# Cool stuff

"Heheheh ... heheheheh ... hmhmhmhm ... cool ... heheheh" - Beavis

### The contraction mapping theorem, in limerick form

Inside a complete metric space,
We know that it is the case
That under the action
Of a contraction,
A single point stays in its place

(Unfortunately the non-empty condition resists attempts to cleanly incorporate it into the limerick. So we have to restrict the definition of a contraction to non-empty spaces to make this work :/)

### The Joy of Secs

log(sec2(x)) + log(cosec2(x)) = log(sec2(x) + cosec2(x))

### Isn't that backwards?

In C++, if a and b are bool variables, it's possible to evaluate (a implies b) by writing "a<=b"

(One way of thinking about this is: if a implies b, then b is at least as true as a)

### Euclid's Algorithm in C

`int hcf(int a, int b){return a?hcf(b%a,a):b;}`

### 22/7

This is in some sense unsurprising. Multiplying the whole thing out will result in a polynomial, plus (Ax+b)/(1+x2). A is 0 by symmetry, and B turns out to be -4. The integral of the polynomial bit must be a rational number, the whole answer must be small, so the rational number must be a good approximation to pi. Similar integrals give better approximations.

### Panama

A man, a plan, a canoe, pasta, hero's rajahs, a coloratura, maps, snipe, percale, macaroni, a gag, a banana bag, a tan, a cat, a mane, paper, a Toyota rep, a pen, a mat, a can, a tag, a banana bag again(or a camel), a crepe, pins, spam, a rut, a Rolo, cash, a jar, sore hats, a peon, a canal, Panama!

### Facetiously

...contains all the vowels exactly once, and in order.

### The Pizza Theorem

The volume of a pizza of radius z and height a is equal to pi z z a