Homeomorphisms between circles and rectangles In our topology class we learn that in $\mathbb{R}^2,$ circles and rectangles are homeomorphic to each others.
I can understand the underline idea intuitively.
But can we find an explicit  homeomorphic between them?
If so how?
Also our professor said that, "we can describe any point in the rectangle $[0,1]\times[0,1]$ using a single coordinate."
I wonder how such thing is possible. 
As I think, for this we need a bijection between  $[0,1]\times[0,1]$ and some (closed?) interval in $\mathbb{R}.$
Can some one explain this phenomena? 

consider $S^1$ parametrized by $(\cos(t),\sin(t))$. Extend this vector from the origin to its first intersection with the square. You can explicate the first quadrant and then just argue by symmetry. For $0 \leq t \leq \pi/4$, you're going to intersect with the $x=1$ line segment, and for $\pi/4 \leq t \leq \pi/2$, the vector will intersect $y=1$ at $\cot(t)$. 
However, the real idea is given in the answer by Fimpellizieri.
