Hutchinson's Theorem (1981)
See below for code This was the most up to date thing we proved in my maths degree. It's a lovely little theorem in the field of Fractal Geometry that enables you to create images like the ones above. But first some primers.... A fixed point of any map f:X\to X is a point x\in X such that f(x) = x. If X=\mathbb{R}^N we can define a contraction as a map f which brings pairs of points closer together, i.e. we can say f is a contraction if there's some \lambda < 1 such that for any x_1,x_2 we have |f(x_1)-f(x_2)| < \lambda |x_1-x_2|. Now, it's easy to see that if f is a contraction then it has a unique fixed point. All you have to do is note that for any x the following sequence converges x, f(x), f^2(x), f^3(x), ... Why's that? Well if we let \epsilon = |x-f(x)| then \epsilon\lambda^{n-1} is an upper bound for the distance between the n^{th} and n+1^{th} members of the sequence. Since $\sum \epsi...
