VSFCF - CVM 1.1

# Brief History

In 1890, Peano ([Pe]), published the first description of a space-filling curve. (All space-filling curves are sometimes known as Peano curves; however, we will reserve this name just for Peano's original curve.) A year later Hilbert ([Hi]) published a description of another such curve, perhaps the simplest of all to describe. Hilbert divided a square into fourths, and connected their four centers as drawn in Figure 1 (upper left) to get the image of a map, h1, from the unit interval into the original square. Then these four squares are subdivided into fourths and the new centers are connected as in Figure 1 (upper middle) to get the image of h2. Continuing, the Hilbert curve will be the limit of the hn. (The image of the Hilbert curve is just a filled-in square; it's the rule that assigns a point of that region to each point of the unit interval that defines the mapping that is the Hilbert curve. For each n the image has finite length and hn describes a constant speed transit starting at the lower left.)

Figure 1. The first six stages of the classic construction of the Hilbert curve. The first three stages include, in black, a grid to aid in construction and viewing. Also available are an animated gif (15kb) version (use browser back button to return) and an animated speed controlled version. These are rather jerky, unilluminating animations since the process is discrete - there are no possible intermediate frames. Most of the other animations in this document will also consist of discrete frames, but of continuous processes; so we will be attempting to show a smooth continuously flowing event. Source code is available.

There were many descriptions given of other space-filling curves. In 1900 Moore ([Mo]) described a variation of Hilbert's curve. In 1904 Lebesgue ([Le]) linearly extended a space-filling map of the Cantor set to get a space-filling curve. In 1912 Sierpinski ([Si]) gave yet another example. In a slightly different vein, in 1903 Osgood ([Os]) described simple curves with positive area. Such curves contain no disks. But, as we will show here, they can naturally converge to any of the classical square filling curves with continuously varying area or dimension. In 1954 Wunderlich ([Wu]) introduced what are now known as iterated function systems to give perhaps the most elegant description of these space-filling curves. The use of iterated function systems has been advanced by the work of Barnsley ([Ba]). A more thorough history of space-filling curves than is given here can be found in Sagan ([Sag]).

Fractals perhaps started with Cantor's description of the Cantor set in 1883 ([Ca]). In 1905 von Koch ([Ko]) introduced his snowflake curve. The name fractal was coined by Mandelbrot ([Ma1]) in 1975, and he later described the Cantor curtain ([Ma2]) - the complete "filmstrip" of a one-dimensional movie showing the variation of a Cantor set from dimension 0 to 1 or measure 0 to 1.

Communications in Visual Mathematics, vol 1, no 1, August 1998.