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,
h_{1}, 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 h_{2}. Continuing, the Hilbert curve will be the
limit of the
h_{n}. (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
h_{n} 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.
Copyright © 1998,
The Mathematical Association of America.
All rights reserved.
Created: 18 Aug 1998 ---
Last modified: Sep 30, 2003 6:18:13 PM
Comments to: CVM@maa.org