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RECENT POSTS IN THIS TOPIC

Vladimir Fedorov: on 2/24/18 at 7:12am UTC, wrote Dear Gary, (copy to yours and mine) Many thanks for the kind words about...

Gary Simpson: on 2/22/18 at 16:26pm UTC, wrote Steven, Thanks for the posting. I have read and commented upon your essay....

Gary Simpson: on 2/22/18 at 16:25pm UTC, wrote Vladimir, Thanks for reading and commenting. I have also read and...

Vladimir Fedorov: on 2/21/18 at 7:06am UTC, wrote Dear Gary, Here we are again all together. I enjoyed reading your...

Dizhechko Semyonovich: on 2/20/18 at 1:51am UTC, wrote Gary, I appreciated your essay on 10 and thinking in order to develop our...

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Steve Dufourny: on 2/19/18 at 19:05pm UTC, wrote You are welcome Gary, hope you are well, I see that you learn piano and...

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FQXi FORUM
August 25, 2019

CATEGORY: FQXi Essay Contest - Spring, 2017 [back]
TOPIC: Four Squares by Gary D. Simpson [refresh]
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Author Gary D. Simpson wrote on Dec. 20, 2017 @ 21:39 GMT
Essay Abstract

This essay discusses the meaning and role of the term “fundamental” as it applies to Math and Physics. The importance of Lagrange’s Four Squares Theorem is also discussed. It is argued that the vacuum is a 5-D Quantum-Space-Time and that the vacuum is fundamental.

Author Bio

I am a retired Chemical Engineer with an interest in Math and Physics. I am especially interested in quaternions. I am attempting to learn to play piano and guitar.

Download Essay PDF File

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Author Gary D. Simpson wrote on Dec. 20, 2017 @ 22:16 GMT
All,

Many thanks for taking the time to read and consider my essay.

This is my third essay dealing with quaternions. My previous essays were "Calculus - Revision 2.0" and "Five Part Harmony". The first of these laid out the basis for a form of Calculus based upon quaternion functions. The second of these laid out the basis for a 5-D geometry based upon quaternions. The present essay presents these ideas as a formal Mathematical Group.

Essentially, I believe that at least some of the difficulties with Physics are due to the Mathematics that we use. I believe that quaternion based Physics will be more compact and that it will allow insights that our present mathematics makes difficult to see.

The most Mathematically significant parts of the essay are the Octonion Group and the matrix inversion associated with it. In "Five Part Harmony", I had not yet had the needed insight to correctly invert this matrix. When I did, I hit myself in the head due to its simplicity. The conjugate of this system causes a 5-D system to collapse into a 4-D system. This made me realize the importance of the Four Squares.

Hopefully you will take away something from my effort that justifies the time you have given me.

Best Regards and Good Luck to All,

Gary Simpson

PS- Merry Christmas

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Author Gary D. Simpson replied on Dec. 27, 2017 @ 13:51 GMT
All,

Here is an example to consider.The relativistic energy equation is:

E^2 = (m_0*c^2)^2 + (p*c)^2

This can be produced from Equation 5.2 by setting Q and Q* as follows:

Q = m_0*c^2 + pc

Q* = m_0*c^2 - pc

Best Regards,

Gary Simpson

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Scott S Gordon wrote on Dec. 21, 2017 @ 00:27 GMT
Hi Gary - I enjoyed reading your essay. In your conclusion you state,

" The author argues that the vacuum is the most fundamental structure in the universe and that the Octonion Group describes the vacuum."

The first part of this statement is profound "that the vacuum is the most fundamental structure in the universe." You also state earlier that we know only 5% of the universe's content.

Do you think it is possible that that spacetime itself can be further broken down into something even more fundamental?

I applaud the math you provided with the Octonion Group BUT is this much different than many other mathematical interpretations in expressing the properties of spacetime? Will your interpretation render a clear picture that leads to everything? The other interpretations have gotten us nowhere. If the properties of spacetime could be derived from something even more fundamental that forms spacetime, then the math of spacetime would be derived instead of the properties of spacetime being interpreted by various mathematical conjectures.

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Author Gary D. Simpson replied on Dec. 21, 2017 @ 11:27 GMT
Scott,

Thanks for reading and commenting upon my essay. I'll answer your questions in order.

I do not think that space-time is fundamental, but I also do not think that space-time can be reduced to something more simple. I realize that this seems contradictory. I think that space-time is EMERGENT from actions upon the vacuum. IMO, that is the meaning of Equation 5.2. I took a pair of 5-D conjugates and multiplied them. The result was a scalar that is equal to the base quaternion's length.

I cannot comment upon how the Octonion Group compares to other mathematical interpretations since I have only recently begun to study Group Theory. I can state that the group includes anti-commutation between the complex i and the unit vectors. This is what makes the 5-D conjugates work. Space-time emerges from the vacuum. The vacuum is more fundamental than space-time and it is more complex than space-time. Therefore, I am arguing that simplicity emerges from complexity rather than the reverse argument:-)

Best Regards,

Gary Simpson

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Anthony John Garrett wrote on Dec. 21, 2017 @ 23:49 GMT
Gary,

I view quaternions as the generalization of complex numbers from 2 to 3 dimensions. Just as complex number manipulation is simpler and easier than vector calculus in 2D, so are quaternions simpler and easier than vector calculus in 3D. The cost in going up a dimension is loss of commutativity when quaternions are multiplied. Clifford Algebra is the generalization to arbitrary dimensions, of which quaternions are the 3D version and complex numbers the 2D version. Dave Hestenes in Arizona has done great work in advancing this point of view. But if you want octonions and have a taste for unconventionality, try Peter Rowland's interesting book "Zero to Infinity: the Foundations of Physics". Good luck!

Anton

Anthony Garrett

Anton Garrett

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Author Gary D. Simpson replied on Dec. 22, 2017 @ 03:38 GMT
Anthony,

Thanks for the comment. You should read my essay. I deal with a hidden variables. I see from your essay that you have an interest in this.

Best Regards,

Gary Simpson

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Anthony John Garrett replied on Dec. 22, 2017 @ 15:54 GMT
Yes, phase is not directly observable; only phase *difference* has an effect in physics. Do you think that there is such a thing as absolute phase?

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Author Gary D. Simpson replied on Dec. 22, 2017 @ 20:53 GMT
Anthony,

I believe that there is an absolute phase angle that is based upon absolute velocity. Essentially, I begin with Euler's Equation and then substitute the Lorentz Transform for the cosine term and (v/c) or the sine term.

You will no doubt ask why this is an absolute rather than relative velocity. I will reply that in order to match the observed mass ratio between the proton and the electron, it is necessary for the v term to be 0.006136 c.

You will then likely ask what this has to do with the Mp/Me ratio and I will answer that the reference frame of the observer is built into the wave function of what is observed.

Best Regards,

Gary Simpson

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John R. Cox wrote on Dec. 22, 2017 @ 19:09 GMT
Hello Gary,

Another well constructed argument in experimental theoretical mathematics.

I can accept your premise that the most fundamental physical condition is the vacuum, IF we firstly distinguish that from the Void. That is to say that if there is existence at all, there existentially exists a vacuum from which the deconstructed elements we parameterize as energy and spacetime emerge. I can then accept that space-time is a parallelized co-existence of stress between a curved line and a straight light and the origin of energy. The rest is math. As you know, I do agree with your "two kinds of Time", represented as a scalar value in parallel with a vector value existing in S0. The hidden variable revealed as the Fifth Dimension. Glad you made the community ratings, Good Luck. jrc

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Author Gary D. Simpson replied on Dec. 22, 2017 @ 20:57 GMT
John,

Many thanks for reading nd commenting on my essay.

I think we are on the same wavelength. I think that the vacuum fills the void. So, I definitely distinguish between those two concepts. In fact, I think that the vacuum might even be a single continuous entity. This would be consistent with non-locality.

Will you be submitting an essay?

Best Regards,

Gary Simpson

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John R. Cox replied on Dec. 22, 2017 @ 21:17 GMT
g.

not anytime soon. Thanks, also I see your suggestion that the existential vacuum is simply continuous. Nice distinction, underlying field theoretics. :-)jrc

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Philip Gibbs wrote on Dec. 23, 2017 @ 10:14 GMT
Gary, I enjoyed your essay. If I understand correctly you see our 4D universe as embedded in a 5D vacuum, with the algebra of quaternions and octonions underlying the physics. You mentioned the cosmic microwave background and the Hubble bubble. Do you accept the big bang and if so how does that look in your model?

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Author Gary D. Simpson replied on Dec. 23, 2017 @ 16:59 GMT
Philip,

Thanks for reading and commenting. You do indeed understand my argument. The basis for this thinking is Equations 5, 5.1, and 5.2.

I consider motion to be the best explanation for the observed red-shifts associated with distant galaxies. Therefore, I think that the observable universe was much smaller in the distant past. Whether it began as (a point followed by inflation then expansion) or as (a homogeneous sphere followed only by expansion) is less clear to me. It is hard for me to imagine a sphere that is so large being almost perfectly uniform. So, I favor the point beginning but I have some doubt.

My knowledge of modern cosmology is fairly limited. As I understand the situation, we observe that almost all of these galaxies are moving away from us. Given the errors of our prior earth-centric thinking, we interpret this to mean that there is no center of the universe and that everything is moving away from everything else. This argument seems to me to be flawed. The simpler explanation is that there is a center to the universe and that everything is moving away from the center but doing so at various velocities. So, the fact that so many galaxies are moving away from us simply means that we are moving slowly compared to most of the universe.

I would envision the expansion of the universe as being driven by the complex quaternion terms present in the Octonion Group. Essentially, I see those terms as being virtual electrons. Since like charges repel, the universe expands.

Best Regards,

Gary Simpson

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Author Gary D. Simpson wrote on Dec. 27, 2017 @ 13:50 GMT
All,

Here is an example to consider.The relativistic energy equation is:

E^2 = (m_0*c^2)^2 + (p*c)^2

This can be produced from Equation 5.2 by setting Q and Q* as follows:

Q = m_0*c^2 + pc

Q* = m_0*c^2 - pc

Best Regards,

Gary Simpson

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Georgina Woodward wrote on Jan. 2, 2018 @ 01:39 GMT
Hi Gary. I liked the introduction, especially where you talk about needing to consider the majority of the universe that isn't the ordinary matter. Please excuse my inelegant paraphrasing. Tied up again nicely at the end where you make clear that you consider 'the vacuum' fundamental.I don't personally think the 5D structure is needed but I like that you clearly explained the reasoning behind your thinking it would make a good model. Lost me rather in the middle where it got 'technical' but overall your essay is readable and clear. Kind regards Georgina

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Author Gary D. Simpson replied on Jan. 2, 2018 @ 17:30 GMT
Georgina,

Thanks for reading and commenting. Will you be submitting an essay? There has been very little activity in the forum thus far.

My apologies for the Mathematics. I have no other venue where I can reasonably present these ideas. So I use FQXi as a sounding board. I realize doing so probably has a negative impact upon my scoring, but I consider it worthwhile to do so. In this case, I considered Equation 5, 5.1, and 5.2 to be noteworthy along with the matrix inversion.

Don't fret too much about that 5'th dimension. It is scalar. It does not have a direction.

Best Regards,

Gary Simpson

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Georgina Woodward replied on Jan. 2, 2018 @ 19:07 GMT
Yes essay submitted awaiting approval.

I won't fret, or argue with you about the 5th dimension -or even the 4th for that matter. We are both just doing our 'own thing', in our own ways,(building sandcastles is my analogy). and trying to get it across to others as well as we can. Good luck with that. Hope you get lots more reviewers. Georgina

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Lawrence B. Crowell wrote on Jan. 7, 2018 @ 22:37 GMT
I have not read your essay yet. I read a couple of your back essays. I will try to get to this soon. I too wonder why the entries are not showing up here.

LC

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Declan Andrew Traill wrote on Jan. 11, 2018 @ 05:55 GMT
Gary,

I just read your essay - interesting as usual.

I noticed your question about the meaning of the complex numbers and I have found in my analysis of the electron/positron wave functions in my paper, the following (quoted from my comment on another essay):

“The reason that the vectors are complex, is that the Schrodinger equation requires them to be, as it relates two vector quantities with a complex 'i' in the equation. The reason for that is that the two quantities are orthogonal - multiplying any complex vector by 'i' has the effect of rotating it 90 degrees around the origin in complex space. The vectors are actually real, but the Schrodinger equation uses this mathematical 'trick' to express orthogonality in a concise way.”

Hope this helps...

Regards,

Declan Traill

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Author Gary D. Simpson replied on Jan. 12, 2018 @ 14:38 GMT
Declan,

See my reply to Dr. Klingman below.

Best Regards,

Gary Simpson

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Edwin Eugene Klingman wrote on Jan. 12, 2018 @ 00:00 GMT
Hi Gary,

I enjoyed your essay, particularly your discussion of Maxwell and quaternions, and that the Octonions group encompasses all of electromagnetism and "something else". I suspect that the "something else" is gravito-magnetism, as represented by equations (5) in my essay. I hope to find time to try to apply the Octonions in this regard, and I think you might find it rewarding to think this through.

As you probably know, the gravito magnetic equations are identical in form to Maxwell's equations, but the gravito magnetic field interacts with itself and is hence non-linear, thus differing from electromagnetic linearity. I do not see this as having any significance from an Octonions perspective, although it is vastly different for the physics involved.

It's also worth noting that Maxwell's quaternions do not imply 4D space-time. That is Einstein's contribution, which I analyze in my essay.

Thank you for reading and commenting on my essay. I am happy to see you pursue Octonion math and it's possible meaning for physics. I believe this is a very important topic.

As for the meaning of 'i', I believe that the best interpretation is given in Hestenes' Geometric Algebra, where 'i' is essentially a duality operator.

My very best regards,

Edwin Eugene Klingman

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Author Gary D. Simpson replied on Jan. 12, 2018 @ 14:37 GMT
Ed,

Many thanks for reading and commenting.

I think you are correct regarding EM and gravity. The Kaluza-Klein Equation is a 5-D model that combines EM with gravity. It was abandoned because AE believed that the implied scalar field was not compatible with GR.

I also think you are correct regarding quaternions and Minkowski space-time. The scalar term in a quaternion can be used to relate the dot product of a vector and the change in that vector to the length of the vector.

Having stated that, I would like to point out that the relativistic energy equation can be produced by setting Q and Q* in my Equation 5.2 as follows:

Q = m_0*c^2 + p*c

Q* = m_0*c^2 - p*c

Here, m_0 and c are scalars and p is a vector.

This implies a velocity quaternion as follows:

V = c + v

Here, c is a scalar and v is a vector.

I think this velocity quaternion is at the heart of the Hertz Equation Galilean Transform that you demonstrated.

If I then integrate that velocity quaternion with respect to scalar t, the following results:

Vt = X = ct + vt

If this is an indefinite integral then there could be a constant in there. If it is a definite integral, then this will be the difference between final and initial conditions.

So, this is a quaternion that represents space-time, but not Minkowski space-time. Whether or not this is actually Physics is another question.

I have given some thought to how to fit all of this together. The main stumbling block that I see is that when two bi-quaternions are multiplied together, the result has four terms (AC, BC, BD, and AD). Each of these must represent something that is physically real and measureable.

I have begun to study Dr. Hestene's work. It will take me several years to build a satisfactory level of knowledge.

Best Regards and Many Thanks,

Gary Simpson

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Author Gary D. Simpson replied on Jan. 13, 2018 @ 01:25 GMT
Ed,

For one of the equations I presented, I should have added the following:

Vt = X = ct + vt = ct + x

Where c and t are scalars and x is a vector.

Best Regards,

Gary Simpson

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Eckard Blumschein replied on Jan. 17, 2018 @ 10:31 GMT
Gary Simpson,

You wrote: "the difference between final and initial conditions.

Do these conditions correspond to a reference that immediately belongs to reality or are they necessarily chosen at will in a model?

By the way, because I am not familiar with Qs and Os, I would appreciate a more easily understandable to laymen answer to Declan Taill's question.

I am still struggeling with the question how quantum theory got complex. Before Schrödinger heuristically introduced a wavefunction, Heisenberg had already used Born's matrices. Square matrices with Hermitian symmetry are equivalent to a representation in complex plane, and they are equally redundant. Forgive me if I guess that elegance and abundance further grow with Qs and even Os.

Regards,

Eckard

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Jeff Yee wrote on Jan. 13, 2018 @ 00:57 GMT
Hi Gary,

I always enjoy reading your works and your love for quarternions. I noticed this time you thank Wikipedia, instead of the brewmasters. :)

For my essay, I put something together based on the wave structure of matter and how it relates a fundamental universe. Hope you like it.

The Fundamental Universe

Jeff Yee

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Author Gary D. Simpson replied on Jan. 14, 2018 @ 16:22 GMT
Hi Jeff,

I'm still brewing beer. And drinking it:-)

Best Regards and Good Luck,

Gary Simpson

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Lawrence B. Crowell wrote on Jan. 14, 2018 @ 20:54 GMT
In response to your comments on my essay page; I am glad you appreciated my work here. I have your essay queued up read to read once I can carve out some time and get to it.

Wilczek advanced the idea of time crystals. They may be in some ways a deep aspect of how nature is organized. They are almost paradoxical, and as I think they are tied in with the holographic principle they share properties similar to the image attached. This is why they are analogous to a thermodynamic system that exhibits dynamics.

I will try to get to yours and other's essay ASAP.

Cheers LC

attachments: 1_mc-escher-waterfall.jpg

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Gene H Barbee wrote on Jan. 15, 2018 @ 18:57 GMT
Gary,

Your equation 4 is very close to the universal wave function I use. P=exp(iEt/H)*exp(-iEt/H). It is what MIT calls unitary evolution based on the Schrodinger equation. You can look it up by searching MIT22 Evolution of Function Chapter 6. It is in Heading 6.1.2, Unitary Evolution. The Hamiltonian can be simply Energy since it is time dependent.

The interesting parallel...

view entire post


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Author Gary D. Simpson replied on Jan. 18, 2018 @ 16:56 GMT
Gene,

Thanks for reading and commenting, Please excuse my tardiness. I had elbow surgery on Tuesday to repair a torn tendon and I am presently wearing a cast. So my typing skill has been reduced by 50%. Plus my motivation is not good at the moment.

I was not aware of the MIT wave function. I simply constructed a function from two exponentials to allow the use of the Separation of Variables Method. I used Euler's Equation for the time function and used a quaternion exponential for the space function.

I have read your essay and will comment in your forum. The similarity that I see is that some of these "things" that you mention are the sum of four squares. The difference that I see is that I am only doing Math but you are doing Physics.

Keep in mind that a dimension might not truly be a dimension. I combined the complex plane with a unit space vector to create that 5-D model. But the scalar component has no direction and the complex term disappears when the conjugate is applied with the result being the sum of four squares. So, the complex plane might represent information.

Best Regards and Good Luck,

Gary Simpson

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Cristinel Stoica wrote on Jan. 15, 2018 @ 19:47 GMT
Hi Gary,

Good to read your essay about division algebras and their applications to physics. So you have four squares twice, in the Lagrange theorem, and in the matrix representation of octonions :). Good luck with the contest!

Cristi

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Author Gary D. Simpson replied on Jan. 18, 2018 @ 23:13 GMT
Thanks Christi!

Best Regards,

Gary Simpson

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Ajay Pokhrel wrote on Jan. 20, 2018 @ 10:25 GMT
Hello Gary,

I must admit that your essay, without any doubt, is well written. Your supporting facts and mathematics used for describing 'the fundamental' are plausible.

In your essay, I like a paragraph which starts with "As examples, let us consider the set of all integers and the subset of all prime integers. For the set of all integers, the values +1 and -1 are fundamental with respect to addition. These values cannot be broken into the sum of two or more smaller integers, and it is possible to generate any integer including zero by beginning with one of them and repeatedly adding either +1 or -1" My essay is something like this and I used the similar type of facts.

Did you notice that you used mathematical equations and patterns to conclude the fact that 'vacuum' is most fundamental? Well, you defined vacuum in term of mathematics and patterns (octonion group), doesn't it show that mathematics and pattern is the root or so-called fundamental of the universe? I don't mean to say your argument is not correct, I just wanted to show what I think.

Anyway, I really enjoyed reading your essay and gaining some knowledge.

You are welcomed to my essay for discussion:

Is Mathematics Fundamental?.

Kind regards

Ajay Pokharel

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Author Gary D. Simpson replied on Jan. 21, 2018 @ 08:09 GMT
Ajay,

Thank you for reading an commenting.

I have a challenge for you. If you believe that Mathematics is fundamental, then construct something from mathematics alone:-) I do not think this can be done. However, I do think that when we finally have a proper understanding of the universe, there will be a one-to-one correspondence between what is physically fundamental and what is mathematically fundamental.

I will read and comment upon your essay.

Best Regards and Good Luck,

Gary Simpson

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Ajay Pokhrel replied on Jan. 21, 2018 @ 11:53 GMT
Dear Gary,

I think the point of being fundamental is being able to create something from that thing alone; the point is whether it defines that 'something' at its root level and that is what mathematics and pattern do. However, I do think that there are many things discovered from mathematics alone; take General Relativity for instance. Why did Einstein felt that he would need those complex field equations and patterns to explain the distortion of space-time? I think it occurred because the universe was made based on those patterns.

And I also agree with your last line in your first paragraph.

Good luck with the competition

Regards

Ajay Pokharel

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Author Gary D. Simpson replied on Jan. 22, 2018 @ 08:30 GMT
Ajay,

I think you might have left out the word "not" from the first sentence.

Best Regards,

Gary Simpson

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Peter Jackson wrote on Jan. 22, 2018 @ 14:04 GMT
Gary,

Thanks for your post on mine. It shows the interference patterns arise from the known unequal femtosecond phase delays between paths. No 'erasure' or backwards causality is then required. That just demonstrates the veracity & power of the main finding.

I very much liked yours to. I agree quaternians and maybe octonians look useful tools for mine as the wave equation is complex and orthogonal. However I'm not a mathematician. Declan has done a great job producing a computer code which matches the (now!) mechanistic sequence ('re)producing' QM.

One part that didn't sound right to me in yours was regimentation to integers. I've published a 'Law of the reducing middle' which when replacing the 'excluded middle' in philosophy & maths removes all current paradox, matches Godels theorem and let to QM's uncertainty. I long ago learnt that in long fibre optic cables our 'square shouldered' signal waves 'degrade' into sine curves so we have to re-square (integerise) them at regular stations. I suggest that's nature being nature!

We know in optics we deal with helices as 'waves', which are anything but integers! Are you happy with a circle that's only a 2D view? Isn't all nature 3D plus time?

I agree with your thesis that "the vacuum is a scalar field of “potential electrons” and that electrons rise up from the vacuum when in the proximity of a proton. That seems very close to my own wider description that ANY particle of matter IN RELATIVE MOTION will propagate 'fermion PAIR' vortices. Do you disagree with any part of that?

But agreement on content not being a scoring criteria ! rate yours very high.

Best wishes

Peter

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Author Gary D. Simpson replied on Jan. 23, 2018 @ 10:14 GMT
Peter,

Thanks for reading and commenting.

I am also not a mathematician. I'm not exactly young either:-) But over 5 years or so I've been able to learn a few things about quaternions.

I'm not bound to integers. However, I found it to be very coincidental that Lagrange's Theorem was composed of four squares and the length of a quaternion is also four squares. Such coincidences make me suspect that something deeper is underfoot. Essentially, things can appear to be quantized (i.e., integer valued) without actually being quantized because both methods reduce to four squares.

Best Regards and Good Luck,

Gary Simpson

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Peter Jackson replied on Jan. 23, 2018 @ 12:58 GMT
Gary,

The 4 square idea does seem analogous to the paired twin inverse states classically reproducing QM's predictions. But Academia is fearful of change so few will dare to do more than ignore new science.

As for polarizers; There are vast numbers and types around for different purposes. They need researching and discussing with makers. You can impose any polarisation you wish, even reverse polarity with 'half wave plate' polarisers. The standard ones in the QM set up were normally linear, but all sorts were used. Circular will couple with rotation (+/-poles) but go to zero at 90 degrees. Linear the inverse, but both then also orthogonally inverse with photomultiplier angles. (measuring an ellipse chord on both axes goes min to max over 90 degrees.)

The Weihs (with Zeilinger) experiment used an 'electro optic modulator' instead for the same result. Interestingly they to found the rotational inconsistencies' Aspect found, seeded out & discarded the data as it didn't match the predictions! Bless their little red/green reversible cotton socks!

Very Best

Peter

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Satyavarapu Naga Parameswara Gupta wrote on Jan. 22, 2018 @ 14:12 GMT
Hi Gary D. Simpson…..

“The importance of Lagrange’s Four Squares Theorem, and the arguments that the vacuum is a 5-D Quantum-Space-Time and that the vacuum is fundamental ”are nicely argued and derived nicely dear Simpson…. Best wishes for your essay….

…..….. very nice idea…. I highly appreciate your essay and hope for reciprocity.

I request you please...

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Jonathan J. Dickau wrote on Jan. 22, 2018 @ 18:55 GMT
Great work Gary!

I'll have to read this over at least twice, to absorb all of what you are saying, but it seems you and I concur on a good many things. One can find close agreement with some of your findings in a formulation called DGP gravity, for Dvali, Gabadadze, and Poratti. This ties in with ideas by Pourhasan, Afshordi, and Mann, regarding a 5-d black hole 4-d white hole scenario.

There is also broad agreement with your work in papers by Merab Gogberashvili and by Vladimir Dzhunshaliev in the context of octonionic inflation theory. And I remember a paper, which I think was by Steven Adler, about the quantum vacuum being necessarily complex valued. Finally; your present work is also highly compatible with recent work by Frank Dodd 'Tony' Smith.

A comment from his essay forum to Lawrence Crowell explains the setup for your scenario. These properties and particles... "emerge from 8-dim Octonionic spacetime of the Inflation Era transition to Quaternionic M4 x CP2 Kaluza-Klein spacetime of our Era." So you might want to check that out, or ask Tony if he thinks the connection is relevant.

All the Best,

Jonathan

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Jonathan J. Dickau replied on Jan. 22, 2018 @ 19:00 GMT
Whoops, I misspelled..

That should be Dzhunushaliev, if you are trying to look up Vlad. He was most willing to discuss his work and forwarded more papers, when I reached out.

All the Best,

Jonathan

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Jonathan J. Dickau replied on Jan. 22, 2018 @ 19:11 GMT
Oh and I should also mention...

Nikodem Poplawski has written extensively on an 5-d --> 4-d scenario arising in the context of Einstein-Cartan by way of Sciama and Kibble. This is perhaps the version of the theory which has gotten the most press, despite the Scientific American article by Pourhasan et al.

I should point out that the braneworld scenarios along the lines of DGP or cascading gravity are equivalent or faithfully represented in work on octonionic inflation. Only the terminology is different. One researcher might talk about branes and another about a thin layer, surface, or membrane; but both mean the same thing.

My advice would be to catalog for yourself which theories have features that are similar, rather than trusting the literature to make the comparison. I think sometimes people put a String Theory slant on their work deliberately, knowing that this will increase their likelihood of publication. But some of those people are being jerks because their work in no way derives from ST.

All the Best,

Jonathan

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Author Gary D. Simpson replied on Jan. 23, 2018 @ 10:31 GMT
Jonathan,

Thanks for reading and commenting.

I am aware of the Kaluza-Klein Theory but the other things you mention are new to me. MANY THANKS! I am pleased to know that I'm not a complete crackpot:-)

The 5-D to 4-D collapse shown by Equations 5, 5.1, and 5.2 was my main point. It is very encouraging to me that several of the theories that you mention have similar ideas.

Best Regards and Good Luck,

Gary Simpson

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Colin Walker wrote on Jan. 29, 2018 @ 18:42 GMT
Hi Gary,

When I read your essay at the time it first came out, the thing that caught my attention (after eq 5.2!) was the matrix in eq 5.3, because that sort of arrangement could be a way to combine 3d quaternion-like polynomials into an octonion whose 3d spectrum has only one zero. It's a bit vague at this point, but I want to try something along that line later this year.

Having just read your essay again, what jumped out was the discussion on the peculiar properties of electrons in neutral atoms, which seems to be a purely wave phenomenon.

I must agree that the "vacuum", whatever it is, is the fundamental structure. And the argument for octonions is ultimately compelling.

Best wishes,

Colin

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Author Gary D. Simpson wrote on Jan. 30, 2018 @ 11:34 GMT
Colin,

Many thanks for reading and commenting.

"Quaternion Dynamics - Part 1" develops the concept of quaternion functions and might be of interest to you. I don't think it is possible for a quaternion or an octonion to be zero, but it is certainly possible for some part of it to be zero.

There is something interesting about Equation 5.3. There is a version of it where the complex i commutes normally. In it, the coefficient matrix contains B rather than B*. B* is the result of anti-commutation between the complex i and the unit vectors. A behavior such as that gives me some confidence that the math is correct. Plus it is really compact.

In "Five Part Harmony" I presented the following hypothesis:

exp(i*omega) = sqrt[1 - (v/c)^2] + i*(v/c)

Essentially, I simply added i(v/c) to the LT to produce Euler's Equation. My motivation for doing so was as a means of explaining the 6*pi^5 coincidence. That gave me a mathematical explanation but I did not have a physical explanation. After thinking about it for awhile, I realized the only way for the idea to work would be if electrons are stationary. And the only way for that to work would be if they are everywhere and simply rise up from the vacuum as needed. That has the extra benefit of providing a medium for action at a distance.

Best Regards and Good Luck,

Gary Simpson

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Lawrence B. Crowell wrote on Feb. 2, 2018 @ 20:14 GMT
I liked the quip you wrote on my page. I have not gotten around to reading your paper yet. I still have yet to read the paper before it. Anyway, I intend to read it tonight if possible.

Cheers LC

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Lawrence B. Crowell wrote on Feb. 4, 2018 @ 01:50 GMT
I read your paper through a couple of time. It appears that in your multiplication table you are treating the octonions as a pair of quaternions. This is legitimate. I was sufficiently impressed with your paper to give a top score. Your matrix multiplication of quaternions is useful. We should take this further though and look at nonassociative multiplication according to a general set of forms.

Cheers LC

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Author Gary D. Simpson replied on Feb. 5, 2018 @ 13:02 GMT
Lawrence,

Many thanks for reading and commenting.

Yes, I am treating an octonion as a bi-quaternion. That is what makes the multiplication table work.

The matrix multiplication is interesting. If the complex i commutes normally with the unit vectors, the coefficient matrix uses B. But if the complex i anti-commutes with the unit vectors, the coefficient matrix uses B*.

I have looked at the question of associative vs non-associative a little but not much. That is on my list of to do's.

I'm glad I gave you something to think about. I'll count this essay as a success.

Best Regards and Good Luck,

Gary Simpson

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peter cameron wrote on Feb. 4, 2018 @ 18:16 GMT
Hello Gary,

Have your essay open now, commenting stream-of-consciousness while browsing...

hmmmm. pondering this

"...in a 4-D geometry, if a right triangle is constructed from an integer

number of basis lengths in each of the four dimensions (a, b, c, and d), then the hypotenuse (f) that traverses through the 4-D space will also have an integer number of the basis...

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Author Gary D. Simpson replied on Feb. 5, 2018 @ 13:33 GMT
Peter,

Many thanks for reading and commenting. You comments are very detailed:-)

I was very taken by the notion of four-squares and its implications. I'm glad you see that too. So, is space-time continuous or discontinuous? I'm betting continuous but do not know for certain.

I am not familiar with the Lasenby Cambridge group. Also, I have not yet studied Clifford Algebra.

6*pi^5 ... is it a coincidence or a clue? Would you bet your life that this is numerology? I have proposed an experiment to answer this. The Mp/Me mass ratio could be determined in the reference frame of a satellite moving with respect to the Earth. The expected result would be that the average value would be the same as on Earth but the standard of deviation of the value would be larger. The burden of proof rests with me. I have no expectation that this experiment will be performed. Therefore, the only way for me to advance the idea is to assume that it is true and see what that will allow me to do with the Math. If it produces something that is true then perhaps 6*pi^5 is more than a coincidence.

You are essentially correct regarding my strategy. I hope to produce a very nifty matrix formulation wherein I can make a one-to-one correspondence between the terms in the matrices and parameters from Physics.

I don't recall making a derogatory comment regarding your Math skills:-( If I did then you have my apology. That is uncharacteristic of me.

Remember, I'm an engineer by education. My new "hobby" is exposing me to concepts in Math and Physics that I did not even know existed when I was in college. I'm doing my best to learn "new tricks". But I am an "old dog".

Best Regards and Good Luck,

Gary Simpson

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Christian Corda wrote on Feb. 5, 2018 @ 17:57 GMT
Dear Gary,

Excellent and entertaining Essay, despite a bit speculative (but speculations are often necessary when we interact with maths and physics).

Concerning the possibility to apply the Lagrange’s Four Squares Theorem to space-time if it is continuous or discrete, my recent research on black hole physics, that you have partially read in my current Essay, enables a very surprising insight: space-time could be neither continue nor discrete! In fact, space-time quantization should be energy-dependent. Then, maybe that Lagrange’s Four Squares Theorem could be only partially applicable.

I agree with your arguing about the fundamental importance of the vacuum in your Universe, but I have a question: If really the 5-D Octonion Group allows sufficient space for both QM and GR to exist, what about their unification?

In any case, you made a very interesting work deserving the highest score. Congrats and good luck in the Contest.

Cheers, Ch.

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Author Gary D. Simpson replied on Feb. 6, 2018 @ 03:56 GMT
Christian,

Many thanks for reading and commenting.

I plead guilty to being speculative. But think about it like this. I am a nobody in the world of Math and Physics. If I say of do something stupid, no one really cares. It does not affect my ability to make a living. Especially since I have already had a nice career as an engineer. BUT, If I propose something that rings true, someone...

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Christian Corda replied on Feb. 6, 2018 @ 08:30 GMT
Dear Gary,

Thanks for your interesting reply.

Being speculative is not a guilty. There is a wonderful statement by the great theoretical physicist S. Coleman. During one of his famous quantum field theory lectures at Harvard, he indeed claimed that “The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.”

Congrats again.

Cheers, Ch.

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Steven Andresen wrote on Feb. 6, 2018 @ 04:37 GMT
Dear Gary D. Simpson

Just letting you know that I am making a start on reading of your essay, and hope that you might also take a glance over mine please? I look forward to the sharing of thoughtful opinion. Congratulations on your essay rating as it stands, and best of luck for the contest conclusion.

My essay is titled

“Darwinian Universal Fundamental Origin”. It stands as a novel test for whether a natural organisational principle can serve a rationale, for emergence of complex systems of physics and cosmology. I will be interested to have my effort judged on both the basis of prospect and of novelty.

Thank you & kind regards

Steven Andresen

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Author Gary D. Simpson replied on Feb. 22, 2018 @ 16:26 GMT
Steven,

Thanks for the posting. I have read and commented upon your essay.

Best Regards and Good Luck,

Gary Simpson

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Jouko Harri Tiainen wrote on Feb. 8, 2018 @ 14:46 GMT
I'm not bound to integers. However, I found it to be very coincidental that Lagrange's Theorem was composed of four squares and the length of a quaternion is also four squares. Such coincidences mEssentially, things can appear to be quantized (i.e., integer valued) without actually being quantized because both methods reduce to four squares.ake me suspect that something deeper is underfoot....

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Jouko Harri Tiainen wrote on Feb. 10, 2018 @ 11:47 GMT
Every time I read your essay I seem to understand, it more and more.

I have a couple of questions about Equation 1

(a² + b² + c² + d²)u² = f²u²

A quote page 3

"The meaning of Equation 1 is that in a 4-D geometry, if a right triangle is constructed from an integer number of basis lengths in each of the four dimensions (a, b, c, and d), then the...

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Author Gary D. Simpson replied on Feb. 10, 2018 @ 15:03 GMT
Jouko,

You present an interesting argument. However, you have misinterpreted Equation 1. The hypotenuse is 4-D rather than 5-D. I do not jump to 5-D until the bi-quaternion form is introduced. The purpose of Equation 1 was merely to show how to construct things in 4-D using integer lengths. I did not mix lengths with areas. Your interpretation does that.

I can bring a 5'th dimension into Equation 1 by using the length of a circle in the complex plane as any one or more of the terms of Equation 1. I mention this near the end of the discussion on integers.

Equations 5, 5.1, and 5.2 do not require Equation 1. The important aspect of Equation 5.2 is that even if the system is 5-D, it will collapse into an equivalent 4-D system.

Best Regards and Good Luck,

Gary Simpson

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Jouko Harri Tiainen replied on Feb. 10, 2018 @ 16:20 GMT
Thanks for clearing it up -- Yours Harri. Will read your essay again.

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Author Gary D. Simpson replied on Feb. 10, 2018 @ 17:15 GMT
Harri,

Not a problem. Think about it like this ... if I make a triangle in the i-j plane, the hypotenuse moves through two dimensions. If I add a vector in the k direction, the hypotenuse now goes through three dimensions. If I then add a time component, the hypotenuse goes through four dimensions.

You can of course interpret Equation 1 as you propose, but you'll need to do so consistently to avoid the mix and match that you identify.

I made a misstatement. Actually, I introduce the 5-D with the wave function in Equation 4.

Don't spend too much time on my essay. There are many more than deserve attention.

Best Regards and Good Luck,

Gary Simpson

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Steve Dufourny wrote on Feb. 16, 2018 @ 12:50 GMT
Hi Gary ,

Beautiful works aboutb these octonions, lie is in you, I liked this essay.

Friendly, good luck

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Steve Dufourny replied on Feb. 16, 2018 @ 12:52 GMT
The vaccuum, the space between sphères that said does not seem existing when we apply a serie of spherical volumes with primes for example and a decreasing of volumes and increasing of numbers, the dark matter and the aether seems gravitational, the space disappears .....

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Steve Dufourny replied on Feb. 16, 2018 @ 12:55 GMT
The not commutativity that said is relevant, these lie algebras seems important when we consider our space time, that said if we have at our pure foubdamental level only matter and energy and that aether is gravitational ? we have new roads of analyses wityh this matter not baryonic .....

Friendly

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Author Gary D. Simpson replied on Feb. 16, 2018 @ 20:01 GMT
Steve,

Welcome back. Thanks for reading and commenting.

Best Regards,

Gary Simpson

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Dizhechko Boris Semyonovich wrote on Feb. 19, 2018 @ 06:35 GMT
Gary D. Simpson, you wrote a very entertaining essay on mathematics. I give the highest rating if you go on my page and we will discuss with you the principle of identity of space and matter Descartes, according to which space is matter and matter is space that is moving. Thus, space is the Foundation for fundamental theories, and the time is synonymous with the universal movementniya. . Look at my page, FQXi Fundamental in New Cartesian Physics by Dizhechko Boris Semyonovich

Do not allow New Cartesian Physics go away into nothingness, which can to be the theory of everything OO.

Sincerely, Boris.

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Author Gary D. Simpson replied on Feb. 19, 2018 @ 19:51 GMT
Boris,

Thanks for reading and commenting. I have read your essay and will comment and rate ASAP. Feel free to rate my essay as you see fit.

You have many ideas that are similar to my own.

Best Regards and Good Luck,

Gary Simpson

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Dizhechko Boris Semyonovich replied on Feb. 20, 2018 @ 01:51 GMT
Gary, I appreciated your essay on 10 and thinking in order to develop our ideas, which coincided, by the use of quaternions.

New Cartesian Physics needs your support to develop further. Visit my page and give it your rating.

FQXi Fundamental in New Cartesian Physics by Dizhechko Boris Semyonovich

I wish you success! Sincerely, Boris Dizhechko

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Vladimir Nikolaevich Fedorov wrote on Feb. 21, 2018 @ 07:06 GMT
Dear Gary,

Here we are again all together.

I enjoyed reading your contribution.

I agree with you, «that the vacuum is the most fundamental structure in the universe». Great!

I hope that my modest achievements can be information for reflection for you.

Vladimir Fedorov

https://fqxi.org/community/forum/topic/3080

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Author Gary D. Simpson replied on Feb. 22, 2018 @ 16:25 GMT
Vladimir,

Thanks for reading and commenting. I have also read and commented upon your essay.

Best Regards and Good Luck,

Gary Simpson

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Vladimir Nikolaevich Fedorov wrote on Feb. 24, 2018 @ 07:12 GMT
Dear Gary,

(copy to yours and mine)

Many thanks for the kind words about my work and for mutual understanding.

The understanding and appreciation are highly valued.

I wish you happiness in your scientific work in search of truth.

Vladimir Fedorov

https://fqxi.org/community/forum/topic/3080

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