Bijection between closed and open disks (of the same radius)?

Question

I am contemplating ways of defining a bijection between closed and open disks of the same radius. By these terms I mean http://en.wikipedia.org/wiki/Disk_(mathematics)

I am aware that the technique of perceiving disk as a set of circles and then mapping circles of the given radius as follows: $$\forall n \in \mathbb{N}: \quad \frac{R}{n} \rightarrow \frac{R}{n+1}$$ appears to work, but how would you personally do it? I mean do you happen to know other elegant and fancy ways?

Answer 1

I would say first that the open disk is homeomorphic to the whole plane and the closed disk is homeomorphic to the standard square. Then finding the bijection is the same thing as finding a bijection between $\mathbb R$ and $[0,1]$.

Answer 2

An alternative is to find a bijection between $^[0,1)$ and $[0,1]$ and then use polar coordinates.