*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:

there exists a unique beth (marred only by the fact that it's not true;- something with a pentagon titled golden ratio tat, a logarithmic spiral, and the Greek letter φ (why is the golden ratio so popular for tattoos?);
- a Mobius strip
- 2
^{5}

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

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

.

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

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

(1+z

^{1})(1+z

^{2})(1+z

^{4})(1+z

^{8})...

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

1 + z

^{1}+ z

^{2}+ z

^{2+1}+ z

^{4}+ z

^{4+1}+ z

^{4+2}+ z

^{4+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 + z

^{2}+ z

^{3}+ ...

and in order for these two expressions to be equal, the coefficient of z

^{n}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.

## 2 comments:

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".

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

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