Rick,

I went through the below and was doing fine but am no mathematician, my skills are elsewhere, so I faltered when I reached "ordered permutation triplet basis product rules."

My question is; will octonians help in a mathematical description from twin pairs of handed (complementary) momenta with inverse Cos distributions? so finding the QM Hamiltonian from the Lagrangian?

As an insight, I replied to your post about barmaids on my string as follows.;;

Rick,

You're right,ish, Classic QM was trickier for barmaids than logical SR, but I've shown it possible with my rotating sphere. Viz; Get her to shut her eyes, spin it on a vertical axis, then;

1. Touch her finger on a pole and ask 'Is it going left or right?' (= 0)

2. Do so on the equator & ask 'Is that clockwise? or anticlockwise? (= 0)

3. Touch it on the N pole & ask 'Is it going clockwise or anti..? (=-1)

4. Touch it on the S pole & ask 'Is it going clockwise or anti..? (=+1)

5. Do so on the equator her side & ask 'Is it going left or right?' (=-1)

6. Same on the other side (or flip the poles) & ask 'left or right?'(=+1)

7. Finally at latitude 45supo & ask is it moving or rotating? (=both)

Now we KNOW the spin AND linear speed both change NON-linearly, by Cos latitude. Rotate the polar axis in any plane and that doesn't change. Three out of five barmaids understand.

Now ALSO tell them each sphere re-emits at 'c' with respect to itself whatever the original 'closing speed', and there are millions on the surface of a lens, and her understanding of SR allows complete unification with QM. There are a number of barmaids around who now understand that (more) logical analysis! Some were impressed enough to... well you'll need to use imagination.

Can you find logical or epistemological fault?

Very Best

Peter

The ordered triplet rule is simple when explicitly stated, which I did not. For a set of three basis elements, say {e1 e2 e3}, they can specify two cyclic product rules based on their order. The ordered rule (e1 e2 e3) is a simple mnemonic for 6 separate products where there are three possible products going through the order shown cyclically left to right and three more going right to left. The basis element at one of the 3 positions multiplied by the next one to its right is the + the next in progression. '"Cyclic" means when you get to the end, the next in progression is on the other end, you wrap around. If you go right to left instead, this is equivalent to flipping (commuting) the order of the multiplication and this instead specifies the result is - the third basis element.

So the ordered permutation triplet basis product rule (e1 e2 e3) is a shorthand expression for all 6 possible basis pair products between two of three elements, implying also the operation is closed for the set, meaning of course the product of two set members is another set member. The explicit results implied by (e1 e2 e3) are

e1 * e2 = +e3, e2 * e3 = +e1, e3 * e1 = +e2 cyclic left to right

e2 * e1 = -e3, e3 * e2 = -e1, e1 * e3 = -e2 cyclic right to left

If we instead did the same rule application on (e3 e2 e1), we would find opposite signs on all 6 products above. This is the only alternate definition. Every possible permutation of the three basis elements will fall into one of these 2 rules.

You could look at (x y z) describing a right hand vector cross product and (z y x) describing a left hand vector cross product. The triplet rule also expresses all products of different non-scalar Quaternion basis elements and the 2 separate ways it can be defined.

On Bell's "theorem", I once had a collegial relationship with someone who thought he had something to say on the subject using essentially the two ways to define Quaternion Algebra as fair coin statistical choices. He made the mistake of singularly enumerating the basis elements but using both rules in the same expression. This "spooky algebra at a distance" gave the desired minus cosine response because simultaneously applying both rules to the same basis names changed the algebra from Quaternion to an algebra with only a scalar product. I tried my level best to help him understand he made what on the surface looked like a simple sign error. A number of other people also tried to make him understand his mistake. Instead of listening and admitting he made a mistake, he and his supporters set out on a years long campaign of personal attacks. My last words on the subject was in Retraction Watch, where he was whining about his published paper with the very same math errors being retracted. After the personal attacks I have zero respect for him and all of his supporters, and will have nothing to do with any of them going forward. I am a little touchy on the subject.

Bell did not touch the metrology of the experimental equipment nor the causal effects on it by the particles. I do not think this can be divorced from the discussion. You seem to be attempting to take a pass with your spherical analogy getting to the requisite +1 and -1 detector clicks. I think the difficulty is going from a sequence of +1 and -1 data values to the minus cosine angular effect. If you take axiomatically the existence of a probability distribution behind each +1 or -1 result, it certainly can be recovered from a large number of samples. This is what QM does, but classical non-stochastic methods can't. QM also seems to force on the subject its credo nothing is known on either side until one side makes a measurement and this forces the distant measurement results to be what they become, but with "spooky" non-communication that has no possible classical description. This pushes the probability analysis in favor of the QM position. The problem of insuring the measurements actually are on "entangled " pairs can easily provide a mechanism to cherry pick the data removing data that may refute the QM position. In short, the game seems rigged. As far as my tackling this, it is so far down the list I will die before it comes up.

Rick

Hello yet again Rick,

I'd like to be kept in the loop once your calculational software is available for use. I've kept in touch with Louis Kauffman in Chicago, who was one of the authors of a book called "Hypercomplex Iterations" about higher dimensional fractals, and I just mentioned your software package in an e-mail to him. So don't be surprised if there are some people interested to see what it can do. I hope you can get a community of people working with the octonions, as a result of your software development efforts.

All the Best,

Jonathan

Jonathan,

I have read your essay a couple of times since it first came out, really do not know how to react to it so haven't. My ideas about Gravitation are a bit more pedestrian, so I have no feel for how the Mandelbrot set might fit in. My ignorance, not my neglect of your efforts.

Thanks for your kind words and support over the years.

On the symbolic algebra software, it was developed to run on freeware Nodejs online downloadable. I have not developed a nice user interface above this since have no time and ok with opening a Windows old style dos command box, typing in command line info to launch node.js pointed at my specific script file that is in a directory with a subdirectory holding my classes and methods that do the work. The command box is the console out, good for errors but not much else, I only look at log file text data generated by the show function output to the console optionally logging to a text file.

I will be putting up a boilerplate script file that has comments saying your stuff starts here and ends here so users can use a text editor to put in calls to the classes to do what they want. Also will be a finished and complete script verifying my conservation equations. It cranks out a ~260k text file with the details.

Maybe some kind person with the time and skills would write something to bring in, edit, save and run user scripts with some contextual type-ahead once you start entering a method giving instant parameter documentation to ease use. Open source would be nice.

I am partially through documentation of classes and methods, someone would have to dig it out of the source code without it but what I put out will have it all fully visible, no library.

Rick

This sounds excellent Rick...

Maybe Phil Gibbs would be up for helping with the interface. I also got a thoughtful reply from Lou Kauffman in Chicago. I'll also reach out to P.C. Kainen. But even with a console-style interface; I'm sure it will be useful and helpful. As I recall; it's the same deal if you want to work with the Atlas of Lie Algebras software. But people take the numerical output and turn that into beautiful graphs anyway. Heck; I still create some of my graphs that way. So I guess development of specific interfaces will be driven by the people who would put these things to use, depending on what the usage would be.

The theory of gravitation I am describing arises most clearly in the context of octonionic inflation. You might find papers by Vladimir Dzhunushaliev and Merab Gogberashvili interesting in that regard. You can find a similar result in recent work from Tony Smith. But it reproduces the phenomenology in braneworld scenarios like DGP gravity and Cascading gravity theories. Inflation goes out to a 5-d volume (which is maximal for hyperspheres anyway), but there is a phase change resulting in a 4-d spacetime bubble. In octonionic theories the bubble is quaternionic.

Anyhow; this corresponds with a situation described by Afshordi, Pourhasan, and Mann, where a black hole in a 5-d volume gives rise to a 4-d spacetime, via a dimension shifting wormhole solution. A similar solution is obtained by Nikodem Poplawski, working from Einstein-Cartan theory by way of Sciama and Kibble. And a result from Frolov and Novikov states that the only stable or eternal black hole is a black hole --> white hole solution. However; this exactly reproduces what octonionic inflation theory says should happen - giving rise to our current universe and spacetime configuration - at least according to some.

Well it turns out the Mandelbrot Set gives the same answer, if used as a chart for cosmology, and that the reason why connects back to the octonions. The rolling ball analogy for G2 proposed by Cartan places one ball on another three times as large. If the Mandelbrot Set is rotated about the real axis; it is pretty easy to see that this analogy is fulfilled. But Cartan's analogy involves a 5-d ball explicitly - duplicating the cosmological scenario outlined above. Anyhow, as shown by Kricker and Joshi in 1995; this is an inflection or bifurcation point in the octonion quadratic. I can forward my copy. The transition from non-associative geometries to quaternionic associativity makes outward facing gravity point inward in the spacetime bubble we inhabit.

All the Best,

Jonathan

Dear Rick Lockyer

Wonderful words about truth in science..."The most fundamental concept in physical reality we generally refer to as Nature is truth. Nature's truths are independent of our existence. Some of Nature's truths are readily apparent, others are hidden by obscurity. Some truths are before us yet await our emergence from ignorance or bias. Pure science is the pursuit of understanding Natute's truths"......

And you are exactly correct in saying that' True science has no political agenda, is not about corroborating preconceived notions, and does not necessarily require conformance with group think or so called "settled science".

I hope you will not mind that I am not following main stream physics...

By the way...Here in my essay energy to mass conversion is proposed................ yours is very nice essay best wishes .... I highly appreciate hope your essay ....You may please spend some of the valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance

Some of the Main foundational points of Dynamic Universe Model :

-No Isotropy

-No Homogeneity

-No Space-time continuum

-Non-uniform density of matter, universe is lumpy

-No singularities

-No collisions between bodies

-No blackholes

-No warm holes

-No Bigbang

-No repulsion between distant Galaxies

-Non-empty Universe

-No imaginary or negative time axis

-No imaginary X, Y, Z axes

-No differential and Integral Equations mathematically

-No General Relativity and Model does not reduce to GR on any condition

-No Creation of matter like Bigbang or steady-state models

-No many mini Bigbangs

-No Missing Mass / Dark matter

-No Dark energy

-No Bigbang generated CMB detected

-No Multi-verses

Here:

-Accelerating Expanding universe with 33% Blue shifted Galaxies

-Newton's Gravitation law works everywhere in the same way

-All bodies dynamically moving

-All bodies move in dynamic Equilibrium

-Closed universe model no light or bodies will go away from universe

-Single Universe no baby universes

-Time is linear as observed on earth, moving forward only

-Independent x,y,z coordinate axes and Time axis no interdependencies between axes..

-UGF (Universal Gravitational Force) calculated on every point-mass

-Tensors (Linear) used for giving UNIQUE solutions for each time step

-Uses everyday physics as achievable by engineering

-21000 linear equations are used in an Excel sheet

-Computerized calculations uses 16 decimal digit accuracy

-Data mining and data warehousing techniques are used for data extraction from large amounts of data.

- Many predictions of Dynamic Universe Model came true....Have a look at

http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.html

I request you to please have a look at my essay also, and give some of your esteemed criticism for your information........

Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.

In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from "http://vaksdynamicuniversemodel.blogspot.in/ "

I request you to please post your reply in my essay also, so that I can get an intimation that you replied

Best

=snp

Jonathan,

My software is designed to throw full Octonion differentiation "del" operators at full Octonion variable not numeric coefficient algebraic elements that begin as the native Octonion 8-potential. My fundamental approach looking for a potential based theory for unifying Electrodynamics and Gravitation is supported by this. It currently does not do algebraic manipulations on numeric coefficient algebraic elements. The structure to cover scaled multiple differentiation product terms is overkill if you just want numeric values, but I guess it would not be too difficult to use the primitive basis product methods and a new numeric only algebraic element type, certainly would execute faster. Why don't you outline how you would use it for your Mandelbrot analysis for me.

Rick

Hello again Rick,

I'd like to recap some of what I said in the hidden comment above. To my mind; inflation and geometrization are automatic in an octonion embedding space, because of the built-in dynamism you describe as the octonions 'needing to drive' and I describe as self-evolving or sequentially evolutive properties of non-associative geometries - applied to Physics - detailed somewhat in my essay from last year about the elephant in the room with quantum gravity researchers.

Well it turns out the same elephant is there with cosmologists too. The exact nature of the inflaton remains a mystery, and the question also remains of why it shuts off when it does. What if the inflaton is non-associativity? This makes the octonions the driver of inflation. If we look at the properties of spheres; we find the 7-sphere is maximally spacious in hypersurface area, but not volume. The maximal volume for hyperspheres (or rather balls - filled spheres) is in 5-d.

These parameters determine that in octonionic inflation, things evolve to a 5-d volume; where inflation ends because there is no further to go and things flip into a quaternionic bubble - as in Cartan's rolling ball model of G2, but where the outer ball is assumed to be 4-d. This transition is effected because the change from non-associative to associative geometry makes the previously outward facing force of gravity turn inward - toward the center of massive objects.

I apologize if this is a distraction Rick, but I was trying to solidify these thoughts in my own mind, and it made sense to elaborate on what I said in the comments above.

Warm Regards,

Jonathan

Thanks greatly Rick!

The complication arises because of the need to use distance estimators rather than firm measures of magnitude, at some point, when calculating hypercomplex fractals. So the implementation of calculus primitives could prove useful or perhaps essential to doing what I want. Ultimately; I'd like to do a fly through of structures in higher-d space via projection (i.e. - 8-d projected onto 4-d then displaying 3-d sections). But that's very ambitious and I want to prove some more basic conjectures first, and build up a vocabulary of working knowledge about the higher-d examples of my butterfly wings and discs - that should be easier to calculate than M itself.

More later,

Jonathan

http://www.scirp.org/journal/PaperInformation.aspx?PaperID=5129

Hi, I took a venture into this area myself in 2011 and have referred to it again as seen in further JHEPGC articles, which have appeared in 2016 and 2017

My take, is that what is crucial is the concept of asociative geometry and when it does and does not break down, and the Octonians are a good vehicle to examine it

Although I did not state it earlier, my work on the cosmological constant was based upon themes of the contributions of Octonionic space time.;

Please feel free to examine my essay, December 21st and interpret it with this in mind

Thank you for your wonderful essay

Andrew

    http://www.scirp.org/journal/PaperInformation.aspx?PaperID=5129

    Has this in it

    Octonionic Gravity Formation, Its Connections to Micro Physics

    Full-Text HTML Download Download as PDF (Size:394KB) PP. 13-18

    DOI: 10.4236/ojm.2011.11002 4,520 Downloads 8,836 Views Citations

    Author(s) Leave a comment Andrew Beckwith

    Affiliation(s)

    .

    ABSTRACT

    We ask if Octonionic quantum gravity is a relevant consideration near the Planck scale. Furthermore, we examine whether gravitational waves would be generated during the initial phase, , of the universe when triggered by changes in spacetime geometry; i.e. what role would an increase in degrees of freedom have in setting the conditions during , so that the result of these conditions can be observed and analyzed by a gravitational detector. The micro physics interaction is due to the formation of a pre Planckian to Planckian space time transition in spatial dimensions at and near the Planck dimensional values, i.e. 10-33 centimeters in spatial dimensions. This transition would be abrupt and arising in micro physics regimes of space time.

    KEYWORDS

    High-Frequency Gravitational Waves (HFGW), Symmetry, Causal Discontinuity

    Cite this paper

    A. Beckwith, "Octonionic Gravity Formation, Its Connections to Micro Physics," Open Journal of Microphysics, Vol. 1 No. 1, 2011, pp. 13-18. doi: 10.4236/ojm.2011.11002.

    References

    [1] L. Crowell, Quantum Fluctuations of Space-time, in World Scientific Series in Contemporary Chemical Physics, Volume 25, Singapore, Republic of Singapore, 2005.

    [2] A. W. Beckwith, "Identifying a Kaluza- Klein treatment of a Graviton permitting a de-celeration parameter Q(Z) as an alternative to standard DE ",Journal of Cosmology, Volume 13, 2011, http://journalofcosmology/BeckwithGraviton.pdf

    [3] F. Li, and N. Yang, "Phase and Polarization State of High Frequency Gravitational waves", Chin Phys. Lett. Vol 236, No 5(2009), 050402, pp 1-4

    [4] F. Li,. M.,Tang, D. Shi, "Electromagnetic response of a Gaussian beam to high frequency relic gravitational waves in quintessential inflationary models", PRD 67, 104008 (2003), pp1-17

    [5] S. Chaturvedi, "Mutually Unbiased Bases", Pramana Journal of Physics, Vol. 52, No. 2, pp 345-350

    [6] John C. Baez, "Renyi Entropy and Free Energy", http://arxiv.org/abs/1102.2098

    [7] A. W. Beckwith, arXiv:0804.0196 [physics.gen-ph], AIP Conf.Proc. 969:1018-1026, 2008

    [8] D. K., Park, H. Kim, H., and S. Tamarayan, "Nonvanishing Cosmological Constant of Flat Universe in Brane world Senarios," Phys. Lett. B 535, 5-10 (2002).

    [9] A. W. Beckwith," How to Use the Cosmological Schwinger Principle for Energy Entropy, and "atoms of Space-Time" to Create a Thermodynamic Space-Time and Multiverse"; accepted for the DICE 2010 proceedings; http://vixra.org/abs/1101.0024, 2010

    [10] M. Maggiorie,. "Gravitational Wave Experiments and Early Universe Cosmology", Physics Reports 331 (2000) pp. 283-367

    [11] G. t' Hooft, Beyond the Quantum, edited by Th. M. Nieuwenhuizen et al. Singapore, World Press Scientific, 2006; http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604008v2.pdf.

    http://www.scirp.org/journal/PaperInformation.aspx?PaperID=79959

    The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors

    Full-Text HTML XML Download Download as PDF (Size:572KB) PP. 727-753

    DOI: 10.4236/jhepgc.2017.34055 93 Downloads 140 Views

    Author(s) Leave a comment Andrew Walcott Beckwith Affiliation(s)

    Physics Department, College of Physics, Chongqing University Huxi Campus, Chongqing, China.

    ABSTRACT

    The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real "instants", is avoided by having initial Octonionic geometry embedded in a larger, nonlinear "pilot model" (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a "big crunch" is picked as the embedding structure, so as to avoid the "instants" of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards "Schrodinger equation" states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to; making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry.

    KEYWORDS

    Octonionic Geometry, Cyclic Conformal Cosmology (Penrose), Modified HUP

    That is the recent iteration as opposed to what I did in 2011

    Again, the methodology, was implicitly used in My December 21st essay

    Hi Rick , thank you for sharing your passion for algebra and in particular octonians.

    At the beginning you state "mathematics is a language of unambiguous truth", I used to think that the certainty of mathematical solutions do make it more reliable than other kinds of communication. Though statistics is notorious for misleading. Having read "Creating perspective on set theory" by Christophe Tournayre, in this competition, I have been re-considering my opinion. I found this paper "When is one thing equal to some other thing?" Barry Mazur (2007) www.math.harvard.edu/~mazur/preprints/when_is_one.pdf Which showed me that even something as common place as use of the equality symbol is not simple and not immune from ambiguity. Also a statement, mathematical or otherwise, can be true, as in not false; and yet not be the whole truth. A more complete truthfulness painting an entirely different picture. An algebraic description can give a singular viewpoint of a phenomenon but does not give all viewpoints, which would be the whole truth. Something Peter is approaching with different observations of the same rotating phenomenon. None of the singular values being the whole truth.

    In a universe where there is sequential passage of time, as needed by QM and fitting the arrows of time, there is non-commutivity. As new Now follows previous Now definitely and so when things happen in the representation is important. Which is something in favour of quaternions ( and maybe their big bother). Though as I see it the real number line is not an actual dimension of the universe where physics is happening, but a history time line without physical counterpart. You mention 4 dimensional space from primary sense perceptions. That has to be a separate 'map' space from the former representing where physics is happening. The sense space has a time dimension, and non simultaneity of events can happen as the time is related to signal transmissions and not the foundational sequential passage of time. The two different forms of universe can not be described by just the one form of the algebra because of the fundamental difference in what time is in each. If quaternions were used for both in one version there is the non physical history time line and in the other the geometric time dimension.

    Thank you for food for thought. I am going to think on about non associativity in nature and whether it is needed in a representation, and about representation of the different kinds of field, that you discussed. Kind regards Georgina

      I posted this on Geoffrey Dixon's essay thread, may be interesting to readers coming here:

      Geoffrey,

      Knowing you are not big on the Cayley-Dickson doubling algorithm, I did neglect to mention another reason my Quaternion triplet enumeration algorithm is cooler than yours (no offense intended). That would be for enumerating the triplets for the sedenions. For Octonion Algebra I used the binary numbers 1 through 7 and partitioned the triplets with the bit wise exclusive or logic function. This can be extended to the sedenions by going to binary 1 through 15, of course representing directly the 15 non-scalar basis elements enumerations. Doing the exclusive or logic operation on binary 1 through 15 partitions them into 35 unique closed triplet sets, and since there are (n-1)*(n-2) = 15*14 = 210 basis products not e0 * en, en * e0 or en * en fixed product rules, and each triplet does 6 rules, 35*6 = 210, covering all 210 basis products with 35 Quaternion triplets.

      For each Octonion subalgebra using this enumeration technique the seven non-scalar basis element indexes will exclusive or to a zero result, as well each Quaternion index triplet. If you take away from the seven Octonion non-scalar basis indexes any set of three which are Quaternion triplet indexes, the remaining four basis elements are called "basic quads" and their indexes will also exclusive or to zero. They are basic quads because given just them, the exclusive or of any two will be the index of one of the companion Quaternion basis triplets and all are provided 2 up with this.

      This gives a method to determine how many Octonion subalgebra candidates there would be, just go through all quads of distinct indexes 1 through 15 for combinations that exclusive or to zero. A simple computer program will show there are 105 such quads. Any Octonion definition has seven Quaternion triplets each with a unique basic quad, so we would expect to see 105/7 = 15 candidate Octonion subalgebras. Each of the 35 Quaternion triplets occurs in 3 Octonion subalgebras 3*35 = 105.

      The rub is you can't make all 15 Octonion subalgebra candidates represent valid normed composition algebras, e.g. division algebras. An algebraic proof the sedenions are not a division algebra? Maybe so.

      Thought you might find this interesting.

      Rick

      Dear Rick

      I agee and like the term Truth is most important, but I am wondering how to measure it or who can measure it?

      Here is my essay that really think there is some fundamental trurh about the nature of particle and gravity, please comment.

      Best wishes

      Bashir

        Dear Rick

        Here is my essay

        https://fqxi.org/community/forum/topic/3143

        Bashir Yusuf.

        Hello Mr Lockyer,

        It seems relevant these octonions not commutative to unify this gravitation, have you already thought about the fact to insert the spherical volumes and their motions and this DM not baryonic.Because we cannot find this weakest force wityh these lie algebras, it lacks something.I read the works of Hestenes with the spinrs and geometrical algebras, it is a good tool also,that said this quantum weakest force have difficluties to appear .....Best regards and congratulations for your essay, good luck :)

        Hello yet again Rick,

        I wanted to pass this on. The Kricker and Joshi paper is behind a pay wall at Elsevier; but a scanned copy of the manuscript was available from Kricker's web page. So here is that paper "Bifurcation Phenomena of the Non-Associative Octonion Quadratic."

        Enjoy,

        Jonathan

          Darn...

          I guess just less than 1 MB is not good enough. I'll send it your way the old fashioned way - e-mail.

          Regards, JJD