Self-Similar Structure in Hilbert's Space-Filling Curve

Abstract

Hilbert's space-filling curve is a continuous function that maps the unit interval onto the unit square. The construction of such curves in the 1890s surprised mathematicians of the time and led, in part, to the development of dimension theory. In this note, we discuss how modem notions of self-similarity illuminate the structure of this curve. In particular, we show that Hilbert's curve has a basic self-similar structure and can be generated using what is called an iterated function system, or IFS. Furthermore, its coordinate functions display a generalized type of self-similarity called digraph self-affinity and may be described using an appropriately generalized iterated function system. The notions of self-similarity used here are described in the text by Edgar [2, Ch. 4]. The definition of a digraph IFS was originally formulated in a research paper by Mauldin and Williams [4], although similar ideas have appeared elsewhere. The author has published a Mathematica package implementing the digraph IFS scheme [6]. A broad introduction to space-filling curves may be found in the book by Sagan [7].