Category Archives: Philosophy

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

VisibLie E8: 3D Perfect Sri Yantra & 2D I-Ching Cloud Demonstrations

I’ve published a version of these demonstrations in the cloud, so no Mathematica installation is needed.

Please click here for the interactive Hindu Sri Yantra demonstration shown below.

Image of the User Interface (UI) of the 3D Sri Yantra demonstration

Please click here for the interactive Buddhist (Tao/Confucian) I-Ching demonstration shown below.

Image of the User Interface (UI) of the 2D I-Ching demonstration.

The Mathematica Cloud version doesn’t work in 3D as well as using a local .NB version with the free Mathematica CDF player. You can File->Download the local demonstration and use it after installing the CDF player found here.

Of course, the full licensed Mathematica Notebook works the best for this. See this file here for the Hindu 3D Sri Yantra and here for the Buddhist 2D I-Ching notebooks.

There is even a full pane 15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria here.

Here is a list of other Interactive Demonstrations within my VisibLie-E8 package:

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

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!

More Fibonacci / Pascal Triangle Patterns

Last year I posted this on Pascal Triangle Modulo 2 thru 9 and their Sierpinski Maps.

I just saw an interesting post by Lucien Khan on some Theological numerology (gematria) as it relates to the Modulo 10 patterns of Fibonacci here.

I verified the pattern and added a few more as it relates to the repeating Modulo 9 and 10 patterns, Pascal Triangle and E8 (as in my original post).

Numbers are highlighted when divisible by:
60 are in red
30 are in green
15 are in blue
12 are in magenta

Fibonacci2

This is all tied to the Pascal Triangle with its 2^n binary (Clifford Algebra) representation, E8 and the number 4!=24 and Octonions.

PascalNew

All of these structures, including the 3 dimensional geometry of the Platonic solids (and its big brother, the 4D non-crystallographic H4 group geometry) are related to E8 via a folding matrix I determined a few years ago here.

For some related interesting reading from John Baez regarding the number 24 and its link to String Theory (M-Theory) updated from last week! (see also related info on his other favorite numbers 5 and 8).

My E8 F4 projection and so called "Sacred Geometry"

Please note, this is about the GEOMETRY used in what has come to be called “Sacred Geometry”. Upon further investigation, along with the Kabbalah’s “Tree of Life”, it seems even the Jewish “Star of David” (Solomon) is medieval and may not be sourced in authentic ancient Abrahamic theology.

As it stands, the so-called “sacred” origin of these, including the pentagon that shows up in the icosahedral/dodecahedral Platonic solids due to the Sqrt(5) Golden Ratio (Phi) relationship within my E8 to H4 folding, is likely medieval and mostly pagan. I am merely pointing out that E8 contains the geometry that many find “interesting” for whatever reason, and am not promoting the underlying cults that use them.

Video (Best in HD):

My E8 F4 projection w/blue triality edges and physics particle assignments:
StarOfDavid1

My E8 F4 projection with 6720 edge lines:
E8-F4mox1-black

My E8 F4 projection rectified:
F4Rect-Sacred-1a-StarOfDavid

Star Of David Overlay:
F4Rect-Sacred-1b-StarOfDavid

Metatron’s Cube Overlay:
F4Rect-Sacred-1a-Metatron

Flower of Life Overlay:
F4Rect-Sacred-1a-Flower

Kabbalah Tree of Life Overlay:
F4Rect-Sacred-1a-Tree

Copyright J Gregory Moxness 2015
Rectified E8 and F4 Triality (Original Work)

WikiMedia Commons source images:
Metatron’s Cube (User:Barfly2001)
Flower of Life (User:Life_of_Riley)
Tree of Life (User:AnonMoos)

7 Perfect Sri Yantras in 2D and 3D

Based on an excellent 1998 paper by C.S. Rao I’ve modeled the Sri Yantra in Mathematica and incorporated it into the VisibLie_E8 demonstrations (I will upload that soon).

This is a reference figure to verify the model is correct for concurrency (intersecting lines) and concentricity (inner triangles and Bindu point are centered). There are 4 of 18 other constraints used in the diagrams below numerically and/or symbolically solved within the Mathematica code.

SriYantra-Refernce

Here are 7 of many possible solutions using the 20 constraint rules to produce optimal spherical and plane forms of the Sri Yantra.

Solution 1: Solving constraints (* 1,2,3,4,8 *)
SriYantra1

Solution 2: Solving constraints (* 1,2,3,10,15 *)
SriYantra2

Solution 3: Solving constraints(* 1,2,4,5,10 *)
SriYantra3

Solution 4: Solving constraints (* 1,2,5,6,19 *)
SriYantra4

Solution 5: Solving constraints (* 1,2,6,14,19 *)
SriYantra5

Solution 6: Solving constraints (* 1,2,8,9,20 *)
SriYantra6

Solution 7: Optimum planar
SriYantra7

3D using pyramids and new clipping and slicing functions to take off the front (and back) portions of the symmetric 3D object.
I will produce a version option with conical shapes just for fun (soon).
SriYantra3D1

This is a list of the details behind each constraint function Fn
ConstraintFunctions