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

5 days later

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

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

    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

      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

      Declan,

      See my reply to Dr. Klingman below.

      Best Regards,

      Gary Simpson

      Hi Gary,

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

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

      The Fundamental Universe

      Jeff Yee

        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

        Hi Jeff,

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

        Best Regards and Good Luck,

        Gary Simpson

        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 LCAttachment #1: 1_mc-escher-waterfall.jpg

        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!

          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

            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

            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

            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

            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

              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

              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

              Ajay,

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

              Best Regards,

              Gary Simpson