CATEGORY:
FQXi Essay Contest - Spring, 2017
[back]
TOPIC:
Four Squares by Gary D. Simpson
[refresh]
Login or
create account to post reply or comment.
Author Gary D. Simpson wrote on Dec. 20, 2017 @ 21:39 GMT
Essay AbstractThis 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 BioI 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
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
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
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.
report post as inappropriate
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
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
report post as inappropriate
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
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?
report post as inappropriate
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
John R. Cox replied on Dec. 23, 2017 @ 16:45 GMT
Anthony,
If I may. The question of an absolute velocity is commonly attached to a Cartesian space in Newtonian time, but that is not what Gary is saying. In any discrete energy field, the spacetime of that field will be 4 dimensional as conventionally held for both SR and GR globally. The Simpson 5th dimension is global which self-organizes to a local 4D spacetime of a particle (or photon). And in that 4D locality the direction vector relates to points within that isolate field, not necessarily to a point in the global 5D universe. Lorentz Invariance still holds across non-locality in the global measurement space, but the absolute velocity of a mass particle can theoretically be ascertained as a vector within the 4D field itself. jrc
report post as inappropriate
Author Gary D. Simpson replied on Dec. 26, 2017 @ 22:18 GMT
John,
Many thanks. It is nice to have someone who understands the Math and Physics to act as an interpreter:-)
Best Regards,
Gary Simpson
John R. Cox replied on Dec. 27, 2017 @ 02:05 GMT
sure thing, g.
I was pleased to see that you got a community member to provide a rating. The essay Topic(s) do seem to get stretched every year, and what you have submitted was at least recognized as a fundamentally organized effort. Not in the mainstream, but not without foundation in both Relativistic and the search for Quantum unification. Take hope in Topology coming into general acceptance as a practical measure to unify mathematics. I think John Klauder might find some agreement in conventional Cartesian space (the shoebox full of cubes) being analytically functional, but not being absolutely essential as the classical measurement space. Good Luck and Happy Holidays, jrc
report post as inappropriate
hide replies
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
report post as inappropriate
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
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
report post as inappropriate
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?
report post as inappropriate
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
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
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
report post as inappropriate
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
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
report post as inappropriate
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
report post as inappropriate
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
report post as inappropriate
Author Gary D. Simpson replied on Jan. 12, 2018 @ 14:38 GMT
Declan,
See my reply to Dr. Klingman below.
Best Regards,
Gary Simpson
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
report post as inappropriate
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
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
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
report post as inappropriate
Author Gary D. Simpson replied on Jan. 19, 2018 @ 14:44 GMT
Eckard,
Thanks for reading and commenting.
Regarding integration limits, mathematically they can be anything although they must be chosen carefully if any Physics is to result.
Regarding Declan's comment, the quotation he provided above was his own. I did not offer an explanation. However, I do think that the complex i is perpendicular to the unit vectors and that it anti-commutes with the unit vectors.
Why is there a complex i in Physics? I don't presume to know but I am willing to speculate. A quaternion can first be viewed as an ordered set of 4 things. These could be anything ... apples, oranges, pears, and lemons for example. Hamilton added a set of axioms to make the ordered set applicable to geometry. These axioms were anti-commutation, ij=k, jk=i, ki=j, and ijk=-1.
Adding the complex i to the system allows the user to have an ordered set of 8 items. And doing so as I have presented makes the ordered set a Group. My gut feeling here is that this allows two distinct entities to simultaneously occupy the same space. The problem will be correctly identifying the rotations and symmetries of the Group.
Best Regards and Good Luck,
Gary Simpson
hide replies
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 UniverseJeff Yee
report post as inappropriate
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
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
report post as inappropriate
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
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 between our work is that we are both evaluating exponents.
P=1 and iEt/H=1 and we have to look inside the 1’s. The equation E=e0*exp(N) that gives the E’s is easily derived from the Schrodinger equation but I have never found any use of the equation in physics. [Barbee, Gene H., Schrodinger Fundamentals for Mesons and Baryons, October 2017, vixra:1710.0306v1].
Below is an excerpt from the proton model described in the reference (quad 2 out of 5 quads). The values of E that satisfy P=1 are 13.797, 5.076, 101.947 and 0.687 MeV. For example 5.076 MeV comes from the equation E-2.02e-5*exp(12.432).
With these 4 E’s, P=1=psi*psi*psi*psi=exp(13.797it/H)*exp(5.076it/H)*exp(-101.
947it/H)*exp(-0.687it/H). The imaginary numbers divide out and each Et/H=1. I labelled the E’s mass, kinetic energy, strong field, and grav field and they describe what I call a quantum circle. There is an equal amount of positive energy and negative energy in the proton model, for example 13.8+88.15=101.947+ 0.687 MeV.
I tried to relate this to your work but you are looking for dimensions and I am looking for information that describes a quantum circle. The circle is also a wave and can be described by your sine and cosine functions. The circle also represents simple properties, spin, parity, charge and fields as indicated in my essay. In my work, the exponents are also probabilities and the dimensions are formed when the surface of a three dimensional sphere is divided into exp(180) individual surfaces with a neutron with kinetic energy on each surface. There is a good discussion of geometry in Principles of Cosmology by P.J.E. Pebbles. I think the use of a surface is justified by dx^2+dy^2+dz^2-(cdt)^2=0. Again, there is a parallel with your four square usage and I like your two time dimensions. Time repeats and counts forward but the time ratio we called gamma gives nature freedom to move (ke=m/g-m).
Thanks for introducing me to quaternions. My question to you is “are we looking at the same thing from a different perspective?” Engineers unite!
view post as summary
report post as inappropriate
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
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
report post as inappropriate
Author Gary D. Simpson replied on Jan. 18, 2018 @ 23:13 GMT
Thanks Christi!
Best Regards,
Gary Simpson
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
report post as inappropriate
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
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
report post as inappropriate
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
Ajay Pokhrel replied on Jan. 22, 2018 @ 12:37 GMT
Gary,
Yes, I missed that 'not' word. Apologies
Ajay
report post as inappropriate
hide replies
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
report post as inappropriate
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
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
report post as inappropriate
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...
view entire post
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 spend some of the valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance
Some of the Main foundational points of Dynamic Universe Model :-No Isotropy
-No Homogeneity
-No Space-time continuum
-Non-uniform density of matter, universe is lumpy
-No singularities
-No collisions between bodies
-No blackholes
-No warm holes
-No Bigbang
-No repulsion between distant Galaxies
-Non-empty Universe
-No imaginary or negative time axis
-No imaginary X, Y, Z axes
-No differential and Integral Equations mathematically
-No General Relativity and Model does not reduce to GR on any condition
-No Creation of matter like Bigbang or steady-state models
-No many mini Bigbangs
-No Missing Mass / Dark matter
-No Dark energy
-No Bigbang generated CMB detected
-No Multi-verses
Here:
-Accelerating Expanding universe with 33% Blue shifted Galaxies
-Newton’s Gravitation law works everywhere in the same way
-All bodies dynamically moving
-All bodies move in dynamic Equilibrium
-Closed universe model no light or bodies will go away from universe
-Single Universe no baby universes
-Time is linear as observed on earth, moving forward only
-Independent x,y,z coordinate axes and Time axis no interdependencies between axes..
-UGF (Universal Gravitational Force) calculated on every point-mass
-Tensors (Linear) used for giving UNIQUE solutions for each time step
-Uses everyday physics as achievable by engineering
-21000 linear equations are used in an Excel sheet
-Computerized calculations uses 16 decimal digit accuracy
-Data mining and data warehousing techniques are used for data extraction from large amounts of data.
- Many predictions of Dynamic Universe Model came true….Have a look at
http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.h
tml
I request you to please have a look at my essay also, and give some of your esteemed criticism for your information……..
Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.
In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from “http://vaksdynamicuniversemodel.blogspot.in/ ”
I request you to please post your reply in my essay also, so that I can get an intimation that you repliedBest
=snp
view post as summary
report post as inappropriate
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
report post as inappropriate
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
report post as inappropriate
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
report post as inappropriate
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
Jonathan J. Dickau replied on Jan. 24, 2018 @ 04:11 GMT
I am now...
Happy to elevate your score a bit.
Good luck!
Jonathan
report post as inappropriate
Jonathan J. Dickau replied on Jan. 24, 2018 @ 05:54 GMT
Oh, and I wanted to mention..
One more theoretical structure for which a 5-d --> 4-d scenario pops out is my own construction inspired by the Mandelbrot Set - presented in my essay. The point of eversion in M is at (-0.75, 0i), where the cardioid and circular area meet and kiss. This location I identify with Cartan's rolling ball analogy of G2 symmetries, where G2 is also the automorphism group of the octonions.
In any case; the set up for Mandelbrot Gravity is to show that octonionic inflation tops out in a 5-d volume, and that G2 symmetries then kick in to carry us into the current era in a 4-d spacetime - which is essentially a quaternionic bubble. I see the Penrose singularity theorem as applicable to the 5-d to 4-d transition, as well, so the Weyl tensor approaches infinity, just beforehand.
This is arguably more similar to Poplawski's work, because he is proposing torsion as an essential component of our current background spacetime. I'm not sure yet where I stand, on that notion. But the braneworld ideas of Dvali, Gabadadze, and Poratti, are no more appealing. I guess I like octonionic inflation best, but it is not well-accepted nor well-understood.
All the Best,
Jonathan
report post as inappropriate
hide replies
Joe Fisher wrote on Jan. 27, 2018 @ 21:58 GMT
Dear Gary D. Simpson,
Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.
All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe consists only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.
Only the truth can set you free.
Joe Fisher, Realist
post approved
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
report post as inappropriate
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
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
report post as inappropriate
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
report post as inappropriate
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
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...
view entire post
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 lengths."
this seems to me to be quite extraordinary, foundational so to speak. Something very simple and basic about 'quantization' in 4D spacetime that i've not seen before. thank you.
then you add a scaling factor that permits you to look at the Planck length. Nicely done. You've brought a form of 4D spacetime quantization to gravity. Starting with the assumption that your present 7.0 is fair, this is a good nudge upwards imo.
excellent beginning of the next section, most appropriate here to ask “What is waving?”. Change of dimension that you mention follows from outer product, likely what you had in mind in taking us by the hand and leading us along. Good to keep in mind tho that in the algebra it comes not from bending things but rather from interactions of the fundamental geometric objects of the algebra.
and then you drift off into curved space. Big mistake imo. The Lasenby Cambridge group showed back in the 90s that gauge theory gravity of geometric Clifford algeba is flat space equivalent of curved space GR (and the only route to quantum gravity imo).
you recover nicely by going to vacuum and electron wavefunction, then spin out by playing the numerology game. Need to stay conscious of the fact that you're playing in an elegant web, where possibilities such as you suggest in attempting to jump to the proton mass have what are 'hidden' connections in the things we do not yet understand. Near as i can tell the goal is to get things in some sort of matrix format such as you show for the octonion, then then try to drop all the coefficents and see what numerical values remain. That's the raw web of interconnections, without pi, spin-related factors of 1/2 and 2, sqrt of 2 and 3 in projection factors,... That's the most basic part. I think if you try to do that in your model and are successful you will find you can take all of them to unity except one of them has to be the number 4. That comes from the fact that dimensionality of the 4D pseudoscalar of the Dirac algebra generated by the product of two 3D Pauli algebra wavefunction.
point is forget the numerology, that's what earned us both a 1 last year and i hope the point is well taken. We both paid the price for it.
i like what you're doing, seems like you have good intuition, am going to push your rating up a bit. Suggest you look at adding the appropriate geometric properties - scalar, vector, bivector, trivector - to the generators of your 'octonion' matrix. It may be that you will then start to see the full power of what you're doing, and hopefully carry it deeper into the math. As you point out, my math sucks. No argument there.
view post as summary
report post as inappropriate
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
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.
report post as inappropriate
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...
view entire post
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 with proper credentials might adopt the idea as their own. So, there is no downside to me speculating and there might be some upside:-)
Regarding your QM and GR question, here is my reasoning. A difference between the two models is the manner in which they treat time. GR allows time to vary but QM treats it as essentially absolute. So, instead of proposing how to modify one or the other of the theories, I propose that they both must be modified. Somehow, a TOE must allow for a scalar time to be acted upon to produce two time-like variables that can then be inserted into QM and GR respectively. Since there are three physical dimensions and two "time-like" dimensions, it follows that a 5-D model should provide enough room to fit them both in.
I propose to achieve this by using a scalar value for time that acts as a "master clock". This would then act upon Euler's Equation to produce the two "time-like" values. The result would then act upon a spatial quaternion. This is essentially what I present as the generic wave-function.
I have asked Dr. Klingman about this and he does not think it is correct. So, it is most likely incorrect. I will simply state that the general idea can produce the relativistic energy equation. See the 2'nd comment in this forum and my reply to Dr. Klingman above. So perhaps there is a shred of truth in here even if it is more general wrong.
Regarding unification, I would see GR using the cosine term. I have previously hypothesized that it is the Lorentz Transform. That would then force the sine term to be used by QM and it would suggest that spin is somehow the result of motion (v/c).
I'm waiting for the one-bombing to start. It has been pretty slow so far. I am very surprised by some of the scores I have received so far.
Best Regards and Good Luck,
Gary Simpson
view post as summary
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.
report post as inappropriate
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
report post as inappropriate
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
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....
view entire post
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. I have to agree totally with this statement since that is what I found as well. My essay
What is fundamental is the area of the imaginary unit" is also based on the four square theorem, though that comes out as a result not an assumption. "The basic idea in the essay is that a number is a square area (not lengths squared), and from this square area we can form four other areas off each side of the number's area. The basic idea is that any number can be viewed as the sum of four squares." And that the imaginary unit can be thought as +i and -i obtained from i
2 the area of the imaginary unit. And yes it uses the fact that "Lagrange's Theorem was composed of four squares and the length of a quaternion is also four squares." In the first post is an attachment
FAQ which has all the details. Very similar ideas. The essay goes into how to form a dual of the purely imaginary unit. Then using the folk-wisdom which in the essay is called "the basic multiplication table for the definitional imaginary units"
1. -ve times -ve gives +ve
2. +ve times +ve gives +ve
3. -ve times +ve gives -ve
4. +ve times -ve gives -ve,
We can show how the quantised (i.e. "the basic multiplication table for definitional units") can produce a continuum as in space-time. Or as you say
"Such coincidences mEssentially, things can appear to be quantized (i.e., integer valued) without actually being quantized because both methods reduce to four squares."Reading your insightful essay has clarified many questions that I had about quaternions and octonians. Your matrix is most ingenious. I think you should read another essay as well
Unified Field which has equations of the form +i
+1 and -i
-1 and goes through a lot of concise ideas towards a "math" unification of reality.
view post as summary
report post as inappropriate
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...
view entire post
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 hypotenuse (f) that traverses through the 4-D space will also have an integer number of the basis lengths."
In Equation 1
Clearly it is the area u
2 that is common to both sides. Since its area's four squares when summed gives a transcendent "number" to both (a
2 + b
2 + c
2 +d
2) and the area f
2. So if we have a 5-d hypotenuse cut from area f
2 within our 4-d space-time based on a well understood four squares geometry with an invariant length "the square root of s
2". How do you avoid this "cut" being s and not the area s
2=(a
2 + b
2 + c
2 +d
2) which what equation 1 is saying. That the total area of (a
2 + b
2 + c
2 +d
2) times the common area u
2 equals the common of area of u
2 times the area f
2. And ever body knows that (the sign of s
2) times (the sign of area u
2) equals (the sign of area u
2) times (the sign of the area f
2).
"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*."
Bi-quaternions are just directed areas, that is, an area with a + and - sign. Clearly the matrix works because we have the invariant area ijk which then allows us to use octonian logic "based on + and - signs" which are attached to the bi-quaternions' areas. Hence in equation 1 the need of the 5-d hypotenuse cut from the area f
2 in our 4-d world which is based on an invariant four squares space-time summation.
Your 5-d area's four squares summation gives us the length of 4-d hypotenuse "the invariant length of the square root s
2" not the total invariant area summation. You have 4-d areas with a 5-d hypotenuse length of the four squares for the area f
2. We have literally have a 5-d hypotenuse length within our 4-d space-time that any four square summation must obey. Since the area of u
2 is the one common transcendental number that bridges both sides of Equation 1, while the 5-d hypotenuse is an invariant 4-d length that any summation must have available to have closure for the geometry of the area of f
2.
A number (which is a perfect square) is the summation of four squares. If the area of f
2 is
n square metres d
2ct, then the physical manifestation of that area is a
n invariant unit lengths of dct in our 4-d space-time. Not an area. We have an area f
2 on the right RHS, then on the LHS, equation 1 has a 5-d hypotenuse cut -- length c(metre) -- an invariant length that, by the 4-S theorem and equation 1 - each and every, any and, all - four square invariant summations must obey within our space-time.
Of course your multiplication matrices Eq 5.4 and Eq 5.5, clearly ties "i" with c(metre), clearly via the common area u
2 which is on both sides, where we have units of the summation of transcendental i if we use the 4-S on both sides at once but using your multiplication rules A,B*,A,B* for – and + sign matrix Eq 5.3, which is, after all, a + and - sign summation using "octonian" logic directed bi-quaternion areas i.e. the column [C,D], using Eq 4.1 about a stationary "ijk" invariant the area f
2, using f a length “the square root of the area of f²” to transverse the equal sign, Equation 1 uses a 5-d length, so cannot be associated 1-1 with a summation of four square labelled A,B,C,D thought of as a “a perfect number as an area” . It is the area u² that is, the common “four square summation” i.e. the perfect square, that spans the equal sign using the 4-S theorem on both sides of Equation 1. A number (which is a perfect square) is the summation of four squares). Your Eq 5.3 is a dance using A,B,C,D where A,B,C,D do integral steps on directed areas ALL on the geometry of the area of ijk. More simply the dance is with the directed areas which have a + or – sign, that is, i and * are not moving, i.e. they don't lead! It is --- i and * --- that are stationary and it is Eq 5.3 that moves areas that equal + or – throughout a basic multiplication table page 6, clearly Eq 5.3 only gives the square root of s², a length not an area for how the multiplication table works in your matrices Eq 5.4 and Eq 5.5.
The full 4-S multiplication "of the areas on both sides of Equation 1" is:-
(the sign of the area (a² + b² + c²+d²)) times (the sign of the area u² on the LHS) = (the sign of the area u² on the RHS) times (the sign of the area f²).
You will find Eq 5.3 octonian area + and – logic uses only the “square roots for the area u²” on the LHS for the bi-quaternions areas plus and minus signs attachment. That is, it is the common area of the transcendent “number” (a summation of four squares) which transverses the equal sign in Eq 1. as perfect numbers). Not your A,B*,A,B*,-,+ matrix dance Eq 5.3. which is after all + and - sign summation using "octonian" logic directed bi-quaternion areas i.e. the column [C,D]; clearly uses Eq 4.1 a stationary "ijk" invariant the area f
2.
More simply the area of f
2 is ijk equals -1 and then we take the square root of the area of ijk that is √-1 the imaginary unit. Clearly the full 4-S multiplication table for the "equal sign" invariant + and - unit count across the equal sign for Equation 1 is a transcendent dimensional process with "a unit of the square root of the area u² (see below)”; we will call the invariant unit of the times table a "sec"" for the area of the total summation of the area of the four squares of space-time. Then the 5-d hypotenuse cut would have a pure number a "transcendental" 5-d number c=i and it's "4-d length" of i(sec). The full 4-S sign multiplication times table used for how the LHS and RHS signs of the area u
2 common area behave across the equal sign, are; same signs on the LHS and RHS give +ve while different signs on the RHS and LHS give -ve. Or the appearance of the bridge (common area) across the equal sign is in units of -- +i and -i -- that is, how we cross the equal sign using the area of u
2 on the LHS and using the area of u
2 on the RHS.
You said in my comments
You have some interesting ideas but they are very speculative. Essay contests such as this are a good place to present such ideas:-) yes you are right about that
I don't think you can set i=c or i=h but I do think you can construct something similar to the following:
PSI = exp(omega) = sqrt[1 - (v/c)^2] + (v/c)i
Then for v=c, PSI=i. I looked at your work instead, to see how you bridged with a common 5-d length (of the square root of f
2) the areas on both sides of of the equal sign. Your method mixes lengths with areas across the equal sign. While in the full 4-S, it is the four sums of +i and -i that are the "invariant count" lengths of the area u
2. The hypotenuse of the area geometry of f
2 is an invariant 5-d length "f" which isn't an area on the LHS.
view post as summary
report post as inappropriate
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
Jouko Harri Tiainen replied on Feb. 10, 2018 @ 16:20 GMT
Thanks for clearing it up -- Yours Harri. Will read your essay again.
report post as inappropriate
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
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
report post as inappropriate
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 .....
report post as inappropriate
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
report post as inappropriate
Author Gary D. Simpson replied on Feb. 16, 2018 @ 20:01 GMT
Steve,
Welcome back. Thanks for reading and commenting.
Best Regards,
Gary Simpson
Steve Dufourny replied on Feb. 19, 2018 @ 19:05 GMT
You are welcome Gary, hope you are well, I see that you learn piano and guitar, I play piano and guitar,do you know for the piano the Hannon method, it is very relevant and important for the fingers and hands , the movements pêrmit to improve the technic.At guitar, it is the same, more you play and make the gamuts, more it becomes easy , good work :) take care
report post as inappropriate
hide replies
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.
report post as inappropriate
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
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
report post as inappropriate
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
report post as inappropriate
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
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
report post as inappropriate
Login or
create account to post reply or comment.