Diophantine Equations Revisited

I wrote a note a while back about shapes produced by Diophantine equations. Specifically, consider an RGB image as a matrix of color luminosities. This is actually how images are typically stored, as a collection of 3 matrixes, where the entry of a given matrix represents the luminosity of one of the three color channels (i.e., red, green, or blue). These are typically integer values from to , corresponding to no luminosity and maximum luminosity, respectively. If all three channels are at a given entry, then the corresponding pixel is black. If all three channels are at a given entry, then the corresponding pixel is white.

The idea I had was to impose a diophantine equation on an RGB matrix set, and see if any matrices actually satisfied it, and it turns out, they do, and the resultant images are aesthetically interesting. You can see a few plotted below, and these correspond to solutions to the equation,

,

Where and are indexes in the three matrices, and B and G are the actual pixel values at those indexes in the blue and green channel matrices, respectively, and is some positive integer. As you can tell, this particular equation disregards the red channel, simply because the color palette I wanted was blue, green, black, and white. Applying this as a practical matter requires a Monte Carlo method to find solutions, and this code does exactly that. This is how I generated the logo for my software, Black Tree AutoML.

In the examples below, all of the the pixels satisfy the equation above. Specifically, each image has a fixed value of , and for a given pixel in a given image, in position , the luminosity for the blue and green channels sum to , when raised to the powers and , respectively. All of the the images below are matrices, with the red channel set to . If there’s no solution for a given , in an image that contains solutions elsewhere, pixel is set to white. There are at times many solutions for a given value of , and the attached code uses the first one found by the Monte Carlo method.

Earlier today, I happened to have turned to a random page in my notes from around the same time I was working on this equation, and I noted only half in jest, that this suggests a more general idea, that could, e.g., imply an equation for pizza. This sounds insane, but it’s obviously not, because the more general idea is a Diophantine equation imposed upon a set of matrices, that in turn produce a representation of a real world object. Intuitively, you would say fine, perhaps, provided that the density of solutions that correspond to real world objects is low. However, looking at the images above, you can plainly see the shapes are symmetrical, and even aesthetically pleasing. This in turn suggests that the density of reasonable representations of real world objects might be more dense than random, and perhaps even meaningfully dense, simply because this particular equation seems to produce fairly sophisticated symmetries.

Though the inspiration was partly motivated by humor (i.e., the equation for pizza), the reality is, this could be a very serious idea, that would allow for the characterization of systems, possibly dynamic systems, using Diophantine equations. This is not how physics, or anything for that matter, is done, but it might work. My intuition is that it will work, and that continuous representations of systems made sense centuries ago, simply because we didn’t have computers capable of solving complex discrete equations, like the one above. Moreover, it is now widely accepted that certain phenomena of Nature are quantized (e.g., electrostatic charge, electron orbitals, and quantum spin), which in turn suggests that this might be how reality is actually organized. If that’s true, it’s not clear to me what continuous mathematics represents, and it really shouldn’t be, because we’re capable only of finite observation. As such, every physically meaningful representation will be informed by a discrete set of observations, even if those observations are used to generate a continuous function. In fact, I developed an entirely new model of physics that makes no use of calculus, and implies all the right equations of physics, including those produced by relativity. This is really puzzling, because Newtonian physics is accurate, yet predicated upon calculus. There should therefore be disconnects, where continuous integration fails, producing an incorrect equation, and the place to look in my model relates to the distribution of kinetic energy in a system (See Section 3.3 of the last link).

There’s also the question of computability, since not all Diophantine equations are computable, and this was shown in response to Hilbert’s Tenth Problem. Therefore, it must be the case, that the set of constraints includes problems that cannot be expressed as solutions to equations solvable by deterministic algorithms. Taking this to its extreme, you could at least posit the possibility that reality itself is the product of such a constraint equation, that restricts possibility, in turn implying physics, as a logical corollary of something even more fundamental. That is, physics would be in this view an approximation of a more fundamental restriction on possibility, given by perhaps a single master equation. If this is true, then it follows that reality could be non-computable, and there is at least some academic work that has found non-computability in existing physics. My personal conjecture is that reality is not Turing-Computable, only some of it is, and this follows from the observation that some people have minds that are plainly superior to machines.

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