Theorem. There exists a continuous curve in that passes through every point of the unit square .
Proof. Define by
Extend to all of by making periodic with period .
By the Weierstrass M-test, both and converges uniformly on . Moreover, it is continuous on . Now define and let denote the image of the unit interval under . We show .
Observe and . Hence . Let . Write
with . Now let
Then since . Now we show and . We show for each … If we can show this, then we have , and this gives , .
Since has period 2, we have
If , then and hence . So in this case. If , then and hence . Therefore, . This proves , . So .
Meaning of this theorem. Observe that is continuous. has dimension 1 and has dimension 2. By the above theorem, the continuity does not guarantee the dimension of spaces. The above curve we constructed is nowhere differentiable. This is proved by Alsina.
It seems that this kind of curve is not useful, but it has quite a lot of applications. One can use this kind of fact to probability theory, topology, etc. It has an application to industry, e.g. Google map.
- J. Alsina, The Peano curve of Schoenberg is nowhere differentiable, Journal of Approximation theory, Vol. 33 (1), 28– 42.
- T. Apostol, Mathematical Analysis
- C. S. Perone, Google’s S2, geometry on the sphere, cells and Hilbert curve