Thanks for sharing your ideas Jonathan. Indeed we don t know what we have at these planck scales, what are these foundamental mathematical and physical objects in fact. We cannot affirm. Have we 1D strings and a 1D main cosmological field creating our geonetries, topologies, matters, spacetime? or a geometrodynamics and points or in my model spheres ? we don t know. I see the generality of this universe like this, but it is an assumption of course, I must prove and I work about this. I consider like you know an universal sphere in optimisation evolution or a future sphere and I consider a central main cosmological sphere sending and coding finite series of spheres playing between these two constants, the zero absolute and the planck temperature. It is a gravitational coded aether made of these finite series of spheres, I consider that they have the same number than this finite serie cosmological of spheres. Oddly I have calculated, it approachs the dirac large number. The relevance of these series if we apply a specific serie is that the space disappears, so the space, the vacuum disappears and is made of particles coded. Take a central biggest sphere and after we applay 3 snmaller around this central sphere after 5 smaller around the 5 and we continue with the number that I have explained. You see that my model is totally different than the strings or the geonmetrodynamics because I consider particles coded and this aether, and so I don t consider that all comes from Waves, fields, oscillations to create this physicality. Now and I work about this , the relevance at my humble opinion is to formalise mathematically these finite series of spheres, a photon is too a finite serie , coded , they are for me like a fuel permitting the electromagnetism, the life Death and the fact to observe, they are just series coded so in resume like all. Now if we fornalise these series, finite with the Hamilton Ricci flow, the cold and heat, the heat equations, the lie derivatives, the lie groups, the topological spaces and euclidian spaces and the poincare conjecture, so it becomes relevant for the distribution and the rankings of our particles and the sortings, superimposings, synchronisations, we can consider the volumes,surface(even the hopf fibrations can be considered here on surfaces of spheres) , the mass, the densities, the angles, the deformations of spheres, the motions, rotations, senses of rotations, the moments and many others. I have even reached this quantum gravitation with this reasoning and in considering a cold Dark matter encoded in nuclei, more a fith force due to a serie of quantum BHs farer than our nuclear forces. I have respected the newtonian mechanics for this quantum gravitation. Like at our cosmological scale this matter non baryonic balances our electromagnetism and heat with this cold. This superfluid gravitational aether is maybe the key to explain all our unknowns and this gravitational aether creates the luminiferous one simply.

Best Regard

9 days later

Jonathan. In my essay "Clarification of Physics--", I introduce a new Successful Self Creation system that adds a new level to the epistemic "horizon". It adds another level to the basis of our knowledge and uses it to develop a complete creation process/result. I think you may find it interesting. Also, I would appreciate your comments on the essay. John D Crowell

    I finally got around to reading you paper. It makes some illustrative points. The associated logistic map is a measure of the chaos on the reals and this terminiates at the Feigenbaum number. BTW, I thought the Mandelbrot set had z в†' z^2 + c.

    The relationship between the reals, complex numbers, quaternions and octonions is subtle. With classical mechanics we have strictly reals, but analysis has behind CM has complex numbers. There is then a sort of covering ПЂ:в„‚ в†' в„ќ where the map is a restriction on this "fibration." With quantum mechanics is more straight forwards. The quantum wave is complex valued and the fibration is from nonabelian groups or Clifford groups for gauge fields onto the complex plane or the phase of a wave function ПЂ: в„Ќ в†' в„‚. Where things get a bit strange is with quantum gravitation, where the spacetime is noncommutative. The similar structure would then be ПЂ:O в†' в„Ќ, where now the base manifold is noncommutative and the fibration is nonassociative.

    Cheers LC

      Thanks for reading my essay Lawrence...

      It is true the Mandelbrot formula can be written z 竊' z^2 + c, and whether you are plotting Mandelbrot or Julia Sets is only a matter of how the c is interpreted. If you choose a single value for every point, what you get is the Julia Set for that point. But if your c is z_0 - the location of the point on the complex plane you are evaluating - then what you get is the Mandelbrot Set.

      Thanks for explicating the subtle relationship between the number types. And indeed it appears that there is an explicit relationship to different aspects of Physics. The quantum gravity regime is the most demanding, in this regard. As we approach the Planck scale, or near a BH horizon, we must give consideration to the non-commutative and non-associative components to know what is happening.

      This brings some interesting and exciting Maths into play.

      All the Best,

      Jonathan

      That is meant to be...

      If you choose a single value of c for calculating every point, and only the z in z^2 c changes, you get the Julia Set for that initial value. But if your c is the same as the location you are iterating (z_0), and changes as you go, this gives you the full Mandelbrot Set. The fun thing is that the Mandelbrot Set is therefore a template for and a table of contents or index of all of the Julia Sets, because each Julia Set seed or initial value delineates the behavior and the character of the form found at that point in the Mandelbrot Set.

      More later,

      Jonathan

      If you read my essay I work with the idea of fractals as outlined by Hossenfelder and Palmer. However, I appeal to a more standard definition of incomputability. The Mandelbrot set in some ways is the mother of all fractals, because the fractal dimension of the boundary is 2 - ε in the limit ε --> 0.

      The role of octonions is something I discuss in my essay, but more in the appendix or supplementary material.

      I am disappointed in how this is going. I entered this contest because the topic has been of interest to me for years. Yet, my essay is not doing well, and I should simply not enter these contests any more.

      LC

      Thanks again Lawrence...

      We are lucky to have you among the entrants. Your essay is of high quality and relevance. I have been pressed for time, and have not had the chance to review and rate essays yet. But I greatly enjoyed skimming yours, and I expect my assessment will raise your score.

      I have to wonder why some decent to fine essays have gotten scores of 2 or 3, but your offering is in the good to excellent range and your score should reflect this. The octonions have been occupying my thoughts a lot lately, and I am contemplating how non-associativity figures into everyday tasks like painting or baking, where cycles are completed in sequence.

      More later,

      Jonathan

      Thank you John,

      I'll get there. It sounds interesting. I have some thoughts along the lines you suggest, but I'll have to read your paper in order to find out if we agree.

      More later,

      Jonathan

      Hi, I composed the following on my side:

      Hi Jonathan,

      Actually that was a general note, However thanks for the kind words.

      I too have only read a handful of papers here. I have been pretty hard at work on this. I have not had much time to read papers or communicate with authors. I am now off work, as we are now into "social distancing." So we have to endure this exercise in Camus' existential angst in his novel The Plague with the expectation this will lessen the impact.

      Stay safe and away from crowds. This virus seems to take older people down a lot. I am not worried about myself much, but I have some concerns with others. My mother is over 90 and I concerns there.

      Cheers LC

      7 days later

      Hi to both of you,

      About these octonions, let s imagine that we consider 3 main E8 and that we consider these finite series of coded Spheres in my model sent from the central cosmological 3D sphere, it is there that sonething codes and transforms the E in matters, coded. I consider these series where space disappears and with a specific number, the same than our cosmological finite serie of 3D spheres, oddly it approachs the dirac large number. Now so consider one E8 in replacing the points or strings by these series and consider 3 E8 , E8xE8xE8 , one for the vacuum space, one for the photons and one for the cold dark matter, so we have main codes inside the E8 vacuum and the two others are fuels and permit the emergence of matters , geonetries, topologies and properties , electromagnetism and gravitation.

      Regards

      Jonathan,

      Wanted to let you know that I updated my essay and uploaded it a few minutes ago. Personally I feel that it is greatly improved. I did rate yours on 2/28, giving it the highest rating, feeling it was the best I have read, even now.

      Please check mine out again and see if you share my own prejudice. Such honest, No BS, reviews are needed by all of us.

      Jim Hoover

      Jonathan,

      As usual, well written essay. Personally I still do not have a intuitive feel for the connection to physics, too busy with Octonions to drill down on what others see.

      On that note, recently added canvas to my symbolic algebra suite, of course Mandelbrot set presentation is an excellent choice to try it out. As such revisited the Mandelbrot symbolic algebra code I did for you last essay. I should have been more thoughtful by making it more computationally efficient by simplifying the z^2 calculation for complex-Quaternion-Octonion math. Did so for the canvas exercise to put out 4096x3072 pixels without too much of a wait. Color palette from counts quite critical as I am sure you know. Big fun.

      The optimizations did bring into focus the fact that doing the iterations in Quaternion algebra and Octonion algebra really does not bring out the non-commutation and non-associative properties of theses Algebras since the iteration is trivially commutative and associative for both.

      The 2d pixel location to 4 and 8 dimension mapping for c is interesting, losing the scalar c term yields circles, rotating complex into other dims tears the complex towards the fully circular. Meaningful? Who knows?

      On my symbolic algebra software I put the limited functionality source code up on my essay blog last year, i was severely disappointed nobody but you made a peep about it. I had hoped for collaboration and others banging on it for debugging, no such luck. Do you know of anyone that has tried it? It is a wonderful pedagogical tool as someone interested in Octonion algebra would find, can't really do anything meaningful with paper and pencil.

      Rick

        Hi Jonathan,

        I hope you're well, and well upstate! Good essay again, right on the money and far more readable than many. This is certainly a case for Mandelbrot set recursions, which as you may recall I agree reflects all of nature. I found your take interesting for symmetry breaking and also phase transitions. We also agree on the import of reconciling CSL and QM with gravitation, a matter on which my own essay identifies a coherent hypothesis I hope you'll look at. A few points and questions;

        1. I like your comment; "This makes cosmology a bit like a process of fractional distillation, where the entirety of the condensed matter universe is only the denser portion of reality with fixed attributes, the lowest fraction." In my observational cosmologist role I've found something very similar, and likely cyclic for consistency with the peculiar CMB anisotropies, and with no halting issue.

        2. Do you think the reductions of the general quintic equation, by Euler or the simplest form; x5 -x + p = 0 can lead to any insights? (I've struggled to see the geometry so far).

        3. One thing I have found is that momentum exchange vector addition in a sequence (so complex) of interactions with rotating spheres with random polar axes does produce the increasing uncertainty found, and of Chaos theory. That's due to uncertainty of +/-'curl' at the equator and linear motion at the poles. Can you rationalise that concept?

        4. You rightly define limits, but not quite Dirac's idea of a 'sharp cut off' to maths validity. I've suggested that limit is physically at the lowest (and strongest!) particle 'coupling' scale for EM energy, the electron, or condensed e+/- 'pair'. (My essay identifies useful implications).

        5. You rightly identify confirmation bias which I find far more common than most realise, but do you think that removing it and embedding of doctrine might lead to understanding without ever larger accelerators?

        6. On the same line; Might that also help resolve what you rightly identify as; "the most vexing problem of all, that we know there is something out there - or in there - waiting to be discovered, but to get the answer would require more waiting time than we have".

        7. But when a good candidate for a coherent set of solutions DOES come along, likely NOT from doctrinal thinking, do you think academics be able to recognise it, or even bother to study it!? From my on experience I suggest not. Do you have a view?

        Nice essay Jonathan, certainly down for a top score from me. Well done.

        Very Best. Stay safe.

        Peter

          Hello all...

          I am sorry for being absent here. Participating in this contest has been a rewarding distraction from my life, as I nervously awaited news of my father's progress, as he slowly got better. Dad fought valiantly toward the end. But he lost the battle and passed last Wed. the 25th, just 1 day shy of his 88th birthday.

          And yes; though he tested negative, he showed the cluster of symptoms characteristic of coronavirus. But since he lost his sense of taste at the end of January, and showed symptoms the first week of Feb.; that would make him one of the first US residents infected. He got better somewhat, then a secondary infection apparently took his life.

          But the spread of Covid-19 must have been ongoing before some who later developed the disease showed any symptoms whatever. Pretty scary! I'll come back to comment more here soon, offer some helpful or commemorative items... I have some things to take care of right now though.

          All the Best,

          Jonathan

            Thanks greatly Rick,

            I will honor him with gifts to humanity.

            Jonathan

            To honor another great soul...

            F.D. 'Tony' Smith passed away last December, and he had much to say that inspired me to investigate various areas of Math I would otherwise overlook or fail to understand. Like Steve Dufourny; I came to realize that the properties of spheres in various dimensions have a relation to Physics that is profound. People expect them to be simple but they are not. One might think adding dimensions would allow you to increase the volume or surface area of a sphere unendingly, but this is not what is real. Instead a sphere has maximal (hyper-) volume in 5-d and maximal (hyper-) surface in 8-d. Tony Smith had this to say:

            Sphere, torus, Klein bottle, Möbius strip, etc are all basic geometric concepts.

            The simplest of these is the sphere.

            When people tried to use math to classify spheres of various dimensions, they found out that classification was not at all simple, but had lots of subtleties. For example, a thing that looks like a sphere from combinatorical/piecewise linear point of view can (in some dimensions) have many different smooth/differential structures:

            sphere - number of possible smooth/differential structures

            S1 - 1

            S2 - 1

            S3 - 1 = 2x1 / 2 (but S3 is a subset of any exotic R4# and there are uncountably many exotic R4 spaces)

            S4 - 1

            S5 - 1

            S6 - 1

            S7 - 28 = 8x7 / 2 = 23 x (23 - 1) / 2

            S8 - 2

            S9 - 8

            S10 - 6

            S11 - 992 = 32x31 = 25 x (25 - 1)

            S12 - 1

            S13 - 3

            S14 - 2

            S15 - 16,256 = 128x127 = 27 x (27 - 1)

            S16 - 2

            S17 - 16

            S18 - 16

            As John Baez has noted,

            there are various distinct questions floating around, including:

            A) how many topological manifolds are homotopy-equivalent to the sphere?

            B) how many PL ( = piecewise-linear = combinatorical) manifolds are homeomorphic to the sphere?

            C) how many smooth manifolds are PL equivalent to the sphere?

            For dimension 3, question A is the Poincare conjecture.

            It was proven by Grisha (Grigori) Perelman.

            For dimension 3, questions B and C are solved and the answer is 1.

            For dimension 4, question A is solved (in the 1980s, by Freedman) and the answer is 1.

            For dimension 4, question C is solved and the answer is 1.

            For dimension 4, question B is open (the smooth Poincare conjecture in dimension 4).

            To try to make sense of this look at spheres by their Homotopy Groups PI(k)(Sn),

            which is roughly the number of ways you can wrap a k-sphere around an n-sphere.

            For example, PI(n)(Sn) is the infinite cyclic group Z, and each element of Z corresponds to a winding number of a wrapping of Sn around Sn.

            If you want to look at homotopy groups from the point of view of all spheres of all dimensions, and take the orthogonal group O(n) as the group of rotations/reflections of Sn, then you can say that O(infinity) is the orthogonal group for infinite-dimensional real space which contains as subgroups all orthogonal groups O(n) for all finite n and is effectively the symmetry of all spheres of whatever dimension.

            Then you find that the homotopy relation is periodic with period 8:

            Bott periodicity PI(n+8)(O(infinity)) = PI(n)(O(infinity))

            The orthogonal structure is directly related to Clifford algebra and Clifford algebra also has the periodicity structure Cl(8N) = Cl(8) x ...(N times tensor product)... x Cl(8)

            So, in some sense the geometry of spheres is described by Clifford algebra which is why I use Clifford algebra as the basis for my physics model.

            Once you describe spheres, you can use that to describe

            torus - sphere with a hole

            Klein bottle - sphere with a twist

            ... etc ...

            so

            I think that Clifford algebras are a nice Math way to describe spheres which are the basic structures of the universe.

            Tony

              Hello dear Jonathan,

              Ulla tells you Hello too , I live in Finland now I have immigrated and we live together.

              I thank you a lot to tell my name, I am honored. I work a lot and study a lot of maths to formalise correctly this Theory of Spherisation, an optimisation evolution of the Universal sphere or future Sphere with quantum and cosmological spheres.

              I have found this universal link maybe 13 years ago in ranking a Little bit of all, animals, vegetals, evolution, minerals, maths, Chemistry, physics, biology and you know how I have had this humble Eureka ? in a book of biology I have seen a simple page where we see the Brains of hominids evolving since the lemurians, we see all the Brains on one page and it was for me incredible, I told oh my god, the spherisation is universal, the particles are spheres and the universe too generally and relativelly,

              I was excited I must say, like I am not a professional, I have improved a lot this theory in studying many works here on FQXi or on arxiv and in the good books.

              I knew that I found in all humility something of relevant, and I was surprised that nobody had thought about this universal link. Even our friend Dr Ray Munroe, I miss him here on FQXi, told me , I don t understand steve the spheric man how I have not thought about this Before lol and he told me the Words of Feynman, one day we shall see all the truth and we shall say oh my god how is it possible that we have not seen a thing so simple Before.

              I beleive that maybe the thinkers have too much focus on detaiuls instead to see th generality, the generality is simple. There I work about the publications and I learn a lot of maths like I told you. I beleive that I have reached this quantum gravitation in changing the distances but I must be sure of course.

              I am happy that several relevant thinkers work about the spheres, that permits to evolve in physics and maths, I beleive strongly that the foundamental mathematical and physical objects are 3D spheres and that we can superimpose 3 main E8 in replacing the points or strings by finite primordial series of spheres playing between the zero absolute and the planck temperature, the main origin is particles coded instead of fields or Waves.

              I like the strings but I consider the spheres at this planck scale and particles coded, I don t consider so that all is made of Waves, fields, branes and from a 1D main Cosmic field, I prefer a kind of gravitational aether made of particles. I didn t know this thinker that you honorate, thanks a lot for sharing, I am going to learn more for the properties of these spheres and the rankings.

              It seems very relevant, it could be very well to have John Baez here also on FQXi and Witten and also Connes, we could make an incredible revolutionary work in complementarity, they are good in maths, I study but alone it is not easy all Days lol I must say, it ios a lot of works and unknowns. In all case we can explain our main unknowns I beleive like this quantum gravitation, the dark matter, the dark energy, the consciousness,...

              I thank you still to have spoken about me, it is very nice dear friend, take care, I find also very well that you have honored this thinker, best regards

              4 days later

              I am happy for you and for Ulla...

              A lot of Cosmology relates to the properties of spheres. But a lot of scientists like to put things in square grids. Maybe that is why the pieces don't quite fit.

              All the Best,

              Jonathan