26 September 2007

Fun with images! (Mostly about math tattoos.)

Stokes' theorem graffiti. In a bathroom. Only in Camberville. (I would have said "only in Cambridge", but this particular graffito is actually in Somerville. Diesel Cafe, to be precise; there was a summer where I spent a lot of time there doing math, so I'm not surprised.)

and a tattoo of the Y combinator, which is part of this collection of images of science tattoos. Other mathematically inspired tattoos in that collection are:

I can think of others. An ex-girlfriend had an infinity symbol tattooed on her (although I believe that was more of a literary thing than a scientific one). A friend of mine has

¬ (p ∧ ¬ p))

tattooed on his leg. A friend of a friend in college had the Taylor series for sine on her arm. It's written out, as

x - {x^3 \over 3!} + {x^5 \over 5!} - {x^7 \over 7!} + \cdots

except with exactly enough terms to go once around her arm; it's not in the more compact form

\sum_{n=0}^\infty {(-1)^n x^{2n+1} \over (2n+1)!}.

I don't have any tattoos; I can't think of anything I'd want permanently inked on me. But if I were to get a tattoo, the equation

\prod_{i=0}^\infty \left( 1 + z^{2^i} \right) = {1 \over 1-z}

would be up there. (I had it written on my wall in college.) I've heard it called the "computer scientist's identity"; it's an analytic version of the fact that every positive integer has a unique binary expansion. The left-hand side expands to


and you can imagine expanding that out; then you get something like

1 + z1 + z2 + z2+1 + z4 + z4+1 + z4+2 + z4+2+1 + ...

where each sum of distinct powers of two appears as an exponent exactly once. The right-hand side is the geometric series

1/(1-z) = 1 + z + z2 + z3 + ...

and in order for these two expressions to be equal, the coefficient of zn in each of them has to be equal. All the coefficients on the right-hand side are 1; thus the coefficients on the left-hand side must be too. That means each nonnegative integer appears exactly once as an exponent, i. e. as a sum of distinct powers of two. The fact that one can disguise this combinatorial fact as a fact about functions from C to C -- and then apply the methods of complex analysis, although they're not incredibly useful in this particular case -- is a fascinating example of the interplay between the discrete and the continuous.


John Armstrong said...

If you like that sort of thing, then you must love the interchange between the sum and product formulas for the Riemann zeta function.

Personally, I never found it all that interesting until Baez wrote about the "free Riemann gas", which is a second-quantized field theory of weird particles called "primons".

walt said...

I saw Stokes' theorem grafitti'd on the wall at a bar in Siem Riep, Cambodia.