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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.

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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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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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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?

Good luck with your essay.

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

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Member Benjamin F. Dribus replied on Sep. 2, 2012 @ 07:12 GMT

Rick,

Thanks for the responses. I can certainly sympathize with the difficulty of the nine-page limit! Regarding quantum information theory, I wasn't referring to quantum gravity or the fundamental scale in that context, but merely pointing out a currently "fashionable" field that someone with your knowledge of the special algebras could contribute to. I'm a mathematician, and I always appreciate when someone takes notice of "obscure" structures or concepts that deserve more attention. Take care,

Ben

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Author Rick Lockyer replied on Sep. 2, 2012 @ 15:38 GMT

Hi Ben,

Having a day job that has involved RF communications for a couple of decades, I have come to learn a thing or three about information theory, the works of Shannon, and error correcting codes. I have read only a small amount about conservation of “quantum information” and also about “quantum error correcting codes”. I presume one is on the cosmological scale and the other on the Plank scale, right or wrong. Anyway, I have to pick my shots with the limited time I have, and this seems on the extremes of a tree limb that already can’t support its own weight.

I am very interested in the quantum character of Nature, I just do not feel it is appropriately covered by today’s quantum theories. I take a more pedestrian view, believing it will naturally occur from a bottom up analysis rather than the long chain of assumptions current theory suffers.

Rick

Steve Dufourny replied on Sep. 5, 2012 @ 12:00 GMT

Jonathan, Tom, Edwin,Brendan and friends, you are bad strategists meriting simply to ba analyzed by the laws. Sort your members Mr Tegmark and MR Aguire.

They have not theior place on this platfrom.They decrease the velocity of evolution of fQXi and its credibility must be universal.Sort these pseudos.

They have simply a strategy for the maney, they are just frusterated vanituious and envious. They must be sorted. They delete in correlation of their strategy of discriminations.In fact they fear that I arrive at New York, so they try with the discriminations. I have faith me, them no !Don't compare a thing which cannot be compared.

Mr Tegamrk and Mr Aguire, don't be troubled by their strategy and their words. These people have simply a heart without faith and reason. They are not scientists, but business man. And you ibm, forget also these persons !with the soa and its superimposings. Be rational and universal.

Steve

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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?

Your essay certainly goes on my re-read list.

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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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 going to zero of the ratio of a surface integral divided by enclosed volume integral. The surface integral is over the differential surface normal vector respectively multiplied by a scalar function, an inner product with a vector function, and cross product with a vector function; for gradient, divergence and curl. He uses this to demonstrate for example spherical-polar representations of these three forms.

I look at this not as an alternate description for n dimensional differentiation, but instead its fundamental definition. Algebra comes into play because multiplication is its dominion. The multiplication on the differential surface normal is an algebraic expression, and if you are working with Quaternions, the three forms of scalar multiplication, scalar result vector -vector products and vector result vector – vector products are all covered by a single operation, the Quaternion product of two algebraic elements, here a 4D differential surface normal and a 4D function. If you were to leg out the Quaternion Ensemble Derivative for a transformation between rectilinear native coordinates and spatial spherical-polar coordinates, you will find proper representations of spherical-polar gradient, divergence and curl, which you may individually isolate with the resultant basis element products. Do it again, you get the second order forms. We all know what they are, so there are no mysteries on whether or not the result is correct as some may argue if the work was done in Octonion 8D space.

There still is the notion of a difference, not simply between two arbitrary points but instead over the full (n-1) dimensional surface, but also over the full set of algebraic products between the surface normal and function to differentiate in order to come up with something transformable. The limit is as the volume approaches 0, arbitrarily close but never touching the point at which we wish to define the differentiation. So there is always a definable surface and a difference between functional values at the point of application and values in a coordinate neighborhood defined by the surface.

This is the genesis of the Ensemble Derivative.

Hope this helps.

Rick

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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

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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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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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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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 of values cannot be made in a bounded length of time") I find that the choice of left and right, physically, is compelled by a continuous function, not an algebraic multiplicaiton rule. Nevertheless, I grok the utility of an axial-polar rotation relation between electric and magnetic fields -- as operations that result in a union of open and closed results in a 3-d coordinate system. (If you are interested, I recently posted a draft paper on my essay site that in fig. 6 pictorially shows a topological interpretation of the phenomenon, a closed external manifold connected to an open internal plane, with all external points mapped to all internal points.)

I think the models that fall out of your, Joy Christian's, Michael Goodband's and my research may converge in a deep sense. However:

"There is an assumption that it is impossible to define analycity within Octonion Algebra." Of course, you know that I am one of those who assume so. And I do agree with you that the tensor calculus is inadequate to the task of a closed logical judgment on wave propagation and electrodynamics in a 4-d continuum. The topological solution still most appeals to me, however -- not as a preference, but precisely because it eliminates the necessity for a preference; everything neatly follows from a free choice of topological initial condition.

Because we are working in the same 8-dimensional space, though, one can't help thinking that our 4-dimensional measure results all originate from a common source. All these mathematical methods may turn out to be dual.

Thanks for a great, forward-thinking essay, Rick! Best wishes in the contest.

Tom

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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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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 framework are algebraic invariants. They are not simply such, they are the full complement of algebraic invariants available, complete. It is really difficult for me to think this is not a very loud statement the concept is a fundamental truth.

Without the application of a suitable analysis process, we have no connection to physical reality. Algebraic invariance can be understood without this connection as pure algebra, which I feel is important since it can be understood without the added complexity of physical reality and our current uncertain mathematical cover of it. After all, we are all hopeful we can improve the math side, whether we believe Octonion Algebra is they way or not. We are ahead in the game if we can separate out components, fully understand them, and then be able to better apply them to the greater whole.

The Ensemble Derivative is the interlocking of analytic and algebraic concepts. It works for the banal transformation of rectilinear coordinates to spherical-polar curvilinear coordinates in a Quaternion setting to the more interesting Lorentz covariance of Electrodynamics in the Octonion setting. Realizing both halves of the Electrodynamic field components transforming in an identical fashion is non-trivial. Algebraic invariance demands the algebraic basis element products for the transformed Electrodynamic field components exactly match those they rotate into. It is all there. So besides the fact that the Ensemble Derivative works, just what is your issue with it?

Thanks for your time and consideration on this Tom, and good luck in the competition. I read yours when it came out, but had no additional comments beyond interchanges we have already had. I still can’t extrapolate as well as you can. I imagine I am missing things. I will try harder.

Rick

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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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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 -> S3 and out falls an electroweak vacuum with a Weinberg angle given geometrically by sin2 = 5/21, which is smack in the experimental range (derivation is in the technical notes of my essay and my paper. I feel like the only one in stunned surprise at this result ... but this isn't the end of it!

The map S7 -> S3 is from a S3 subspace of the S4 base-space of S7 to the other S3 space, and this picking out of the S3 from S7 gives an effective sphere decomposition that is locally S3*S3*S1. In the context of field theory, this amounts to a symmetry breaking from the space S7 to a space containing S1, which by homotopy group relations means that there *must* arise topological monopoles in the 3-space of the spatial universe. The spectrum of these topological monopoles will be given by the number of ways of picking out the S4 base-space from the S7, which is given by the homotopy group for the map S7 -> S4 and gives a 3 by 4 table of possibilities. The charge eigenvalues for this 3 by 4 table is given by identifying the spheres S3*S3*S1 with group spaces, which for the map S7 -> S3 must locally be SO(3)*SU(2)*U(1) and this gives a 3 by 4 spectrum of topological monopoles which the same charge eigenvalues as the particles - specifically including 1/3 electric charges.

Topological monopoles in field theories have generally been thought of as only bearing magnetic charges, but the underlying structure says not - there are two distinct topological maps for these monopoles, one would be expected to have magnetic charges, but the other would give electric charges. So one spectrum of these 12 topological monopoles would bear electric charges, and so look just like the 12 fundamental particles. Am I the only one in stunned surprise?

These homotopy group results obviously come from the underlying quaternion and octonion spaces in which the spheres S3 and S7 reside. I am amazed that the simple consideration of a map from one to the other yields the correct electroweak vacuum and the correct spectrum of particle charges and NO more. The context for this map S7 -> S3 that I consider is GR extended to 11-dimensions as it seems the scenario that makes most physical sense.

Can you explain how the algebraic structure of the octonions and quaternions is ultimately responsible for these homotopy group results? It obviously is, but I currently don't understand how.

Michael

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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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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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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 subset of the O parent, and the subset forms a proper algebra all by itself. This means it abides by the three rules of algebraic element addition, multiplication by a scalar, and algebraic element multiplication closed for the subset of basis elements.

Geoffrey Dixon mentioned in his latest book his lack of enthusiasm over the doubling process, and I must say I fully agree with his position. This process has been the genesis of the “made from” mentality. As he mentions, you can double through the division algebras, but you can also double O to the Sedenions, which are not a division algebra, so the doubling does not conserve this very important characteristic. I further contend it has also led to a one O algebra mentality because many missed the fact doubling a commutative algebra (R to C or C to H) is not the same as doubling a non-commutative algebra as with H to O. The subalgebra perspective works end to end since R can have no subalgebra since it has only a single basis element.

Sedenion algebra defines 35 perfectly valid H subalgebras, and 15 Octonion-like subalgebras of which only 8 can be made into normed composition algebras. The latter is why Sedenions can’t be made into a division algebra.

C has but one choice for definition of basis element multiplication, so no variability impact on the definition of H. But H does have 2 choices analogous to 3D right handed and left handed vector products. The subalgebra connection for H from O must accommodate these 2 choices, which is why there are 16 choices for proper O Algebra. The 16 choices, that is the full variability in O definition, is determined by handedness choices for the 7 H subalgebras that leave O a normed composition algebra. If as you question, O was simply Hx7, there would be 128 valid O for all possible H choices, but there are not.

The proper H selections are less important from the H perspective than they are from the O perspective, since things only get interesting when we make O algebraic element products. Then one “H” in a way multiples another “H”, and the ins and outs of this sets the algebraic variance and invariance characteristics of fundamental importance.

Kind of a long answer, but hopefully you will find it satisfactory.

Rick

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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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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 insist furthermore with the compactification and the geomatrical algebras. If you you understand the 0, the 1, the infinity, the numbers, complexs, naturals, reals, R C O H or this or that.If you you understand the finite groups and the real infinity. Me really I am the queen of England you know. If you pondered intresting between us, ok, but no, even like that you insist on your stupidities for I don't know me.Probably a problem of vanity or a kind of play just due to your unconsciousness.In fact,you are not really skillings.Because If I learn here on fqxi , it is not with your team you know.

In fact you are not general, here is the probelm.And even your details are not good.So you imagine my pity , you can delete, betrween us, you understand, isn't it ? your hate increases, logic, your strategies are just a simple bad play of a kind of super team , but in fact it is a team who makes pity.really. You are even ready for all in fact. You are bad persons simply. When I see all this story since the begining. It is incredible in fact with your false universalism and false patriotism, I am a better american that you furthermore because me I am a real universalist loving his fellowman. A real christian. And you have made all this just because you are vanitous , envious and full of hate against people who critics universally. It is a sad team with bad tools, and bad strategies. You are not universal. Fqxi merits more than this kind of comportment. The integrity, universal is essential.

How can you have this kind of comportment in fact.What is your heart to make that. I pray for you, I have pity. All my pc is checked.All my platforms where I discuss.Linkedin,xing,fqxi also,facebook,....it is really bizare.

I forgive always. It is sad this story in fact. How people can make that ? the world is sick, if already the imrpotant systems of foundamental sciences are bizare.Where are we going? It is bizare simply.

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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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Author Rick Lockyer replied on Sep. 8, 2012 @ 23:44 GMT

Jonathan,

Happy to hear you agree with my premise. Sadly there are physical religion bigots that would not give the essay a look, much to their loss.

Algebra, analysis, topology and groups are interlocking parts. The most fundamental is the algebra, for it sets the tone for the remainder. For O, nobody has put it all together yet. My work is the easy part in many respects, for it is clear cut. Right is right and wrong can be demonstrated. I put a high value on the work of both Joy and Michael.

Rick

Steve Dufourny Jedi replied on Sep. 20, 2012 @ 14:01 GMT

always weak and furthemore too much focus on 2D.In fact you are not competent for this play.Buy people or please increase your team because there it is too easy. I d say even that it is time that you really kill me.Because I don't understand how you can support all this teaching from me. In fact probably that you have still more hate against me.It is logic for this kind of persons. Aahahah even...

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always weak and furthemore too much focus on 2D.In fact you are not competent for this play.Buy people or please increase your team because there it is too easy. I d say even that it is time that you really kill me.Because I don't understand how you can support all this teaching from me. In fact probably that you have still more hate against me.It is logic for this kind of persons. Aahahah even your strategy is weak, it is easy to find the holes and errors.ahahah a lesson, who gives a lesson to who band of comics. in all cneters of interest I eat your sciences at my breakfast.I am understanding that my arrogance increases your hate, but frankly I have pity.An advice don't speak about that inside the global sciences community.you know it exists the trues and the falses. But I understand your comportment.In fact it is logic in this time of crisis. You fear to loose investments or to be less credible.But you know it is the life. The strategists even with all the tools and with a lot of friends cannot win against he message of God.Am I Crazy? Yes , it is evident. I am insisting you know dear bad team, your only one solution is to be murders. And what ? you think what ? my universal faith is so above yours that I am laughing in fact. The revolution spherization has already began. You do not imagine the number of persons knowing my theory. and it is a begining. Alone with an old pc and without net protection.and you with your strategy of superimposings with the SOA adapted for specific selected strategists. Incredible no, even like that I give you the informations and I so optimize your quantum systems. Ahahaha I have the syndrom of the savor,I beleive that I am the elected ahahah wait I see spheres everywhere and you know what they speak me, I am not alone in my head band of comics.

ps please the team is weak,please dear directors of this team, buy and pay others people, thay have not really good discussions with relevance.Even the entropy they do not know, even the gravitation they do not understand. Really it is frustrating to see these weak posts.You have not better competitors please ? because they repeat always the same.

wait, they are going to invenbt a time machine to travel towards the aether. :)

wawww impressing their maths ahahah so incredible that even God is without words ahahah wait he says me, Steve, my son, they are simply not aware of the truth, keep faith my son.....

amen

Regards

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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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The Spherical Jedi replied on Sep. 11, 2012 @ 12:26 GMT

Ahahahah and they insist furthermore.You are not real christians. Me yes.Don't try with things that you do not understand.

How can you converge if you have not understood the reall message.It seems not possible. In fact your goodband is just a team without real innovations. A goodband, yes of course, a goodband of businessmen yes. And you try with bad strategies.Like poor thinkers frustrated and jealous and envious and full of hate. My universal love will eat your hate. Dead or alive, I will continue. even a gun on my head, I will continue.

You know ; kill me , it is better you know and more quick for your strategy.

Aahahah I am crazy, I have god with me , Jesus Christ also is with me and Buddah. You want really to try with the faith. Let me laugh band of bad men.

Irritating that I forgive you all no?You would in fact that I discriminate people and country. Let me laugh, I love more than you the USA. the United States of the Sphere you know. No? don't worry you shall understand on the entropical arrow of times. I love USA !

A goodband, yes of course.a good band for monney yes.

Regards

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Author Rick Lockyer replied on Sep. 12, 2012 @ 16:36 GMT

Jonathan and Michael,

On O “thwarting efforts…”, the problem is not with Octonion algebra. The issues you and others have voiced are all the outcome of trying to force-fit O into some other mathematical structure or approach rather than immersing oneself within the algebra and looking at the world through its perspective. This is precisely what I was getting at in my essay with “you...

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Jonathan and Michael,

On O “thwarting efforts…”, the problem is not with Octonion algebra. The issues you and others have voiced are all the outcome of trying to force-fit O into some other mathematical structure or approach rather than immersing oneself within the algebra and looking at the world through its perspective. This is precisely what I was getting at in my essay with “you do not get to drive”, you are a passenger. You need to put trust in the fact the algebra knows they way, and let it take you there.

I posted in Roger Schlafly’s essay blog a rebuttal stating the mapping from only a subset of mathematics to physical reality is surjective. Most of us can agree there is only one physical reality, and all mathematical models must make a proper accounting for what we perceive this reality to be. So we naturally look from the one physical reality to the many plausible mathematical theoretical explanations for it.

Many people pick one of these choices and run with it. In the big scheme of things this is an optimal approach, for the collective will succeed faster by leaving no stone unturned. The uber-intelligent types with superior uptake skills often fill their pumpkin heads with so much stuff they overreach their capacity to sort through it all, their intelligence becomes an obstacle to achieving more than an understanding of the status-quo. The rest of us that *must* cut loose of alternatives by specializing in only a subset of plausible mathematical approaches perhaps have a better chance for advancing understanding. So while the collective is optimized, the individual is not completely from the risk position. The plausible choices are not all assured of leading to a complete description of physical reality, and we will not know which do and which do not until we get there. Several choices may end up being true to reality but some of these may require a deeper understanding a priori for us to appreciate it. This is to say they are not optimal paths for working things out, but provide a deeper perspective in the after-glow of achieved knowledge.

There are several examples of true but difficult paths. My number one is Lagrangian methods. I have no doubt once we know the destination to be described by an Euler-Lagrange equation, a suitable Lagrangian will be possible. I just can’t see the process at this point of being more than a successive guess approach, for it provides little guidance. Another is the Standard Model. I can’t see it showing the way without something more down to earth providing some assistance. I assume this assistance will be mutual.

Me, I am somewhat risk-adverse. I want the method to tell me how it is. I do not want to tinker with things to mold it into my personal belief on how reality should look like for I understand my beliefs will be imprecise. The *fun* is restored when you come to a fundamental appreciation for what O algebra is all about, and how it is fully capable of telling you how it *must be* in complete and definite terms. You may need to walk back your position on Gravitation as intrinsic curvature, on a fundamental stochastic character for nature. It just might work out that “space” is immutable; that the clay reality is sculptured from is potential functions. I am sure I alienate many readers with this position, probably minimizing my chances for success in this essay competition. So be it.

Rick

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Thomas Howard Ray replied on Sep. 13, 2012 @ 18:30 GMT

Hi Rick,

You're leading me to understand some of the historical rivalry between analysts and algebraists that had me puzzled in the past. :-)

I just don't understand the aversion you seem to have in translating one mathematical method to another. For example, 16 degrees of octonionic freedom is identical to 16 redundant points of a Minkowski space tensor. Hestenes, with spacetime algebra, gets smoothly from the algebra to the analysis by deriving Minkowski space. If he couldn't get there, I would be prone to dismiss your arguments, because I wouldn't understand them.

I truly admire your ability to get seamlessly to your destination by letting the algebra drive, as you say. Some of us lesser mortals, however, walk on crutches, and have no chauffeur.

Tom

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Author Rick Lockyer replied on Sep. 14, 2012 @ 06:12 GMT

Hi Tom,

The history of the Quaternions is a case study in human behavior. It got a bit nasty over the latter half of the 1800’s. I guess some things will never change. Quaternion Algebra was approachable for the sensibilities of the time, but Octonion Algebra was not. I think part of the problem many people have with appreciating O comes from the tedious nature of their manipulation when...

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Hi Tom,

The history of the Quaternions is a case study in human behavior. It got a bit nasty over the latter half of the 1800’s. I guess some things will never change. Quaternion Algebra was approachable for the sensibilities of the time, but Octonion Algebra was not. I think part of the problem many people have with appreciating O comes from the tedious nature of their manipulation when all you have is pencil and paper, which is all there was back then. So maybe it worked out OK for the advancements that were made in the first half of the 20th century.

We have computers today, but available software like Maple and Sage do an insufficient job with O in my opinion. I was fortunate to have math, physics and software development skills that enabled me to write my own symbolic algebra software, which I have modified as needed. Because of this, I have had a clear advantage over the pre-computer generations, and a lesser but still significant advantage over many of today’s folks. I appreciate the challenges people have wrapping their minds around O without proper tools to play and explore. I try my best to explain things I have found, but learning by doing is very challenging for many and will be until the available tools improve.

I do not have any aversion to translating one mathematical method to another if I believe it would be important, it is more that I rarely see the utility in doing so. I feel no obligation to derive Octonion matches for much more than Electrodynamics, which I believe I have done an admirable job on. I will accept any portion of the remainder of today’s less certain orthodoxy if and only if it is suggested by or copasetic with Octonion Algebra.

I get a chuckle out of hearing people insist physical reality is 4D, happily accept the use of tensors, and dump on any algebra with more than 4 dimensions. Tensors above rank 1 are algebraic structure added to overcome shortcomings. Their use is no different than formulating everything in flat higher dimension algebra. I would not go so far as saying there is equivalence between these two representations, for the flat algebra is more general. Be it simple matrix or tensor, the equivalent separate basis element for every position multiplication table will have zero entries since each position does not multiplicatively interact with every other as it can with the general algebra of same dimension. Then we have the fact that matrices are associative for multiplication but O is not. So I ask you, why should we look to the less general for all of the answers? The answers may not be describable in these restrictive structures.

Rick

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Rick Lockyer wrote:

"I call The 3:4 Morph Rule."

I call 3:1 Yuri Rule

See please

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

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

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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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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 isotropic algebraic elements. You do not get S7 with split Octonions that are not even a division algebra. Perhaps you could explain this sentiment to me. I can see where you are coming from since it is good politics, just do not see how you are going to get to where you seem to want to go.”

In the event you are unfamiliar with split Octonions, they provide a norm with mixed signature, my guess on how a Minkowski like split signature metric space might have anything to do with Octonions of some form. Perhaps you could review the exchanges between Jens and myself and opine here.

I liked your essay quite a bit, even though I do not think we are in agreement on some things, just as Edwin pointed out. But as I mentioned to him above, there are aspects of your work that are to be taken notice of. Hopefully you read my essay and have a sense about where I am coming from. It will take a lot for me to back away from my position that Gravitation is not intrinsic curvature when expressed in a suitable framework such as provided by Octonion Algebra; it is described by potential functions. I cannot possibly think it is mere coincidence that the proper form for Electrodynamics from potentials to conservation of energy and momentum is *mandated* by the structure of Octonion Algebra, as is the Lorentz Transformation, all by applying Ensemble Derivatives and The Law of Octonion Algebraic Invariance, and no Minkowski metric space in sight. I believe you are keen on GR, perhaps you might benefit from a change of opinion on this position. It just might be one of those “fundamental assumptions” the essay contest is all about.

Regards,

Rick

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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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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

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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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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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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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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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 this with the charge force in a 4D setting is a bit problematic. I can’t say this was Einstein’s main motivation for looking for something completely different, but I have never heard a convincing argument for using curvature on a first principle basis.

If you think about 4D tensors of rank > 1, this is entirely an algebraic structure that adds dimensional count, as it must for say, the 4D EM field tensor, since the electric and magnetic fields have distinctly different component transformation characteristics. We often say they are manifestations of the same thing, but this same thing has more than 4 dimensions to it, or more precisely additional degrees of freedom. The robust character of mathematics gives us more than one way to address the need for additional degrees of freedom. Whether or not you call the additional degrees of freedom “physical dimensions” is a semantics choice.

When we talk about “observed 3D space”, just what does this mean? I think this is typically interpreted as observed with our primary sense of sight and perspective. God gave us this ability to enable us to function in the world long enough to procreate, not to be able to fundamentally observe the nuances of the nature of things. We should not impose three spatial dimensions just because we can “see” it. We should look to span the space in a purely mathematical way. This requires more dimensions, and obviously I think the number is 8.

We all understand tensors fall short of a full deck of cards. This is why spinors are in fashion. Both are algebraic structures that are not fully general. This may be why they have had their share of success while leaving so much unrevealed, and it looks like the missing pieces must be hand crafted. This is not the case for Octonion Algebra. It is the most general of the division algebras, and as I have stated in the essay, fully descriptive when the concept of algebraic invariance is applied with the proper calculus. Clearly things are not yet theoretically complete, but this is fertile ground.

Rick

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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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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

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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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 its secrets. For years we thought he just wouldn't associate with the other family members at all, but somehow we worked out how to do it safely. You see; Octonius is very persuasive, and can make people do almost anything - so he can't be trusted, or rather no one person can ever see him alone. And when we send two, they always disagree on what was said. Therefore; we always visit Uncle Octonius in committees of three. But; the first time a group of us visited, he insisted that he must see all the family members - with equal frequency - and that there always be someone in common between any two visits. Luckily; this worked out, because there are seven of us.

The thing is; Octonius is incredibly wealthy and knows the secrets of the universe, but we were all so afraid of him that we never knew why he seemed so crazy. You see; he always liked to break the laws of algebra - or insist on things being backwards sometimes - whenever we tried to use the associative and commutative rules to simplify expressions for him. But we never understood why that was, until we attempted to rank ourselves - thinking that both the greatest and slightest within our family needed to be included, within each committee, to assure trust. Then Octonius explained that committees follow a rule that is non-commutative, and then if you include everyone at once things become non-associative, because there can be disagreements between members or committees - but there is also a hierarchy or ordering of and within any committee.

Though we are still not sure we can trust him, Uncle Octonius tells us this is as fair as it can be, and now he is teaching us the secrets of the universe. So who could complain? I'm glad cousin John didn't let the others lock him up in the attic, or we would never have learned of his vast wealth and untold secrets.

end of story

Jonathan

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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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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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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 in team and furthermore with algorythmic tools.The pity is still a weak word for your poor team. It is even not an antithesis in fact.It is just business and marketing. I don't know me, make an other job or I don't know me, study for example.

You are not competent , so why you insist with your octonions ? just to make surf and to live in california with a lot of dollars.Let me laugh . If you were skilling, yes , but no, you repeat always the same stupidities.In fact you are not general, so how can be your details ? even the entropy you do not understand, so how can you understand its entropical distribution on the arrow of times. In fact you confound waht is the objectivity and the subjectivity. In fact you do not understand the determinism and its physical laws. So how can you understand the 3 vectors and the scalar? the evolution is unknown for you, you do not understand neither the principle of equivalence, nor the newtonian mecanic and its universal proportions. In fact you superimpose with pseudo parallelizations and you insert irrational reversibilities. All these extrapolations are not rational and dterministic. It is evident that the game is even not interesting because you repeat always the same things.So please increase your team or buy scientists, general please because there I don't evolve :).

Jonathan , you are just skilling for the strategies, not for the rational part.Increase your team please, just for me, too easy there dude.

Regards and sphericaly yours, Steve Dufourny Jedi of the Sphere and inventor of the Theory of SPHERIZATION ......eureka !!!

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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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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

attachments: R_Lockyer_book_chapter_10_preliminary_snippet.pdf

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

Please use the link help page for web links so the blog home page doesn't get messed up.

[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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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 number of non-scalar basis elements to 7*30. There is nothing algebraically new provided by any of the 30 possibilities. An analogy would be my calling you basis element Greg today, but deciding to call you Larry tomorrow, it does not change what you essentially are, just how I choose to enumerate you in order to distinguish between you and someone else. This is why I suggest picking one of the 30 and running with it, and see little physical distinction/importance provided by the full set of aliases.

The E8 connection to all 30 does nothing to convince me otherwise. Physical reality *is not* symmetry groups, physical reality *exhibits symmetry*. I think this distinction is lost on many SM aficionados. Most symmetry abstractions have a multiplicity of physical/geometric application, making it suboptimal to look outward from a symmetry perspective in one’s methodology for discovery of a TOE.

I looked at your link. Your (presume) description of the construction of the 16 ways to roll out O for a given set of 7 triplets was not complete enough to judge, not knowing how you enumerated them. Perhaps there is additional information off the entry page. The simple essential and thus fundamental method is what I called in my essay the 3:4 Morph Rule. Starting with any valid O definition, you can create another by changing the handedness of all 3 H subalgebras that include a common basis element, or changing the handedness of all 4 H subalgebras that exclude one basis element. There is a plethora of ways to represent any single O algebra with a directed Fano plane. So if you are trying to gleam the variability in O through categorizing the various ways to draw a Fano depiction, you are over-complicating things.

Rick

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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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