November 11, 2011

Unary Day

I wouldn't have spent this special day with unary sequence 11h1111s at 11/11/11 without a terrible pun, mentioning it as the anniversary of Attila, since it's "an invasion of the Huns" ("huns" for "ones", uttered with an awful accent); even made a Facebook page for that: Attila Fest. Not proud :) Yet this dull (not in the Ramanujan sense) sequence reminded me of two nice formula for the golden ratio (aka $\phi$, or the divine proportion): one rational, one radical. So it seems $\phi$ could be tamed (or approched) quite easily. Indeed, it is one of the simplest non-rational numbers, as a root of a basic equation of degree 2. Thus, far from being transcendental. Surprisingly, $\phi$ bears some kind of transcendence (in the religious sense) as it is, somehow, beyond the grasp of the human mind, meaning it cannot be approched easily in a "rational" way, i.e. worse that any other number, as stated in a theorem by Adolf Hurwitz (1856-1919). In the following formula, there exist infinitely many $m$ and $n$ for any irrational $\xi$, and the constant $\sqrt(5)$ cannot be improved, due to $\phi$.
The radical formula for $\phi$ above is the key: with divisions by ones, denominators increase very very slow. Contrary to common knowledge (in Age of Empires) that Huns are faster. Thus being, in think i'd better go back to a paper on applications of unary filters, instead of making dull puns.