Category Archives: Physics

Hyperbolic Dynkin Diagrams

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

The Comprehensive Split Octonions and their Fano Planes

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

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.

Octonions-E8-Particles-Elements

Created a new 4D Stowe-Janet-Scerri Periodic Table

I’ve replaced the standard periodic table in the 7th “Chemistry Pane” of my E8 visualizer with a 2D/3D/4D Stowe-Janet-Scerri version of the Periodic Table.

Interestingly, it has 120 elements, which is the number of vertices in the 600 Cell or the positive half of the 240 E8 roots. It is integrated into VisibLie_E8 so clicking on an element adds that particular atomic number’s E8 group vertex number to the 3rd E8 visualizer pane.

The code is a revision and extension of Enrique Zeleny’s Wolfram Demonstration

A few screen shots….
PeriodicTable_3D

PeriodicTable_3Ds

PeriodicTable_2D

Stowe-Janet-Scerri 2D Periodic Table

ToE_Demonstration

ToE_Demonstration-new-stowe-i10

ToE_Demonstration-new-stowe-i10a

Stowe-Janet-Scerri 3D Periodic Table

2D and 3D electron orbitals for each element
eOrbitals2D

eOrbitals3D

eOrbitals3D-1

eOrbitals3D-2

eOrbitals3D-3