This evening I had an interesting idea for a new way to look for patterns in the distribution of numbers such as the prime numbers and the digits of Pi. In a nutshell I propose that there may be patterns in these number sequences that might not be evident to a computer but could be evident to the human eye and human intelligence, which among other things is tuned to find order in chaos, even when that order is "fuzzy." In this article I propose a new class of rules that are similar in some respects to cellular automata, for generating visualizations of the distribution of numbers, and for leveraging distributed human intelligence to evalute those visualizations for meaningful patterns.
Let's use the prime numbers as an example for this essay. We can represent the distribution of prime numbers visually as a line of black dots that is interspersed with white dots wherever primes occur. We can also represent primes that belong to various classes with various corresponding colors if we wish. In any event, it's just a line of pixels. Now let's define a program that generates rules for displaying this line of pixels in various ways.
(Note: After writing this article, I discovered some software that illustrates one instance of this idea beautifully, based on work by Stanislaw Ulam, one of the originators of cellular automata theory. The "Ulam Spiral" generates interesting visualizations of the distribution of primes that illustrate that they are clearly non-randomly distributed.
The basic difference between my conception and Ulam's is that I do away with the constraint that the primes should be arranged in a spiral -- that is just one way to pack the primes into a plane and why should it be the one we choose? My approach generates different packings of primes in space in the hopes that one of them will reveal even more interesting hidden structure to the distribution of primes, that structure being expressed by the rule that generates the packing. It's interesting that I came up with this idea without first knowing about Ulam's work on this, but it's not surprising that Ulam was onto this; he was one of my inspirations in the early years of my interest in cellular automata theory).
In my approach the rules for generating layouts are simple -- they are based on the analogy of the "Etch-a-Sketch" (a drawing toy in which you make picture by drawing a single line and simply turning it left and right) -- a pixel is drawn and then depending on various variables (such as the number of pixels already drawn, the number of pixels drawn before of a single color, the color of the most recent pixel, and the colors of pixels about to be drawn, to name a few possibilities) the next pixel is placed in a certain direction relative to the previous pixel (any direction, depending on the number of dimensions -- in a 2D visualization there could be at least 9 directions in which to draw the next pixel -- same place, north, northeast, east, southeast, south, southwest, west, northwest).
A more general, and ultimately better, version of my rule for laying out the numbers is to simply say that the location on the screen in which to place the next number in the number sequence, is determined as a function of the number itself and/or preceding numbers (and/or some set of numbers that follow it, if desired). This allows for all types of layouts to be generated, from layouts in which the numbers are arranged contiguously in space to those in which adjacent numbers do not appear in adjacent locations in space.
My hypothesis is that there may be a hidden structure in the distribution of prime numbers that is only evident when they are arranged on the right type of surface, according to the right algorithm. For example, consider arranging the number line on a sphere, then where do the primes appear and does that reveal hidden structure? What about higher-dimensional layouts?
One rule that such a system could generate might wrap the line at every n pixels, in order to fit it in some sort of a rectangular array. Another rule might wrap the line in a much more complex way, making right and left turns to wind it through a complex path in two dimensions. Still other rules could potentially wrap the line to fill a 3D space, or even higher dimensional spaces. Because these rules can also potentially draw pixels on places where pixels were once drawn in the past, they can even appear to be animated over time -- for example, suppose a rule draws a square of pixels and then redraws a new square over that square -- if this runs fast enough it would appear to be a movie of sorts and this opens up the possibility that the pattern in the primes might be visible as a sequence in time and space -- which actually makes sense given that our universe consists of space-time. Who knows, perhaps our universe IS simply the pattern generated by such a rule running on the distribution of primes in a certain number of dimensions.
In any case, using such simple rules, the line displaying the distribution of primes can be displayed according to an infinite variety of geometric visualizations. My hypothesis is that while the vast majority of such visualizations will appear to be random, some perhaps may display unexpected structure -- for example, suppose that in some the primes line up in a certain way, or form a complex but recognizable geometric shape or tiling.
The human eye is capable of recognizing even the slightest traces of order in such chaotic fields, and this is key to my approach. There may be patterns in the distributions of primes that are not precise, but are rather "fuzzy" or "nearly precise" in the same chaotic way that Nature is. Computers are not good at recognizing "fuzzy patterns" -- but human senses and brains are tuned specifically for this. In fact, a variant on the above visual pattern scheme might be to render these distributions with sound -- for example by interpreting a Rule as a specification for a soundwave or rhythm for example. In any case, the key is to leverage the ability of the human brain to find order in chaos, especially when that order is only semi-ordered. So how can we accomplish this, given that the search space of possible patterns is infinite?
One way might be to leverage the minds of millions of people at once -- for example by making a distributed screensaver -- similar to SETI@home -- on which randomly selected visualizations of prime number sequences are rendered according to randomly selected rules. As these visualizations flash across thousands or millions of computer monitors, users can click to rank the ones they see as "more patterned" or "less patterned" and this feedback goes back into the system to affect the rules being used to generate further visualizations. In this manner perhaps we can guide the evolution of rules towards more interesting regions of the search space. We might also just get lucky -- perhaps someone sitting at their computer terminal one evening will see a surprisingly ordered pattern flash in front of them -- perhaps it will be a complex geometric tiling, or a shape, an animation, or a gradient -- and this pattern, may reveal a hidden structure to the primes that could open up incredible new vistas in our understanding of mathematics, science and even perhaps our universe itself.
I often wonder whether there is something potentially cosmic hidden in the distribution of prime numbers -- perhaps something related to the deepest structure and dynamics of our universe in fact. Is the key to chaos hiding there -- the meta-pattern that explains all natural patterns, including for example, the chaotic fluctuations in the weather, in population dynamics, and in stock markets? Or will it turn out that the pattern we find requires a space with 11 dimensions, for example -- and what might that imply about the universe we live in? Or will the pattern by dynamical -- a changing sequence that has certain repeating or at least self-similar properties over time and space? And wouldn't it be interesting if the best way to find that structure was with human (not computer) intelligence, with all its inherent fuzziness and imprecision?
And now for the Big Idea in all of this. I have a hypothesis that there exists a particular manifold on which the prime numbers "line up" perfectly to form some sort of structure, and that this particular manifold is something fundamental -- it will tell us something fundamental about the structure and nature of spacetime, number theory and physics -- in a sense it will unify them. You might call this Platonic Unification in that it unifies abstract number theory with physical space and time -- it bridges the gap between "the Forms" and the world of "shadows" that we live in, to put it in Platonic terms. But as long as we are speculating, let's not stop there -- what if the structure of the primes on this manifold is actually of interest as well -- for example, suppose it is a map of something -- a map of the universe, or perhaps the key to chaotic system dynamics.
In other words, what I am suggesting here is that the fact that the distribution of primes is non-random is a "clue" to some Big Secret that we are "supposed to discover" -- a clue left for us to find by "God" so to speak, that will lead us to the Ultimate Secret. We can approach this problem iteratively, starting from a small number of dimensions, and trying every packing in spaces of a certain shape and volume. By using a genetic algorithm approach we can tune our search to focus on the packings that yield the most promising structure. It's like solving a high-dimensional "Rubik's Cube" problem. But actually it's really a "Reverse Rubik's Cube" -- that is we start from the assumption that the colors on the Cube already line up and now given a sequence of colors we have to figure out the shape of the "Cube." Of course it may not be a cube at all -- it may be a torus, or hypercube, or some other complex topology.