George,

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,

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

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

Wilhelmus

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,

Jonathan

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,

Colin

    Colin,

    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.

    Colin,

    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

    Phillip,

    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 viXra.org. I continue to use your website.

    Best Regards and Good Luck,

    Gary Simpson

    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)

      Boris,

      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

      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

      6 days later

      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

        Willy,

        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

        Gary,

        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.

        Pete