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...
