# Experimental Math

There’s a great divide within the physics community — you’re either an experimental physicist, or a theoretical physicist.  If you’re experimental, then you know how to solder wires, make things out of metal on a lathe, you’ve dipped things in liquid nitrogen, and you probably have not the best personal hygeine.  If you’re a theorist, you do your work with a pencil and paper, pride yourself on not knowing how to do anything with wires, you’re not even sure what solder is, you secretly believe to be truly a mathematician instead of a physicist, and you probably have not the best personal hygeine.  Theorists generally work much more rigorously — deriving things from first principles, using theorems and lemmas, carefully proving and cross-checking mathematically-based ideas.  Experimentalists have the chance to be more exploratory — you can start dipping things in liquid nitrogen, and see what happens.  In experimental work, you can publish papers describing interesting things your experiments do, without knowing why — in fact, that’s the really fun part, to stumble across something really strange, that isn’t explained by any theory.  You poke around, try a little of this, a little of that, and maybe find something completely new.  The theorists then come in, poke at your new experimental results, and mutters, “no no, this won’t do, get a theory up and going here immediately!”

Theoretical physics of course inherits this rigorous, proof-based approach from mathematics, where virtually everything is rigorously proven from very basic fundamental axioms.  Every branch of mathematics grows from the same roots — calculus is built up from algebra and geometry, algebra and geometry from arithmetic, and arithmetic from very basic ideas about numbers.  You don’t come across a topic that doesn’t spring from more fundamental topics — virtually all of mathematics comes about by building on current ideas, by expanding some prior idea, or by combining several ideas together.  There’s not much room in math for the kind of free-wheeling, exploratory work you find in experimental physics, where you poke around and try stuff, see what happens.  There’s not much room, but there is a little room.  The proliferation of computers has made it easy these days to explore, run experiments, try stuff out, to in essence discover a fact or idea, without being able to say why it is true.  These days, anyone can conduct experimental mathematics.

The most famous examples of these live in the suburbs of mathematics frequented by chaos theory, fractals, cellular automata, and the like.   The famous fractal patterns (including the  Mandelbrot set, shown here to the right and immortalized on countless dormitory walls) are examples of mathematical ideas that were more discovered experimentally than proven theoretically — many of them started as playing around on a computer.  The Mandelbrot set itself arises from a surprisingly simple math equation, one that an elementary student could understand and play with.  Such amazing, complex structure arises from this simple equation, structure that would be very hard to predict beforehand.   You take the iterative equation

Z (n + 1)     =     Z(n)2    +    Z(n)

(where “iterative” means you plug in the answer coming out of the equation back into the equation), and start computing it for every point in space.  You then color-code that point in space depending on how big the numbers get.  I’m criminally oversimplifying, but that’s the gist — a very simple idea that gives rise to a strange crinkly-edged shaped black thing.  Who would have guessed that something this complicated could arise from such a simple idea?  You’d never know until you started plugging numbers in, and try it out — a true experimental discovery.  How many other simple, easy-to-understand equations hide such complicated patterns?

Another example would be the logistic map, a very simple algebraic equation, that turns out to contain completely chaotic behavior.  Many people cover this idea much better than myself, so I won’t go into extreme detail, but this is another iterative equation, easy to write down, doable on a pocket calculator in fact, that generates such strange behavior that it had to be discovered experimentally, as no one could have predicted or proven it beforehand.  It sometimes behaves “chaotically”, in the dorky sense of the word, in that it spits out numbers that appear completely random, but aren’t.  No pattern can be found in the numbers generated, yet the numbers are completely deterministic and reproducible.  Compare this to other phenomena such as weather, or rolling dice, or the state of my kitchen floor, which are sometimes called “chaotic” but are really random.  These phenomena aren’t deterministic, in that there are really random effects that can’t be predicted beforehand.  For the logistic map, however, if you use the same numbers in the equation, you’ll get out exactly the same random-looking sequence of numbers.  They look random, but they’re completely determined by the equation.

If my bias isn’t blatantly obvious by this point, my training was in experimental physics, but I like to think I could have been a rigorous, theoretical, no-nonsense mathematician if I had wanted to.  Of course I’m completely wrong — I don’t have the mathematical talent or the attention span to prove anything, particularly at my age.  I could have played in the majors, but I hurt my knee on a lemma in college ball.  So experimental math is a fun place to stretch the old mathematical muscles, to throw around a derivative or two.  Using computers these days, it’s easy to try a very simple idea and come up with something very strange.

This tour through experimental mathematics describes a handful of cool little math and math-like explorations — some ranging through very basic number theory, others into language, but all bonded by a common feature — they’re all virtually useless, no practical value whatsoever.  And that’s part of the fun — don’t we all do practical, useful stuff all day at work?  Well, no, but that’s what the boss thinks.  These little forays are very basic in terms of the math and computer programming involved, don’t really lead anywhere in terms of a conclusion or a point, and will almost certainly never wind up being used for any concrete purpose.  But in their own small ways, they contain ideas that are completely new — they’re meant to inspire you to maybe try some of this stuff yourself.

Read on to hear about my forays into experimental math.  Anyone who develops a PhD thesis out of any of this stuff, you owe me a six-pack of Guinness.

Experimental Math Part I:  Fun With Number Theory

In this experiment, we have a little fun playing around with the integers.  Remember prime factors?  Every integer can be factored into the product of a bunch of primes.  What if you took the sum of the prime factors for a number?  Well, not much — it’s a pretty useless thing to calculate.  But let’s take a look at it anyway.