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  • Post #21

I think it will be worth your time to get through this. Doing an equivalent to the Octonion 8-force-work expressions that have an outside differentiation that are analogous to the divergence of the Electrodynamics stress-energy-momentum tensor allows the conservation of energy and momentum equations to be formed in the Octonion Algebra framework. The 8-force-work is 9 pages of differential equations and the outside differentiation equivalent is 24 pages, tough to pull out of the 9 pages without some help. The help is looking at all 9 algebraically invariant terms of the form below for for d_j representing partial differentiation, A_j the Octonion 8-potential coefficients and bases e_j

d_i e_i * {( d_j e_j * A_k e_k) * (d_l e_l * A_m e_m)}

which of course is determined by basis products e_i * { (e_j * e_k) * (e_l * e_m) }

The Electrodynamics portion is exactly as it is with 4D tensor representations but the whole includes more fields and forces, including Gravitation and a number of not EM or Gravity related rotational fields. Of course this can be written much simpler inside a half page by representing things in terms of time rate of change and gradients of rotational and irrotational energy densities, time rate of change and divergence of the 7D Octonion Poynting vector, rotational and irrotational dyadics, etc just like the Electrodynamics approach but in more dimensions, necessary to span more stuff.

When I first derived this with the aid of my home grown symbolic algebra software, I was blown away that this many terms actually balanced out as an equality. THIS WAS NOT A SIMPLE COINCIDENCE. The power and truth of Octonion Algebraic Invariance if formidable.

As for my choice of triplet enumeration technically I do not think it matters. I already mentioned the "exclusive or" logic operation bit wise on binary values 1 through 7 naturally provides 7 closed sets perfect for the triplets:

1^2^3 = 7^6^1 = 5^7^2 = 6^5^3 = 5^4^1 = 6^4^2 = 7^4^3 = 0

So 1^2 = 3, 2^3 = 1 and 3^1 = 2 fully commutative closed set, same for other triplets.

I think this is more "cool" not to mention advantageous within my symbolic algebra software since all processors to the exclusive or as a native uP instruction.

Spend sometime with the Invariance/variance issue, it is all there in the table and my thread note. It is critical to mathematical physics within the bounds of Octonion Algebra.

Rick

What a piece of something, last was me. By time I got though entry and several attempts at the I am not a robot, it signed me out.

I must cnofess that you are too frank a person who can criticise one's own writings. Truth is quite illusive though it can also be termed as simple in nature! Mahatama Gandhi often said that his is not that much a struggle against British rule over India as it is a search for truth! That gave rise to Peaceful Non-cooperation Movement for India's freedom. Easy said than done, today if we look critically as to how India got freedom, several factors come on the scene, like rebellion among the soldiers of native Indians in British army, Netaji Subhash Chandra Bose movement with the help of Japanese, etc. British found it difficult to continue their rule. Also, they thought dividing India into two through creation of Pakistan will be enough for them to let the continent difficult to manage! British were not all wrong in their suppositions as Pakistan is faced with several unsurmoutable problems and India continued to remain poor and insufficiently developed. Only recently, leadership of current PM Modi that hopes have arisen for India to progress towards a status due to it on account of population and talent it posesses intinsically.The wisdom behind lies both in friendly foreign relations and effective internale fforts towards development and wealth generation. May i say that just Truth is not enough, its practice in actions and character building of individuals and community together are needed during implementation....

    Kindly also elaborate on the wrod ' Octonian '. in your words as language used by others have not helped me comprehend this word adequately!

    Mr. Rick Lockyer,

    Very nicely written, thank you for some insights.

    Do you see any similarities between these 3 essays? {link: fqxi.org/community/forum/topic/2951]1[/link], 2, 4

    Maybe you read them or maybe not, but it seems to me that all of them are pointing in the same direction, with different approaches. Don't take me too serious though, I'm just playing with "images leftovers" from the essays.

    Silviu

      *no relation other than this contest between me and any of the above authors

      My earlier post on my Algebraic Invariance and Variance Sieve seems to have missed the mark of being presented in an understandable way since Geoffrey Dixon did not follow it, telling me I did a poor job. Let me try to improve things. First, what the heck is this all about?

      In my essay I state the different ways Octonion Algebra can be defined is fully covered by select orientation changes within the seven Quaternion subalgebra triplets, a significant simplification over changes to 64 element multiplication tables. There is no reason to believe one of the 16 is preferred over the others, but the outcome of changes between definitions is the possibility of a result sign change. So if the result is our attempt to describe something we can observe to be one sign and never the other sign, the algebra change would be problematic if the result might change sign.

      Fortunately there are Octonion product terms that maintain consistent signage for all algebra definitions. These should be considered invariant symmetries of Octonion Algebra. If we can't assign a preference to one algebra definition over another we must insure the results of any theoretical application of Octonion Algebra to physical reality must fully reside inside the set of invariant product terms. But how can we simply determine whether or not a product term is an Algebraic Invariant?

      Product terms that are not invariant to Octonion Algebra definition changes will have one sign for eight of the definitions and the opposite sign for the other 8. These should be considered anti-symmetric variants of Octonion Algebra. All Algebraically Variant product terms can be sorted into one of 14 sets, where set members all change signs in exactly the same way for all Octonion Algebra definition changes.

      Every product term for any possible series of Octonion algebraic element products can be classified as a member of the invariant set or one of the variant sets. Which set is fully determined by the product history from initial native algebraic element definitions through the final results after some number of multiplications have been performed. Product order matters, and at each step of the product history there is a multiplication between two basis elements with an algebra specific rule determining the result. After the first multiplication in a sequence, one of these elements is a composite result of all earlier products and the other is the next up in the history. The algebraic rule is going to be determined by one variable definition Quaternion triplet rule, or the consistently defined products including the scalar basis or between like basis elements.

      For the sieve algorithm applied to a product history, at each basis element pair product, a move is prescribed from the current row in the Hadamard Matrix table in my essay to either itself or a new row. Which move is entirely determined by the next basis pair product. If the rule this product is governed by is not a variant triplet rule, remain at the current row. If not, the new row is found by doing the row composition described in my essay between the Hadamard elements of the current row and the Hadamard elements for the row labeled by the triplet called out by the basis element product. One could consider the stay on the same row as a composition move specified between the current row and the "none" row which is the composition identity element.

      Procedurally you start a product history on the identity "none" row. The first basis element product rule composition is applied, and you change rows or don't as described above. Next determine the rule between the result basis element from the last product with the next basis element up in the product history sequence, use it to make the next row change or stay. Repeat until the product history sequence completes. If you end up on the "none" row, the product sequence is an Algebraic Invariant. Otherwise you will be on a row with a triplet label and the product sequence is an Algebraic Variant partially described by the label, let's call it {abc}. If the sequence involved an even number of triplet compositions, the variant set will be what I call V{abc}, if odd count, it will be in the set V-{abc}.

      Hopefully this clears things up.

        Did you read my essay or are you simply resting to the title?

        My concept of truth here has nothing to do with one's opinion, and certainly not one's behavior.

        Consult Wikipedia but spell Octonion correctly for answers to your next post.

        Rick

        My inability to understand something frequently has little to do with how well an idea has been presented. You may have done a great job presenting the ideas; and I may have been (and likely was) too lazy to dig into it adequately. I grasp ideas through a kind of osmosis, which takes time.

        I put a comment in my essay section trying to prove something that is maybe irrelevant to your discussion, viz., all representations of O are isomorphic. I really have no idea if that is pertinent. Quite possibly not. It doesn't change your notion - which I share - that left and right representations are in some ways distinct. In my work, as I recall, the distinction arises from the X-product. Using our ea, a=1,...,7, type of representation of O, there are two distinct categories. Two representations are in the same category if one can get from one to the other via an X-product revision. Using this method of mapping one rep to another, it becomes clear that the collection of reps is divided in two, all reps in each half can be linked by an X-product variation, but there is no way to do this from one half to the other. I am hoping that this means we are talking about the same thing, at least mathematically. I'm not yet close to figuring out your connection to physics. Again, likely my fault.

        I need to reread your essay. Soon. For that matter, I also need to reread my own material on all this stuff. As a wise man once said (you), so little time.

        Excellent essay Rick!

        What can I tell you? You hit all the bases. You speak directly to the question of what is fundamental. You present a compelling argument for your choices; though I might add 'and this also'... I am in general agreement with your premise and I look forward to working with your octonion calculation software, possibly for creating some higher-d fractals. I will have to re-access some of your prior work too, since I see hooks into what I am now working on that didn't pop out before.

        I will read this again, but offer my rating now while I am still impressed.

        All the Best,

        Jonathan

          I should add this...

          My brief conversation with Tevian Dray at GR21 affirms what you are saying about the fact that non-associativity is a necessity and not something that should be avoided. I like the way Geoffrey said it on another thread; non-associativity is not a bug, it is an amazing feature that comes into play in the perfect way. P.C. Kainen also shares in the conviction that non-associative algebra and geometry must play a part in Physics.

          And your statement that the octonions need to drive or drive the process is spot on. They are the Big Daddy in terms of evolutive processes, and made so by their non-associativity, so that is most certainly not a defect - and instead it is quite possibly the most powerful attribute an algebra can have. So I applaud your efforts to make the octonions more accessible for people working in Physics and Cosmology.

          All the Best,

          Jonathan

          Rick,

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

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

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

          Rick,

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

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

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

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

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

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

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

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

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

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

          Can you find logical or epistemological fault?

          Very Best

          Peter

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

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

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

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

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

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

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

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

          Rick

          Hello yet again Rick,

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

          All the Best,

          Jonathan

          Jonathan,

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

          Thanks for your kind words and support over the years.

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

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

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

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

          Rick

          This sounds excellent Rick...

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

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

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

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

          All the Best,

          Jonathan

          Dear Rick Lockyer

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

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

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

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

          Some of the Main foundational points of Dynamic Universe Model :

          -No Isotropy

          -No Homogeneity

          -No Space-time continuum

          -Non-uniform density of matter, universe is lumpy

          -No singularities

          -No collisions between bodies

          -No blackholes

          -No warm holes

          -No Bigbang

          -No repulsion between distant Galaxies

          -Non-empty Universe

          -No imaginary or negative time axis

          -No imaginary X, Y, Z axes

          -No differential and Integral Equations mathematically

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

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

          -No many mini Bigbangs

          -No Missing Mass / Dark matter

          -No Dark energy

          -No Bigbang generated CMB detected

          -No Multi-verses

          Here:

          -Accelerating Expanding universe with 33% Blue shifted Galaxies

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

          -All bodies dynamically moving

          -All bodies move in dynamic Equilibrium

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

          -Single Universe no baby universes

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

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

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

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

          -Uses everyday physics as achievable by engineering

          -21000 linear equations are used in an Excel sheet

          -Computerized calculations uses 16 decimal digit accuracy

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

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

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

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

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

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

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

          Best

          =snp

          Jonathan,

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

          Rick

          Hello again Rick,

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

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

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

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

          Warm Regards,

          Jonathan

          Thanks greatly Rick!

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

          More later,

          Jonathan