The $n$-dimensional Peano Curve

Abstract

One of the most startling mathematical discoveries of the nineteen century was the existence of plane-filling curves. As is well known, the first example of such a curve was given by the Italian mathematician Giuseppe Peano in 1890. Subsequently, other examples of plane-filling curves appeared, with some of them having $n$-dimensional analogues. However, the expressions of the coordinates of the Peano curve are not easily extendable to arbitrary $n$ dimensions. In fact, the only known extension of the Peano curve to an $n$-dimensional space-filling curve, made by Stephen Milne in 1982, is rather geometric and makes it difficult to establish basic properties of these curves, as continuity and nowhere differentiability, as well as more advanced properties, as uniform distribution of the coordinate functions. Here, we will introduce in a completely analytical way the $n$-dimensional version of the Peano curve. More precisely, for a given integer $n\ge 2,$ we will define (by means of identities) the $n$ coordinate functions of a continuous and surjective map from a closed interval to the unit $n$-dimensional cube of $\mathbb{R}^n,$ which, for the particular case $n=2,$ agrees with the original Peano curve. With this description, as we shall see, one can easily establish all the properties we mentioned above, and also calculate the Hausdorff dimension of the graphs of the coordinate functions of this curve.