Good question John!

Squaring a pure imaginary number gets you a negative real number, which then gets added to the original pure imaginary number, and this becomes a complex number - as it has both real and imaginary components. If we start out with a complex number, it will almost always give us a complex result, although sometimes summing makes terms cancel out - and we end up with a pure real, a pure imaginary, or a null result (0,0i).

If we take out the addition step, and just iterate the squaring function, we find that any initial value whose distance is greater than 1 from the origin will grow unendingly, and for initial values whose distance is less than 1 from the origin, the successive values approach the origin or shrink monotonically. Of course; for a value or distance in C of exactly 1, the function remains at the boundary forever, neither growing nor shrinking.

However; when I tried to use a similar shortcut for the Mandelbrot algorithm, by looking for a result that shrinks over 3 successive iterations; I didn't get the Mandelbrot Set at all, and what I found was the Mandelbrot Butterfly instead. Pretty weird, huh?

Regards,

Jonathan

Dear Mr. Dickau,

I have no wish to be disrespectful to you or your essay, but I think abstract mathematics and abstract physics have nothing to do with how the real Universe is occurring for the following real reason:

Do let me know what you think about this: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of abstract NOTHING. Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

    Thanks for writing Joe,

    I find value in at least some of what you have to say, and I agree even if for very different reasons from yours, because I too think the uniqueness of individual units of form is often unappreciated or trivialized - and indeed like a snowflake each one is different. To assume otherwise is problematic, and prevents one from seeing what is real sometimes.

    On the other hand, there is a pattern where every quartz crystal has similar properties - though each one is certainly unique and different from all the other crystals in existence. They are not identical, but different pieces of quartz are obviously the same in many ways. Similarly; all snowflakes display a hexagonal symmetry, though none is absolutely symmetrical.

    I think Science advances, Joe, when people notice consistent patterns over time. So while we can get into trouble by trying to apply abstract principles too broadly, it is better if you are not looking into the mouth of Old Faithful geyser at the start of an eruption, and therefore it is wise to know how often eruptions come - on the average - so that one can stand back and watch it happen, rather than getting a face full of boiling hot water.

    All the Best,

    Jonathan

    Dear Jonathan,

    Always something to learn from your essays. Well done!

    I have my biased perspective just like others have theirs, hence I have partly pondered the below questions in my essay and I think you should too:

    - If there is indeed a Planck lower limit to size ~10-35m, how will the Mandelbrot Set pattern confront this limit?

    - On "Planck-sized atoms of space", if indeed space is of such nature, what will separate one atom from another?

    - Are the 'laws of nature' or 'atoms of space' eternally existing or can they perish and cease to exist?

    Best regards,

    Akinbo

      • [deleted]

      • Post #32

      Dear Jonathan,

      Thank you for not reporting my comment to FQXi.org as being inappropriate in order for it to be classified as Obnoxious Spam.

      Each quartz crystal has a real surface, and rather than trying to arrange abstract numbers of abstract crystals in a pretty pattern the only thing one needs to know is that the surface of all quartz crystals travel at the same constant speed as all other surfaces do.

      Man is the only creature who does not have a natural habitat. Living near a volcano is unnatural. Birds have enough sense not to do so.

      Warm Regards,

      Joe Fisher

      Jonathan,

      It's all weird to me! As I said in an exchange with Constantinos Ragazas on Steve Sax's page, I often wonder if mathematicians realize how phenomenal their memory capabilities are. I rely on people like you to gain some insight of what the myriad maths are about, and often find myself completely out of my depth.

      Intuitively though, it seems that if the addition operation is short-cutted, a moderating value would be lost, and the butterfly comes out of chrysalis in the round, as one direction lengthens without restraint. I still can't grasp the morphology of quarternions and division algebras that project a 4-D shape, and lost track of where I read a post of yours about 3-D imagining being an approximation. Like many I suppose, I find the images beautiful and can only wonder how that fills space over time.

      And of course there is the ever present counter-intuitiveness of QM that over-shadows just how counter-intuitive Relativistic spacetime is itself! I for one greatly appreciate that yours is a genuine kindness in patient explanations of the rudiments of mathematical physics, thank-you. jrc

      Thanks for sharing Joe,

      I think I am starting to get the gist of what you are saying.

      All the Best,

      Jonathan

      I have considered these questions, Akinbo...

      1st - the Mandelbrot's cusp at (.25,0i) is the minimum extent and highest energy represented in the Mandelbrot figure. But the theory would indicate that this translates into a minimum time step. However; for anything to persist longer than the Planck time, in this theory, it must have a non-zero size.

      2nd - particles act as probes of the properties of a given space, retaining and conveying information about separability and separation. I would say that once forms exist as self-contained independent units, which can move relative to each other, this defines or helps determine the dimensionality of space as well.

      3rd - I think part of the meaning of Math is that it preserves some features of natural law that are persistent, from cosmological era to era, from inception to its demise or the beginning of a new cycle, or from universe to universe in a multiverse scenario (more below).

      As for atoms of space, however; that concept speaks mainly to how the fabric of spacetime emerges, and one can't discern individual unit cells after that. If space and time are relativistically indistinguishable; then there is a lower limit of around 10^-13 cm - where particle separability is possible - in which Relativity is defined. And item 2 answers this.

      The Cosmology based on the Mandelbrot Set does not tell us whether a cold dark end is the universe's ultimate fate, or whether a new cycle would begin, as I can show you the graphical representation of both scenarios. Likewise; it supports the idea that the universe is singular and allows for the possibility of multiple universes. This suggests these possibilities coexist equivalently.

      All the Best,

      Jonathan

      Hi Jonathan,

      You get my high mark, even while I disagree with your view of Tegmark's view.

      You and Max both approach your respective frameworks for a unifying physical theory with personal, subjective accounts of your journey through mathematics -- Max's hypothesis is not philosophy, however; he explicitly holds forth a way to refute the physical framework.

      That's why I have a hard time getting my mind around a particular mathematical structure, such as the Mandelbrot set (or Julia, or Koch or ...) as fundamental to a unifying theory. (Same goes for Lisi's E_8 symmetry.)

      For if we allow the fundamental reality of such structures, we lend more meaning to the calculating machinery that creates them, than to relations between and among the quantities and qualities that dominate our physical experience. The former is static and discrete; the latter is dynamic and continuous.

      All best,

      Tom

        Thanks Tom,

        This same question was raised by Lorraine on the general discussion page for the contest (Brendan's thread); she asserted that the Mandelbrot Set is 100% boring, and I presented a different view. To me; the static nature of the object belies what's going on beneath the surface (so to speak), where every point on the set is associated with a different flavor of dynamism.

        In fact; we can look at the Julia Set for any one point, and study its properties as a dynamical system. This can get boring too, though it leaves room for variations. But when the entire Mandelbrot spawning these Julias is considered, the evolutive properties of the dynamism become apparent. It is this evolution of dynamism that is my primary area of research and interest in M.

        All the Best,

        Jonathan

        Well, of course I don't think that the Mandelbrot set, or any of the variety of self similar sets are boring. The Mandelbrot set, in fact, has earned its title as the most complex object in mathematics.

        What I mean, is that the initial conditions for any of these structures are arbitrarily chosen and cannot be shown to be generated from any first principle more general than the spatial assumptions that precede them.

        Best,

        Tom

        4 days later

        Dear Jonathan,

        Thank you for your generous comments on my blog. I did not attempt to be on the ridge, there are many snipers! For sure, we will contnue to interact and learn from each other as I experience from many authors at this contest.

        Best.

        Michel

        6 days later

        Jonathan, Sorry I hate this essay. It is entirely too hagiographic, in that it makes mathematics into a Godlike hero. That is entirely unjustified by the facts. What I see is a human illusion that mathematics is effective in physics and a lot of propaganda to justify making mathematics take over the role of GOD. Sorry I am not buying this modern mythology of mathematics as the new GOD.

          4 days later

          Dear Jonathan,

          I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

          All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

          Joe Fisher

            Dear Jonathan,

            Thank you ever so much for the terrific you left about my essay.

            All real things have a real surface. Real light does not have a real surface. Real light does not consist of abstract photons, or abstract plasmons. It is physically impossible to create a real light by means of manipulating an abstract photoelectric effect.

            Proof that real light did not have a real surface was established by the slit test. When the pre-light emission struck the first surface, real light appeared on all of the area of the surface, except of course where the slit, or slits had been cut. The pre-light emission flew through the slit or slits and when they struck the surface behind the slits they had to produce a real light effect that was different than the real light showing on the first surface.

            Joe Fisher

              Dearly Beloved Jonathan,

              I am so excited about your absract,which constitutes a significant commentary on the inseparable coefficiency between Maths and Physics.I am even much more elated with your lucid explanation of the phenomenon in the body of your essay.

              Keep on flourishing.

              Lloyd Tamarapreye Okoko.

              Jonathan,

              Time grows short, so I am revisiting essays I've read to assure I've rated them. I find that I had questions which leads me to your Mandelbrot algorithm: http://www.fractaldesign.net/FractalAlgorithm.aspxrated. I find that the imaginary number discussion and orbits represented on a Cartesian plane are similar to machinations of Euler's Identity that I use. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

              Jim

              Hi Jonathan,

              It is a pleasure to meet you in FQXi Essay Contest. Once again, you made an excellent work through a very interesting and enjoyable Essay. Here are some comments:

              1) Your idea that Nature has been shaped by all of the applicable Maths since the beginning of time is intriguing. In this way Nature seems a unique sentient being.

              2) I am fascinated by fractals. I see that you were a pioneer of fractal cosmology. Is it a coincidence that your original work on this issue is dated 1987, i.e. the same year that Luciano Pietronero and his team attempted to to model the distribution of galaxies with a fractal pattern?

              3) I am pleasured to know that you entertained a proposition similar to Tegmark's MUH long before his framing of it. Congrats, this must be popularized.

              4) I did not know Gibbs' statement that "the laws of physics are a universal behaviour to be found in the class of all possible mathematical systems." I completely agree with Phil. Now, I am going to read his Essay.

              5) It is not a coincidence that you have found that the concepts and entities most central or fundamental to Math also have the greatest relevance to Physics.

              6) Gerard 't Hooft's answer "We don't need atoms of space or whatever, because the laws of nature do the calculating for us" is intriguing and I agree with your interpretation that this means the laws of nature are inherently mathematical.

              Finally, I find your Essay extremely intriguing. Thus, I am going to give you a deserved highest score.

              I hope you will have a chance to read my Essay.

              I wish you best luck in the Contest.

              Cheers, Ch.

                Thank you Christian,

                Your comments and appreciation are gracious. I first started your paper a day or two ago, but have not finished reading and digesting yet. I do expect to read it and rate you before the deadline.

                All the Best,

                Jonathan

                Thank you Lloyd,

                I appreciate your taking the time to read and comment, and also your gracious remarks. I am glad my essay and its message have pleased you, and I look forward to reading yours.

                All the Best,

                Jonathan