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Thomas Ray: on 3/25/17 at 14:35pm UTC, wrote Hi Gary, When Kaluza-Klein theory came up, Einstein acknowledged that a...

Gary Simpson: on 3/23/17 at 1:02am UTC, wrote Peter, Thanks for having a read and the comments. FYI, I read and...

Gary Simpson: on 3/23/17 at 0:39am UTC, wrote Vladimir, Many thanks for reading and commenting upon my essay. I have...

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Peter Jackson: on 3/22/17 at 14:09pm UTC, wrote ...oops, wrong button! I'm sure I once made some sense of the above but...

Peter Jackson: on 3/22/17 at 13:49pm UTC, wrote Gary, Thanks for the score & post which I've responded to. I've read yours...

Vladimir Fedorov: on 3/22/17 at 13:10pm UTC, wrote Dear Gary, With great interest I read your essay, which of course is...

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March 27, 2017

CATEGORY: Wandering Towards a Goal Essay Contest (2016-2017) [back]
TOPIC: Five Part Harmony by Gary D. Simpson [refresh]
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This essay's rating: Community = 4.7; Public = 4.1

Author Gary D. Simpson wrote on Jan. 10, 2017 @ 21:50 GMT
Essay Abstract

This text demonstrates that the complex i can be combined with a Hamilton style quaternion to produce a 5-D mathematical structure. Essentially, the complex plane is combined with an arbitrary unit vector. The complex i is shown to anti-commute with the unit vectors i, j, and k. The resulting geometry is shown to be an extension of Hamilton’s quaternions based upon the complex plane rather than real numbers. This new geometric structure is presented in Figure 1 and Equations 3 through 3.3. This configuration makes it possible to calculate the diameter of the proton at rest with the estimated value being 1.668 x 10-15 meter. This is within the accepted measured range of the proton diameter at 1.755(102) x 10-15 meter as given by the NIST, and it is very close to the proton diameter at 1.68174(78) x 10-15 meter as measured at the Paul Scherrer Institute in 2010 by using muonic hydrogen.

Author Bio

I am a retired chemical engineer. I graduated from Texas A&M University with BS and MS degrees many years ago. I am interested in quaternions and wave functions.

Download Essay PDF File

Author Gary D. Simpson wrote on Jan. 10, 2017 @ 22:28 GMT
Thank you for taking a look at my essay.

The essay topic seems to me to be very challenging. The fact that it took so long to post the first set of essays seems to support this belief. Regrettably, the only contribution that I can make concerns the "arrow of time". My essay presents a different way of thinking about the "arrow of time". I argue that it is the complex i that has direction rather than time and that time is simply a scalar.

You might ask "Since they are presented together as i*c*t, what difference does this make?" The difference is that if time is the dimension then the result is a single 4-D spacetime. But if the complex i is the dimension then the complex plane can be combined with 3-D space to produce a 5-D space with a continuous group of 4-D subsets. I think this extra degree of freedom can allow QM and GR to be combined into a single entity.

For an inertial reference frame, the only physical dimension that is needed is the direction of motion. This model seems to say this and thereby allows for the other two spatial dimensions to be used for GR and QM. It is not clear to me which should be scalar and which should be the complex i.

Paul Dirac once commented that spin is the square root of geometry. I think this 5-D geometry is consistent with his thoughts on the subject. The reason is that the square root needs a direction to point just as the complex i needed a direction to point.

My hope with this essay is that someone who is knowledgable in Physics might recognize some similarity between the presented structures and known Physics. This might give me a clue as to how to proceed.

Best Regards and Good Luck to All,

Gary Simpson

Joe Fisher replied on Jan. 13, 2017 @ 16:26 GMT
Dear Gary,

As I have thoughtfully pointed out in my brilliant essay, SCORE ONE FOR SIMPLICITY, the real Universe consists only of one unified visible infinite surface occurring in one infinite dimension, that am always illuminated by infinite non-surface light. Invisible complexity has never existed, so it would be physically impossible for invisible complexity to combine with any other sort of invisible miasmatic state.

Joe Fisher, Realist

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Steve Dufourny wrote on Jan. 18, 2017 @ 08:58 GMT
Hi Gary ,

Congratulations for your papper about geometrical algebras.It is very relevant these algebras and these diamaters and others Tools.Thanks for sharing and good luck in this contest.You are going to have a prize in logic.You merit it.


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Author Gary D. Simpson replied on Jan. 18, 2017 @ 12:07 GMT

Thanks for taking a look and reading my essay. I think the geometric algebras will eventually explain everything to us. What is most interesting to me about what I have presented is that under some circumstances, an 8-D octonion can simplify to a 5-D space. I think this is very significant. It is necessary to take all of the concepts together as a single entity to understand fully what is happening with these algebras.

Best Regards,

Gary Simpson

Steve Dufourny replied on Jan. 18, 2017 @ 18:19 GMT
Hi Gary,

You are welcome.I liked your method in playing with octonions and this 5D.I like also these geometrical algebras now.I try to formalise my theory of spherisation with 3D quant and cosm sphères with the spherical geometrical algebras and my equation about matter and energy E=mc²+ml² It is a big work Gary I search the good method.I like the quaternions, and others with the 3 vectors and the scalar and this and that, but we consider points for these algebras.If now we insert the spherical volumes and the motions more the geometrical algebras, that become relevant.Even for computing we can correlate with the bloch sphere for qbit.The strings are a beautiful tool for computing also.The convergences can appear after all.good luck in this contest.:)

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Jose P. Koshy wrote on Feb. 1, 2017 @ 10:23 GMT
Gary Simpson,

Your finding that the mass ratio of proton to electron is related to 'pi' is interesting; the observed deviation is attributed to absolute motion. In my opinion, the deviation may be due to quantum- continuous disparity.

For example, make a circle using small spheres. What is its effective circumference? A continuous line passing through the outermost points, or a continuous line passing through the centers of the spheres, or something in between. Here I think the Feynman diagrams may help, because these are used in the case of wave- particle duality, which is a case of quantum- continuous disparity. I regard the Feynman diagrams as approximation techniques. However, I have not used them till now.

As my approach is Newtonian, I take that the integration of grainy (quantized) matter follows "symmetrical spherical packing", and so 'pi' has a crucial role in deciding the masses of proton and neutron. The mass of neutron is slightly grater than 1838 electrons. This is not an arbitrary number; it represents the minimum number of spheres required to make a nearly spherical structure having perfect symmetry, when the spheres come in pairs (like electron-positron pairs). Here also, 'pi' decides the number. The surprise is that Fine Structure constant remains inbuilt in such a structure.

So the equation you obtained for proton radius cannot be a mere coincidence; it has something buried inside (however, in my model proton and neutron have sizes proportional to that of electron).

Jose P Koshy

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Author Gary D. Simpson replied on Feb. 2, 2017 @ 00:52 GMT

Thank you for commenting on my essay. I was beginning to feel a bit ignored. It is clear to me that you read and understood.

The coincidental value 6pi^5 is what set me upon a long journey. To the best of my knowledge, it was first noted by Friedrich Lenz in the April 1951 issue of Physical Review. I independently discovered it in April 2012 and I have been trying to make sense of it since that time.

You mention an alternative method of explaining the deviation of the empirically observed value of Mp/Me from 6pi^5. Your thinking seems to be very similar to that of Don Hotson. He authored three papers in the fringe web journal "Infinite Energy". The first two of those essays are excellent reading and very thought provoking. Regrettably, he has passed away. I would expect you could find his work using a Google search.

My present thinking is that the pi^5 term is the result of the 5 dimensions and that the coefficient 6 is the sum result of a scalar one value for each of the three real dimensions vector i, vector j, and vector k, plus a scalar one value for each of the complex dimensions (complex i)x(vector i), (complex i)x(vector j), and (complex i)x(vector k).

Best Regards and Good Luck,

Gary Simpson

Steve Dufourny replied on Feb. 3, 2017 @ 10:19 GMT
Hello to both of you?

All these reasonings are relevant.Do you know if we could find the correct spherical volumes and the correct serie with the good probablistic method.The serie and its finite number and the volumes are the keys in fact logically.I read the works of Feynman about électrons.It is a wonderful discussion.The fine structure constant is relevant also.I study a little this constant and its meaning.Feynamn in all case was incredible, when we read his papper and his words ,it is so deep and general.He was an incredible thinker.Do you remember that he said these words " one day we shall see all the truth and we shall say all oh my god but how is it possible that we have not seen a thing so simple before ?" sometimes the complexity returns to simplicity :)


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Author Gary D. Simpson replied on Feb. 3, 2017 @ 22:16 GMT

Thanks for the continued interest. Perhaps you will submit an essay?

Best regards,

Gary Simpson

Declan Andrew Traill wrote on Feb. 4, 2017 @ 21:09 GMT
Hi Gary,

I like your work involving complex numbers in relation to particle physics. I too believe particles can be constructed or modeled with a field of complex number vectors, though I think 3 + 1 dimensions is sufficient. I have such a model for electrons/positrons here:

It would be good if you could explain how your work fits into the essay topic of wandering towards a goal resulting in goal oriented structures in the Universe. Clearly the formation of particles is the most essential step in this process (as I have discussed in my essay) so it would be good to explain this in the essay.

Overall: nice work and a well written essay. Good luck!!

Best Regards,

Declan Traill

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Author Gary D. Simpson replied on Feb. 5, 2017 @ 01:21 GMT

As I state in my essay, I address the "arrow of time". Essentially, I eliminate it and replace it with scalar time that operates upon the complex plane. This simplifies the methods whereby any system may evolve with respect to time.

Both GR and QM are 4-D models. They share the same physical space. However, they treat time differently. The time in GR is not the same as the time in QM. Therefore, there is no 4-D model that can combine them. The least number of dimensions tat can do that is 5 which is what I present.

The fact that I have accurately calculated the size of the proton should be considered a very strong piece of circumstantial evidence in support of a 5-D model.

Best Regards,

Gary Simpson

Declan Andrew Traill replied on Feb. 5, 2017 @ 05:17 GMT
Hi Gary,

Can you explain what is different about time on GR and QM?

It seems to me that formulating GR in terms of a scalar field where space is fixed and the speed of light and rate of time are variable (by the same amount - thus giving the impression to any observer that the speed of light doesn't change) fits in nicely with QM, and in the same space 3D particles can be modeled that agree with QM.

What would the fifth dimension be?

Best Regards,

Declan Traill

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Author Gary D. Simpson wrote on Feb. 5, 2017 @ 19:34 GMT

Please read my essay slowly and carefully. It is not light reading. Figure 1 presents the 5-D geometry. It does so in a way that can be construed as 3-D geometry for an inertial reference frame.

In both GR and SR, time and space are presented together as a 4-vector. The 4'th dimension is presented as i*c*t. Essentially, i*c*t is added to a space vector and the result becomes Minkowski space-time. GR is presented taught using tensors. Thus far, I have not learned to use tensors competently.

Time in QM is more simple. It is presented as a scalar rather than a pseudo-vector component of space time. Space in QM is also treated differently ... more like a scalar than a vector. The Schrodinger Equation presents time coupled with the complex i and it presents the space variables x, y, and z presented as scalars rather than as vectors containing the unit vectors i, j, and k.

Wikipedia is a wonderful resource. I give Wikipedia a modest amount of financial support. But it is not a substitute for other sources.

BTW, Einstein himself specifically rejected the idea of a scalar field. I mention this near the end of the essay. Space-time is not a vector plus a scalar as in a quaternion. Space-time is a vector plus a "time-vector" based upon the speed of light. That is not the time of QM.

Best Regards and Good Luck,

Gary Simpson

Lawrence B. Crowell wrote on Feb. 6, 2017 @ 02:29 GMT
I see you have an essay here. I remember an essay last contest with similar content, it dealt with quaternions. I presume you are the same fellow. I will have to read it tomorrow.

As for spelling, I often write from and form and similar errors, drop s from plural words and so forth. It is just a glitch.

Cheers LC

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Lawrence B. Crowell wrote on Feb. 6, 2017 @ 22:41 GMT
I just gave your paper an 8. The reason I did not give a 10 is that I think things go awry at the end. Up until equation 4.2 and page 8 things look very interesting. It is at the end when you write that the time is a scalar and not GR or SR and so this departs from GR. Actually this is a form of Clifford algebra CL_{3,2}(C) ~ SO(3,2) based on the interval

s^2 = i^2 + ω^2 - (\bf i}^2 - (\bf j}^2 - (\bf k}^2,

where for one of the "time" components constant (say i) this defines the anti-de Sitter spacetime AdS_5 ~ SO(3,2)/SO(3,1). This is general relativity! Spacetime can be described according to the spinor variables A and b with the interval

s^2 = a·b + axb, x = times,

so the scalar a·b plays the role of time and the "vector part" axb is spatial part.

The equations 3.3 and 3.3.1 are a "biquaternion" version of the electromagnetic or Yang-Mills field tensor. The biquaterion form comes from the 2x2 matrices A_0 and A_i etc in eqn 3.3. I worked out a couple of years ago how the derivative

d(f{\bf i} + g{\bf j} + h{\bf k}))


d(x{\bf i} + y{\bf j} + z{\bf k}))

leads to the field tensor. This approach leads to a bi-quaterionic form that I think is connected to the Petrov-Penrose tensor.

So there is a lot of good stuff here. Given that you are not trained to be a math physicists it is pretty remarkable.


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Author Gary D. Simpson replied on Feb. 7, 2017 @ 14:41 GMT

Many thanks for taking the time to read and comment on my essay. In the previous essay contest, I had an entry that dealt with using quaternions as a basis for Calculus. There were several other essays that dealt with quaternions also. Perhaps mine is the essay that you remember.

Can you recommend a suitable textbook(s) for Clifford Algebra and de Sitter Space? Perhaps the time has come for me to learn these things. Keep in mind that I am an engineer by education and engineers do not study things like Group Theory ... perhaps there are some prerequisites necessary prior to Clifford and de Sitter ...

Do you think that 6*pi^5 and the resulting proton diameter are coincidence, or could they have a physical significance? Or is it simply wrong?

The concept of scalar time that I am presenting is a little different from what you have interpreted, but I did not really make it very clear. I am proposing that there is a "master" scalar time that is multiplied by Euler's Equation. This produces two time-like values. The product with the cosine is hopefully the time used by SR & GR. The product with isine is hopefully the time used by QM. I then interpret all of this geometrically by having the "master" scalar time operate on the complex plane. That is where the 2x2 matrices become essential and the bi-quaternion form of Hamilton's work presents itself.

Best Regards and Good Luck,

Gary Simpson

Lawrence B. Crowell replied on Feb. 8, 2017 @ 12:34 GMT
The coincidence you cite I am not sure of. Where things went I think a bit awry is with the interpretation of time as a scalar being different than that of relativity.

I am going to try to carve out some time to work on this. I will present a bit here.

The book by Hestenes and Sobczyk is good. Hestenes writes in a straight forwards way that avoids overly pedantic stuff.

Cheers LC

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Declan Andrew Traill wrote on Feb. 7, 2017 @ 12:15 GMT
Hi Gary,

I don't recall seeing the complex 'i' in GR for the time dimension, it is just c*t from my experience and thus a scalar time. Have you introduced the i in your work? I know it is in the Schrodinger wave equation and thus features in standing waves (i.e. for particles), but I don't see why the same notion of time cannot be used in an understanding of GR and QM (from a WSM standing wave perspective of the Universe).


Declan Traill

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Author Gary D. Simpson replied on Feb. 7, 2017 @ 14:47 GMT

Your statement is basically like saying that you do not see the unit vectors i, j, and k in Euclidian Geometry. It might be true, but it also misses the point. Minkowski space-time is based upon 4-vectors and that means i*c*t. The complex i is there whether or not you see it presented in writing. This is a prime example of how Wikipedia can unintentionally mislead people.

Best Regards and Good Luck,

Gary Simpson

Declan Andrew Traill replied on Feb. 8, 2017 @ 05:50 GMT
Not necessarily. There is nothing magical about Tensors, they are simply 4 dimensional matrices - a way to neatly encode equations involving 4 dimensions. The time dimension need not involve the complex i. If you say that i, j and k are unit vectors in the 3 physical dimensions, you could say that complex i is a unit vector in the time dimension, but it need not be a vector at all. The equations work just fine treating time as a scalar even though it occupies a 'dimension' (in the mathematical sense, rather than a physical sense) in the 4D matrix.

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Declan Andrew Traill replied on Feb. 8, 2017 @ 06:09 GMT
I rather think that the complex i vector comes into play when 3D standing waves are involved (i.e. in the case of particles comprised of 3D standing waves). In standing waves the medium moves in circles. In the case of Electromagnetism these circles are traced out by complex number vectors (this is where the complex i comes in) and so time itself is scalar, but as it determines the rate of these complex vector rotations in particles the complex i and t are closely coupled.

Anyhow, I have given your essay a community score of 10 because your essay is well written and I like your work on complex numbers and quaternions and I think there should be more of it.

Best of luck in the competition...


Declan Traill

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Branko L Zivlak wrote on Feb. 7, 2017 @ 20:42 GMT
Dear Mr. Gary D. Simpson

As you said:

Even if you don't think you are a top thinker (hey, I'm not for sure) it is worth participating. You just never know how your thoughts might affect someone else who might then affect someone else ...

If someone has only ten percent of good ideas, it's better than prosaic essay that supports the prevailing opinion. Your article is full of ideas and is useful and supported with math. Your math is certainly not numerology. But section, Proton Diameter leads to wrong conclusions. Your Calculated value is close (but not in the middle of the interval) from the CODATA value which is known to very few of significant digits. This means that the calculated value is likely wrong, but true.

In my article “Cycle towards Methodology of Everything“,
/6731 I got value for the proton radius 8.764-16m.

I have it determined, as well as all other relations without using dimensions.

Therefore, I argue that the debate about the number of dimensions is waste of time.

In this year's essay I confirmed the importance of Mach principles and values of Proton shift, which I calculated. I would appreciate if someone finds errors in formulas.



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Author Gary D. Simpson replied on Feb. 8, 2017 @ 15:40 GMT

Many thanks for reading an commenting upon my essay. Perhaps I have given you an idea? If so, please use it wisely and freely.

Regarding the proton diameter, the value that I estimate is in the lower part of the NIST data range. It is very close to the value determined using muonic hydrogen. Having said that, I should also say that my value is outside the range published by the Paul Scherrer Institute as part of their study. I also claim that the Earth's reference frame is in absolute motion. I have not yet determined the correction for this motion.

I have briefly read your essay but not carefully enough to make any sensible comments. I'll study your work more closely and comment.

Best Regards and Good Luck,

Gary Simpson

Satyavarapu Naga Parameswara Gupta wrote on Feb. 8, 2017 @ 10:55 GMT
Dear Simpson

Your Harmony essay is very good. You envisaged, 5 axis coordinate system. , which are x,y,z,t and i the imaginary. You used it to derive the Diameter of proton and other results nicely.

Is that really necessary to have that i axis? Why should we need imaginary things to derive real things…..? Both the classical Physics and QM are for real matter at different scales.

Hope you will clarify me….

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Satyavarapu Naga Parameswara Gupta replied on Feb. 8, 2017 @ 10:59 GMT
Dear Simpson

I modified the text and FQXi server did not take that change(modifications) even after 10 times trying and restarting my computer.... So I am putting another post...

"Is that really necessary to have that i axis? Why should we need imaginary things to derive real things…..? Both the classical Physics and QM are for real matter at different scales."

Hope you will clarify me….

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Author Gary D. Simpson replied on Feb. 8, 2017 @ 23:16 GMT

Thank you for taking the time to read and study my essay. Do we really need imaginary things? I will simply say this ... if the universe is truly 5 dimensional, then the extra two dimensions are not imaginary. The evidence that I offer in support of this is the proton diameter calculation. Admittedly, this is only circumstantial evidence, but if correct then it seems to be pretty good evidence.

In some ways it is unfortunate that we use the term "imaginary" when speaking of a certain set of numbers. It is better to think of the complex i as an operator that causes direction to be reversed when applied twice.

Think of it like this ... in the Michelson-Morley experiment, no aether drift was observed. The presumption is that motion with respect to the aether is the same as motion between two physical objects. But what if we really don't know how to describe or measure this motion. What if motion with respect to the aether is in the direction of the complex i? What if we are moving in such a way that we do not even notice the two extra dimensions? I realize this seems very speculative. I will simply ask how does the proton diameter work if there are not 5 dimensions?

Best Regards and Good Luck,

Gary Simpson

Satyavarapu Naga Parameswara Gupta replied on Feb. 12, 2017 @ 22:54 GMT
Dear Garry Simpson

Thank you for nice reply , which made me to go into deep thinking.... Best wishes....

I am reproducing the reply for the question you asked on my essay...

see there for the attachments....

Thank you very much for studying my paper so thoroughly and giving esteemed questions. I am just giving two reported cases of Galaxies / Clusters of Galaxies which are being generated after Bigbang

[35] Rakos, Schombert, and Odell in their paper ‘The Age of Cluster Galaxies from Continuum Colors’ Astrophys.J., 677 , 1019, DOI: 10.1086/533513, e-Print: arXiv:0801.3665 [astro-ph] | PDF arXiv:0801.3665v1 [astro-ph] 23 Jan 2008


See the CANDLES web pages also for simple language explanations.

There are many other papers and websites also if want them I will give them,

By the way, see the attachments to this post, to see these files for your quick reference…

Best Regards

SNP. Gupta

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Edwin Eugene Klingman wrote on Feb. 10, 2017 @ 05:27 GMT
Hi Gary,

Your work with quaternions is fascinating. You begin by noting that time is viewed as either a scalar or a vector ("dimension"), and choose scalar, with complex 'i' viewed as a dimension.

Since we have discussed Geometric Algebra, I thought you might be interested to know that a recent book Understanding Geometric Algebra for Electromagnetic Theory, by John W Arthur, treats time both ways. He first develops Maxwell's equations viewing time as a scalar. After this he treats time as a vector and shows how this leads far more naturally to the fully relativistic formulation compatible with Hestenes 'Space-Time Algebra'. In neither of these is the complex 'i' a vector.

What's so special about geometric algebra is that every term has both a geometric and an algebraic definition, unlike every other branch of mathematics. In this scheme it's probably more correct to say that the complex 'i' does not appear in GA, since, although it has the algebraic value of the square root of -1, it is a pseudo-scalar, which is a different beast. Even so,in 2D it reproduces the complex plane very nicely.

I think of the 'i' in GA as an operator, and specifically an analog of the Hodge duality operator of differential geometry. It has the effect, when operating on a term, of transforming the product into the dual of the term being operated on. This is quite different from the scalar 'i'.

I don't know if this is any help at all, but I find viewing 'i' as the operator in GA to be quite helpful to my own understanding. I don't know that it applies to your immediate work, but since GA includes quaternions, and since you seem to be moving in the direction of GA, I thought I'd put my two cents in.

And thanks for your comments on my paper.

Best regards,

Edwin Eugene Klingman

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Author Gary D. Simpson wrote on Feb. 11, 2017 @ 00:15 GMT

Many thanks for reading my essay. I am planning to post one more paper to to show the inversion of the matrix that I present in this essay. Then I plan to spend as much time as needed to learn GA and Dr. Hestenes' work. The extra reference will prove useful I am sure. I also think of the complex i as an operator.

I think there is a connection between the consciousness field that you propose in your essay and the scalar field that I propose in mine. It might be a way to add the observer to the picture.

Do you think that the 6*pi^5 observations are coincidence, or might they contain some truth? Or is the proton size calculation simply wrong?

Best Regards and Good Luck,

Gary Simpson

Lawrence B. Crowell wrote on Feb. 11, 2017 @ 01:55 GMT

I worked this up a couple of years ago. It is not difficult to understand and is related to your work. It also has content in your equations 3.

Your paper is not at all related to my paper here, but if you are interested you might find it interesting. It has little bearing on quaternions in a direct was. If you are interested I can send a paper I published which does illustrate a Clifford algebraic format for the equivalency of spacetime geometry and the Tsirelson bound of quantum mechanics.

Cheers LC

attachments: 2_quaternion_notes.pdf

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Author Gary D. Simpson wrote on Feb. 11, 2017 @ 12:49 GMT

Many thanks for the post. Yes, please send a copy of the file that you mention to the email address on the cover sheet of my essay.

I've studied the note already and part of it is helpful to me. However, part of it is not consistent with Hamilton's definitions. You state that ij = jk = ki = ijk = -1. That is not what Hamilton stated. He stated that i^2 = j^2 = k^2 = -1 = ijk. As a result, ij = k, jk = i, and ik = -j. These identities impose handedness onto the system. We agree regarding anti-commutation of the unit vectors.

BTW, does anti-commutation of the complex i with the unit vectors seem reasonable to you?

Best Regards and Good Luck,

Gary Simpson

Lawrence B. Crowell wrote on Feb. 11, 2017 @ 22:19 GMT
Here is the paper I published a few months ago. I intend to publish an extended version of this in something like Annals of Physics.

You are right in that ij = k, jk = i and ki = j, it is cyclic. I wrote wrongly there, even though I know otherwise. As for commutation, you do have ij = k and ji = -k. So this looks like commutators, not anti-commutators. One could potentially of course look at Jordan products or a graded quaternion system with

E^i = e^i + iσ^_{ab}(\bar θ^aψ_b + θ^a\bar ψ_b) + F_{ab}\bar θ^bθ^a,

which would be a sort of supersymmetric version of this.

attachments: LawrenceCrowell_V7N13.pdf

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Colin Walker wrote on Feb. 14, 2017 @ 00:24 GMT
Hi Gary. It occurs to me that the Higgs field is supposed to be a product of complex and quaternion fields, so perhaps the multiplication in your first equation is relevant there.

I have become comfortable with the idea of 3 temporal dimensions from working with Fourier analysis of plane waves, but your essay reveals another possibility - complex time. Quite an interesting idea. Had to chuckle at your comment about nature being devious, having asked myself that a few times.

But I have to question the hypothesis of an absolute speed relating to the Mp/Me ratio in the way you suggest. If the cosmic microwave background defines the rest frame, the speed of the solar system is about 0.0012c, which is about 1/5 the absolute speed, 0.006136c, required to produce the Mp/Me ratio. A practical researcher does not ignore hunches based on numerical coincidence, but sometimes that's all it is. In any case, failure of this final speculation would not affect the rest of the work.

By the way, I think I first read about three time dimensions in Milo Wolff's book about space resonance and matter waves which seems to reflect some aspects of the Higgs mechanism - the book with Milo (presumably) and his motorcycle on the cover. His tripod website is still there. Sad to hear of his passing.

Anyway, nice work! - cw

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Author Gary D. Simpson replied on Feb. 15, 2017 @ 04:08 GMT

Thanks for reading and commenting upon my essay. If my memory is correct, you are familiar with quaternions and such. Hopefully this was an easy read for you.

Also, thanks for the heads up regarding the Higgs. That is exactly the sort of clue that I was hoping to get by posting this essay. I'm not sure where it will lead me but I will study the idea.

I think nature is more devious than we can even imagine ... I'm glad you liked the humor:-)

I was not aware that the cosmic background radiation implied a velocity of 0.0012 c. This is another good clue. There are certain types of average whereby a value is divided by the number of degrees of freedom that a system possesses. Since I argue for 5 dimensions, then it is possible that both arguments are true. I will need to refresh my memory on this. I seem to remember it from statistics. In any event, as you note, the mathematical structure does not depend upon the hypothesis presented as Equation 2.

You can definitely get three time dimensions simply by multiplying the complex i by the three spatial dimensions. But that implies to me that the complex i has the dimension and that time is simply a scalar that operates on the complex i.

Thanks again.

Best Regards and Good Luck,

Gary Simpson

basudeba mishra wrote on Feb. 18, 2017 @ 12:20 GMT
Dear Sir,

You have correctly said that this topic is about mindless mathematics and that time is not a dimension. But do extra-dimensions exist? We are hearing about it for over a century. But it has never been found. In fact, the term dimension is still to be unambiguously and scientifically defined. Quaternions are a complex number of the form w + xi + yj + zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions. Is it a physical description? If yes, why cannot complex numbers be used in computer programming? If i stands for square root of -1, what does j, k stand for? This type of mindless mathematics are being addressed in this contest. Kindly clarify.



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Author Gary D. Simpson replied on Feb. 18, 2017 @ 14:24 GMT

Thank you for reading and commenting upon my essay. I will answer your questions to the best of my ability.

You agree that time is not a dimension and you question the existence of extra dimensions. You ask for proof of extra dimensions. I have presented an accurate calculation of the size of the proton based upon a model that uses 5 dimensions. I argue that the accuracy of...

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basudeba mishra replied on Feb. 20, 2017 @ 00:28 GMT
Dear Sir,

You say: “The unit vectors i, j, and k represent the x, y, and z axes respectively. The unit vectors make it possible for the axes themselves to be a part of computations”. In that case why complicate things by adding terms i, j, and k? The x, y, and z axes could have been sufficient by treating them as unit vectors. After all, vectors are different only because they have movement (energy) and direction. The axes provide direction. The axes have no meaning without something to represent. We also use mobile coordinates. Thus, what is the justification of adding i, j, and k? Further, x, y, and z are real, whereas i, j, and k satisfy i^2 = -1, which means complex. Why should we use complex numbers at all? They do not have physical presence. Anything that has no physicality cannot be a part of physics. Your statement that you have tried to “combine physical space, in the form of an arbitrary unit vector, with the complex plane…” presupposes that both do exist physically. Is there any proof in its support? Can you give examples?

Your statement: “Let x = 1 and let y = 1. It follows that x + y = 2. This is a simple scalar result. Now let x = i and let y = j. It follows that x + y = i + j. This is a vector from the origin of length sqrt(2) at an angle midway between the x and y axes” only conforms our views. Addition is linear accumulation, which is possible between similars. Here x and y have the same value and belong to one class. But x = i and let y = j shows that they belong to two different classes. You cannot add 5 oranges and 3 apples. You can add them only as fruits. We have submitted an essay to physically explain 10 dimensions. Unlike your 5 –D inferred space, we have shown direct correlation, where we have used the same logic as you have shown here.



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Author Gary D. Simpson replied on Feb. 20, 2017 @ 03:55 GMT

Let me see if I understand this ... you don't think I am correct to argue for 5 dimensions, but you have submitted an essay that uses 10 dimensions? That is puzzling to say the least. BTW, I do not see an essay associated with your name.

I will simply restate my evidence. The calculation of proton size is based upon a 5-D model. The resulting value is within the accepted NIST range. If you wish to challenge this calculation, then please do so. Perhaps your 10-D model makes predictions or calculations?

You continue to misrepresent the unit vectors as complex numbers. They are not. They represent physical space. You will not understand Hamilton until you abandon that thinking.

Actually, I can add apples and oranges. A quaternion can contain 4 separate types of things. That is one way of interpreting the scalar, i, j, and k. Although treating them as something non-spatial would destroy identities such as ij = k. It would also destroy the i^2 = j^2 = k^2 = -1 identities.

I look forward to your essay.

Best Regards and Good Luck,

Gary Simpson

Jonathan J. Dickau wrote on Feb. 18, 2017 @ 18:49 GMT
Fun stuff Gary!

Your work appears to have tie-ins with recent work by Stephen Adler. Here he talks about complex-valued spacetime foam..


Here he suggests we need to test for quaternionic values in QM experiments.


Also notable is recent work by Hyun Seok Yang asserting that non-commutative spacetime is inherently emergent. See these papers for starters.



I already know I want to give you a high score, but I also know that elevating you now will make you a target. So I'll wait until your rating dips down a bit, before I rate your essay, and then boost it back up.

All the Best,


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Author Gary D. Simpson wrote on Feb. 18, 2017 @ 20:23 GMT

Many thanks for reading my essay and for the numerous references for further study. That will probably take me awhile.

Dr. Klingman also chooses to vote late. You are both wise men.

Best Regards and Good Luck,

Gary Simpson

Jeff Yee wrote on Feb. 19, 2017 @ 01:33 GMT

Your work starts with the wave function and ends with a very nice acknowledgement to the late Dr. Milo Wolff, founder of the Wave Structure of Matter. In between, your understanding of math to explain the proton is what this FQXi contest is all about - well done and good luck to you.

A quick note about Dr. Wolff for those who are not familiar with his work. Dr. Wolff started a revolution for those that are working on a theory of matter that can be explained by wave energy. There's a lot of work remaining to prove this theory, but it has been an inspiration to some, and hopefully a call to action to others to explain the mysteries of the universe.


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Author Gary D. Simpson replied on Feb. 19, 2017 @ 03:08 GMT

Many thanks for reading my essay. Yes, Dr. Wolff was a major influence in my thinking. He is sorely missed.

Best Regards and Good Luck,

Gary Simpson

Colin Walker wrote on Feb. 22, 2017 @ 23:23 GMT
Hi again Gary. Not sure if this turns out to be useful, or even makes sense, but it jumped out because there is a chance it might relate to the factor of 1/5 which seemed to be missing from the speed in the CMB frame in your final conjecture.

I was just reading Jonathan Dickau's essay and was intrigued when he mentioned that the greatest hypersphere is produced in 5 dimensions. I don't know whether your 5-d construction is a hypersphere, but he refers to a Wolfram page Ball which has a formula V=S/n (Eq.2) relating the volume and surface area of a unit hypersphere to the dimension n. For n=5, the surface would be 4-dimensional. So it looks like there could possibly be a factor of 5 floating around in the math going between 5 dimensions and 4 dimensions.

I am still working on my essay involving the derivative of a conjugate quaternion, which I started fiddling with after reading your Calculus 2.0 essay last year. -cw

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Author Gary D. Simpson replied on Feb. 23, 2017 @ 01:08 GMT

I am flattered and delighted that my previous essay gave you something to think about. Supporting work for that essay and this essay are posted to The paper names are "Quaternion Dynamics Part 1 and Part 2". They can be found here:

I'm not sure yet what to think about the statement that you mention is Jonathan's essay. He speaks of the hyper-volume of a hyper-sphere being maximized for n = 5. I was not aware of this when I hypothesized a 5-D geometry. He also uses that as a rationale for dimensions becoming small for n > 5.

I look forward to reading your essay ... you better hurry, the deadline is approaching.

Best Regards and Good Luck,

Gary Simpson

Jonathan J. Dickau replied on Feb. 24, 2017 @ 03:46 GMT
I can mention here..

Somewhat paradoxically; while the (hyper-)volume curve is maximal around 5.25-d, the (hyper-)surface is maximal at ~7.25-d. There is some interesting material to access here:

2.1.3 A Brief Look at "Complicating" from 'Mathematics Itself: Formatics - On the Nature, Origin, and Fabrication of Structure and Function in Logic and Mathematics' by David M. Keirsey.

It would seem that Hilbert's hotel results from using hypercubic expansion, instead of building on spheres. People who build brane-world models often forget that branes are a generalization of spheres.



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George Kirakosyan wrote on Feb. 25, 2017 @ 04:36 GMT
Dear Gary

I am thankful that you are with me! We must to join our efforts to push ahead what we believe are the right. I see one of important criterions of our rightness in that the different brains in the different times and in the different places may come to similar conclusions. So, I just felt myself very obligated to read your work to say something.

About of decay of freely neutron you are fully right. I have tried to explain it and I even calculated its time, based on my model (you can find it in good time in my ,,Rethinking ... (I),, )

I will answer you within short time!

Good wishes

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George Kirakosyan wrote on Feb. 25, 2017 @ 07:22 GMT
Dear Gary,

You have used the the complex vector representation and the Euler's beautiful formula in your attempts to describe proton, in this case. You have the definite success on this. It shows just that you are on the somewhat right way. I am very agree with you that the dynamics and harmony should be the base to understand the microcosm. By the way, the solutions of Maxwell's equations (in macrocosm) and Schrodinger's equations (in microcosm) with its different modifications correspond with this. The main questions however, has become there - how need to interpret these solutions, since a what of physical values must to put there as the this or that members of equations? That is why I am calling to put the ideas first before of math! You know of course the merits of Faradei as well as Nikola Tesla ..... who was very weak in math! So, the math does not disturb them to RIGHT THINKING and to find the right answers by the same! Then their job was continued by whom who was more well with math ....!

So, I welcome your work and I will happy to help you.

I wait that we can be agree each with others.

Good wishes

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Author Gary D. Simpson replied on Feb. 25, 2017 @ 18:17 GMT

Thanks for taking the time to read and comment upon my essay. I will study your other works when time permits. I am very curious to read your thoughts regarding neutron decay.

You are generally correct that empirical observations should come before mathematical analysis. Faraday was a very great empiricist and his work and ideas were formalized by Maxwell. And Tesla changed the world with his work with electricity.

Around the year 1895, there was a great debate among mathematicians regarding the direction that mathematics should proceed with respect to geometry. Some favored Grassmann and some favored Hamilton. The ideas of Grassmann prevailed. It seems to me that since that time, mathematics has been unable to satisfactorily describe the more complicated observations made in physics. Therefore, I am revisiting the work of Hamilton with the intent being to describe the physics that has been observed since that time.

Best Regards and Good Luck,

Gary Simpson

Wilhelmus de Wilde wrote on Feb. 25, 2017 @ 16:11 GMT
Dear Gary,

I read your interesting essay and understood your intention.

You lose me when formula's are introduced, It is a language that I just have problems with.

You say : It is certainly possible – perhaps even likely – that these conclusions are false. However, if that is the case then the reader is left to explain the proton diameter calculation.

So in order to discuss your conclusions I have to calculate ? In my humble opinion the diameter of an instananeous excitation from the past called "proton"

is just trying to nail an idea onto a board with a screw.

That is why I like your essay , you are aware of its relativity...

I hope that you can find some time to read and maybe rate my essay : "The Purpose of Life"

best regards

Wilhelmus de Wilde

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Author Gary D. Simpson replied on Feb. 25, 2017 @ 18:36 GMT

Thank you for reading and commenting upon my essay. My apologies for the mathematics. Since I lack proper credentials, this is the only forum wherein I can present these ideas.

I present several ideas that must all be accepted or all rejected as a group. That is why I make that statement in the Conclusions. Since my calculation is based upon the 5-D model and the calculation is in agreement with the measured value, the calculation is circumstantial evidence that supports a 5-D model. But the Mp/Me ratio is a problem because of its high known accuracy. That is what necessitates Equation 2 as a means of explaining the observed difference from 6*pi^5. It is also entirely possible that the Mp/Me ratio and my calculation are both simply coincidences ... but I have trouble believing nature would be so.

I have read your essay but not yet scored it or commented upon it. I will need some more time but will do so.

Best regards and Good Luck,

Gary Simpson

Wilhelmus de Wilde wrote on Feb. 26, 2017 @ 16:56 GMT
Thank you Gary for your instructive comments on my essay "The Purpose of Life".

Next time I will try to avoid acronyms (perhaps I like to mimic formula's...)...

Indeed there is more to explain but I could not realise it in nine pages. I have solutions for time-travel, black hole information loss and more that you can find in my article published in The Journal of Consciousness Exploration and Research Total Consciousness in Total Simultaneity

Your "Complex Plane" is indeed almost the same idea as Total Simultaneity, maybe I need your advise to mathematically explain my idea of Total Simulaneity which is both a not in our reality existing singularity without time and space as well as it contains ALL Eternal NOW Moments and contents also a "field" named Total Consciousness.

best regards


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Jonathan J. Dickau wrote on Feb. 28, 2017 @ 04:35 GMT
In regards to questions about non-integer dimensionality..

Hi Gary,

Non-integer dimension arises in causal structure theories of quantum gravity, which is referred to simply as a running D - compared to the case where D = n, where n = 0,1,2,3... Of course; this gives space a fractional dimension, and makes it a fractal - along the way - which is simply how surface roughness evolves into a new (whole) dimension or extent. So briefly; fractional dimension arises because of folding of space in the microscale. As Lawrence alludes, the Hausdorff dimension evolves.

One can also think of this as relating to emergent spacetime, because if the observed properties of space and time, this means that intermediate values are accessible between the onset of geometrogenesis and the current era. One can also see this as connected with a different root dimension for the microscale and macroscale, as with Rainbow Gravity (which was explored by Magueijo, Amelino-Camelia, and others). If space is 2-d at the Planck scale and 3-d at the common scale; what is it in between?

Lastly; this is a broad feature of what is called bi-metric gravity. There are many formulations in that family. There's too much to say simply, but as the name implies there are two co-existent descriptions of space - to deal with the weak-field and strong-field, low-energy and high-energy regime, or common scale and microscale, and so on.

All the Best,


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Colin Walker wrote on Mar. 3, 2017 @ 20:20 GMT
Hi Gary. I think I see now what you are getting at with your equation 3.1 - it is a combination of two quaternions but having complex coefficients instead of real in the real quaternion basis. I recall that what you would get is a biquaternion. It turns out that a biquaternion 2x2 complex matrix is not a quaternion, and any 2x2 complex matrix can be expressed in biquaternion form. Like an octonion, a biquaternion has 8 independent real variables. A quaternion is a biquaternion with a specific combination of symmetries that allows only 4 independent real variables. I cannot see getting an octonion unless your 'complex i' was the octonion 'l' in the sequence of seven octonion imaginaries i,j,k,l,m,n,o - and then it would have to be checked for (or arranged to have) the appropriate symmetry.

Best to you,


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Author Gary D. Simpson replied on Mar. 3, 2017 @ 23:36 GMT

There are 8 ordered terms ... that makes an octonion:-)

What this essay presents is a 5-D subset of the octonions ... I state that in the essay. The 5'th dimension for me is the complex i with the scalar an vector terms being from the quaternion. I also specify what the various scalar coefficients must be in order to satisfy the geometry that I present.

Dr. Crowell also referred to what I have presented as a bi-quaternion although the term is new to me. It does make sense though since I add a real quaternion to a complex quaternion.

Best Regards and Good Luck,

Gary Simpson

PS - I have scored your essay.

Author Gary D. Simpson replied on Mar. 4, 2017 @ 00:14 GMT

One other thing ... if I describe the physical universe with 5 terms from the full set of octonions, then there are still 3 terms left to describe the physicality of you an me ... the observers. This is why I titled my essay "Five Part Harmony" and it is why I refer to the structure as a pentuple.

Best Regards and Good Luck,

Gary Simpson

Philip Gibbs wrote on Mar. 5, 2017 @ 08:33 GMT
Hello Gary, You say that the mass ratio depends on absolute velocity and suggest a satellite test. Would the ratio also vary as the Earth changes direction round the sun? If the absolute velocity is the sum of the sun's motion and the Earth's motion round the sun, would it change with the seasons?

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Author Gary D. Simpson replied on Mar. 5, 2017 @ 09:14 GMT

Many thanks for reading my essay and commenting.

The short answer to your question is that I think the mass ratio is dependent upon the Earth's motion around the Sun.

Here's the long answer. The Earth's motion is the sum of its motion around the Sun, the Sun's motion in our galaxy, and our galaxy's motion through space. There might even be another structure to consider. I think that the major velocity component is the motion of our galaxy through space. I think the motion of the galaxy is parallel to the axis of the galaxy. The Earth's motion around the Sun is in the ecliptic which is perpendicular to the axis of the Sun. Therefore, if the axis of our Sun is parallel to the axis of the galaxy, then the Earth's motion around the Sun will not affect the mass ratio. On the other hand, if the Sun's axis is not parallel to the axis of the galaxy, then the Earth's motion around the Sun should be a factor. My guess is that stars that are clos to the galactic core have rotational axes that are parallel with the galaxy's axis of rotation. Stars - like ours - that are out on an arm could have any orientation. They might even be more stable if their axes are in their plane of motion through the galaxy.

Keep in mind that Equation 2 was necessitated to explain the difference between the observed Mp/Me ratio and 6*pi^5. There could be alternative hypotheses. At least this thinking is TESTABLE:-)

Also, thanks again for I continue to use your website.

Best Regards and Good Luck,

Gary Simpson

Jonathan J. Dickau replied on Mar. 7, 2017 @ 19:37 GMT
On this topic..

There was some research, a few years back, showing that nuclear binding and half-life varied with the solar cycle and could potentially be used to predict sunspots. I think Ephraim Fischbach was part of the research team. I hope this helps.



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Dizhechko Boris Semyonovich wrote on Mar. 6, 2017 @ 17:09 GMT
Yes, the quaternion is fascinating, a lot of promises and makes you wander before you come to a path leading to the goal. This goal - the recognition of your merits by most mathematicians and getting your name in the books, at least for the fact that you are using the quaternions of his own invention, calculated the size of the proton.

I wish you success on this path, at least in this contest!

Because you went deep and put a lot of work to that branch of mathematics which looks to be a dead end, I'll put your essay the highest score.

New Cartesian Physic, which is based on the equivalence of space-matter, needs the theory of complex numbers not only on the plane, but mostly in space. After all, Descartes himself believed that the basis of physics must lie geometry.

From New Cartesian Physic great potential in understanding the world. To show this potential in his essay I gave materialistic explanations of the paranormal and supernatural. Probably, I made a mistake that has bound New Cartesian physics with the paranormal and supernatural, because it does not attract the attention of others. Visit my essay and you will find something in it about New Cartesian Physic. Note the drawing of geometric relationships in the atom.

Sincerely, Boris Dizhechko. (Note that I did not know English and use online translator)

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Author Gary D. Simpson replied on Mar. 6, 2017 @ 19:24 GMT

Many thanks for reading and commenting upon my essay. I lack proper academic credentials. Therefore, I have no expectation of recognition. I simply hope to make a few people ask themselves a few questions ... is my calculation also a coincidence? Is there an alternative interpretation to the wave function other than the Copenhagen Convention?

I'm not so sure that I would dismiss quaternions and octonions as a dead end. The Maxwell Equations can be formulated using the quaternions. Therefore, the paradoxes that occur with Relativity might simply be the result of Einstein attempting to preserve Euclidian geometry when it is more appropriate to use a Geometric Algebra.

I will read and score your essay soon.

Best Regards and Good Luck,

Gary Simpson

Karoly Kehrer wrote on Mar. 7, 2017 @ 16:26 GMT
Hello Gary


Karoly (It took 3 - 4 hours and finally after keeping the site open for an hour your I seen by accident it shown up. Is this site overloaded???

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Author Gary D. Simpson replied on Mar. 7, 2017 @ 16:36 GMT
Thanks Karoly,

I don't think the site is overloaded ... it probably does not get a lot of traffic compared to most sites. I wonder if there is a problem with your hardware or connection. If I have a problem accessing something, I usually close all my windows and re-open them. Sometimes I also reboot my computer and/or my internet hub.

Best Regards and Good Luck,

Gary Simpson

Willy K wrote on Mar. 14, 2017 @ 06:27 GMT
Dear Simpson

I loved the way you closed out the essay, “It is certainly possible – perhaps even likely - that these conclusions are false. However, if that is the case then the reader is left to explain the proton diameter calculation.” I would have used a similar closing for my essay as well, had I been able to come up with an exacting calculation similar to what you have done.

Having only recently discovered octonions via Yanofsky’s and Dickau’s essays, I cannot pretend to have suddenly gained the capacity to validate the calculations you have presented in the essay. But I can recognize simplicity and elegance when I see it, and your work has it. While I can appreciate Dickau's patience with regard to voting (he has more experience with this contest), I would much rather vote right now for an essay that I like and you are certainly one of them. All the best!

Warm Regards, Willy

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Author Gary D. Simpson replied on Mar. 14, 2017 @ 13:47 GMT

Many thanks for reading and commenting upon my essay.

Physics is very demanding. You must get the right answer and you must do so for the right reason. You are correct to focus upon the proton diameter calculation. As I see it, either a coincidence has led me to a calculation that is also a coincidence, or the Mp/Me ratio and my calculation are not coincidences. I can believe a single coincidence, but two coincidences that are linked seems too much to believe.

You have begun a new journey. Quaternions, Octonions, and the Division Algebras are very interesting indeed. I have been studying them for a few years now and have barely begun to understand and appreciate them. There is also a related field known as Geometric Algebra. David Hestenes has written several books in this field.

Essentially, the question that I ask is "What does it mean to move in 5-D space if time is not a dimension?". I'm not certain that Euclid type thinking is applicable. Much of the controversy of Einstein's work was because he imposed Euclid's Geometry onto Relativity ... but what if Euclid is just a subset of the geometry of space? I think this is the path that we now need to follow.

I will read and comment upon your essay soon.

Best Regards and Good Luck,

Gary Simpson

peter cameron wrote on Mar. 18, 2017 @ 21:52 GMT

Suggest you take a look at the work of David Hestenes on geometric Clifford algebra, if you haven't already. Appears to me much of what you are doing is a subset of that algebra, to pick up on the rest of it might be very helpful to you.

Several of the contributions to our contest are in that geometric language, including those of Michael Manthey Manthey FQXi essay and Michaele Suisse Suisse FQXi essay.


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Author Gary D. Simpson replied on Mar. 18, 2017 @ 22:39 GMT

Thanks for the advice. I have a text by Hestenes but I have not read it yet.

Best Regards,

Gary Simpson

Author Gary D. Simpson replied on Mar. 18, 2017 @ 23:39 GMT

BTW, do you have an essay in the contest? I have looked through the entries and do not see you listed. I always try to read and comment upon everyone's essay who comments upon mine.

Best Regards,

Gary Simpson

peter cameron replied on Mar. 20, 2017 @ 17:00 GMT

Yes, I have an essay in the contest. Michaele Suisse is my co-author, it is submitted in her name.

"Aims and Intention from Mindful Mathematics: The Encompassing Physicality of Geometric Clifford Algebra"

Surprised by your comment of earlier today on my thread, that even tho I steered you in the right direction, here it is two days later you still don't get it.

The only thing i can conclude from that comment is that you didn't understand what you read. The algebra of the physicist's S-matrix of figure 3 of our essay is GA. The wavefunction is comprised of the fundamental geometric objects of the geometric Clifford algebra of 3D space - one scalar, three vectors, three bivectors, and one trivector. This is pure Hestenes from his 1966 book, Spacetime Algebra, very basic stuff.

Similarly, if one models interactions by taking the geometric products of wavefunctions, that yields a 4D Dirac algebra of flat Minkowski spacetime. Again this is basic spacetime algebra.

If you take a look at this paper co-authored with Michaele

"Geometry and Fields: Illuminating the Standard Model from Within"

you might find it a little easier to see the connection between the math and the physics. It is deep, diverse, and incredibly precise - at the ppb limit of experimental accuracy on many calculations.

Haven't rated your essay yet, hope to get to that soon.


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Thomas Howard Ray wrote on Mar. 20, 2017 @ 18:29 GMT

Just a quick note. I promised comprehensive commentary -- working on it. Meanwhile, you might want to look into the work of this author:

All best,


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Akinbo Ojo wrote on Mar. 21, 2017 @ 14:32 GMT
Hi Gary,

The mathematician in you has brewed another lovely essay, but I personally still prefer ‘Calculus 2.0’. I will not claim expertise on quaternions so I will pick my comments.

Yes, I agree time is scalar. This is in keeping with the physics of Galileo and Newton.

But calling the complex i, a dimension may be okay mathematically, but physically it will be a hard sell to me who have become wary of the wares that mathematical physicists peddle these days.

What is the ‘v’ in your equation 2 and how is the value to be determined? You say it represents motion, motion with respect to what?

Then, you give a role to the observer in the determination of mass in your ‘prediction’ segment. This looks like a Special relativity effect and must share in the flaws of SR theory.

Finally, when you say that the Earth is moving at 0.006136c with respect to a true rest frame, is this a new prediction? The value is so much higher than the 370km/s speed relative to CMBR.

Best regards and all the best in the competition,


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Author Gary D. Simpson replied on Mar. 22, 2017 @ 00:25 GMT

Thanks for reading and commenting and hopefully for voting. The one-bombers have been significant in this contest.

The previous essay actually led to this one. I tried to evaluate some problems in mechanics and found the results unsatisfactory. So I revisited the problem and multiplied by Euler's Equation one more time. That gave me this subset of octonions.

Your skepticism regarding the complex i as a dimension is wise. I don't know if it is or is not. The only evidence I can provide is that the proton size calculation is accurate and that is only circumstantial evidence. Perhaps it is simply another coincidence or perhaps it actually means something else.

The v in Equation 2 is absolute motion with respect to a fixed background. It is a variation on the abandoned aether concept. However, relative motion is more complex that what was previously considered and the motion of A with respect to B is not the same as B with respect to A. They are more like inverses ... by that I mean they are more akin to division ... 1 divided by 2 vs 2 divided by 1 as an example. So this puts SR into the hypothesis but in a different form than what we usually have. Maybe it is better, maybe it is worse. But it does contain a fixed reference frame.

You are correct regarding the value of the velocity. Colin Walker also pointed this out. The value I present is almost exactly 5 times the CMBR value. My best guess here is something that I remember from statistics. There is a certain type of average where you divide a value by its degrees of freedom. In this case, that would be 5. I have not found the statistical reference that I am looking for regarding this yet but I am certain I have read it before. That will provide another clue as to the correct form of the math.

Best Regards and Good Luck,

Gary Simpson

Vladimir Nikolaevich Fedorov wrote on Mar. 22, 2017 @ 13:10 GMT
Dear Gary,

With great interest I read your essay, which of course is worthy of high praise.

You are absolutely right that «The q0 term must be added to the four-vector to produce an object or structure … with a scalar term. The author will speculate that the scalar term is related to the vacuum energy/ The author will also speculate that it is possible for the scalar term to be a function of time.»

I believe that it is provisions of yours that are the key to the answer about the self-organization of matter and to the question set by this contest.

Your essay allowed to consider us like-minded people.

You might also like reading my essay .

I wish you success in the contest.

Kind regards,


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Author Gary D. Simpson replied on Mar. 23, 2017 @ 00:39 GMT

Many thanks for reading and commenting upon my essay. I have read through your essay but wish to study it a bit more before commenting. I will do so soon.

Yes, adding the scalar term to a 4-vector is really the mathematical essence of what I discuss. It produces a structure that fits the Kaluza-Klein Theory which combined electro-magnetism with gravity. Unfortunately, Einstein himself eventually concluded that Kaluza-Klein was not compatible with General Relativity.

Best Regards and Good Luck,

Gary Simpson

Peter Jackson wrote on Mar. 22, 2017 @ 13:49 GMT

Thanks for the score & post which I've responded to. I've read yours again. My maths isn't adequate (I prefer dynamic geometry) but I do think I like scalar values for time, and I'm sure there's really no co-incidence but an important physical truth. May it be hiding in the windings of toroid pairs (3D helices when in relative motion) and the Gottfreid-Jackson angle?


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Peter Jackson replied on Mar. 22, 2017 @ 14:09 GMT
...oops, wrong button!

I'm sure I once made some sense of the above but now struggle a bit, can you?

What I also struggle with is the 'absolute motion' you invoke. As an astronomer we've rather found better sense from the most recent data (including the Planck survey) in 'hierarchical' relative motions. Indeed George Smoot first derived that from WMAP for his Nobel.

To pick just one as a start; Our galaxy is in motion within the local group (itself in motion in the cluster and that revolving around the filament), our local arm is in motion in the galactic rest frame and the sun is moving rapidly through it (witness the strong 'bow shocks' we find at each transition and the 'anomalies' NASA finds & allows for from Pioneer etc) and onwards both up and down. Background frames will then be 'local' not absolute, so the transform depends on how many stages 'up' the emitter was. Is that a problem? or helpful?

On cosmic redshift, your prediction should be part correct, but with intervening frames to account for. a derivation without needing accelerating expansion and with helical expansion is here; Redshift Video.

I hope you'll read my essay if you haven't. That also contains an important finding but what it really need is a mathematician with your knowledge to collaborate. Could you enumerate the dynamics identified? (3D extensions of pythagorus theorem).



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Author Gary D. Simpson replied on Mar. 23, 2017 @ 01:02 GMT

Thanks for having a read and the comments. FYI, I read and commented upon your essay fairly early although I did not score it until yesterday.

Regarding the copy paste in the first of your posts, I suppose it could be true ut my thinking regarding quaternions is not really based upon toroids or similar structures. Rather, a quaternion is exactly what Hamilton said it was ... the ratio between two non-linear vectors. That can then be applied in any number of ways with your description being one.

Regarding absolute velocities. Equation 2 is a testable hypothesis. Its truth or lack of truth requires experimentation. It is possible that the 6*pi^5 value is coincidence. If it is not a coincidence, then there must be a way to get from 6*pi^5 to the observed Mp/Me ratio. Motion was the easiest thing that came to my mind and since the equations need to contain the Lorentz Transform, Equation 2 seemed like a good bet. I'm sure there are other corrections possible.

Your present essay uses quaternions quite well although perhaps you don't realize it. All of the vector rotations you present can be represented by quaternions ... i.e., what do I multiply vector a by to get vector b (Q = b/a).

Regarding 3-D extensions of the Pythagoras Theorem, I do not understand your question. You can build a theorem with as many dimensions as desired, you simply do it two terms at a time. Going beyond three dimensions simply does not make physical sense. Whether or not ct can serve as a 4'th term, it is not something you can measure with a yardstick. That means either c or t can be whatever you want it to be to make the four square sum be correct.

Best Regards and Good Luck,

Gary Simpson

Peter Martin Punin wrote on Mar. 22, 2017 @ 20:25 GMT
Dear Gary,

It is a pleasure for me to meet you again on fqxi. On my own forum, I posted a reply to your kind comment.

I read your essay for the first time, needing ca. two days before posting a reply.

Best regards,

See you soon


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Thomas Howard Ray wrote on Mar. 25, 2017 @ 14:35 GMT
Hi Gary,

When Kaluza-Klein theory came up, Einstein acknowledged that a five dimensional theory was a promising direction to field unity. But he cautioned that extra dimensions (beyond the 4 of Minkowski space-time) should be entertained “only when there are good physical reasons to do so.”

Lisa Randall’s “warped spacetime” has five spacetime dimensions (the fifth may be...

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