Tag Archives: E8

Physics

E8 folding to H4+H4/φ

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I found the rotation matrix that shows the E8 Dynkin diagram can indeed be folded to H4+H4/φ.

The H4 and its 120 vertices make up the 4D 600 Cell. It is made up of 96 vertices of the Snub 24-Cell and the 24 vertices of the 24-Cell=[16 vertex Tesseract=8-Cell and the 8 vertices of the 4-Orthoplex=16-Cell]).

It can be generated from the 240 split real even E8 vertices using a 4×8 rotation matrix:
x = (1, φ, 0, -1, φ, 0, 0, 0)
y = (φ, 0, 1, φ, 0, -1, 0, 0)
z = (0, 1, φ, 0, -1, φ, 0, 0)
w = (0, 0, 0, 0, 0, 0, φ^2, 1/φ)

where φ=Golden Ratio=(1+Sqrt(5))/2

It is also interesting to note that the x, y, and z vectors project to a hull of the 3D Rhombic Triacontrahedron from the 6D 6 cube Hexaract (which then generates the hull of the Dodecahedron and Icosahedron Platonic solids).

Here’s a look at the Dynkin Diagram folding of E8 to H4+H4/φ:

E8-H4-Fold

I find in folding from 8D to 4D, that the 6720 edge counts split into two sets of 3360 from E8’s 6720 length Sqrt(2), but the combined edges and vertices recreate the E8 petrie diagram perfectly.

Some visualizations of this in 8D:
E8-8D-Polytopes-2

to 4D:
E8-4D-Polychora-2

and also showing the Rhombic Triacontrahedron folding from 6D:
E8-6D-StarPolytopes-2

to 3D:
E8-3D-Platonic-2

Physics

Another look at integrating the Pascal Triangle to Clifford Algebra, E8 Lie Algebra/Groups, Octonions and Particle Physics Standard Model

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Pascal-g

Modified Lisi split real even E8 particle assignment quantum bit patterns:

Lisi_Particle_Assignments

Assigning a specific mass, length, time, and charge metrics based on new dimensional relationships and the Planck constant (which defines Higgs mass).

ToEsummary

The split real even E8 group used has been determined from this simple root matrix (which gives the Cartan matrix upon dot product with a transpose of itself):

DynkinE8Full.svg

This Dynkin diagram builds the Cartan matrix and determines the root/weight/height with corresponding Hasse diagrams.

E8Hasse

E8HassePoset.svg

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

Physics

The 480 octonions, their Fano planes and multiplication tables

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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.

Physics

Update: Quaternions, Octonions, Time, Particle Mass/Charge and Reality

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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.

Physics

Complete Integration of Octonions with E8

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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.

Physics

E8 to 4D (3D+T)

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E8 8D to 4D (3D+T) Animation

Shown in this animation are the 240 vertices of E8 with shape, size, and color assigned based on theoretical physics of an extended Standard Model (eSM). It is made up of three sets of 120 frames, each with a different algorithm for calculating perspective and orthogonal, rotational and translational 8D flight paths. It is interesting to note that it is the 8D camera that is moving through 8D space and the vertices remain in their same 8D position.

The 30 blue triangles represent E8 triality relationships using an 8D rotation matrix based on 2Pi/3 (or 120 degrees). Each vertex in a blue triangle is transformed into an adjacent one by the dot product with the matrix. A second transformation transforms it to the next, while the third recovers the original vertex.

The 28 red and green triangles are created from a subset of the 6720 (shortest) edges of 8D norm’d length Sqrt(2). These are filtered to represent the particle sums (linked by a red line) for a common (clicked) vertex (linked by 2 green lines). It is interesting to note that all sums for a given vertex are only found in adjacent vertices.

Higher definition (2 sets 60 frames each):