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CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: The Algebra of "Everything" by Rick Lockyer [refresh]

Author Rick Lockyer wrote on Aug. 31, 2012 @ 12:07 GMT
Essay Abstract

Next year we will mark the 170th anniversary of John T. Graves’ discovery of Octonion Algebra. Since its discovery Octonion Algebra has for the most part languished in relative obscurity. Almost everyone that studies Octonion Algebra revels in the beauty of its algebraic structure, yet precious few have come to believe it has any connection to physical reality. Why is this? Historically, Quaternion Algebra H got a bum rap by tinkerers that sadly assumed the forms for gradient, divergence and curl were individually fundamental, instead of structured sub-components of a more fundamental H. This hindered due consideration for Octonion Algebra throughout the 20th century. Internally, there are many fundamental assumptions about Octonion Algebra that are counter-factual. Externally, physicists have been jaded by the success of alternate mathematical systems, none of which can be demonstrated as necessary, at best only sufficient for their degree of success. The bedrock foundation for a Theory of Everything should be an Algebra of Everything. I submit it is Octonion Algebra.

Author Bio

I received a B.S. in Physics from Stanford University in 1973. I was convinced at that time what I wanted to learn was not part of any graduate program in Physics. I had marketable engineering skills and lived in Silicon Valley, so I opted for an engineering career instead of continuing with formal education. I never stopped thinking about foundational Physics. The essay material presented is the result of decades of private research.

Jonathan J. Dickau wrote on Aug. 31, 2012 @ 21:39 GMT
Hello Rick,

I too feel that octonions are a good candidate for an algebra of everything. In fact; I'm inclined to believe they are the 'actual' or 'true' numbers, by being the most general case, where reals, complex numbers, and quaternions are special cases.

I look forward to reading your essay, which I have already downloaded and have open in another tab. I give some favorable attention to the octonions in my own contest essay "Cherished Assumptions and the Progress of Physics." I hope you do well in the contest.

Regards,

Jonathan

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Jonathan J. Dickau wrote on Aug. 31, 2012 @ 22:08 GMT
Hello again, Rick;

Very interesting on first read through, but I'll need to re-read some parts several times to understand fully. I like the proof that 480 possible tables reduce to only 16. I guess that's 8 each for left hand and right hand. Geoff Dixon claims on his web-site that only 4 are in common usage, where his and Cederwell's plus and minus conventions are cited. But doesn't Okubo do things differently? Anyhow; it was a nice job.

I especially liked the part at the end where Electrodynamics comes out in its complete form, with no fudging for the SET. As you say; when the octonion formulation gives us terms that are unobserved bits, it doesn't necessarily mean that they are useless. Rather; it means we have yet to find out what they tell us.

All the Best,

Jonathan

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Author Rick Lockyer replied on Sep. 1, 2012 @ 04:07 GMT
Hi Jonathan,

Thanks for taking a look. I did look at yours when it came out, and found it very interesting. As you know from our past interactions, we agree on much but not everything, notably dimensional stability. For me if Octonions ever were appropriate, they will always be so. Liked your paper with Ray: In Defense of Octonions.

Dixon likes to generate different Octonion Algebras with arithmetic rules on the indexes. Problem is it produces different triplet sets that disguise the fundamental structure embodied by the Quaternion subalgebra triplet chiral choices only visible if the same seven triplets are used. This is critical to algebraic invariance, which is a fundamental law of physical reality from my point of view. Common use of less than all 16 is precisely the problem that has held back some smart people that have looked long and hard on Octonion Algebra.

Interesting that as shown in the endnotes, the non-observable variants come in three product term sets. Kind of quirky, or is that quarky?

Good luck to you too.

Rick

Jonathan J. Dickau replied on Sep. 2, 2012 @ 00:49 GMT
Thanks Rick,

I appreciate the universality factor for octonions, and I agree. Though we may appear to live in a lower dimensional space, octonions are a fundamental reality. Yes, algebraic invariance is the crucial property to be preserved or conserved, indeed. That's what makes all the nice symmetries possible. I'll think on the 16 distinct variations all being important question.

Regards,

Jonathan

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Member Benjamin F. Dribus wrote on Sep. 1, 2012 @ 00:01 GMT
Rick,

Interesting essay. You might be interested in some of the connections between quaternions, octonions, Hopf fibrations, and quantum information theory. Unfortunately I don't have a completed paper on this, but there are some partial treatments you can find on the web. Some people also believe that quantum information theory is more deeply connected to quantum gravity than it's usually given credit for; I mention this briefly near the end of my essay

On the Foundational Assumptions of Modern Physics[link]

Take care,

Ben Dribus

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Member Benjamin F. Dribus replied on Sep. 1, 2012 @ 01:13 GMT
(sorry for the link format; I left out the slash!)

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Author Rick Lockyer replied on Sep. 1, 2012 @ 04:24 GMT
Ben,

Thanks for taking a look. The connection between Quaternions and Octonions is at the heart of the concepts behind the essay. The choices for the Quaternion subalgebra chirality that maintain a normed alternative composition algebra for O defines the full range of O variability. If you notice I used up every bit of 9 pages, and had a tough time shoe-horning it in. I could have said much more about many aspects of R to C to H to O without the length limitations.

I agree with you on Hopf fibrations. My current intellectual diet has a high fiber content. I did read your essay and liked many portions of it. I think it would be difficult to cover Electrodynamics without including the notion of time within the manifold. It works for 4D and equivalently within an Octonion framework as I presented in my essay. Have you had any success leaving it out without implicitly having it in?

Rick

Author Rick Lockyer replied on Sep. 1, 2012 @ 14:19 GMT
Ben,

Rereading my response to you, I was distracted by a young house guest my wife and I just received that I was really being a bad host for by being on the computer at all last night. I do not know much about quantum information theory but it would seem to be going the route of quantized spaces. The algebra and analytic tools do not seem to need this, a continuum seems to fit just fine. If I am missing your point please let me know.

Rick

Edwin Eugene Klingman wrote on Sep. 1, 2012 @ 07:27 GMT
Dear Rick Lockyer,

I enjoyed your essay immensely, beginning with your observation of the essentially religious aspect of our assumptions. I too believe General Relativity and Quantum Mechanics are not fundamental, and that Electrodynamics and Gravitation should be united.

I have read all of the material on your website, but it was a year ago and that material hardly sticks in one's mind. I wish that you could write more explanatory material. For example, the notion that "divergence, gradient, and curl are not standalone forms" seems an ideal topic to expand on.

Since the weak field approximation to GR has the form of Maxwell's equations, I've used this in my current essay and would appreciate any comment on the feasibility of reformulating it in terms of Octonions. In particular, because the energy of gravito-magnetic fields have mass, the fields interact with themselves, in Yang-Mills fashion. Do Octonions handle this aspect of gravity?

Edwin Eugene Klingman

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Author Rick Lockyer replied on Sep. 1, 2012 @ 15:02 GMT
Hi Edwin,

I read your essay when it came out just because I know you from your posts and previous interactions we have had. I wanted to know your perspective on wave functions even though the subject does not resonate with me.

I am in progress on a book about my work, with a fair amount of content not on the website. This has allowed me to expound more on the philosophy motivating the mathematics as well as providing more detail. I particularly like the Sedenion chapter where I extend the Boolean triplet generators from 1-7 to 1-15, and employ them on basic quads (Octonion seven minus Quaternion triplets one at a time) to show the ways to roll out Sedenions in valid and not so valid Octonion subalgebras, and exactly where the 168 terms in N(A*B) – N(A)N(B) come from. I think you will find the book up to your desire to see more explanation.

As for the gravito-electromagnetic fields, all there can be is presented, both in the field algebraic elements and the dynamics of force-work and conservation. The big question is what are the other rotational fields, and how do they fit in to nature. I expect them to be the glue so to speak. The optimal coordinate system will not be the rectilinear native u in the essay. It will likely be some curvilinear system that pops the symmentries.

Rick

Edwin Eugene Klingman replied on Sep. 1, 2012 @ 19:35 GMT
Rick,

I look forward to your book. Let us know when it's available.

Another thing I would like to see in more detail is the algebraic 'equivalent' of calculus. As I recall, derivatives are essentially 'delta'-elements and integrals are sums of such. But I would really like examples and explanations that assume a good knowledge of calculus and a minimal knowledge of Octonions.

Edwin Eugene Klingman

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Author Rick Lockyer replied on Sep. 1, 2012 @ 21:11 GMT
Edwin,

The concept is not an algebraic equivalent for calculus, it is algebra working in harmony with calculus. I do not know about newer texts, but if you look up “Mathematical Methods for Physicists” by Arfkin, in chapter one on vector analysis, he mentions an integral definition for gradient, divergence and curl as limits for a volume with the point of application an interior point...

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Author Rick Lockyer wrote on Sep. 1, 2012 @ 14:31 GMT
I omitted one reference I meant to include. It is "A History of Vector Analysis" by Michael J. Crowe. This book gives the story of the "bum rap" I allude to in the abstract.

Rick

Author Rick Lockyer replied on Sep. 13, 2012 @ 15:48 GMT
Due to the 9 page length limitation for the essay, I was unable to discus Jens Koeplinger’s paper arxiv:1103.4748 on my Octonion Algebraic Invariance principle, opting instead to present in terms of my perspective using the Hadamard matrix based Iso() connection as previously discussed between us. It was an oversight on my part not to include it in the Reference section. I recommend reading his paper for additional insight.

Rick

Lawrence B Crowell wrote on Sep. 1, 2012 @ 23:30 GMT
Rick,

I attach a paper I published earlier this year. It discusses octonions and E_8 within the setting of computing states of a black hole. I have been less concerned with trying to employ it directly, but am trying to come to some understanding on how O might naturally occur.

My current essay is also directed in part this way. This leads to an argument for quantum states as modular or a part of the Eisenstein series and θ functions. The E_8 lattice is computed with the Jacobi θ functions. In the context of the Eistenstein series these form so called Mock θ functions. You can read some of the comments I make on my blog page for details that lie outside my essay, which connect more with these issues than my actual essay.

Cheers LC

attachments: 1_Crowell_EJTP_counting_states_in_ST.pdf

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Author Rick Lockyer replied on Sep. 2, 2012 @ 15:36 GMT
Lawrence,

If you did actually read my essay, you would have gotten my opinion on your question about coming “to some understanding on how O might naturally occur”. While this might not cover your immediate concerns narrowly related to your perspective on things, it fundamentally answers the question. O provides mandated structure that I show in the essay covers Electrodynamics soup to nuts as only a subset of the formalism. The remainder is explicitly provided, and IMHO explains the remainder of physical reality.

None the less, you probably should read my essay if you haven’t. You might change your mind on believing GR is what needs to be unified with QM. If there is an Octonion tie in with QM, you will have a better shot at unifying “Octonion Relativity”.

Who knows? You might even have a life changing experience reading it. Perhaps you will have a change of faith and come to realize the path to an understanding of the quantum nature of things is down here on earth, and not in the cosmos.

Rick

Lawrence B Crowell replied on Sep. 2, 2012 @ 21:15 GMT
I have given your essay a read through, which is to say that I have not focused on details and depth. I always at first read a paper that way. You have constructed a differential geometry which expresses a gauge theory according to octonion algebra.

To be honest I see the octonions as a representation of E_8 or the E_8 lattice and its extended role in the Leech lattice and quantum error correction codes.

Cheers LC

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Jayakar Johnson Joseph wrote on Sep. 3, 2012 @ 03:47 GMT
Dear Rick Lockyer,

Theta functions also describe tetrahedral-branes of eigen-rotational strings,, in that theory of elliptic function is applicable for the conformal mapping of eigen-rotational phase-transform, with holomorphic functions.

With best wishes,

Jayakar

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Thomas Howard Ray wrote on Sep. 4, 2012 @ 13:34 GMT
Hi Rick,

I always enjoy reading your well-crafted and cogent arguments. I appreciate the beauty of reducing 480 multiplication tables to a manageable 64 and its identity with the 8 X 8 Hadamard matrix.

Why algebraic invariance over analytical covariance, however? Comparing your law of invariance with Lamport's Buridan's principle ("A discrete decision involving a continuous range...

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Author Rick Lockyer replied on Sep. 4, 2012 @ 15:30 GMT
Hi Tom,

I do not at all put “algebraic invariance over analytic covariance”. If you realize algebra and analysis are interlocking components, you see both from a more fundamental perspective. Algebraic invariance as I have defined it is a natural and simple principle that matches observation. All currents, forces, work, energy, energy flux, stresses and strains described in an Octonion...

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Thomas Howard Ray replied on Sep. 4, 2012 @ 16:25 GMT
Well, now that you put it that way ... :-)

I could be persuaded. No matter -- I think we're on different pages of the same book with the same ending. Thanks again, Rick.

Tom

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Michael James Goodband replied on Sep. 6, 2012 @ 10:50 GMT
Hi Rick

I think that Tom is definitely right and our work is related at a deep level. My considerations have just been at the level of the homotopoy groups of spheres, initially for a map from a particle gauge space of S7 to a closed spatial universe of S3 - the underlying structure in question is obviously that of the octonions and quaternions.

Consider a non-trivial map S7 ->...

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Paul Reed wrote on Sep. 4, 2012 @ 14:18 GMT
Rick

“We must look for the connection between… and physical reality”

Indeed, any representational device must correspond with physical reality. So the question is, what underpins this mathematical system, and does that have proven correspondence with physical reality?

To put that question in context, I would suggest that physical reality does not have three dimensions, this is just the conceptual minimum. And the number of possible dimensions in physical reality is half (because dimension relates to a direction, either way) the number of possible directions that the smallest elementary particle could travel from a given spatial point. Dimension/size being an expression of ‘physical presence’ which can be conceptualised in terms of spatial footprint.

Paul

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Jonathan J. Dickau wrote on Sep. 5, 2012 @ 05:35 GMT
Hello again, Rick;

I just left a comment on Lawrence's page that bears mentioning here. He said that 'octonions are really a system of quaternions' (7 of them) which relates to a statement I made in the paper "In Defense of Octonions" with Ray Munroe. I wrote to Lawrence that while octonions could be represented that way, they do have to be resolved in an orderly fashion, and it's not the same as saying O is really H x 7.

In a paper I'm working on now; I suggest that working with octonion algebra is similar to assembling a watch. "Every layer or sub-assembly must mesh correctly, and then the layers must fit together in the correct relationship, for the watch to function. The same metaphor aptly describes what is required to do multiplication and division with octonions, as you must perform seven ordered groups of three operations in sequence."

Is this an apt characterization? My guess is that Lawrence's approach would treat the component quaternions in the same way that Physics folks normally treat grad, div, and curl - as independent or fundamental quantities, where in reality (or as you demonstrate) they are structured components of the quaternions. I suggested his statement might be made true if octonions are treated as an ordered or nested system of seven quaternions. Is this essentially correct? Do we also need an extra scalar value, for the Real component?

Enlighten us.

all the best,

Jonathan

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Author Rick Lockyer replied on Sep. 5, 2012 @ 16:33 GMT
Jonathan,

The Octonions are at the end of the R to C to H to O chain, and their discovery order as well as the doubling process both follow simple to more complex direction. I prefer to think of the Octonions as the most fundamental since there is no higher dimension normed division algebra, and each of the more simpler division algebras are all O subalgebras, meaning their basis set is a...

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Steve Dufourny replied on Sep. 5, 2012 @ 17:22 GMT
Weak reasoning ! :)

Edwin, Lawrence, Jonathan,Joy,Tom,Rick and friends.

It is weak, even the strategies are weak, it is easy to find the holes. 0 really.And me alone, integre, transparent, without tool,just with rational sciences. Between us ,it is ironical no? In all case me I laugh. Because even like that I have teached you so many things ahahah incredible no?

And they...

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Jonathan J. Dickau replied on Sep. 8, 2012 @ 18:08 GMT
Hello Rick,

I agree the octonions are more an absolute endpoint or an ultimate starting point, rather than some obscure way station in a process of infinite doubling. It is far easier to make sense of things by asserting that octonions are the fundamental starting place from which the H, C, and R subalgebras are special cases, or steps in a sequential limiting of degrees of freedom.

Real numbers are the most common, and a lot of folks feel that the whole concept of number comes out of the natural or counting number system. First, of course; we must distinguish none or zero from one. Then there is the concept of many. So to imagine that the reals are a subset of an 8-dimensional number system is to some people rather far-fetched.

But if the topological anatomy of the universe is something like what's being discussed by Joy Christian and Michael Goodband, then Octonions are far more fundamental than the so-called natural numbers, and they are indeed the algebra of everything.

Regards,

Jonathan

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Jonathan J. Dickau wrote on Sep. 10, 2012 @ 00:17 GMT
Rick,

The octonions are amazing. They are perhaps the Holy Grail of Mathematical Physics. But you have to admit they are ball busters dude. People who like algebra because they have memorized the rules of simplification hate the Octonions, because they thwart their best efforts at every turn. They are the epitome of difficulty, in that sense, because they are the most demanding of all the well-behaved algebras, in terms of keeping track of the order and/or syntax of mathematical statements and procedures. As I say in the above comment; it's like putting together a watch, to do proper algebra with octonion variables - at each stage, or with each cycle or operation. For some people, that takes the fun out of Math.

You and I are different, that way. The very thing that makes octonions demanding - their sequential or procedural ordering property - is what makes them fascinating to me. But when this is respected; algebraic invariance is preserved, and equivalently so are the physical symmetries such reversibility principles represent. In my mind; this makes octonions a kind of minimal starting place, as an octonion background space is what must be assumed if there are no added evolutive or limiting conditions. That is; when considering the question of what the minimum conditions are, to generate the universe of form we observe, the Octonions are likely as simple as you can get.

All the Best,

Jonathan

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Michael James Goodband replied on Sep. 10, 2012 @ 15:10 GMT
I just wanted to second Jonathan on this one. The Octonions thwart efforts at simplification or principles to keep it together. But is this really the case? Especially for the decomposition into subspaces (S7->S3*S4->S3*S3*S1) of O that are likely to turn out to be relevant?

Michael

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Jonathan J. Dickau replied on Sep. 11, 2012 @ 01:16 GMT
Thanks Michael, for the point of agreement;

It really depends how the subspaces relate to the octonionic background or bulk. Hans Van Leunen's 'Hilbert book model' uses quaternions to link us with the microscale and quaternions again to link us with the cosmic scale - and with other systems in general - through probability amplitude distributions. He calls them QPADs. But if in your theory; we are living in a quaternionic topological space - S3, that is embedded in or contiguous with octonionic space - S7, what does that imply conceptually? And how do things fit together? The linkage is all important.

My guess is that the orderliness of the octonions actually comes to the rescue there, or for helping reconcile your theoretical ideas with Joy Christian's, in terms of helping to sort out the way things link together and the order in which things must evolve.

I've been working for some time on a universal protocol for measurement or determination, and it turns out to have strong tie-ins with the octonion algebra's procedural aspect, as it requires precisely seven stages. I keep returning to points made by Connes, in papers from 12 years ago, regarding how we define smooth, topological, and measurable spaces. He spells out some of the precursors or prerequisites of measurability. But that's just a starting place. However; I think the octonions suggest and endpoint to that process.

More later,

Jonathan

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Michael James Goodband replied on Sep. 11, 2012 @ 12:14 GMT
Jonathan and Rick

Just to clarify. In my model the S3 of space is *not* a subspace of S7 but the two S3 and S7 form a 10 dimensional product space. This can either be viewed as a torus - if we pick the S1 fibre from the S3 - with an S7 cross-section, or a sphere - if we pick out the S2 base-space of S3 - which has a S7 surface cross-section. The absolutely critical element of my model is that there is a twist in the S7 cross-section in going around the S3 (like a higher dimensional mobius strip or twisted torus) - this is the electroweak vacuum and it picks out the sub-spheres of S7, but in a way that mixes S7 with the S3 spin space. In my model, this gives the reason for why the electroweak charges of the particles are chiral. This is also why my results with a 10 dimensional product space are *not* actually in disagreement with the S7 of Joy's correlation results. Spin (space S3) is correlated with isospin (space S3) and hypercharge (space S1) which are sub-spaces of S7, but colour charges are independent and not correlated with the other observables.

So my model says that viewing S7 as all of space - as Joy and Rick do - will work for spins and electroweak charges - which are correlated in particle physics - but that leaves the colour space unaccounted for. Joy's correlation results imply the *independent* colour space would be S3 (as S7 isn't a group space). The twist in my model of a physical gauge space S7 in going around the space of a closed S3 universe has the peculiar effect of switching S3 spaces in the *measurement* space - the S3 spin space is swicthed for the S3 colour space. The linkage of the subspaces is all important - as is the distinction between physical spaces and measurement spaces - and just how 'the orderliness of the octonions' actually does the rescuing is the piece of the puzzle I'm looking for.

Michael

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Yuri Danoyan wrote on Sep. 16, 2012 @ 22:53 GMT
Rick Lockyer wrote:

"I call The 3:4 Morph Rule."

I call 3:1 Yuri Rule

http://www.fqxi.org/community/forum/topic/946

http://fq
xi.org/community/forum/topic/1413

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Edwin Eugene Klingman wrote on Sep. 17, 2012 @ 00:39 GMT
Rick, Jonathan, Joy, Michael, and Tom:

As another author observed in a comment, "It's so hard to change others minds." Obviously this is related to the investment others have made in pushing their own model of understanding.

Rick described it beautifully [12 Sep @16:36]: "Many people pick one of these choices and run with it [which is optimal because] the collective will succeed...

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Author Rick Lockyer replied on Sep. 17, 2012 @ 15:41 GMT
Hi Edwin,

The diversity presented by people’s different view of how Octonion Algebra fits into the nature of things is completely a good thing, not at all a problem by the same “leave no stone unturned” thinking.

I have no problem with people continuing to work on extending current thinking on relativity, quantum theory, cosmology, string theory, etc. etc. None of these have anything to do with Octonion Algebra, and each runs contrary to my own intuition on how nature is structured at its most fundamental level. I am better though with the people that are looking for “Octonion stones” to turn over, like Joy and Michael.

Michael is coming from the standard model view much like Geoffrey Dixon. Both lean towards a tensorized increase in complexity beyond straight up Octonions because they do not see everything this perspective seems to indicate within the algebra itself. I am not confronted by this at all, for I too believe there is something within that model that rings true. I just don’t think it is an on the surface of the algebra kind of thing. I think the family of solutions for potential functions within the dynamics I have laid out will show the connection.

Good to have you back in the discussion.

Rick

Michael James Goodband replied on Sep. 19, 2012 @ 21:16 GMT
Hi Edwin and Rick,

I think we have some agreement on "it's so hard to change others minds". That might also be in part because it is often so hard to change our own minds with our own ideas. It took me quite a while to change my own mind about the relevance of Godel to physics and bypassing it through representational change. Joy has managed to change my mind about Bell's Theorem, but then I was in a very receptive frame of mind for such a result having independently arrived at the same conclusion about QT. Mine and Joy's work are actually compatible, to what extent is yet to be agreed. For my part having read Joy's book I see no current disagreement - there is some about future direction.

Geoffrey Dixon and Cohl Furey consider R*C*H*O and get the Standard Model group eigenvalues for SU(3)*SU(2)*U(1), whereas I consider the physical manifold S0*S1*S3*S7 and get a colour group dispute - but QT not being fundamental. In either case, the octonion structure is likely to prove to be required reading, but we are going to want it from the particle physics perspective in order to present it to others - sorry, but it won't just be about us seeing it the "correct" octonion way.

Michael

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Author Rick Lockyer replied on Sep. 20, 2012 @ 02:35 GMT
Hi Michael,

I asked Jens Koeplinger a question on his essay blog that also included you. Best to present it to you:

“I am a bit puzzled by both you and Michael Goodband talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath. The metric for O and its subalgebras is the norm, which is positive definite all + signature so O has no...

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Hoang cao Hai wrote on Sep. 27, 2012 @ 08:27 GMT
Dear Rick Lockyer

"The bedrock foundation for a Theory of Everything should be an Algebra of Everything."

Seems like you already have the equations and formulas for Theory of Everything.

Maybe that's what I not yet do (I just had a theory in philosophy)

Please check out the - topic / 1417 - to see the general point that we can complement each other?

Kind Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Jonathan J. Dickau wrote on Oct. 4, 2012 @ 02:30 GMT
Hello Rick,

While the story goes that the Octonions are the weird and crazy old uncle whom everyone would like to hide away in their closet or attic; that is only partly true, and misses the point somewhat. Instead; the octonions are like the wealthy and eccentric old uncle, who almost nobody else in the family associates with, because they are intimidated by his freedom and power. Of course; the rest of the family is jealous, but they would never admit it, because then they would end up feeling a little less powerful themselves.

But we would all be a whole lot less wealthy, if it were not for the Octonions - which are the great granddaddy of all the other number types. However; the wonderful properties of those amazing octonions assure that the rest of the family emerges and persists as well. I rated your essay highly, because I believe in what you are saying - and that you said it well. Of course; I fully understand that you can't talk meaningfully about your subject without getting too technical sometimes, which you did, but so it goes.

All the Best,

Jonathan

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Author Rick Lockyer replied on Oct. 4, 2012 @ 17:06 GMT
Jonathan,

One must make an intellectual commitment to reach an appropriate respect for Octonion Algebra and just what it can tell us about physical reality. The premise of this essay contest is there may be fundamental assumptions that may need rethinking. Octonion Algebra tells us the list is long and distinguished, and thus it intimidates many who take this as a fontal assault on their core beliefs. You will have a tough time convincing a Southern Baptist that they should study the Koran. It is just about as bad for Octonion Algebra; so many very intelligent people do not make the necessary open minded intellectual commitment.

I try my best to present my findings in an approachable way, and in the setting of this essay contest I can only hit the highlights. I believe I did a very good job with my essay given the limited format. For years now I have been both surprised and disappointed that so few people have developed the enthusiasm I have for the potential of this powerful algebra, so the lack of dialog and moderate community rating is nothing new for me. As they say, you can lead a horse to water, but you can’t make it drink.

Rick

Jonathan J. Dickau wrote on Oct. 4, 2012 @ 02:39 GMT
Hmm,

Thinking on the above comment; I imagine you could make a detailed storyline out of it - where, for example, the rest of the family will only approach uncle Octonius in committees of three, but there is always a member of each committee in common. Such a dramatic approach might illustrate the kind of special respect the octonions demand. Just a thought.

But I like the idea of the Octonions being a wealthy and eccentric uncle (who knows untold secrets) better than a weird and crazy one, any day.

All the Best,

Jonathan

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Jonathan J. Dickau wrote on Oct. 4, 2012 @ 04:20 GMT
To continue;

Good old Uncle Octonius was one crafty old genius. The family insisted that the only way anyone in the family could be trusted to meet with him is if they went in committees of three. And of course; he insisted that - in order to be fair - there must be one member in common each time they send a committee. But how did he know that there would be seven committees? That crafty old guy must have planned it that way.

Of course; the committees are called quaternions - which is short for "It's our turn again." Anyhow; I'm just thinking out loud on your page.

All the Best,

Jonathan

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Sergey G Fedosin wrote on Oct. 4, 2012 @ 05:35 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
$R_1$
and
$N_1$
was the quantity of people which gave you ratings. Then you have
$S_1=R_1 N_1$
of points. After it anyone give you
$dS$
of points so you have
$S_2=S_1+ dS$
of points and
$N_2=N_1+1$
is the common quantity of the people which gave you ratings. At the same time you will have
$S_2=R_2 N_2$
of points. From here, if you want to be R2 > R1 there must be:
$S_2/ N_2>S_1/ N_1$
or
$(S_1+ dS) / (N_1+1) >S_1/ N_1$
or
$dS >S_1/ N_1 =R_1$
In other words if you want to increase rating of anyone you must give him more points
$dS$
then the participant`s rating
$R_1$
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

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M. V. Vasilyeva wrote on Oct. 5, 2012 @ 17:41 GMT
Mr Lockyer,

a very interesting essay. I remember a few years ago I was looking for math that would allow the multidimensional nature of space to be expressed with ease. I wish I had found Octonion Algebra then, but all I saw was quaternions. I imagine how hard it was to pack it all in 9 pages -- well done! I hope your essay will get the recognition it deserves.

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Author Rick Lockyer replied on Oct. 5, 2012 @ 18:48 GMT
Ms. Vasilyeva.

Thank you for your kind words. I must say I enjoyed your essay also. The structure of space is important for us to think about. But we must not over think it, and apply more structure to the fundamental idea of space in favor of structure built on top of space. Potential functions over a simplistic notion of space whose purpose more is to set the algebra, dimensionality and allow us to distinguish one point from another has always resonated with me more than say intrinsic curvature of space itself to describe Gravity. But mathematics is robust enough to permit useful activity either way, perhaps both a blessing and a curse at the same time. The map reality to mathematics is one to many, some choices will be more optimal than others. I tend to look for the choice that provides fewer options while speaking loudly with authority about how nature must be. Octonion Algebra does this quite well in my opinion.

Good luck with your fine essay,

Rick

M. V. Vasilyeva replied on Oct. 6, 2012 @ 07:09 GMT
Mr. Lockyer,

Congrats to us all on having it over with!

Thank you for your response. You wrote, " Potential functions over a simplistic notion of space whose purpose more is to set the algebra, dimensionality and allow us to distinguish one point from another has always resonated with me more than say intrinsic curvature of space itself to describe Gravity. "

-?? did I understand you right?...If it does not curve, what's gravity?

I am very curious about your take on space. Like, there are many topological, geometrical, algebraical spaces whose characteristics and properties are mathematically defined. There is only one real thing. What do you think defines its characteristics and properties?

I imagine you must think that space has more dimensions than the observed 3. Otherwise, why Octonian Algebra?

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Author Rick Lockyer replied on Oct. 6, 2012 @ 18:17 GMT
Ms. Vasilyeva,

Gravity without curvature? Yes, you do understand me correctly. Einstein showed that gravity *could* in a 4D setting be described by intrinsic curvature. He neither showed that it *must be*, nor did he or anyone else show physical reality *must* be 4D. Think about how gravitation was handled prior to Einstein, with a potential function. As I stated in my essay, integrating...

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Jonathan J. Dickau wrote on Oct. 7, 2012 @ 01:18 GMT
Hello Rick,

Good old Uncle Octonius would be proud! Assuming no further vacillation; I wish you the best of luck in the finals. You deserve to be there!

All the Best,

Jonathan

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Joy Christian wrote on Oct. 7, 2012 @ 15:27 GMT
Congratulations, Rick.

You have brought our crazy old uncle to the party. I hope he unleashes unimaginable havoc.

Joy

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Author Rick Lockyer replied on Oct. 7, 2012 @ 16:24 GMT
Thanks Joy,

I hope to unleash unimaginable clarity, not "unimaginable havoc". But as you know well and most essay authors have stated time and again, it is a tough sell to get acceptance for ideas that suggest they rethink positions long held. This is however what the essay contest is all about.

All I can do is point out what I have come to know, and ask the reader to decide for themselves if the results are merely coincidences or if they point to fundamental truths that show us a different and better way to proceed.

Rick

Jonathan J. Dickau wrote on Oct. 9, 2012 @ 23:19 GMT
Hello Rick,

I'm cross-posting this comment I just left on Ben's page, with due respect to John Baez, who is not my cousin in real life, because it deals with octonions.

A story:

They say my uncle is crazy, and cousin John tells me some family members wanted to lock old Uncle Octonius up in the attic, but I think he is only eccentric because he's seen the universe, and knows...

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Author Rick Lockyer replied on Oct. 10, 2012 @ 04:14 GMT
Hi Jonathan,

Cute story. The only problem I have with it is "You see; he always liked to break the laws of algebra...". Not the case. The general algebraic rule for multiplication e_i * e_j = sum k: p_ijk e_k encompasses commutative, non-commutative, associative and non-associative algebras. It all comes down to the structure constants p_ijk.

Rick

Joy Christian replied on Oct. 10, 2012 @ 04:23 GMT
Or structure variables p_ijk(x) in my variable torsion picture, at every point x in S^7.

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Jonathan J. Dickau replied on Oct. 10, 2012 @ 19:27 GMT
Excellent!

You two certainly know how to spice up a story. I like both of your answers. And though it might be tough to work into the story, that's part of what makes our 'crazy' old uncle so interesting. But if we had decided he was crazy, and just left it at that, we never would have learned of his incredible wealth.

But I think I clearly conveyed that he likes to be in charge, and makes the others do things his way. And as you both just said, in the final analysis he never really broke the law. He just bent the rules, to have a little more play, and ultimately made the others play the game by his rules.

All the Best,

Jonathan

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Steve Dufourny wrote on Oct. 11, 2012 @ 14:10 GMT
always weak dear Jonathan Dickau and the bad team so Rick,Lisi,Tom,Christi,Florin,and the friends .Your strategy show us your weak generality.In fact you are not intresting and your reasonings still less. You utilize bad startegies just for the monney and the vanity.You are not scientists.You are just business men thinking that all is permitted just because you are limited.Fortunally that you are...

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Steve Dufourny wrote on Oct. 14, 2012 @ 23:07 GMT
and Mr Durham who is lost now in the meanders of irony.

Jonathan and Mr Durham are in a bar of 11 dimensions, the team of bad angels is lost in its own stupidity. Mr Dufourny Steve, him, Jedi of The Sphere, inventor of the Theory of Spherization fights the bad with universal wisdom and universal sincerity.:) revolution spherization !!!

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Author Rick Lockyer wrote on Dec. 5, 2012 @ 20:40 GMT
I had to breeze through the conservation of Octonion energy and momentum in my essay due to the length restrictions. I attach below a snippet from the book I am writing that covers the derivation of the conservation equations in more detail.

It sure would be nice to get some grant money to finish the book and further my ideas, especially now that I am recently unemployed and would have the time to work on it if I do not get another engineering job.

Dreamingly yours,

Rick

J Gregory Moxness wrote on Jan. 1, 2013 @ 23:55 GMT
Please see http://theoryofeverything.org/wordpress/?m=201301 for links to a Mathematica demonstration that completely links E8 and octonions to AESToE (AG Lisi) extended SM particle assignments. It generates all 480 permutations of octonions and links them 2:1 with E8.

Greg

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Fred Diether replied on Jan. 2, 2013 @ 00:02 GMT

[this post can be deleted]

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Author Rick Lockyer replied on Jan. 2, 2013 @ 16:47 GMT
Greg,

I do not put much thought into E8 because its O connection is made to the 480 representations. As I said in my essay, the 30 ways to roll out 7 H subalgebra triplets from 7 distinct non-scalar basis elements are nothing more than aliases. But beyond this, there is no proper algebraic method for their simultaneous application (E8 single reality?), unless you want to increase the total...

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Thomas Howard Ray replied on Jan. 2, 2013 @ 17:11 GMT
" Physical reality *is not* symmetry groups, physical reality *exhibits symmetry*."

That is an excellent and profound statement, Rick. Physically it agrees with the relativistic prohibition against a privileged frame of reference.

Tom

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