Mon nouveau tatouage

Between the time I decided to make an entry about this and the time I hit the first key just now, I went through about six different themes in my head. First, I was going to talk about why I got a tattoo. Then I thought, who honestly gives a damn? My reasons for getting a tattoo? How arrogant to think that this would matter to anyone else. Then I thought, well you know what's interesting? It's the reactions people have had. I have two kinds of acquaintance, scientists and normal people. The scientists all flip out because I got a tattoo. They stare with their mouths open. The normal people all think I'm weird for getting an equation tattooed on my arm.

But so what. Moderately interesting. Then I thought, I'll talk about how people choose their tattoos. If you ever get one, you will discover a world of skeletons and snakes, dragons and skulls, vines and naked women. What in the name of Buddha has created this culture? This is the ugliest stuff I've ever seen. And people brand it permanently on their skin. Do you see Picasso, Mondrian, Pollack tattooed on people? No, it's skulls, dragons, butterflies and knives. Do you see the prose of TS Eliot, Yeates, Shakespeare? No, you see "Hasta la Muerte" and "Born to Fuck".

I thought of a few other themes but then I forgot. Really. I have memory problems. Really.

Anyway I realized, the only interesting thing is the tattoo itself. In fact, what inspired me to make a blog entry about my tat was my brother in law, who made up some goofy statement about what the tattoo means. Here it is:

Equation of state and collision rate tests of parton cascade models?

Goofy bastard. I have no idea what he's talking about but it's funny. He's really smart. Still trying to figure out why he married my sister. Actually, I'm still trying to figure out why Ms. Faury married me. Anyway, the tattoo shows the Fourier transform, one of the most extraordinary accomplishments in mathematics. It is due to Jean Baptiste Joseph Fourier, 1768-1830. I won't put his picture here because he's butt ugly. Anyway JB realized that in a way, a different universe lives beside us. We are accustomed to x, y, z and t. x is horizontal, y is vertical, z is near-far, and t is time. Everything we can feel and intuit is expressed in these terms. But entangled in this Cartesian universe is another one, a jiggly world of images and sounds and signals and waves. Just consider the wave below:

Usually when you look at a picture of something, the picture tells you a lot about it. How's this working for you? Can you tell what it is from looking at it? Well, the image contains every bit of information you need. It's someone saying a word. (Click on it an you can listen to it.) You might say, "The picture is just two dimensional!!!" Well, partner, the ear is one dimensional. So you've got more information here than you need. No, the problem is, the the information here is given to you in the frequency domain. And this is outer space for us. We are lost.

Or at least we were, until Ja-Boopie came along. He realized that one must think in the "frequency domain." What does this mean? Well, consider this excerpt from the waveform above. We're going to zoom in on the time segment from 0.44 to 0.445 seconds:

See the very fast oscillations? There's about 50 of them across the chart. They go rapidly up and down. These are the high frequency components, and they make the voice sound bright. The fastest components account for the sibilants (the "s" sound). But notice also that there is a slower frequency hidden within. It oscillates about 5 times across the chart. I'll highlight it below:

The idea I'm getting at is that complex waveforms can be broken down into "frequency components." In fact, Fourier proved mathematically that any waveform can be reconstructed with (or equivalently, broken down into) the sum of a set of sine and cosine functions with different frequencies and amplitudes. These are called frequency components. If the complex waveform in question is a periodic one, like a sound, then the frequency components are called harmonics. Harmonics are multiples of the fundamental frequency. So if you hit a tuning fork, you'll get 440 Hz. If you pluck the 5th string on a guitar you'll get the same frequency, plus 880 Hz, 1320 hz, 1760 Hz, etc. If you only got 440 Hz, it would sound very weird, like a warning sound from a space-age computer. Musicians call these harmonics "timbre."

The equation for finding the harmonics of a periodic waveform is this:

What's really amazing, in my thoroughly unqualified opinion, is the way Fourier generalized this to any waveform, not just periodic ones. So, as it turns out, any waveform--for example, a picture of a face--can be decomposed into its frequency components. This called a Fourier transform, and the equation for it is one of the greatest accomplishments in the history of Western civilization. It is pervasive in math, physics, biology, and other disciplines, and it has widespread influence within each of these areas. And probably everyone who uses the transform is amazed at what it does. It tells you what something is in terms of its frequency content. Thus we can look at two things, not in terms of space and time, but in terms of how much of each frequency is present. Consider the first waveform I showed you above (the one you can click on). That's my son Lucas's voice. Now here's my voice saying the same word:

Ok, maybe it doesn't look the same. But what do you make of this? Nothing, I'll bet. It's meaningless to you. Now let's look at the Fourier transforms of the voice. Lucas's will be on top, mine on the bottom.

The x axis shows the frequency, and the y axis shows you how much of that frequency is present in the voice. Lucas has a big hump in the 100-1000 range, which corresponds to the "throaty", open voice range. There is another, large peak in the 2000-8000 range. These are the frequencies that give the voice sparkle. Ever heard a Trent Reznor recording where he sounds like he's singing in a closet? He just uses a low-pass filter to cut out this upper frequency range. The lower range is entirely intact.

Anyway, sound is a frequency-domain phenomenon. You can't understand it in the space-time domain. Some things are strictly space-time, and other things can be seen either way. You might be amazed at how useful it is to look at the visual world in the frequency domain. In fact, that's how our brain does it. Everything passes through a kind of Fourier transform before the higher visual processing centers do anything with it. Visual computation, for the most part, follows frequency analysis. You have to admit--The Foomeister was definitely the man.

But the best thing about his work, hands down...

was the beautiful equation he created:

Say hello to my little friend: the Fourier transform.