Sunday, July 29, 2012

Ulam's Spiral Revisited

In a previous post, I compared Ulam's Spiral (below) to something I erroneously called the "fundamental theory of the universe."

I apologize for this. There is no such thing as the "fundamental theory of the universe." Besides, even if there was, it would more likely apply to physics than astronomy (think fundamental interaction i.e of particles).

No, indeed, it makes much more sense to compare the theory behind the Spiral to, perhaps, the second law of thermodynamics.

That, of course, states that an organized system tends to move towards disorder - or, in a word, entropy.

Let us entertain the idea that we can expand upon this long-established law of physical science with terminology used by Newton himself:

"An organized system tends to move towards disorder unless acted upon by an outside force."

But naturally, this does not apply to the Universe! The Universe is moving towards disorder not in spite of the forces acting upon it, but because of them!

This is where defining the intersection between number theory and astronomy becomes a bit tricky. Number theory, you see, is an invented science focusing on the study of numbers and how they behave.

It is rather strange, if you give it some thought, for "numbers" and "math" are simply things we humans invented and then imposed upon the natural world - seemingly, without a full understanding of how they really work. And yet, in the same breath, it can be argued that everything in science is simply a label to help us categorize and understand things about the natural world.

If we decided to start calling physics "scisyhp," nothing about the concepts that field studies or explains would change in the slightest. Similarly, if I said there are 1A trees in my backyard, that does not change how many trees are actually there (twenty-six).

With that said, let's return to Ulam's Spiral, a fascinating example of "strongly nonrandom behavior."

When Ulam drew his spiral, he noticed the diagonal lines that appeared almost entirely predictably. This is one of the things people cite to back up theories that the amount of prime numbers is infinite. In my previous post, however, I seemed to imply through comparison to the Milky Way that the distribution of stars in the Galaxy is "strongly nonrandom." and that, by extension of the Spiral theory, there are an infinite amount of stars.

For one, the amount of stars in the Galaxy is finite. Granted, there are billions upon billions, but it is still a measurable number. At the rate old stars die and new stars are born, it might be difficult to obtain an exact number, but it is certainly possible.

Additionally, the distribution of stars in the Galaxy is, in fact, strongly random. Herein lies another disconnect between Ulam's Spiral and the Galaxy - the Galaxy is a three-dimensional space, whilst the Spiral is not (although yes, technically both are indeed spirals). The positioning of a star s on an (x,y,z) co-ordinate space is random but influenced by a variety of things such as gravity and mass (which in turn affect each other). Star s affects the position of star t because of its mass, whilst both are affected by the black hole most educated people believe is the center of the Galaxy.

This means that you cannot expect to find a star at a certain location (x,y,z) relative to another location (x,y,z). In Ulam's Spiral, however, it is entirely possible provided you can figure out a trend line for one (or several) of the diagonal lines and, from there, predict a certain location (x,y) where a prime number p will fall with 95% confidence.


Remember that "theory" I referenced? It is, in fact, a real theory, one which is widely accepted amongst scientists and educators. I will call it, for lack of a better name (or locatable Wikipedia article), the "No Special Place" theory. To refresh your memory, this is the belief that there is no "unique" or "special" place in space; that it's all pretty much about the same in terms of the "stuff" any given part of space contains. The only reason we see more in one part of the sky than another is because of things blocking our view in the atmosphere (particulates, humidity, light pollution) or far beyond us entirely (interstellar dust).

Why does this happen?

Assuming that it does happen, the reason there is "no unique place in space" is in fact due to entropy. Reason: nature is lazy. There is a reason trees grow straight up. When you blow a bubble, there is a reason it is always spherical. On that note, it's the same reason stars are always spherical. Things expand to fill a space evenly until some outside force prevents them from doing so - in this case, stars only stop growing because the energy they produce only allows them to expand to a certain point before they can no longer hold themselves together.

Thus, stars in a galaxy are distributed both randomly and non-randomly. Their initial positioning is decidedly non-random, since each influences another and all are influenced by something more powerful (i.e black holes). Within a certain region of space, however, a star's position given in terms of (x,y,z) is random, but predictable to a certain degree.

That is, in a cubic AU of space (if space was ever measured in cubic AUs), there should be x stars. This just makes sense considering the scientifically accepted theory that no place in space looks or is composed incredibly differently than any other place in space.

Humans did not invent stars, but we did invent numbers. Since this is by design a predictable system superimposed on the natural order of things, it does not innately abide by the laws of entropy - rather, it affirms them.

As one moves further and further from zero, there is less and less likelihood that a number n will be prime. However, consider the design of Ulam's Spiral - comparing it to the Universe, or at least the Galaxy, is still entirely feasible in this context. As it moves further from zero (the origin), there should be fewer primes as numbers increase in size, right? Yes and no. There is less likelihood that n will be prime as n increases in size, but this is compensated for (though not negated, given the patterns that form in the Spiral) by the fact that each 360-degree path past the origin contains progressively more and more numbers.

Basic probability states that with more values of n, the likelihood of n being prime remains about the same. Compare this to the theory that with 21 people in a room, it is likely two will have the same birthday,

Ulam's Spiral confirms the first statement - in fact, it seems to provide visual confirmation that n is nearly always prime at a certain point in the spiral!

Consider the picture below, on which I've imposed red lines (click to see the full resolution with the lines more visible).

Lines can be drawn practically anywhere, but the most pronounced are diagonal, vertical, and horizontal perpendicular lines, as well as diagonal parallel lines. Notice also the existence of tangential (and sometimes chordal) lines for each diagonal line.

There is too much of a pattern to be ignored. Numbers, although we made them up, do in fact represent real and quantifiable things.

Why would we invent a system to keep track of things, to maintain order in our world in a way we can understand, if that system was not predictable?

Granted, we do not fully understand the way "numbers" (especially prime numbers) work. This is why I write little prime-testing programs and why we have large-scale distributed-computing projects to find ones with millions and millions of digits.

Ulam's Spiral can theoretically be extrapolated to infinity. I don't know when that will happen, but I can guarantee you it will look like the picture above.

In the meantime, the great mathematical and analytical thinkers of our day are likely close to a breakthrough in prime number theory as more and more are discovered - each of which, as demonstrated above in two different formats, maintains the integrity of Ulam's Spiral and seems to point to a single ultimate conclusion - there are infinite primes...

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