[contravariant]

Well some numbers expose patterns when written as a continued fraction. In particular e becomes pretty regular.

You can modify the continued fraction slightly to make pi regular as well, but the normal continued fraction sequence doesn't give much of an insight. Other than the fact that 3 + 1/(7 + 1/16)) is a damn good approximation (7 digits, pretty good for something that can be written using only 4 digits total: [3;7,16]).

[azeemba]

Phi/golden ratio also has a cool continued fraction sequence...it's only 1's all the way down

[thechao]

Larger integers in continued fractions mean you get 'more information' out of the limb. That means not only is Phi "1s all the way down" it is the continued fraction that converges the slowest. If you've ever used the iterated matrix product (which is a specific edge-case of the algorithm to convert continued fractions to decimals), you'll know how slow it is!

[contravariant]

Square roots in general have periodic patterns. Which isn't too surprising, something like z = a/(b+cz) is pretty much a quadratic equation after all.

But phi is indeed especially interesting because of what its sequence implies for rational approximations of phi.