Let $D$ be the unit (closed) disk $$D=\{(r,\theta)\mid 0\le r\le 1\}\subseteq\mathbb{R}^2$$ and let $H$ be the subset of $D$ consisting only of the rays from the origin with irrational slopes $$H=\{(r,\theta)\mid \theta\notin\mathbb{Q}\cap [0,2\pi)\}$$ kind of like the "spikes" of a hedgehog.
There is a bijection $D\to H$ because they have the same cardinality but 1) I can't find any and 2) Even If I could, I suspect it wouldn't be simple to describe.
So the question is: can you find a bijection that is easy to describe and preferably "nicest".
I guess the question is equivalent to finding a bijection $[0,2\pi)\to[0,2\pi)\setminus\mathbb{Q}$ but can't do that either.
Thanks!
Define $f\colon[0,2\pi)\to [0,\pi)\setminus \Bbb Q$ as $$ f(x)=\begin{cases}(x+7)\bmod 2\pi&\text{if }x\equiv a+7b\pmod{2\pi}\text{ for some }a\in\Bbb Q\cap[0,2\pi), b\in\Bbb N_0\\x&\text{otherwise}\end{cases}$$
Note that if $x$ is of the form mentioned, then so is $f(x)$, and only if it is of this from with $b=0$ (and so necessarily $k=0$) it is not $f(x')$ for some other $x'$.