# Amplituhedron Visualizations!

Playing around trying to visualize the latest in the theoretical high energy particle physics (HEP/TH) determining how particle masses might be predicted from geometric principles (based on a theory by Nima Arkani-Hamed, with some Mathematica code from J. Bourjaily for the positroid diagram).

Positive Grassmannian Amplituhedrons

Compare this Mathematica(l) basis to the artistic representation by Andy Gilmore

Positroid Diagrams:

# Updated demonstrations now include split octonions

I’ve updated the .NB, .CDF and interactive demonstrations to include the ability to visualize the split octonions.

# Detail explanation of E8 Integration with Octonions, Particles, and the Periodic Table

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).

# 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….

2D and 3D electron orbitals for each element

# MyToE updated from 2013 Planck Spacecraft and 2012 LHC results

With the recent publication of results for the Planck spacecraft results and the recent LHC Higgs results for the discovery of a 125(+/-1.5)GeV Higgs boson, I thought I would publish the minor tweaks to my PDF ToE summary or see the Mathematica notebook here. Enjoy!

It is interesting to note that my original paper from ’98 (updated ’01) Constants-A_New_Look.pdf contains this new Planck CMB probe result for a Hubble constant of H0=67.133 km/s/Mpc, which is at odds with WMAP, COBE, and prior estimates.

This old paper has a Higgs mH=98.137 GeV, which is around the minima of the ChiSquared value for Higgs. This was quickly excluded by FermiLab results around that time. The basic Higgs prediction relies purely on the dimensionality of the model and the Planck constant. Adding a factor of 2^1/2 gave an aesthetically pleasing mH=147.98 in a ’07 paper here. This was close to a June ’11 FermiLab CDF bump at 148 GeV. Unfortunately D0 at FermiLab didn’t confirm it.

Interestingly between the ’01 and the ’07 papers, one of the Higgs mass models I used 2^1/4 (not 2^1/2) and puts mH=124.43 GeV – right where LHC ATLAS and CMS seem to have found it. This same model also (fairly) accurately integrates this to the Fermi Constant and vacuum expecation (VEV).

The bottom line, my model describes results that are within the most current and accurate experimentally measured data, including the LHC Higgs and Planck CMB probes.

ToE Summary

# 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.

# Neutrinos and electrons are the same particle separated by a yaw spin!

A long and technical post to document a New Years Day (early morning) 2013 discovery!

The 2:1 cover of the 480 octonion permutations over the 240 vertices in E8 provides for a potential approach to resolve the issue of fitting the Standard Model (SM) into the beautifully symmetric structure of E8. This post (to be followed by a paper) describes the basics of that approach. For reference, see this Mathematica visualization demonstration tool for what follows (http://theoryofeverything.org/TOE/JGM/ToE_Demonstration.cdf).

But first, a bit about E8 and particle assignments:
Each vertex in E8 has been mapped (by AG Lisi with minor modifications in assignment) to a fundamental particle in an extended SM. This mapping is controlled in the above referenced demonstration by 8 bitwise quantum parameters which index the 256=240+16 “excluded” particles.
These 8 bits are:
(a)nti – which inverts the binary and negates the E8 vertex and gives the anti-particle of each particle
(p)Type – which differentiates the Neutrinos from electrons (and up quarks from down quarks, etc.)
2 (s)pin – up/down (or x-y or pitch rotation) and left-right (or y-z or roll rotation) spin
2 (c)olor – white (no color) and red-green-blue (e.g. the quark colors)
2 (g)eneration – 0 (no generation for bosons) and 1,2,3 for fermions

A bit about octonions:
The 480 octonion permutations are generated with a list of 30 sets of triples (3 numbers from 1-7 with no two triples with two numbers in common). These can be visualized as a Fano plane. The Fano plane is easily constructed by taking the first 3 (of 7) triads and deleting the 4th and 7th (of the 9) numbers. This gives a “flattened” list of numbers (1-7) in some permutation order. While there are 7!=5040 possible, only 480 create proper octonions. But how to pair up the 480 to mesh with E8?

There are 16 possible +/- sign reversal (or sign mask (sm) bits also shown as hex number) permutations available in each of the 30 Fano planes (30*16=480). These sm bits also “invert” like the “anti” bits in E8 (which sorts it into a top half/bottom half canonical list built using the 9th row of the Pascal triangle and Clifford algebras). So the 16 sm possibilities split into two sets of 8. This changes the direction of the Fano plane arrows. It turns out with some skillfull pattern matching in constructing the sign masks, the 8 sm bits pair up into two sets of four {1,4,6,7} and {2,3,5,8} giving a kind of “spin” match to E8. These pairs of 4 also operate as a left-right symmetry pattern seen in the structure of E8.

So far so good with 3 bits clearly paired between octonions and E8. The E8 excluded vertices (assigned 0 color and 0 generation bits – bosons) should also be excluded in the octonions. This would indicate the color and generation E8 bits would be indicated by those bits in octonions as well. But how? One hint is the generation 0 boson vs. fermion split of E8. These split E8 into a C8 group of 128 1/2 integer vertices and the 112 D8 integer vertices.

The first discovery several months ago:
In building the visualization for Fano planes, I noticed a pairing was required in order to algorithmically construct the Fano plane for each permutation of octonions. That pairing was aligned with the need to “flip” sequential pairs of numbers in one of the first three triads (the ones that create the flattened list). It also turns out that these pair into 15=7+8 pairs that correlate to the E8(240)=D8(112)+C8(128) groups of vertices.

The 7/8 (or D8/C8) split also provided the indicator on where to assign the “excluded” permutations (they get the 0 color and 0 generation assignment referenced above). It becomes trivial to apply a naive sequential assignment of 30 into two sets of 7/8 into the generation and color bits.

To reach the doubling of the 512=480+32 “excluded” octonion permutations, I added what I call a 9th “flip” bit parameter. This correlates well with A Zee’s approach to grand unification using a 9D SO(18) group theory. It turns out that the pattern required for flipping works in unison with the a and p bits in a binary logic relationship. As the demonstration clearly shows, the p bit flips the two sets of four sm bits. (Hint: this is the key to the eureka moment). Also note: Charge-Parity-Time (CPT) conservation law (symmetry) considerations also strongly link the a, p, and spin bits.

The binary logic relationship with the “flip” bit and the sm bits clearly create two 4 bit quaternion sets (anti+3 spin bit quaternion and a generation+color quaternion). While this double set of quaternions was already obvious to me from the E8 construction (noted in http://theoryofeverything.org/TOE/JGM/DynkinParticleReductions.cdf), it was good to confirm it with octonion integration! It seems that this pair of quaternions could be visualized as Hopf Fibrations on 7-manifolds (see http://www.nilesjohnson.net/seven-manifolds.html for an excellent review of this).

It correlates well with my previous post for the relating of time to octonions, charge to quaternions and 3-space to complex (imaginary) numbers. This idea also validates the dimensional considerations that started me down this path 15 years ago (documented in http://theoryofeverything.org/TOE/JGM/ToE.pdf).

Last night’s eureka moment:
Viewing the p bit as a kind of in/out (or z-x or yaw rotation), it becomes clear that the thing that differentiates electrons from neutrinos (pType) is a third intrinsic spin type. This rotation is well known in 6 degree of freedom (6DoF) models, but is typically ignored due to it not being “required” for 3D rotations.

What this implies is that both the charge and mass differences in these particles can be assigned to the third intrinsic spin (rotation) type. It should be deeply related to the long known Higgs SM mechanism and its recently discovered Higgs particle near my predicted mass of 124.443…GeV. It should also help resolve the Distler and Garibaldi argument with Lisi related to incorporating the 2nd and 3rd generation of fermions into E8.