Thank you Domenico...

I am glad my comment was useful or helpful. I look forward to further interaction.

Best,

Jonathan

Thanks for the thoughtful comments David...

I find value in the work of Joy Christian as well. People do not appreciate the subtle features of higher-dimensional spheres. I draw your attention to the work of Nikodem Poplawski, who is using the Einstein-Cartan model to derive results similar to what DGP gravity suggests (see Pourhasan, Afshordi, and Mann). As it turns out; Christian has recently written papers with Fred Diether exploring Einstein-Cartan via Sciama and Kibble, to derive a modified Hehl-Datta equation.

I used to imagine that the only way mainstream folks would consider Joy Christian's work valid would be if he came up with a full-blown Quantum Gravity theory to explain the context. But I think a large class of models with a higher-d origin are in the bulk compatible with Christian's work. I cite a few of the papers relevant to this discussion in my essay. Thanks for reading it.

All the Best,

Jonathan

Thanks for the high praise Christian,

Coming from you, it means a lot, because I know you are demanding about proper technical rigor. I am glad my hard work on this essay was appreciated Sir.

All the Best,

Jonathan

Glad to hear from you again, Jonathan,

"r=1 also defines a unit sphere"

given that there exists no universal reference of scale for either time or space, that deceptively simple statement is key to rationalizing a proportional base volume in theoretical criteria that can be then scaled to the standards of empirical measure chosen (I like cgs). best as always jrc

Thanks so much John!

Yes it's true. I have been using r = 1 in my lectures as a talking point for more than 20 years, and it never grows old. People's eyes light up when they see that such a simple mathematical statement can lead to a host of generalizations.

All the Best,

Jonathan

Just a thought...

It is provable that the Misiurewicz point M3,1 is a repeater, so long as you use the exact algebraic value derived in the endnotes of my essay. However; no numerical value, even the 600 digit result I provide, will remain stable or repeat indefinitely. This is because M3,1 is a repelling point, which means that any point in its local neighborhood displaced from that spot is divergent. So it is not possible to enter a numerical value in that neighborhood that will repeat indefinitely.

Therefore we have a known property that is not provable or decidable through numerical calculation, because its exact value is a transcendental number. I just thought this fact was curious enough to be worthy of mention.

All the Best,

Jonathan

Erratum:

I think that my statement that M3,1 is a 0th order branching point is wrong, and it should be considered a 1st order point instead, because it stands to reason we would designate terminal Misiurewicz points to be of 0-order in branching, while M3,1 has one outgoing branch, which makes it 1st order instead. I was not able to find this information in the literature during a quick review, but I will research this further.

In the meanwhile; I think this correction should stand.

All the Best,

Jonathan

Dear Jonathan J. Dickau, I was impressed with your statement, "Questions arise both in the physical and abstract reality of what is decidable or undecidable, what is computable or uncomputable, and which events are predictable or unpredictable." So I argue that it is necessary to distinguish geometric space from physical space. Geometric space is an abstraction of physical space. Physical space moves relative to itself, because according to Descartes it is matter. Arguing in this way, I showed that the probability density of states in an atom depends on Lorentz abbreviations: length, time, mass, etc. I believe that this is the unifying principle of modern physics, which will reduce the level of unsolvability, uncomputability and unpredictability in it.

I invite you to discuss my essay, in which I show the successes of the new Cartesian generalization of modern physics, based on the identity of space and matter of Descartes: "The transformation of uncertainty into certainty. The relationship of the Lorentz factor with the probability density of states. And more from a new Cartesian generalization of modern physics. by Dizhechko Boris Semyonovich »

    It interesting your essay. A good essay.

    You use a simple formula to study a simple condensation (simple to obtain using a computer), then you can use the results on this set to obtain general results for condensation.

    I think that each condensation phenomenon, with a general use of critical exponent, that come out from a statistical mechanics study of simple system (for example Ising model), and experimentally tested, to obtain same results (condensation of cluster from short range interaction); then, I have a problem: in a Mandelbrot set there is not a interaction from points, in a lattice, so that a statistical analysis is not possible; it could be possible using different initial points, and consider a swarm of moving points, but there is not interaction (so that the statistical phase transition is improbable); but it is possible that I am wrong.

    I understand that the Mandelbrot set is interesting because of the dynamics is unpredictable a priori, but a Conway's Game of Life (a hypothesis) or the Ising model (a classic analysis), with many different interacting patterns have the semplicity and the statistical complexity of a physical system.

    I tkink that it is possible to use the chaos theory to reduce the dimension of a space dynamics (for example Hausdorff dimension for chaotic system), but almost every differential (or discrete) dynamics system reduce the dimension of the space.

    Domenico

      Thank you for your enlightened feedback...

      Your comments really make me think Domenico. I know there are ways to resolve the Mandelbrot Set from a chaotic initial value, that jumps all over the complex plane and eventually resolves into the Mandelbrot Set we know. You can set the graininess of what is going on behind the screen, in that case, so that what emerges is an averaged value for the outcome of nearby points. And I know there has been some research into fuzzy Mandelbrots, where exact trajectories are uncertain.

      What you seem to be looking for is a procedure like the inverse of the distance estimation method used in ray-tracing, ray marching, and ray forcing algorithms. In that case, we are looking from the outside shooting at the body of the Set from a distance, and bouncing off the repeller sets that surround it. This is used extensively for higher-d fractals. It would indeed be interesting to see if the Mandelbrot outline could be obtained shooting from the inside instead, where if you hit a repeller edge or surface you know you have gone to far.

      Thanks again,

      Jonathan

      Thank you Dizhechko...

      I invite you also to read my essay for more detail, and to discuss the relative merits and faults of my approach. I will definitely get around to reading yours soon, and I appreciate the heads up about what is in your paper. Descartes had a lot to offer, and reviving a Cartesian approach might be the shake-up Physics needs. We will discuss this more later.

      Best,

      Jonathan

      Dear Jonathan J. Dickau.

      In the time of Descartes, they did not think that speed had a limit. Now that we are confident that the speed of light is the highest speed and nothing can move faster than it, we should consider how space, which is matter, resists its movement relative to itself. Therefore, the physics arising from this should be called new Cartesian. I will read your essay again to comment on it from the point of view of a new Cartesian generalization of modern physics, which is based on the identity of Descartes's physical space and matter, in order to bring our views on physics closer.

      However, I noticed this: "This is because M3,1 is a repelling point, which means that any point in its local neighborhood displaced from that spot is divergent. So it is not possible to enter a numerical value in that neighborhood that will repeat indefinitely. " A similar situation arises when I gave the opposite meaning to the Heisenberg uncertainty principle, which has now become the principle of definiteness of points in physical space. According to the new principle, an infinitely large momentum is needed to separate a point from other points.

      Boris Dizhechko

      Oh Jonathan,

      I was cogitating on your observation that transcendental numbers are non-computable (in proofs) and last night there was a mention of Fermat's Last Theorem on a TV Sci-Fi program which made me wonder if Fermat's cryptic note in the margin wasn't referring to a proof of the non-calculable form of his theorem; but that the incalculable identity of the theorem was itself a proof that such mathematical entities as transcendental numbers were non-computable? Mathematics is, after all, an artifact of human intellect. best - jrc

      Dear Jonathan,

      your essay touches on many interesting themes. You start off with what I've heard described as the 'edge of chaos'---the site of interesting complexity where the strictly and sterilely lawful and the completely unstructured random meet, and where consequently interesting phenomena occur. Wolfram has extensively written about his 'Class 4'-cellular automata, which are capable of showing persistent, nonrepetitive, novel behaviors. It's conjectured that such behavior is necessary for supporting universal computation, for instance, and, as you note, may be what spurred the origin of life, or even supports its persistence.

      Furthermore, you investigate the notion of fixed points: where maps, repeatedly applied, leave their objects invariant. There is an interesting connection here: fixed points---or rather, their absence---are at the heart of results like Gödel's theorem, or the undecidability of the halting problem (for which connection, if I am permitted some self promotion, see my own essay). It's not hard to see how they are related to self-reference: a map, placed on a desert island, if detailed enough, will have to contain a copy of itself, and a single (fixed) point where map and territory, so to speak, coincide.

      You combine these notions via the fractal geometry of the Mandelbrot set. This, too, has regions of great complexity at the boundary of more 'boring' regions---an edge of chaos---, and whether a point lies within that region is decided by application of a self-referential formula.

      It would be interesting, to me, to see this connection explored more in depth. When does self-reference facilitate the emergence of nontrivial complexity, of novel structures that nevertheless do not degenerate into mere chaos?

        I can see that you read for detail Jochen,

        It is a pleasure to hear your expert analysis and to come up not wanting more than I offer. I too like the edge of chaos notion. I tried to bring that out in my overlay showing the alignment of bifurcations with M at the same location I showed condensation happening earlier. Had I more space to write; I'd have devoted some discussion to Susskind's ideas about complexity being maximal at event horizons, but it was one more thing than I could add.

        I remember fondly a book by Briggs and Peat "Turbulent Mirror" where they do a very nice job with that idea and also that there is a far shore of the chaotic realm. I don't know if that was their intended meaning, but it seemed strongly implied. In any case the analogy of the band-merging point at M3,1 with gravity is strong. The trajectories gather to a point at the gravitational horizon/critical point of condensation, but behind it the degrees of freedom rapidly multiply as the trajectories expand.

        I would love to explore the connection you bring up in more depth.

        Warm regards,

        Jonathan

        I have begun to read your essay Boris...

        More about it on your page. It seems you have some interesting ideas. I'm not sure I see it all fitting together seamlessly. But such is the path of progress.

        We may well agree that Descartes was ahead of his time, and that some parts of his methodology still have value for today's Science.

        All the Best,

        Jonathan

        As I have commented below (Just a thought)...

        Some numbers have an exact algebraic value that can be written out. The long expression in the Endnotes is a number not a varying relation, and yet it would require an endless string of digits to render it in its pure form - so it is indeed a transcendental number in decimal form. That is; its algebraic form is finite though its numerical form is infinite.

        Cool stuff.

        Have Fun!

        Jonathan

        Hello again folks,

        For those who are interested to know more about or do some exploring themselves in the Mandelbrot Butterfly figure and its family of figures, I offer these links to content at Fractalforums.

        Jonathan

        The Mandelbrot Butterfly is a Spectral Manifold

        The Butterfly's inverse is also Spectral

        Formula files for Mandelbrot Butterfly family figures on Chaos Pro

        I hope this is helpful.

        More later,

        Jonathan

          Jonathan,

          Are you familiar with Keenan Crane's GPU code for ray-tracing the Julia set?

          https://www.cs.cmu.edu/~kmcrane/Projects/QuaternionJulia/

          About inverse ray-tracing, I'm not sure that would work well. You'd have to assume that the ray can penetrate the mesh many times, and that it's only the last (the outermost) penetration which counts. This is inverse of the regular method, where the ray only has to march to the first penetration point. I hope this makes some sense... I'm not a mathematician.

          Just for fun, I'd like to mention something about the Julia set: Normally one uses the quaternion magnitude to determine if a point remains within the set. The path followed by the quaternion not only has a magnitude, but also a total length and total displacement. These three properties of a path are all different in value (histogram). However, they all produce the same fractal shape. I would have thought for sure that the shapes would be different, but they're not. Weird.

          - Shawn

          P.S. Haha! You have The Beauty of Fractals too! Page 61 is my favourite version of the Mandelbrot set!