Tag Archives: ToE

The Isomorphism of E8, STA Hodge Star Octonion/BiQuaternions, and the [CU;AG] Codon GenoMatrix

The palindromic E8, STA Hodge star Octonion/BiQuaternions, and [CU;AG] Codon GenoMatrix can be integrated into a possible isomorphism between an E8 based ToE (i.e. unifying a 3 generation Grand Unified Theory (GUT) of the Standard Model (SM) with General Relativity (GR)) and the biochemistry of Life itself. It does this from a single mathematical 240+8 dimensional Exceptional Lie Algebra / Group / Lattice / Polytope in one Grand Objective Design structure with a Trinity of triads and triality.

The Palindromic Isomorphism of E8, STA Hodge Star Octonion/BiQuaternions, and the [CU;AG] Codon GenoMatrix

The above figure is a matrix integration of my E8-H4 Folding (Rotation) matrices, the STA Hodge Star Octonion/BiQuaternion multiplication matrices, and the [CU;AG] Codon GenoMatrix. The GenoMatrix is a modified form of that presented by Sergey V. Petoukhov, Head of Laboratory of Biomechanical System, Mechanical Engineering Research Institute of the Russian Academy of Sciences, Moscow in a 2011 paper arXiv:1102.3596 “The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts”.

The H4 Cell-First {5,3,3} 120-cell (J), dual of the 600-cell, showing the hull of the chamfered dodecahedron. It is also called a truncated rhombic triacontahedron, constructed as a truncation of the rhombic triacontahedron. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated. It is shown with physics particle assignments.

The H4 Vertex-First {5,3,3} 120-cell (J’) with physics particle assignments

The linkage between the GenoMatrix and E8 is not surprising when understood in the light of how E8 folds to 4 copies of the H4 4D duals of the 600/120-cells, which have the convex hull of the chamfered dodecahedron. This 3D shape is similar to the truncated icosahedron (aka. soccer ball) used in the (2003 abandoned patent) mnemonic created by Mark White and documented in 2007 arXiv here “The G-Ball, a New Icon for Codon Symmetry and the Genetic Code”. There is an interesting blog post in American Scientist about that here. The germ of this idea (pardon the pun) seems to have come from Petoukhov in 1981 (as per his list of works here) in a paper ” Investigations on Non-Euclidean biomechanics. Projective geometry, Fibonacci numbers and the kinematic scheme of human body.”

Truncated Icosahedron (aka. soccer ball) Codon by Mark White (image linked from Bryan Hayes “Ode to the Code”).

It is interesting to note that the particular octonion (or Hodge Star BiQuaternion) multiplication matrix used by Petoukhov is the same one as highlighted in my recent papers and blog posts regarding the most beautiful of the 480 octonion symmetry pattern, which I introduced in my “Isomorphism of H4 and E8” arXiv paper here and discussed in a bit more detail here “Space-Time-Algebra (STA) Octonionic Illustration”, with this summary:

This was originally introduced in my (always updated/corrected) last few papers here and here (or if you prefer to get the originals off arXiv here and here). It happens to be the first canonical triad (with Fano plane index fPi=1) with 16 sign-mask set of sm=5 taking the first sign-mask in that set being hex 08H (which reverses triad 4 from 246→264). This not only naturally produces a palindromic RCHO Standard model with E8 vertex counts, the STA Hodge star elements are in the reverse diagonal made up of e7‘s, its non-null Derivation has en with n= {1,2,3,4,5,6,7} with the unique split derivations being the 7 triads in order. It is the only octonion of the 480 that has that property! There are only 6 others (out of 48 total starting with the canonical quaternion first triad of {1,2,3}) that have the triad derivations, but the triad node orders are mixed along with the multiplication table +/- signs in the last row being mixed across {0,1,2,3} and {4,5,6,7}.

8×8 E8 and Octonion Matrices
E8 with physics particle assignments shown in triality projection

For completeness, we show below the 4D projection of the Atomic Element Periodic Table of Chemistry below:

3D/4D Octahedral/4-Orthoplex projection (Stowe) of a 4D Periodic Table with angular momentum color coding and spherical harmonic element orbitals

Updated ToE Summary of Fundamental Constants and a “more” Natural Unit-of-Measure (UoM)

In a post from 2013, I summarized the findings of my original 1997 paper (updated in 2007) documenting a new FineStructureConstant (α) based set of definitions for the fundamental constants (c, ħ, GN, and H0) . This post was due to updates from 2013 Planck spacecraft CMB (H0) results and the 2012 LHC confirmation of finding the mass of the Higgs boson (mHiggs).

About the same time, Wolfram Mathematica deprecated their PhysicalConstants package in order to provide for more robust curated datasets (e.g. CODATA ParticleData) which includes Quantity (exact or with experimental uncertainties) and UnitConvert capability.

Until now, I had maintained my own version of this package in order to add a custom UoM with conversions between it and the SI UoM based on my analysis of a possible unification of the fundamental constants and dimensional relationships of (L)ength, (T)ime, (M)ass, and (C)harge. These strongly indicate an 8D (E8) fine structure linked a posteriori particle mass prediction of the Higgs boson mass=124.44 GeV/C2 and cosmological constants of gravity (GN) and H0.

This post provides for the integration of that newer Wolfram Mathematica capability as well as the 2019 redefinition of the SI UoM which replaced some measures with exact unit definitions. These changes are described on Wikipedia here and by Wolfram’s Chief Scientist Michael Trott here.

My Powerpoint overview is here with the first slide shown below:

Mass-Length-Time (MLT) UoM with Fundamental Constants compared with SI UoM with 2019 exact and experimental error bars.

This new Time Unified UoM identifies a UnitLength= α/Rydberg Constant (R) associated with atomic mass spectroscopy and UnitAcceleration that is 4π(cH0=Acceleration AssociatedWith CosmologicalExpansion Rate) as a a dimensionless unit with UnitLength=(complexified UnitTime)2. It links the macro (cosmological) to the micro (nuclear/chemical) scales via integer exponents (dimensions) related to the FineStructureConstant (α). Other notable defining constants are a 4D (TimeUnit4) UnitCharge and 8D c=α-8 UnitVelocity, h-8 UnitAction, and Newton’s GravitationalConstant (GN~4πH08/UnitTime).

See below a list of the constants and UoM elements for SI, Natural (Planck or geometrized) units, and my own “more Natural” Time Unified UoM, which includes the Higgs mass (a posteriori) postdiction. The list is sorted by n where (Tn) is the unified (reduced) time dimension of the physical constant or UoM.

Mass-Length-Time (MLT) UoM with Fundamental Constants compared with SI UoM with 2019 exact and experimental uncertainties.
Mass-Length-Time (MLT) UoM code snippets used to create the above chart.

Cloud Based VisibLie_E8 Demonstration

The cloud deployments don’t have all the needed features as the fully licensed Mathematica notebooks, so I included a few of the panes that seem to work for the most part. Some 3D and animation features won’t work, but it is a start. Bear in mind that the response time is slow.

Link to the demonstration.

A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:

1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein

2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle

3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize

4) Math: Group Theory: Dynkin Diagram Algebra Create

5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize

6) Physics: Quantum Elements: Fundamental Quantum Element Select

7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity

8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays

9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation

10) Chemistry: Atomic Elements: 4D Periodic Table Element Select

11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select

12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding

13) Biology: Human Neurology: OrchOR Quantum Consciousness

14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori

15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria

16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier

17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life

18) CompSci: Human/Machine Interfaces: nD Human Machine Interface

My VisbLie E8 demonstration system for Mathematica v13

The newer version of the VisibLieE8-NewDemo-v13.nb (130 Mb) will work with those who have a full Mathematica v13 license. It is backward compatible to earlier versions. There are a few bug fixes from the older version of ToE_Demonstration.nb (130 Mb), which should work on v13 and older versions as well.

For more detail on the modules, see this blog post.

Please be patient, it is very large and can take 10 minutes to load, depending on your Internet connection, memory and CPU speed.

Updated Analysis of RCHO Bi-Octonion Standard Model (Cohl Furey Papers)

For the latest (as of 09/23/2024) on the topic see this post.

I’ve updated an analysis I did on the work of Cohl Furey’s papers from several years ago. Since then, she added another paper: https://arxiv.org/abs/1910.08395v1

The short pdf version of my analysis (with some detail cells collapsed) is here (34 pages), and a longer version is here (no collapsed cells and 51 pages). These pdf’s are a direct output from my Mathematica (MTM) Notebook. I will follow up with a LaTex paper on the topic soon.

This notebook has code built in to operate symbolically on native MTM reals, complexes, and quaternionic forms, as well as my custom code to handle the octonions, and now the bi-octonions (which doesn’t assign the octonion e1 to be equivalent to the complex imaginary (I)). That change also applies to the native quaternion assignments where of e1=I, e2=J, and e3=K in order to work with quater-octonions. This was a fairly trivial change to make since it simply involves removing the conversion of complex (and quaternion) operators from being involved in the octonionic multiplication.

Please note that my previous analysis here (from Feb. 2019) made the mistake of not commenting out these operations. As such, it was operating on octonions (not complexified bi-octonions), so some of my concerns were resolved based on correcting that error.

The bottom line is that I did validate much of the work presented in the referenced papers, with the exception of some 3 generation SM charge (Q) assignments in that latest paper (Oct. 2019).

I am very interested here in the suggestion at the very end of that paper [5] in the Addendum Section IX(B/C) on Multi-actions splitting spinor spaces, Lie algebras/groups, and Jordan algebras. I suspect having the ability to create a machine (i.e. a symbolic engine such as MTM) to operate on and visualize these structures as hyper-dimensional physical elements is critical to making progress in understanding our Universe more thoroughly.

While I have had some success in replicating quark color exchange, as well as flavor changes (e.g. green u2 to d3 quark exchange using g13), there doesn’t seem to be a complete description of how to construct each of those color and flavor exchange actions from the examples given. So for reference I present all possible combinations of these actions across the particle/anti-particle definitions (see the image linked in the last paragraph of this post). This comment about limited examples also applies to replicating the 3 generation charge (Q) calculation using the sS constructs mentioned above.

I welcome any help or advice or additional examples.

Below is an example image of the 3 generation SM from the 2019 paper built from bi-octonions (with my octonion multiplication table reductions applied. The anti-particles (not shown) are simply the complex-conjugate of these. While I show in string form of Q, I am not showing the commutations based evaluations for them due to the questions / issues I have on how to get it to work.

The image below shows more detail of the 3 generation SM from 2014 with my code implementing the reductions. This leaves off the charge (Q) which was not defined as above in 2014 (AFAIK).

The image below shows a simple construction of the 0-V to 6-V splitting of the Mf Clifford algebraic structures, which I generated using MTM Subsets:

The rather large (long) image here checks all SM particle color and flavor changing actions and includes the anti-particles. The output is extensive and given my open questions on the formalism presented, the accuracy likely deviates from the intent of [5], but it is interesting to show how everything transforms. If no transform is found for a particular action, it outputs an * for that action. If a color or flavor changing transformation action is found, it identifies that action with the list of particles to which the transformation applies. Note: it only identifies a transformed particle if the source particle has a non-zero reduced value and the resulting match is exact (red) or a +/- integer factor of that particle (blue).

Updated My ToE Demonstrations to Wolfram Language (aka. Mathematica) 11

Please see the latest in .nb, .cdf demonstrations files and web interactive pages.

ToE_Demonstration-Lite.cdf Latest: 08/15/2016 (10 Mb). This is a lite version of the full Mathematica version 11 demonstration in .CDF below (or as an interactive-Lite web page) (4 Mb). It only loads the first 8 panes and the last UI pane which doesn’t require the larger file and load times. It requires the free Mathematica CDF plugin.

This version of the ToE_Demonstration-Lite.nb (13 Mb) is the same as CDF except it includes file I/O capability not available in the free CDF player. This requires a full Mathematica license.

ToE_Demonstration.cdf Latest: 08/15/2016 (110 Mb). This is a Mathematica version 11 demonstration in .CDF (or as an interactive web page) (130 Mb) takes you on an integrated visual journey from the abstract elements of hyper-dimensional geometry, algebra, particle and nuclear physics, Computational Fluid Dynamics (CFD) in Chaos Theory and Fractals, quantum relativistic cosmological N-Body simulations, and on to the atomic elements of chemistry (visualized as a 4D periodic table arranged by quantum numbers). It requires the free Mathematica CDF plugin.

This version of the ToE_Demonstration.nb (140 Mb) is the same as CDF except it includes file I/O capability not available in the free CDF player. This requires a full Mathematica license.

(The CDF player from Wolfram.com is still at v. 10.4.1, so still exhibits the bug I discovered related to clipping planes/slicing of 3D models).

A ToE should…

An interesting post re:qualifications for a ToE prompted me to jot down my initial list of requirements. A ToE should inform, expand on, or rationalize, in a mathematically self consistent and rigorous way, the state of the current SM & GR confirmed experimental data:

1) Prescription (aka prediction, retrodiction or specific rationalization) for 3 generations of fundamental fermion and boson particles (and their resulting composite particles), including: charge, spin, color, mass, lifetime, branching ratio, and scattering amplitudes (aka. S-Matrix)
2) Prescription for CKM and PMNS unitary matrices and CPT conservation
3) Framework for the integration of QM and GR, including e/m, weak, strong and gravitational forces
4) Explanation for dark energy and dark matter in proportion to visible matter
5) Solution to the hierarchy problem
6) Provide a realistic computational model based on the above for the evolution of the Universe from BB to present
7) Explain an arrow of time that is consistent with GR and QM CPT conservation symmetries

Non-specific general appeals to the anthropic principle, landscapes, and/or multiverses tend to excuse or avoid prescription and thus become a benign point (or possibly even meta-physical or philosophical), such that they are not considered supportive of an actually verifiable (aka. scientific) theory.

If the theory says “we can’t know” or “we can’t measure” or “it just is that way” – it isn’t science or part of a ToE. again – my opinion and definition of “science”.

There is redundancy in this list, that is expected (even required). Of course, the beauty of the theory would be in conclusively demonstrating that throughout!

Until we can study an actual ToE that is put on the table – the list is only a guide to what might be needed. I am working on a ToE, but it doesn’t yet meet all the criteria (it’s hard work 😉

IF we have a ToE and really understand it, we should, as Feynman suggested, be able to explain it in plain language to anyone. But in the current state of physics, a completed ToE does not yet exist.

IMO, a ToE is about knowing the Universal Laws of Physics (ULPs). It isn’t, in detail, involved in knowing the Universal Initial Conditions UICs).

If you believe that the laws of “climate science” are known (don’t get me started…), then the only problem with predicting the weather is not so much about NOT knowing the laws – it is about not knowing with sufficient accuracy the initial conditions (location & momentum) of enough particles in the system. We’re missing the “butterfly flapping its wings in the Pacific” data points.

The Copenhagen interpretation of QM suggests that is impossible in principle to know any quantum system ICs (vs. the deterministic formulation of QM by DeBroglie-Bohm). Either way, my view is ToE=ULP.s w/o UICs. So prediction of all long term events specifically (like what I will think about next) is NOT the goal.

We just need enough of an idea about the UICs to initiate the computer model so it comes out close enough to get Earth like planets with weather and life forms thinking about this topic.

Easier, but NOT easy!

Universal History Simulations Added

I’ve added more features to the Universal History N-Body Gravity OpenCL GPU Simulation. Here are a few snapshots…

Big Bang – Planck Epoch
outNBody_1a (7)

Inflationary Epoch(s) w/Pre & Post (These are built into my ToE as an accelerating space-time, varying FineStructure, c and hBar)

ElectroWeak Symmetry Breaking Epoch
outNBody_1a (10)

Quark-Gluon Plasma Epoch
outNBody_1a (11)

Meson Epoch
outNBody_1a (5)

Baryon Epoch
outNBody_1a (4)

Lepton Epoch
outNBody_1a (6)

Big Bang NuceloSynthesis Epoch- Nuclei of Hydrogen/Helium/Lithium
outNBody_1a (2)

Photon Epoch- Atoms of Hydrogen/Helium/Lithium
outNBody_1a (3)

Matter Dominated Stellar NucleoSynthesis- Quasars
outNBody_1a (9)

ReIonization- Galaxies
Galaxy

Dark Ages- Large Scale Structure
outNBody_43a

Recombination- Solar Systems/Chemistry/Biology/Sociology
outNBody_1a (8)