Euler's constant *e* can be expressed very prettily as an infinite continued fraction, following the pattern [1 0 1 1 2 1 1 4 1 1 6 1 1 8 1] where the (3*n*+2)th term is 2*n* and all other terms equal 1. This describes the following fraction:

1 + 1
_______
0 + 1
_______
1 + 1
_______
1 + 1
_______
2 + 1
_______
1 + 1
_______
1 + 1
_______
4 + 1
_______
1 + 1
_______
1 + 1
_______
6 + ...

Euler was probably the first to discover this (he used the expansion to prove

*e*'s

irrationality, as well as that of its square). The zero up the top is of course somewhat redundant, however the alternative is to express the fraction as [2 1 2 1 1 4 1 1 6 1 1 8 1], which is not quite so homogenous a pattern and somewhat conceals the identity's

beauty. (This "zero trick" was invented by

Bill Gosper to smooth out what he saw as a glitch in the original representation.)