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Introduction

Common assumptions about the concept of dimension seem to be violated by the existence both of fractals and of space-filling curves. Here we use a continuum of iterated function systems (IFS's with a time parameter) to produce families of fractal curves that in the limit become a space-filling curve. This procedure is especially useful in viewing a space-filling curve not as the limit of a discrete sequence of polygonal, one-dimensional curves (as is usually done), but rather as the limit of curves that continuously increase in dimension (or in area) to that of the space-filled region. This leads naturally to the visually elucidating production of animations showing the construction of space-filling curves.

After some brief comments on the history of space-filling curves and fractals, we discuss the definition of dimension. Also, we give a brief definition of iterated function system (IFS). Then we show how to get a family of curves including the (fractal) von Koch snowflake curve that in the limit give the (space-filling) Sierpinski-Knopp curve. In this family the dimensions increase smoothly from 1 through log34 (the dimension of the standard snowflake curve, about 1.26186 ) to 2. Then we consider the Hilbert curve, introducing the idea of connecting segments to show how a variation of IFS's can give us a description of the classical (Hilbert's original) approximating curves. We produce a family of curves that end at the Hilbert curve with dimension increasing from 1 to 2. Additionally, we look at the Cantor set (the original fractal) and discuss it in this context. We give a family of IFS's that produces Cantor sets of smoothly increasing dimension from 0 through log32 (the dimension of the standard Cantor set, about 0.63093) to 1. Further, Peano's original space-filling curve is given this same treatment as is a space-filling curve due to Sierpinski. As a final example of fractals converging to planar space-filling curves, we come back to the Sierpinski-Knopp curve. Instead of viewing it as a limit of snowflakes, we use IFS's with attractors and connecting segments giving converging curves. Then we briefly look at two other ways of viewing space-filling curves: using area increasing curves and three-dimensional visualizations, and discuss the drawing algorithms used to produce our pictures. Also, we show some generalizations to curves that fill in 3 dimensional space. Finally, some philosophical questions involved in representing fractals are discussed.

All of these are demonstrated by showing several frames of "movies": figures that are animated gif-format images. By analogy with the artist Calder's stabiles and mobiles, perhaps our figures should be differentiated as stafigs (which appear in this level of this electronic document) and mofigs (which are accessed by clicking on links given in the stafig descriptions, and require the browser back button to return). To best view with a browser, so that the complete figures are visible, one may need to hide the toolbar or to scroll to center the movie. There are also java versions that allow the user to controll the speed of the animation. We include links to source codes and to software sites.

The author would like to acknowledge Jenny Harrison and William D. Withers with whom discussions helped motivate some of these ideas. Also, thanks are due to Thomas J. Sanders who prepared the java code that was adapted to show speed controlled animation.


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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: 18 Aug 1998 23:59:59
Comments to: CVM@maa.org