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**Jonathan Dickau**: *on* 4/2/18 at 0:58am UTC, wrote Oh well... Should be 'thank you!' JJD

**Jonathan Dickau**: *on* 4/2/18 at 0:57am UTC, wrote I got this update.. And I think you Rick! I'm starting to have some fun...

**Rick Lockyer**: *on* 3/30/18 at 15:16pm UTC, wrote I had a discussion on sedenions with Geoffrey Dixon related to our...

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FQXi FORUM

May 23, 2019

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. 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.

B.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.

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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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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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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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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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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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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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.

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.

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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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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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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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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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

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

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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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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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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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...

view entire post

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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"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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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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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;

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;

Excellent job. Please do ask any questions of mine if not clear.

Very Best

Peter

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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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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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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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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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To make sure everyone...

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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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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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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 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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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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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, 4

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

Silviu

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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, 4

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

Silviu

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

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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...

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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 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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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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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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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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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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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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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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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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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

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

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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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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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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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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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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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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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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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=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...

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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...

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

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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...

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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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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

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

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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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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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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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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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

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

Dear Rick

Here is my essay

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

Bashir Yusuf.

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Here is my essay

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

Bashir Yusuf.

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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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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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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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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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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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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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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 Equations

Enjoy, JJD

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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 Equations

Enjoy, JJD

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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

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

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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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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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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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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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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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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"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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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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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

So, maybe it can, can’t say with any authority.

Rick

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

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

well did not work, unsure if zip files not possible of just was how I did it.

One more try

attachments: 1_OctonionScripting.zip

One more try

attachments: 1_OctonionScripting.zip

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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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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Hmm, 2 one-bombs

Anonymous votes with no comments to back them up is an obvious deficiency of the “community” rating.

Going to be cowards?

Going to be cowards?

"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, 2011

9 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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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, 2011

9 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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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

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

"There is plenty to look at just within Octonion Algebra itself ..."

Is the following relevant?

Bisht, P. S., Tianjun Li, and O. P. S. Negi. "Octonion and Split Octonion Representation of SO (8) Symmetry." arXiv preprint arXiv:1212.2536 (2012).

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Is the following relevant?

Bisht, P. S., Tianjun Li, and O. P. S. Negi. "Octonion and Split Octonion Representation of SO (8) Symmetry." arXiv preprint arXiv:1212.2536 (2012).

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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

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

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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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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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

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

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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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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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

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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