It was during a boring trip back home at the end of some summer vacation with my parents (destination Colorado or Wyoming, or the like) about four years hence that I first taught myself how to play polyrhythms. To space out two beats on one hand and three beats on the other within each measure, you just have to count six beats total: Start repeating to yourself, either aloud or loudly in your mind:

1-2-3-4-5-6-1-2-3-4-5-6

Then hit your leg with one hand on 1's and 4's

1-2-3-4-5-6-1-2-3-4-5-6

Then stop that hand, and hit the other hand against the other leg on the 1's 3's and 5's

1-2-3-4-5-6-1-2-3-4-5-6

And then try the synthesis:

1-2-3-4-5-6-1-2-3-4-5-6

And since you won't get it (at least not very fluently) the first time, go back and practice each part individually some more, and try to accent the beats you would be playing with the other hand in your mind. Try shifting your awareness of the accents from the 1-3-5's to the 1-4's and back and forth and back again. Also, be sure to switch which hands plays which rhythm. You'll get it after a little effort, and it'll be stuck there forever.

Three against four was much more of a challenge for me, but I'm very grateful now that I had so much time to waste practicing way back when. The symmetry is a lot harder to grasp at first, and you have to skip over more beats. Perhaps having it typed out as it is here would have made it easier:

1-2-3-4-5-6-7-8-9-10-11-12-1-2-3-4-5-6-7-8-9-10-11-12
1-2-3-4-5-6-7-8-9-10-11-12-1-2-3-4-5-6-7-8-9-10-11-12
---------------------------------------------------------------------
1-2-3-4-5-6-7-8-9-10-11-12-1-2-3-4-5-6-7-8-9-10-11-12

It was quite an epiphany when my young mind discovered that the two rhythms grow further apart, then get back closer together to unite back at the 1, and do so completely symmetrically. Once you get this down, you're practically set to teach yourself any other straight n-against-m polyrhythm you can fathom... just keep in mind that as you skip more and more counts between beats, the natural feel gets harder and harder to attain, and a lot more practicing is required to embed the rhythm into your muscle memory. But after long enough, you won't have to bother counting out all 12 beats, and it'll be completely natural to switch between counting 1-2-3-4-1-2-3-4 and 1-2-3-1-2-3. I've recently started subconsciously tapping my leg 4 times for every 3 steps I take. It's actually very fun.

If you've studied any modular arithmetic, you might have noticed that in general, n against m polyrhythm requires the appendage producing n beats to perform on any count number x such that x = 1 (mod m), and the other (which produces m beats/measure) to perform on x = 1 (mod n). So when you imagine a ring of integers (mod mn), each hand is traveling around the ring at the same rate, and making sound when it is "above" an integer congruent to 1 (mod m) or 1 (mod n), respectively. They both sound together at 1 (mod nm)... kind of reminds you of the Chinese Remainder Theorem, doesn't it?

If you're not familiar with modular arithmetic (and even if you are, because this is slightly different), imagine a circle, with a dot at the top. Each time you produce a beat, imagine that a point (or a bug or a rabbit or something) has traveled all the way around the circle back to the top dot, with constant velocity, since the last time you produced a beat with that hand. Notice that your two hands have different speeds. If you're playing two against three, one hand makes two revolutions in the time it takes the other hand to make three revolutions. If the your speed is such that the downbeat happens to be once every second, then one hand completes 2 cycles per second (aka, 2 Hz), while the other travels at 3 Hz.

If you know much about music theory, then you know that musical intervals can be defined not only by how many half-steps or whole-steps are between the two tones that compose them, but also by the ratio of the frequencies of the two tones. For example, a note that is an octave higher than another will, by definition, oscillate at twice the frequency of the lower note. An octave is defined by the ratio 1:2, and is the most stable interval there is. The next most stable is the perfect fifth, defined by the ratio 2:3. If I play an A at 440 Hz, then the perfect fifth is at (440Hz)*3/2 = 660Hz. What's cool about this is that it means if you could slow down your perception enough, you would hear a 2 against 3 polyrhythm in exactly the same way as a normal person hears a piano playing an interval of a perfect fifth. On the other hand, if you could speed up your perception enough, you'd be able to hear the polyrhythms formed by the strings oscillating at 440Hz and 660Hz (although actually, a vibrating string at 440Hz also produces tones at 880Hz (1:2 = one octave up), 1320Hz (1:3 = 1:2*2:3 = one octave+perfect fifth up), 1760Hz (1:4 = 1:2 * 1:2 = two octaves up), etc, with decreasing audibility as the frequencies go higher; andthe string at 660Hz also produces tones at 1320Hz (common tone responsible for why a perfect fifth sound stable), 1980Hz, etc).

And when you stop and think about it, all life processes are examples of harmonic oscillators moving at different frequencies, producing harmony and stability when they move with each other. For humans, walking, dancing, breathing, pumping blood, engaging in regular sleep cycles, and ovulating are the the most obvious examples I can think of. When someone walks with a limp, we notice it and wonder what's wrong with them. When someone can't keep rhythm in dance or music, they don't fit in with the rest of the artists. When you don't sleep on a regular cycle, you feel bad. If your heart won't beat regularly, you'll die. All these different rhythms move at different speeds, but form harmony in allowing your body to live and do things.

And that, so I hear, is the main concept underlying polyrhythm for African musicians. The opposing yet symmetrical rhythms represent life forces, existing harmoniously and forming a greater pattern. This is why music and dance are perfect ways to unite people. By moving at the same rate as others, you share an experience of creating a natural pattern with them. The experience transcends humanity and taps right into a basic function of life - rhythm. You're quite literally on the same wavelength.

As a final note, I believe that there are other forms of polyrhythm that aren't so straightforward and mathematical as the n against m examples I've given, but I'm not familiar with them. Check back later, or go hunting yourself.