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FQXi Essay Contest - Spring, 2017
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Truth by Rick Lockyer
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Author Rick Lockyer wrote on Feb. 5, 2018 @ 19:31 GMT
Essay AbstractThe 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. 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”. From a position of partial understanding, there are many plausible paths scientists may take towards a goal of more complete knowledge. There is utility in stepping back, examining what is truly fundamental, then using it to optimize the process, perhaps going in a different direction.
Author BioB.S. in Physics, Stanford University many moons ago. Correctly determined what I wanted to learn would not be taught in an advanced degree, so began a successful Engineering career designing with hardware and firmware for microprocessors and software. Kept my passion alive for Physics, got sucked into Octonions ~25 years ago, no escape in sight.
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John Hozy Hozy wrote on Feb. 5, 2018 @ 20:24 GMT
I really inspired by this post. I would like to say that Nature always wears the colors of the spirit. According to
students writing services there is required look deep into nature as well then anyone can able to understand everything better.
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David Lyle Peterson wrote on Feb. 5, 2018 @ 23:54 GMT
Dear Rick
Your papers indicate that you have mastered what we consider to be an exceedingly difficult arena – the algebra of Octonians. I have a hunch that a next big breakthrough in physics (apart from string theory) will finally use this algebra for representing higher relevant Lie groups and perhaps as something similar to taking the “square root” of the Dirac equation (like he did from the Klein-Gordon equation). When that happens, you will be in a great position to understand it.
Best Wishes
Dave
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Author Rick Lockyer replied on Feb. 6, 2018 @ 05:28 GMT
Dave,
Thanks for reading my essay. My preferences and what I see as more fundamental is Octonion Algebra over any associative algebra which would include all groups which by definition are associative. This is not to say group theory is not important to Octonion analysis, it most certainly is. Octonion Algebra just can't be used for "representing higher relevant Lie groups..." since it is...
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Dave,
Thanks for reading my essay. My preferences and what I see as more fundamental is Octonion Algebra over any associative algebra which would include all groups which by definition are associative. This is not to say group theory is not important to Octonion analysis, it most certainly is. Octonion Algebra just can't be used for "representing higher relevant Lie groups..." since it is generally non-associative for multiplication.
Octonion Algebra does infer group structure, as I discuss in my essay. The master of this is Geoffrey Dixon, and I would recommend you take a look at his essay and buy his books also. Not wanting to speak too loudly for him, he is all about applying group theory, using division algebra structure as a guide for initial "right and left action" matrix based group member construction and using them as the means to the end.
For me it is the other way around. The division algebras are fundamental, and the group theory connections are suggestive of added structure that helps us understand the division algebras. The division algebras, mostly Octonions are the means to the end. It is all about which speaks more loudly to us. Octonion Algebra mandates structure that corresponds to physical reality, whereas group theory is completely general and widely applicable, so gives back little more than what we put into it, which to a large extent in the standard model is "by hand". I do not want to tell the math what to do, I want it to tell me how it must be.
Take note of my comment that the generally non-associative Octonion Algebra does not need non-associative physics to justify its use, and more important my point that Octonion analysis can go where matrix - tensor - matrix based spinor analysis can't because of the non-associativity for multiplication. My efforts will be exclusively Octonion Algebra based. Any connection to group theory will be welcomed if it is a nature outcome of the algebraic structure, but used within the structure intrinsically.
Rick
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Geoffrey Dixon replied on Feb. 9, 2018 @ 18:00 GMT
Hmm. Am I "all about applying group theory"? Groups arise as secondary structures from the mathematics inherent in tensored division algebras. This was first noted in conjunction with color SU(3) 45 years ago at Yale by Feza Gürsey and company. His starting point, as is mine, is the division algebras: specifically, C⊗O. Clearly his group had SU(3) as a goal, and were initially very pleased with how naturally it arose from the mathematics. But they tried to shoehorn this elegant connection into QM. Their failure in this regard made them turn rather vehemently away from trying for any further connections of O to physics. They had reputations to protect.
I chose to ignore QM and just pay attention to the algebraic structure. You say: "I do not want to tell the math what to do, I want it to tell me how it must be." We are on the same page here. Yes, I did assume that the mathematics would lead from C⊗H⊗O to U(1)xSU(2)xSU(3). And in fact, even in the absence of any knowledge of physics it does. So, cool. But in my paper "Seeable Matter; Unseeable Antimatter" I take the mathematical structure well beyond what we all know and love. I shan't be taking those ideas any further, but to get to that point, if the mathematics had ever not presented an easily followed path I would have given up on the whole thing. In particular, the mathematics never had anything to say about gravitation that I could see, so I never tried to incorporate it. And if that above mentioned paper is in fact correct (which of course it is ... ahem), then gravity has a whole new playground - one I freely admit I am not competent to enter.
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Author Rick Lockyer wrote on Feb. 6, 2018 @ 00:10 GMT
About 25 years ago while looking to integrate a potential based unification of Electrodynamics and Gravitation, I stumbled across Octonion Algebra, which looked very promising.It became immediately apparent its application would not be possible using a pencil and paper. I started development of a symbolic algebra software tool since nothing that met my needs was available. Frankly, there still isn’t. Over the years it has developed into something quite useful. I want to encourage more interest and appreciation for Octonion Algebra and it’s utility for mathematical physics. To help this along, I will be putting up the latest rendition written in JavaScript open source on my website after I get some time to document it better. It was developed for use with Nodejs.
Included as an implementation example will be my verification script for my derivation of the Octonion conservation of energy and momentum, matching the classical Electrodynamics divergence of the stress-energy-momentum tensor terms exactly while including gravitation and a number of new rotational fields in proper juxtaposition.
Stay tuned, let you know when it is up.
Joe Fisher replied on Feb. 6, 2018 @ 16:16 GMT
Dear Rick Lockyer,
FQXi.org is clearly seeking to confirm whether Nature is fundamental.
Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.
All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.
Only the truth can set you free.
Joe Fisher, Realist
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Joe Fisher replied on Feb. 6, 2018 @ 16:18 GMT
Dear Rick Lockyer,
FQXi.org is clearly seeking to confirm whether Nature is fundamental.
Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.
All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.
Only the truth can set you free.
Joe Fisher, Realist
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Joe Fisher replied on Feb. 6, 2018 @ 16:19 GMT
Dear Rick Lockyer,
FQXi.org is clearly seeking to confirm whether Nature is fundamental.
Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.
All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.
Only the truth can set you free.
Joe Fisher, Realist
post approved
Joe Fisher replied on Feb. 6, 2018 @ 16:23 GMT
Dear Rick Lockyer,
FQXI.org might be seeking to know if there could be a fundamental nature.
Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.
All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.
Only the truth can set you free.
Joe Fisher, Realist
post approved
Joe Fisher replied on Feb. 6, 2018 @ 17:03 GMT
Dear Rick Lockyer,
My research has indicated to me that there must be a fundamental Nature.
Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of the Universe allowable.
All objects, be they solid, liquid or vaporous have always had a visible surface. The real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.
Joe Fisher, ORCID ID 0000-0003-3988-8687.
post approved
Steve Dufourny replied on Feb. 16, 2018 @ 12:08 GMT
Hello Mr Lockyer,
It seems relevant these octonions not commutative to unify this gravitation, have you already thought about the fact to insert the spherical volumes and their motions and this DM not baryonic.Because we cannot find this weakest force wityh these lie algebras, it lacks something.I read the works of Hestenes with the spinrs and geometrical algebras, it is a good tool also,that said this quantum weakest force have difficluties to appear .....Best regards and congratulations for your essay, good luck :)
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Joe Fisher wrote on Feb. 7, 2018 @ 16:41 GMT
Seeing that the surface of the earth was formed millions of years before man and his complex finite informational systems ever appeared on that surface, it logically follows that Nature must have permanently devised the only single physical construct of the real visible Universe allowable.
..........
Joe Fisher, Realist
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Gary D. Simpson wrote on Feb. 8, 2018 @ 16:15 GMT
Rick,
After reading your essay, I don't know what to say other than simply ... "Thank You". You have given me a road map. Your ideas will be very helpful to me.
Best Regards,
Gary Simpson
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Author Rick Lockyer replied on Feb. 9, 2018 @ 00:01 GMT
Thanks Gary,
Octonion Algebra not only provides a roadmap, it follows naturally Electrodynamics. I extensively used the 4D tensor cover of Electrodynamics as my roadmap to derivation of Octonion conservation of energy and momentum. I have already skimmed your essay and I am familiar with your previous interest in Quaternion. I will comment later.
Rick
Geoffrey Dixon wrote on Feb. 8, 2018 @ 20:40 GMT
Hi Rick
FYI, in my Windmill Tilting division algebra book I have a section with the following heading:
"480 Octonion Products: Renumberings"
Cheers, GD
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Author Rick Lockyer replied on Feb. 9, 2018 @ 00:35 GMT
Geoffrey,
Have had this book and your earlier book since first learning they were available. Know your index processes for coming up with the Quaternion triplets. They are “cool” but not as cool as my favorite which uses binary numbers 1-7 and the computationally friendly exclusive or logic function. This naturally partitions 1 through 7 into 7 closed sets of three, perfect for the...
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Geoffrey,
Have had this book and your earlier book since first learning they were available. Know your index processes for coming up with the Quaternion triplets. They are “cool” but not as cool as my favorite which uses binary numbers 1-7 and the computationally friendly exclusive or logic function. This naturally partitions 1 through 7 into 7 closed sets of three, perfect for the seven triplets. These are what I use and it ripples into my Right and Left Octonion enumerations and morphs between.
Your O+3 is a “Right” Algebra as is O-5, O-3 and O+5 are “Left” Octonion. Take any index, cyclically shift the three triplets containing it until the chosen basis is in the middle position. If the 3 indexes on the right side are another triplet, you have a “Right” Algebra, otherwise the left side will be another triplet and you have a “Left” Algebra. If this is not consistent for all 7 indexes, you are not representing a normed division Algebra.
Now think about that index in the central position of the Fano Plane representation, and envision how there are 2 distinct forms, either the vertex bisectors all come out or all go in. Different structure.
You and I discussed the 480 forms back at the turn of the century. My position was and still is there are 30 different ways to partition the Quaternion triplets, and any one will do, with the others being nothing more than aliases. So you have 480/30 = 16 meaningful differences, 8 Right, 8 Left. If I call you Geoffrey today and Jimmy tomorrow, you will still be the same self-described curmudgeon.
Liked your essay, having read your books you could have left your name off it and I would have guessed it was you without the clear give-aways. Will comment later.
Did you actually read mine? Check out the 2012 essay also.
I first heard about the Hadamard connection from your 1994 book, thanks for that. I am sure it was subconsciously there when I explored the valid chiral changes on the triplets and the Hadamard pattern they describe.
Rick
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Geoffrey Dixon replied on Feb. 9, 2018 @ 17:26 GMT
I remember years ago contact, but not context. Anyhum, I'll try to prove I've read the essay (I am somewhat of a slacker in this regard):
"Most noticeable was J. C. Maxwell some 30 years later. He saw within individual Quaternion differentiation sub-forms all of the differentiation product forms he needed for his famous 4 equations,but could not work it out with full 4 dimensional...
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I remember years ago contact, but not context. Anyhum, I'll try to prove I've read the essay (I am somewhat of a slacker in this regard):
"Most noticeable was J. C. Maxwell some 30 years later. He saw within individual Quaternion differentiation sub-forms all of the differentiation product forms he needed for his famous 4 equations,but could not work it out with full 4 dimensional algebraic elements."
Part of the reason I was awarded a PhD was that at the end of two years of not very stellar grad school there was an oral qualifier exam. I was asked a question about Maxwell's equations. I responded that to answer it I'd have to write the equations in a form the professors had never seen before. They were skeptical, but I had all 4 written as one using quaternions. They had in fact never seen anything like it. I passed the qualifier. :) I should add that it is not clear to me what "Quaternion differentiation sub-forms" means. That kind of thing makes reading difficult for me.
Speaking of quaternions: "It has three separate Complex sub-algebras." Well, ok, if we just look at the basis units, but really there are infinitely many sub-algebras isomorphic to C, one for every element of norm 1 with no real part, so topologically, S
2. This is just a quibble.
"One should expect Octonion Algebra to fundamentally speak to us louder and perhaps with more authority than this Geometric Algebra." Expect? Authority? The problem with this in my view is that in having an expectation of this sort you are in a sense instructing the maths where you want it to go. It may not listen. (On the other hand, you're not wrong, IMO.)
"For some, the fact that the above Geometric Algebra is associative for multiplication makes it more attractive. This reluctance towards the use of Octonion Algebra is a fundamentally false criticism."
Well, yes. But you're glossing over the fact that the algebra of actions of O on itself is an associative geometric algebra for a 6-D space. Anyway.
"If division must be a characteristic of Nature's Algebra, we can stop at Octonion Algebra, since it is the end of the line."
I of course agree that O is the end of the line, but rather because there are only 4 parallelizable spheres, and only 4 sequences of classical Lie groups. Any higher dimensional algebra you may concoct will have none of the properties and associations that make R,C,H and O exceptional, generative, and resonant. If you start with spheres, no further arguments are needed.
"We have useful three dimensional items like the magnetic, electric and gravitational fields that we must find homes for in the eight dimensional structure of the Octonions."
Let me say that I heartily approve of pursuing this line of thinking, for whether or not it is physically correct, it can generate mathematical insight that can lead to better physical applications. If, that is, one listens to the mathematics. I question whether trying to find homes for gravity and EM within the relatively small structure of O you are allowing your ideas of what the mathematics ought to do lead you. It may have other ideas that your preconceptions blind you to seeing. Maybe.
I am intrigued my your thoughts on the applicability of left-right versions of O. They are, of course, isomorphic, but ...
Anyway, in conclusion, you have real ideas here, and, as suggested, "so little time" to develop them all. But what better way to use that time than by listening to the whispers from the universe.
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Peter Jackson wrote on Feb. 9, 2018 @ 10:16 GMT
Rick,
Hmmm. Very interested. You'll recall I've supported your view in principle before but I'm no mathematician. I've invoked Maxwells 4 states mechanistically without ever understanding quaternions. You conclude;
"Nature's choice will surely be Octonion Algebra." I think I've found it already has done! Are you familiar with Dirac's QM twin stacked orthogonal inverse pair equation? I'd like you to check out a very important finding for me (with non-linear Chirality) advise, and maybe sign up to help with the algebra!
It's all in my essay (with matching computer code and plot in Declan Traill's) amazingly appearing to reproduce QM's predictions classically, shocking enough but all barriers to compatibility with 'SR' are then also lifted!!
Thanks for yours, very hopeful and helpful I hope and pencilled in for a possible 10. I don't think it was surprising physicists;
"had difficulty working full four dimensional algebraic elements into physical theories. But I'm now convinced the inverse of that will work! So Octonians may indeed be prove a critical reality.
Excellent job. Please do ask any questions of mine if not clear.
Very Best
Peter
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Peter Jackson replied on Feb. 12, 2018 @ 14:27 GMT
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
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Author Rick Lockyer replied on Feb. 12, 2018 @ 17:55 GMT
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...
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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
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Author Rick Lockyer wrote on Feb. 10, 2018 @ 17:26 GMT
Having the table of ordered permutation triplet basis product rules for Right Octonion Algebra in the body of my essay, it can be used to visualize my algebraic variance/invariance sieve, and demonstrate that all product terms for any number of Octonion algebraic element products will fall into an algebraically invariant set, or one of 14 algebraically variant sets.
To make sure everyone...
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Having the table of ordered permutation triplet basis product rules for Right Octonion Algebra in the body of my essay, it can be used to visualize my algebraic variance/invariance sieve, and demonstrate that all product terms for any number of Octonion algebraic element products will fall into an algebraically invariant set, or one of 14 algebraically variant sets.
To make sure everyone gets the drift on the ordered permutation triplet basis product rule, going cyclically left to right, the consecutive product of two basis elements is + the third element, and commuting the product order going cyclically right to left, the consecutive product of two basis elements is - the third element. For any three unlike non-scalar basis elements there are two possible definitions, starting with one order then forming another rule exchanging any two elements which is fully equivalent to ordering them in the opposite direction. This singular rule change has the effect of changing the sign on all 6 basis product pairs. Very important that it is just a sign change.
It will not be of any importance that I use a table for Right Octonions and not Left Octonions, nor that the table is relative to algebra R0. Any other choice will sieve the same product terms into the same sets, the only difference will be the relative signs within the variant sets.
I will do a proof by example using a sequence of basis element products describing the product history of the final resultant basis element. The product e1 * e2 is ruled by the ordered permutation triplet including the set {e1 e2 e3}, using curly braces to not imply a sign rule quite yet. For R0, R1, R2 and R3 the sign rule is (e1 e2 e3) and for R4, R5, R6 and R7 the rule is the opposite: (e3 e2 e1). The correspondence between algebra enumerations and triplet enumerations is no accident, they were purposefully enumerated to make it such. Anyway, for the former 4 the result is +e3, and for the latter 4 it is -e3. This would be a simple example of an algebraically variant product term; it can change signs when a change of algebra definition is made.
Now take the e3 result and multiply on the left by e2, effectively doing e2 * (e1 * e2). This is based on the same triplet rule {e1 e2 e3}, so any algebra change induced negation will be done twice, meaning for every possible Octonion Algebra choice, the result will always be +e1. This is an example of a non-trivial (not singularly defined e0 * en, ej * ej etc.) algebraically invariant product term.
If instead we did e5 * (e1 * e2), the second product would be using the rule for {e6 e5 e3} so the sign on the final result basis element e6 would be dependent on both {e1 e2 e3} and {e6 e5 e3} rules, and they can change in different ways for specific changes in algebra. Relative to R0, which has all +1 values in its column and results in +e6, the e6 sign for some other Right algebra will be determined by the product of the row {e1 e2 e3} and {e6 e5 e3} values for that algebra's column. If the row product is +1, the result in that algebra will be the same as in R0: +e6, and if the row product is -1, the result for that algebra will have opposite sign as R0 indicates, or -e6. These column products are precisely the compositions mentioned in my essay. {e1 e2 e3} and {e6 e5 e3} have e3 in common, and the only other triplet including e3 is {e7 e4 e3} which indeed is the resultant row from the composition operation. So we can look at this row and see for R0, R3, R4 and R7 the final result for e5 * (e1 * e2) will be +e6, and for R1, R2, R5 and R6 the result will be -e6. Doing the same row composition on our first example above, the composition ends up on the all +1 row, which is where every algebraic invariant product term will end up, independent of the number of products. If along the way we needed to multiply by the scalar e0 or the very same current result basis, these rules are singularly defined, so one must stay on the current row.
You can see now 7 algebraically variant rows to land on, and might be questioning how I get to my claim of twice this number. We must bring in the Left Octonion Algebras into the discussion, or more precisely the anti-automorphism Right to Left morph which negates all seven triplet rules. If you ended up on a particular variant row through the application of an even number of variant products, changing every one would not change the final result. However, if you did an odd number of variant products, the anti-automorphism would change the result sign. Thus the variant count parity doubles the 7 to 14. My V+{abc} is even count, and V-{abc} is odd count. Since the composition rule is closed, any number of basis products may be done with comparable results.
Now, what happens in each of these variant sets is that for any legitimate Octonion Algebra definition change, every product term in a set will either change sign, or not change sign. My "Law of Octonion Algebraic Invariance" states any observable described by Octonion Algebra must be an algebraically invariant form. The corollary to this is any algebraically variant form is not observable. If we individually add/subtract per sign all product terms in a variant set and force a zero result for each set, a mixed bag of variant and invariant product terms now becomes fully invariant since +0 = -0. These are my "Homogeneous Equations of Algebraic Constraint". It is easy to believe experimentation will not show everything that needs to be seen. I am thinking these equations of constraint are extremely important.
Octonion Algebra is not talking softly here, it is shouting!
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Geoffrey Dixon replied on Feb. 10, 2018 @ 22:23 GMT
Very hard to grasp this without a blackboard. I think we have likely both experienced the frustration of an audience failing to immediately grasp ideas with which we are so familiar that we can no longer imagine everyone does not see what we see.
Why do you use Porteous's (e1,e2,e3) quaternionic triple? In 1993 at the first Octoshop (which I organized in hopes of finding meterial to finish my first book), everyone was using tables for which the set of quaternion index triples was invariant wrt index doubling and cycling. At the time mine was based on the index triple, (1,2,6), and Martin Cederwall (inventor of the octonion X-product, and whose university hosted the Octoshop (Porteous was there, too)) started with the triple (1,2,4). Conway and Sloane also used (1,2,4), and I eventually succumbed to peer pressure and now also start with (1,2,4). Anyway, if you are not already familiar with it, you should get familiar with the X-product, and my extension of it, the XY-product). It's very cool stuff.
However, I have not played with this stuff in a number of years, so even to follow my own work would now require some effort.
I need to see your stuff with fewer words, and more symbols.
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Anonymous replied on Feb. 10, 2018 @ 23:47 GMT
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...
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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
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Author Rick Lockyer replied on Feb. 10, 2018 @ 23:50 GMT
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.
Narendra Nath wrote on Feb. 11, 2018 @ 02:22 GMT
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....
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Author Rick Lockyer replied on Feb. 11, 2018 @ 19:04 GMT
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
Narendra Nath wrote on Feb. 11, 2018 @ 02:25 GMT
Kindly also elaborate on the wrod ' Octonian '. in your words as language used by others have not helped me comprehend this word adequately!
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corciovei silviu wrote on Feb. 11, 2018 @ 17:41 GMT
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,
2,
4Maybe 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
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corciovei silviu replied on Feb. 11, 2018 @ 17:45 GMT
*no relation other than this contest between me and any of the above authors
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Author Rick Lockyer wrote on Feb. 11, 2018 @ 18:43 GMT
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...
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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.
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Geoffrey Dixon replied on Feb. 11, 2018 @ 19:46 GMT
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.
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Jonathan J. Dickau wrote on Feb. 12, 2018 @ 02:23 GMT
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
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Jonathan J. Dickau replied on Feb. 12, 2018 @ 02:44 GMT
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
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Jonathan J. Dickau replied on Feb. 12, 2018 @ 18:20 GMT
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
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Author Rick Lockyer replied on Feb. 12, 2018 @ 20:56 GMT
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
Jonathan J. Dickau replied on Feb. 13, 2018 @ 17:50 GMT
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...
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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
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Author Rick Lockyer replied on Feb. 14, 2018 @ 16:09 GMT
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
Jonathan J. Dickau replied on Feb. 14, 2018 @ 16:43 GMT
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
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Satyavarapu Naga Parameswara Gupta wrote on Feb. 14, 2018 @ 13:10 GMT
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...
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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.h
tml
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 repliedBest
=snp
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Jonathan J. Dickau wrote on Feb. 14, 2018 @ 16:19 GMT
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
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Andrew Beckwith wrote on Feb. 14, 2018 @ 21:35 GMT
http://www.scirp.org/journal/PaperInformation.aspx?PaperID=5
129
Hi, I took a venture into this area myself in 2011 and have referred to it again as seen in further JHEPGC articles, which have appeared in 2016 and 2017
My take, is that what is crucial is the concept of asociative geometry and when it does and does not break down, and the Octonians are a good vehicle to examine it
Although I did not state it earlier, my work on the cosmological constant was based upon themes of the contributions of Octonionic space time.;
Please feel free to examine my essay, December 21st and interpret it with this in mind
Thank you for your wonderful essay
Andrew
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Andrew Beckwith replied on Feb. 14, 2018 @ 21:36 GMT
http://www.scirp.org/journal/PaperInformation.aspx?PaperID=5
129
Has this in it
Octonionic Gravity Formation, Its Connections to Micro Physics
Full-Text HTML Download Download as PDF (Size:394KB) PP. 13-18
DOI: 10.4236/ojm.2011.11002 4,520 Downloads 8,836 Views Citations
Author(s) Leave a comment Andrew...
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http://www.scirp.org/journal/PaperInformation.aspx?PaperID=5
129
Has this in it
Octonionic Gravity Formation, Its Connections to Micro Physics
Full-Text HTML Download Download as PDF (Size:394KB) PP. 13-18
DOI: 10.4236/ojm.2011.11002 4,520 Downloads 8,836 Views Citations
Author(s) Leave a comment Andrew Beckwith
Affiliation(s)
.
ABSTRACT
We ask if Octonionic quantum gravity is a relevant consideration near the Planck scale. Furthermore, we examine whether gravitational waves would be generated during the initial phase, , of the universe when triggered by changes in spacetime geometry; i.e. what role would an increase in degrees of freedom have in setting the conditions during , so that the result of these conditions can be observed and analyzed by a gravitational detector. The micro physics interaction is due to the formation of a pre Planckian to Planckian space time transition in spatial dimensions at and near the Planck dimensional values, i.e. 10–33 centimeters in spatial dimensions. This transition would be abrupt and arising in micro physics regimes of space time.
KEYWORDS
High-Frequency Gravitational Waves (HFGW), Symmetry, Causal Discontinuity
Cite this paper
A. Beckwith, "Octonionic Gravity Formation, Its Connections to Micro Physics," Open Journal of Microphysics, Vol. 1 No. 1, 2011, pp. 13-18. doi: 10.4236/ojm.2011.11002.
References
[1] L. Crowell, Quantum Fluctuations of Space-time, in World Scientific Series in Contemporary Chemical Physics, Volume 25, Singapore, Republic of Singapore, 2005.
[2] A. W. Beckwith, “Identifying a Kaluza- Klein treatment of a Graviton permitting a de-celeration parameter Q(Z) as an alternative to standard DE “,Journal of Cosmology, Volume 13, 2011, http://journalofcosmology/BeckwithGraviton.pdf
[3] F. Li, and N. Yang, “Phase and Polarization State of High Frequency Gravitational waves”, Chin Phys. Lett. Vol 236, No 5(2009), 050402, pp 1-4
[4] F. Li,. M.,Tang, D. Shi, “Electromagnetic response of a Gaussian beam to high frequency relic gravitational waves in quintessential inflationary models”, PRD 67, 104008 (2003), pp1-17
[5] S. Chaturvedi, “Mutually Unbiased Bases”, Pramana Journal of Physics, Vol. 52, No. 2, pp 345-350
[6] John C. Baez, “Renyi Entropy and Free Energy”, http://arxiv.org/abs/1102.2098
[7] A. W. Beckwith, arXiv:0804.0196 [physics.gen-ph], AIP Conf.Proc. 969:1018-1026, 2008
[8] D. K., Park, H. Kim, H., and S. Tamarayan, “Nonvanishing Cosmological Constant of Flat Universe in Brane world Senarios,” Phys. Lett. B 535, 5-10 (2002).
[9] A. W. Beckwith,” How to Use the Cosmological Schwinger Principle for Energy Entropy, and “atoms of Space-Time” to Create a Thermodynamic Space-Time and Multiverse”; accepted for the DICE 2010 proceedings; http://vixra.org/abs/1101.0024, 2010
[10] M. Maggiorie,. “Gravitational Wave Experiments and Early Universe Cosmology”, Physics Reports 331 (2000) pp. 283-367
[11] G. t’ Hooft, Beyond the Quantum, edited by Th. M. Nieuwenhuizen et al. Singapore, World Press Scientific, 2006; http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604008v2.pdf.
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Andrew Beckwith wrote on Feb. 14, 2018 @ 21:55 GMT
http://www.scirp.org/journal/PaperInformation.aspx?PaperID=7
9959
The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors
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DOI: 10.4236/jhepgc.2017.34055 93 Downloads 140 Views
Author(s) Leave a comment Andrew Walcott...
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http://www.scirp.org/journal/PaperInformation.aspx?PaperID=7
9959
The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors
Full-Text HTML XML Download Download as PDF (Size:572KB) PP. 727-753
DOI: 10.4236/jhepgc.2017.34055 93 Downloads 140 Views
Author(s) Leave a comment Andrew Walcott Beckwith Affiliation(s)
Physics Department, College of Physics, Chongqing University Huxi Campus, Chongqing, China.
ABSTRACT
The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial Octonionic geometry embedded in a larger, nonlinear “pilot model” (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a “big crunch” is picked as the embedding structure, so as to avoid the “instants” of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards “Schrodinger equation” states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to; making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry.
KEYWORDS
Octonionic Geometry, Cyclic Conformal Cosmology (Penrose), Modified HUP
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Andrew Beckwith wrote on Feb. 14, 2018 @ 21:56 GMT
That is the recent iteration as opposed to what I did in 2011
Again, the methodology, was implicitly used in My December 21st essay
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Georgina Woodward wrote on Feb. 15, 2018 @ 00:36 GMT
Hi Rick , thank you for sharing your passion for algebra and in particular octonians.
At the beginning you state "mathematics is a language of unambiguous truth", I used to think that the certainty of mathematical solutions do make it more reliable than other kinds of communication. Though statistics is notorious for misleading. Having read "Creating perspective on set theory" by...
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Hi Rick , thank you for sharing your passion for algebra and in particular octonians.
At the beginning you state "mathematics is a language of unambiguous truth", I used to think that the certainty of mathematical solutions do make it more reliable than other kinds of communication. Though statistics is notorious for misleading. Having read "Creating perspective on set theory" by Christophe Tournayre, in this competition, I have been re-considering my opinion. I found this paper "When is one thing equal to some other thing?" Barry Mazur (2007) www.math.harvard.edu/~mazur/preprints/when_is_one.pdf Which showed me that even something as common place as use of the equality symbol is not simple and not immune from ambiguity. Also a statement, mathematical or otherwise, can be true, as in not false; and yet not be the whole truth. A more complete truthfulness painting an entirely different picture. An algebraic description can give a singular viewpoint of a phenomenon but does not give all viewpoints, which would be the whole truth. Something Peter is approaching with different observations of the same rotating phenomenon. None of the singular values being the whole truth.
In a universe where there is sequential passage of time, as needed by QM and fitting the arrows of time, there is non-commutivity. As new Now follows previous Now definitely and so when things happen in the representation is important. Which is something in favour of quaternions ( and maybe their big bother). Though as I see it the real number line is not an actual dimension of the universe where physics is happening, but a history time line without physical counterpart. You mention 4 dimensional space from primary sense perceptions. That has to be a separate 'map' space from the former representing where physics is happening. The sense space has a time dimension, and non simultaneity of events can happen as the time is related to signal transmissions and not the foundational sequential passage of time. The two different forms of universe can not be described by just the one form of the algebra because of the fundamental difference in what time is in each. If quaternions were used for both in one version there is the non physical history time line and in the other the geometric time dimension.
Thank you for food for thought. I am going to think on about non associativity in nature and whether it is needed in a representation, and about representation of the different kinds of field, that you discussed. Kind regards Georgina
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Author Rick Lockyer replied on Feb. 23, 2018 @ 16:54 GMT
Georgina,
Thanks for reading my essay. I did scan through yours, not with enough detail to comment. As in other posts here I am apologizing profusely for not having time to study the many wonderful essays in this contest.
I have always appreciated the defense of your positions in your blog participation here. You have more patience than I for a certain individual who will remain nameless. Keep up the good fight.
Rick
Author Rick Lockyer wrote on Feb. 15, 2018 @ 03:18 GMT
I posted this on Geoffrey Dixon’s essay thread, may be interesting to readers coming here:
Geoffrey,
Knowing you are not big on the Cayley-Dickson doubling algorithm, I did neglect to mention another reason my Quaternion triplet enumeration algorithm is cooler than yours (no offense intended). That would be for enumerating the triplets for the sedenions. For Octonion Algebra I used...
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I posted this on Geoffrey Dixon’s essay thread, may be interesting to readers coming here:
Geoffrey,
Knowing you are not big on the Cayley-Dickson doubling algorithm, I did neglect to mention another reason my Quaternion triplet enumeration algorithm is cooler than yours (no offense intended). That would be for enumerating the triplets for the sedenions. For Octonion Algebra I used the binary numbers 1 through 7 and partitioned the triplets with the bit wise exclusive or logic function. This can be extended to the sedenions by going to binary 1 through 15, of course representing directly the 15 non-scalar basis elements enumerations. Doing the exclusive or logic operation on binary 1 through 15 partitions them into 35 unique closed triplet sets, and since there are (n-1)*(n-2) = 15*14 = 210 basis products not e0 * en, en * e0 or en * en fixed product rules, and each triplet does 6 rules, 35*6 = 210, covering all 210 basis products with 35 Quaternion triplets.
For each Octonion subalgebra using this enumeration technique the seven non-scalar basis element indexes will exclusive or to a zero result, as well each Quaternion index triplet. If you take away from the seven Octonion non-scalar basis indexes any set of three which are Quaternion triplet indexes, the remaining four basis elements are called “basic quads” and their indexes will also exclusive or to zero. They are basic quads because given just them, the exclusive or of any two will be the index of one of the companion Quaternion basis triplets and all are provided 2 up with this.
This gives a method to determine how many Octonion subalgebra candidates there would be, just go through all quads of distinct indexes 1 through 15 for combinations that exclusive or to zero. A simple computer program will show there are 105 such quads. Any Octonion definition has seven Quaternion triplets each with a unique basic quad, so we would expect to see 105/7 = 15 candidate Octonion subalgebras. Each of the 35 Quaternion triplets occurs in 3 Octonion subalgebras 3*35 = 105.
The rub is you can’t make all 15 Octonion subalgebra candidates represent valid normed composition algebras, e.g. division algebras. An algebraic proof the sedenions are not a division algebra? Maybe so.
Thought you might find this interesting.
Rick
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Bashir Yusuf wrote on Feb. 15, 2018 @ 19:39 GMT
Dear Rick
I agee and like the term Truth is most important, but I am wondering how to measure it or who can measure it?
Here is my essay that really think there is some fundamental trurh about the nature of particle and gravity, please comment.
Best wishes
Bashir
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Author Rick Lockyer replied on Feb. 23, 2018 @ 16:46 GMT
Thanks for reading my essay. My “truth” was meant to imply truth that exists independent of us or our limited knowledge opinions. It is out there for us to learn, if we listen.
As I have said a number of times and apologized for, I am working full time and have had time just to scan a number of essays here. Hopefully people will not hold this against me.
Rick
Bashir Yusuf wrote on Feb. 15, 2018 @ 19:47 GMT
Dear Rick
Here is my essay
https://fqxi.org/community/forum/topic/3143
Bashir Yusuf.
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Jonathan J. Dickau wrote on Feb. 16, 2018 @ 19:49 GMT
Hello yet again Rick,
I wanted to pass this on. The Kricker and Joshi paper is behind a pay wall at Elsevier; but a scanned copy of the manuscript was available from Kricker's web page. So here is that paper "Bifurcation Phenomena of the Non-Associative Octonion Quadratic."
Enjoy,
Jonathan
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Jonathan J. Dickau replied on Feb. 16, 2018 @ 19:51 GMT
Darn...
I guess just less than 1 MB is not good enough. I'll send it your way the old fashioned way - e-mail.
Regards, JJD
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Jonathan J. Dickau replied on Feb. 16, 2018 @ 20:03 GMT
But wait!..
I've found it is available as a scanned document download from the CERN library. So here is a link to
Bifurcation Phenomena of the Non-Associative Octonion Quadratic. This way; anyone interested can access the document.
All the Best,
Jonathan
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Jonathan J. Dickau replied on Feb. 17, 2018 @ 00:24 GMT
Hello yet again Rick,
While looking for something else; I happened upon these papers by a researcher I mentioned above, which should be of great interest to you, given the subject matter. I hope it adds something, rather than detracts, and I have not read these for detail enough to know. From Merab Gogberashvili
Octonionic Electrodynamics and
Octonionic Version of Dirac EquationsEnjoy, JJD
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Author Rick Lockyer wrote on Feb. 17, 2018 @ 16:30 GMT
Jonathan,
Added a new class to my scripting called NUMBERS, for doing Octonion math on 8D arrays of numeric values. Methods are so far add, subtract, multiply with Algebra choice and selective conjugation, return the norm, multiply and divide all by real number, output to console/text file. You should be able to create your own complex number, Quaternion number or Octonion number Mandelbrot sets with it. Tried something as a test, iteration Z_n+1 = Z_n ^2 + c, bailing on stationary norm with assigned value of 0, else iteration loop count to have the norm go past 1000000, staying in Quaternions. We all minutes to do a 1000 x 1000 grid of 2 Quaternion non-scalar starting points with increments .00001 with the starting points fixed for other 2 and c being fixed same non grid elements. Got some kind of pattern, have not plotted. What do you use to plot colored plots?
Rick
Author Rick Lockyer replied on Feb. 17, 2018 @ 16:33 GMT
Wow, autocorrect sucks, need to read first. Within a few minutes, like 10 to do 1000 x 1000 grid
Jonathan J. Dickau replied on Feb. 17, 2018 @ 21:20 GMT
Far Out!
I've used a purpose-built program written in Pascal for years to convert arrays of integers into 24-bit color values (3 5-bit values for R,G,&B). It's pretty cumbersome, and I have it making files in an obsolete format then converting that to TIFF, but I can deconstruct and re-construct that code to modern specs in fairly short order. It breaks the number palette into ranges, separates odd and even values, and then uses an algorithm to produce color gradients within each range. Your progress gives me a good reason to get back into the programming end of things.
All the Best,
Jonathan
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Jonathan J. Dickau replied on Feb. 20, 2018 @ 22:58 GMT
Hi Rick,
I just sent you an e-mail with a snippet of code which is the subroutine for color assignment using a 24-bit palette. The choices for colors and ranges created by that code looks like this image for the standard Mandelbrot Set plot.
Good Luck!
Jonathan
attachments:
MandelCrop.jpg
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Steven Andresen wrote on Feb. 22, 2018 @ 06:23 GMT
Dear Rick
If you are looking for another essay to read and rate in the final days of the contest, will you consider mine please? I read all essays from those who comment on my page, and if I cant rate an essay highly, then I don’t rate them at all. Infact I haven’t issued a rating lower that ten. So you have nothing to lose by having me read your essay, and everything to...
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Dear Rick
If you are looking for another essay to read and rate in the final days of the contest, will you consider mine please? I read all essays from those who comment on my page, and if I cant rate an essay highly, then I don’t rate them at all. Infact I haven’t issued a rating lower that ten. So you have nothing to lose by having me read your essay, and everything to gain.
Beyond my essay’s introduction, I place a microscope on the subjects of universal complexity and natural forces. I do so within context that clock operation is driven by Quantum Mechanical forces (atomic and photonic), while clocks also serve measure of General Relativity’s effects (spacetime, time dilation). In this respect clocks can be said to possess a split personality, giving them the distinction that they are simultaneously a study in QM, while GR is a study of clocks. The situation stands whereby we have two fundamental theories of the world, but just one world. And we have a singular device which serves study of both those fundamental theories. Two fundamental theories, but one device? Please join me and my essay in questioning this circumstance?
My essay goes on to identify natural forces in their universal roles, how they motivate the building of and maintaining complex universal structures and processes. When we look at how star fusion processes sit within a “narrow range of sensitivity” that stars are neither led to explode nor collapse under gravity. We think how lucky we are that the universe is just so. We can also count our lucky stars that the fusion process that marks the birth of a star, also leads to an eruption of photons from its surface. And again, how lucky we are! for if they didn’t then gas accumulation wouldn’t be halted and the star would again be led to collapse.
Could a natural organisation principle have been responsible for fine tuning universal systems? Faced with how lucky we appear to have been, shouldn’t we consider this possibility?
For our luck surely didnt run out there, for these photons stream down on earth, liquifying oceans which drive geochemical processes that we “life” are reliant upon. The Earth is made up of elements that possess the chemical potentials that life is entirely dependent upon. Those chemical potentials are not expressed in the absence of water solvency. So again, how amazingly fortunate we are that these chemical potentials exist in the first instance, and additionally within an environment of abundant water solvency such as Earth, able to express these potentials.
My essay is attempt of something audacious. It questions the fundamental nature of the interaction between space and matter Guv = Tuv, and hypothesizes the equality between space curvature and atomic forces is due to common process. Space gives up a potential in exchange for atomic forces in a conversion process, which drives atomic activity. And furthermore, that Baryons only exist because this energy potential of space exists and is available for exploitation. Baryon characteristics and behaviours, complexity of structure and process might then be explained in terms of being evolved and optimised for this purpose and existence. Removing need for so many layers of extraordinary luck to eventuate our own existence. It attempts an interpretation of the above mentioned stellar processes within these terms, but also extends much further. It shines a light on molecular structure that binds matter together, as potentially being an evolved agency that enhances rigidity and therefor persistence of universal system. We then turn a questioning mind towards Earths unlikely geochemical processes, (for which we living things owe so much) and look at its central theme and propensity for molecular rock forming processes. The existence of chemical potentials and their diverse range of molecular bond formation activities? The abundance of water solvent on Earth, for which many geochemical rock forming processes could not be expressed without? The question of a watery Earth? is then implicated as being part of an evolved system that arose for purpose and reason, alongside the same reason and purpose that molecular bonds and chemistry processes arose.
By identifying atomic forces as having their origin in space, we have identified how they perpetually act, and deliver work products. Forces drive clocks and clock activity is shown by GR to dilate. My essay details the principle of force dilation and applies it to a universal mystery. My essay raises the possibility, that nature in possession of a natural energy potential, will spontaneously generate a circumstance of Darwinian emergence. It did so on Earth, and perhaps it did so within a wider scope. We learnt how biology generates intricate structure and complexity, and now we learn how it might explain for intricate structure and complexity within universal physical systems.
To steal a phrase from my essay “A world product of evolved optimization”.
Best of luck for the conclusion of the contest
Kind regards
Steven Andresen
Darwinian Universal Fundamental Origin
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Author Rick Lockyer replied on Feb. 23, 2018 @ 16:40 GMT
Unfortunately I am still working full time doing consulting. This has prevented me from reading many essays in enough detail to reasonably respond or grade, so I haven’t.
Rick
David Brown wrote on Feb. 23, 2018 @ 09:18 GMT
"One can only imagine the reluctance, which continues to this day, for the applicability of Octonion Algebra for mathematical physics." How is octonion algebra related to the Leech lattice?
Wilson, Robert A. "Octonions and the Leech lattice." Journal of Algebra 322, no. 6 (2009): 2186-2190.Can 3 copies of the Leech lattice be used to explain why there are 3 generations of fermions?
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Author Rick Lockyer replied on Feb. 23, 2018 @ 16:37 GMT
Not saying the Leech lattice is not important, just that it has not been important enough to me to put it in front of other things on a list too long for me to possibly get through. I still have a full time job consulting right now having nothing to do with physics or Octonion Algebra which I do on my time, my dime. Sadly this has severely limited my time to participate in reviewing other essays.
So, maybe it can, can’t say with any authority.
Rick
Author Rick Lockyer wrote on Feb. 23, 2018 @ 16:29 GMT
As advertised attached is my JavaScript symbolic algebra tool, along with my verification script for my derivation of the Octonion conservation of energy and momentum equations. Also just for fun is a Mandelbrot set script that can do complex, Quaternion and Octonion iteration loops.
It requires you to download the free Node.js runtime and has only been used on a Windows 7 machine by me. Expand the attached zip file (hopefully it will be there), read the pdf, have fun.
Rick
attachments:
OctonionScripting.zip
Author Rick Lockyer replied on Feb. 23, 2018 @ 17:14 GMT
well did not work, unsure if zip files not possible of just was how I did it.
One more try
attachments:
1_OctonionScripting.zip
Author Rick Lockyer replied on Feb. 23, 2018 @ 17:16 GMT
Seems to have worked, use second one
Sorry,
Rick
Jonathan J. Dickau replied on Feb. 24, 2018 @ 05:35 GMT
Thanks Rick...
I have downloaded the ZIP file from the second link and will report back after a while. I also got the e-mail before, and I will try to process or utilize soon. I have been working on my proceedings submission for FFP15. I'm also trying to read the max # of essays before the end.
All the Best,
JJD
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Author Rick Lockyer wrote on Feb. 25, 2018 @ 16:18 GMT
Hmm, 2 one-bombs
Author Rick Lockyer replied on Feb. 25, 2018 @ 16:26 GMT
Author Rick Lockyer replied on Feb. 25, 2018 @ 16:30 GMT
Anonymous votes with no comments to back them up is an obvious deficiency of the “community” rating.
Going to be cowards?
Author Rick Lockyer replied on Feb. 25, 2018 @ 16:36 GMT
Solution to this problem would be for FQXi to publish all votes each participant made, by name.
David Brown wrote on Feb. 27, 2018 @ 10:15 GMT
"Anonymous votes ..." According to Pavel Kroupa, "Science is not a democracy."
What are the 4 most important mathematical structures? Could the answer be the real numbers, the complex numbers, the quaternions, and the octonions?
The octonions Author: John C. Baez Journal: Bull. Amer. Math. Soc. 39 (2002)"Division algebras and quantum theory" by John C. Baez, arXiv.org, 20119 copies of the octonions = 64 dimension of virtual particle paths + 3 dimensions of linear momentum + 3 dimensions of angular momentum + 2 dimensions of quantum spin ???
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Author Rick Lockyer replied on Feb. 27, 2018 @ 16:25 GMT
I think the only “science” exhibited by the community voting system here is social science. The mass down-voting on Sunday on highly rated essays had nothing to do with the quality of these essays, it had everything to do with selfish people trying to elevate their own ranking.
Too bad, it could lead to the ultimate downfall of FQXi providing the opportunity in the future. If they have to pay people to sort out the worthy papers for awards, because the community fails to do so in a fair and unbiased way, it will consume more money than the total award purse.
Regarding my essay, I barely got it in before the deadline, so it was not up long enough to gather a significant number of votes such as to dilute any 2-bombs let alone 2. Those votes were not on the merits of my essay. Took me down 75 places. I was an easy mark.
If people were different than they are, communism would have actually worked.
Clearly I think the division algebras are the most important, and the lack of associativiy for multiplication is nature’s way of telling us others will only give methods that will only provide close approaches.
There is plenty to look at just within Octonion Algebra itself, this is priority to me, not sticking Octonions in other structures or using it as a means to another end.
Rick
David Brown replied on Feb. 28, 2018 @ 22:00 GMT
Author Rick Lockyer wrote on Feb. 28, 2018 @ 15:03 GMT
Just curious if anyone was successful getting Node.js running with my symbolic algebra tool.
Even if you are not computer literate enough or don’t care to use the tool, you really should download the files and open my provided text file giving the results you will get if you did run the verification script for my Octonion conservation equations.
The verification script goes through the very same math three times. First it has all 8 Octonion potentials functionally dependent on all 8 Octonion differentiating variables. This is the full complete cover. Next pass the potential functions are limited to those for just the Electrodynamics cover, and you will see exactly what the 4D tensor approach yields but in an Octonion framework. The last pass is restricting the potential functions to those for Gravitation. You will of course see both the Electrodynamics and Gravitation product terms in the full functional dependency first pass. The other terms in the first pass are the glue that holds things together.
This is not non-associative physics, yet it requires non-associative Octonion Algebra.
Rick
Jonathan J. Dickau replied on Mar. 9, 2018 @ 17:19 GMT
I have just begun to play Rick...
I downloaded the Node JS package last week, then got sidetracked by reading about code execution, blocking vs non-blocking code, and other preliminary bull. I hope to get some time to fiddle this weekend.
All the Best, Jonathan
P.S. - I was dismayed to see how your essay and several of my other favorites were bombed at the last minute. I too was bombed and lost more than 20 places in 48 hours, but then I got one or two good ratings afterward so I got lucky. Know that your work is valued, whatever the contest results show. - JJD
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Author Rick Lockyer wrote on Mar. 30, 2018 @ 14:45 GMT
I made some editorial and brain fade fixes to the PDF file describing my Octonion symbolic algebra tool. I neglected to document the seven Quaternion subalgebra non-scalar triplet sets used. All class/method definitions are the same but the documentation of their use has some editorial improvements.
If you already have downloaded this, download again and move over just the PDF to your current OctonionScripting folder.
Seriously folks, if you have ANY interest in Octonion Algebra, it will be worth your time.
Also since I provided my script verification of my Octonion conservation equations, this will make more sense having the textual documention backing up the math. The chapter from my in-progress book is included for you to download.
Rick
attachments:
chapter_10.pdf,
2_OctonionScripting.zip
Jonathan J. Dickau replied on Apr. 2, 2018 @ 00:57 GMT
I got this update..
And I think you Rick! I'm starting to have some fun playing and I'll have some questions or comments before long.
Best,
JJD
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Jonathan J. Dickau replied on Apr. 2, 2018 @ 00:58 GMT
Oh well...
Should be 'thank you!'
JJD
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Author Rick Lockyer wrote on Mar. 30, 2018 @ 15:16 GMT
I had a discussion on sedenions with Geoffrey Dixon related to our differing mechanisms for enumerating the seven Quaternion subalgebra non-scalar basis element triplet sets for Octonion Algebra. He also is not big on Cayley-Dickson doubling typically used to generate the path from reals through sedenion algebra. The attached document shows a different way that easily delineates the Octonion subalgebra candidates, which cannot be all made legitimate Octonion subalgebras due to the fact that each of 35 Quaternion triplets must be singularly defined and each appears in three Octonion candidates, causing some to not follow the Right-Left orientation required to be a normed division algebra.
This discussion inspired me to revisit the algebraic proof the sedenions are not a division algebra documented in my in-progress book as an appendix. It had a bit too much hand-waving.
Download this and take a look, the exclusive-or method works out quite well for Quaternion, Octonion and sedenion Algebras. Pretty much makes an algebraic proof sedenions are not a division algebra possible. To the best of my knowledge nobody else has provided an algebraic proof. Correct me if wrong about this.
Rick
attachments:
appendix_b.pdf
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