Please see Integrated E8, Binary, Octonion for a tutorial that explains the detail of the content of Fano.pdf. It outlines the relationships in the integration of E8 with Octonions, Binary, Particles, NKS, and the Periodic Table of Elements. Other formats are also available (.ppt or .pps and .pdf).
Tag Archives: Octonion
Created a simpler Fano plane and cubic .pdf
A simpler version of Fano.pdf is here FanoOnly.pdf (15MB).
Improved complete index of E8, Binary, Octonions, Particles, NKS CA & Periodic Table
The main change was an improved association of the 120 periodic table elements to the 120 root vectors weighted by their simple root grading. See the full 241 page reference 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.
Octonion Math Notebook
In responding to a query on StackExchange, I created a small notebook that includes a Cayley-Dickson doubling of the built-in Mathematica quaternions to perform general octonion math. It is the foundation for some of the visualizations used here.
The 480 octonions, their Fano planes and multiplication tables
I am pleased to announce the availability of Fano.pdf, a 241 page pdf file with the 480 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). For each split real even E8 vertex, the algebra root, weight and height are listed along with the Clifford/Pascal binary and physics rotation coordinates. On each page, the E8 particle number, symbol, and assigned 2D/3D shape are shown along with the (a)nti, (p)Type, (s)pin, (c)olor, (g)eneration bitwise quantum assignments. Also included is the particle experimental mass and lifetime along with my ToE theoretically calculated mass. (30MB)
I believe this is the only comprehensive presentation of all 480 Fano planes with their multiplication tables available.
Update: Quaternions, Octonions, Time, Particle Mass/Charge and Reality
I am improving and tightening the octonion to E8 to particle symmetry assignments which should inform the particle mass/charge assignment (prediction).
Keep looking at the .CDF .NB or interactive pages – it is always up to date.
I am thinking about how the quaternion trialities within the non-associative algebra of octonions relate to time (and differentiate particles). So it is less about E8 and more about how octonions (linked to E8) affect our 4D reality.
Complete Integration of Octonions with E8
I’ve updated all .CDF, .NB and older .NBP demonstration code with Mathematica v.9
This now includes the completed integration of octonions with E8 & the extended Stadard Model particle assignments. This means the 480 octonions are in fact in a 2:1 cover of E8’s 240 vertices (with their association to Lisi’s particle assignments). This creates the opportunity for a self-dual type of “super symmetry” where all three generations emerge from the octonions.
Octonions, Quaternions and the Fano plane mnemonic – Conquered!
See:
http://theoryofeverything.com/TOE/JGM/ToE_Demonstration.cdf
Of course, it requires the free Mathematica CDF plugin.