nodes and edges, and for those with full Mathematica licenses the notebook allows for added Dynkin line types.

The Lie group names are calculated based on the Dynkin topology. The geometric permutation names are calculated (to rank 8) based on the binary pattern of empty and filled nodes. The node and line colors can be used as indicators in Coxeter projections and/or Hasse diagrams.

Rank3CompactHyperbolicDynkins1-31bw.svg

Rank3NonCompactHyperbolicDynkins32-75bw.svg

Rank3NonCompactHyperbolicDynkins76-123bw.svg

Rank4HyperbolicDynkins124-176bw.svg

Rank5HyperbolicDynkins177-198bw.svg

Rank6HyperbolicDynkins199-205bw.svg

Rank6HyperbolicDynkins206-212bw.svg

Rank6HyperbolicDynkins213-220bw.svg

Rank7HyperbolicDynkins221-224bw.svg

Rank8HyperbolicDynkins225-229bw_svg

Rank9HyperbolicDynkins230-234bw_svg

Rank10HyperbolicDynkins235-238bw.svg

Physics

The Comprehensive Split Octonions and their Fano Planes

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Please see this post for updates to these graphs.

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.

splitFano

Personal

Connecting the Octonion Fano Plane to the Atomic Elements

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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.

FanoLegendre

Physics

Integration of the Atomic Elements to E8 and Octonions

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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.

Octonions-E8-Particles-Elements