A space-filling curve

In mathematics classes where the natural numbers are studied, several surprising results can be shown about the relative “sizes” of infinite sets, such as the fact the there is a oneto-one mapping between the natural numbers N and the even numbers, showing that they are both countably infinite, with the “same size”. Similar logic also shows that there is a mapping from the natural numbers N onto the product space N×N of all pairs of natural numbers. When dealing with real numbers, it should therefore be somewhat expected that there is a mapping from the unit interval [0, 1] onto the unit square [0, 1]2. However, what is surprising is that there continuous mappings of this form. These are frequently referred to as space-filling curves, or Peano curves, after the Italian mathematician Giuseppe Peano (1858–1932). Here, a specific space-filling curve due to Schoenberg [1] is described. As can been seen in the graphs below, this curve has a complex and overlapping structure, and it is possible to construct curves with a much more regular structure. However, the curve by Schoenberg has the advantage of being relatively straightforward to analyze. To begin, a continuous function f : R → [0, 1] is introduced, which is even and is periodic with period 2, so that f (x) = f (x + 2) for all x. On the interval from [0, 1],