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# Crinkly Curves

Some curves are so convoluted they wiggle free of the one-dimensional world and fill up space

# Cantor’s Conundrums

Applications of space-filling curves are necessarily built on finite examples—paths one can draw with a pencil or a computer. But in pure mathematics the focus is on the infinite case, where a line gets so incredibly crinkly that it suddenly becomes a plane.

Cantor’s work on infinite sets was controversial and divisive in his own time. Leopold Kronecker, who had been one of Cantor’s professors in Berlin, later called him “a corrupter of youth” and tried to block publication of the paper on dimension. But Cantor had ardent defenders, too. Hilbert wrote in 1926: “No one shall expel us from the paradise that Cantor has created.” Indeed, no one has been evicted. (A few have left of their own volition.)

Cantor’s discoveries eventually led to clearer thinking about the nature of continuity and smoothness, concepts at the root of calculus and analysis. The related development of space-filling curves called for a deeper look at the idea of dimension. From the time of Descartes, it was assumed that in d-dimensional space it takes d coordinates to state the location of a point. The Peano and Hilbert curves overturned this principle: A single number can define position on a line, on a plane, in a solid, or even in those 11-dimensional spaces so fashionable in high-energy physics.

At about the same time that Cantor, Peano and Hilbert were creating their crinkly curves, the English schoolmaster Edwin Abbott was writing his fable Flatland, about two-dimensional creatures that dream of popping out of the plane to see the world in 3D. The Flatlanders might be encouraged to learn that mere one-dimensional worms can break through to higher spaces just by wiggling wildly enough.

# Bibliography

• Bader, M. 2013. Space-Filling Curves: An Introduction with Applications in Scientific Computing. Berlin: Springer.
• Bartholdi, J. J. III, L. K. Platzman, R. L. Collins and W. H. Warden III. 1983. A minimal technology routing system for Meals on Wheels. Interfaces 13(3):1–8.
• Dauben, J. W. 1979. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge: Harvard University Press.
• Gardner, M. 1976. Mathematical games: In which “monster” curves force redefinition of the word “curve.” Scientific American 235:124–133.
• Hilbert, D. 1891. Über die stetige Abbildung einer Linie auf ein Flächenstück. Mathematische Annalen 38:459–460.
• Moore, E. H. 1900. On certain crinkly curves. Transactions of the American Mathematical Society 1(1):72–90.
• Null, A. 1971. Space-filling curves, or how to waste time with a plotter. Software: Practice and Experience 1:403–410.
• Peano, G. 1890. Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36:157–160.
• Platzman, L. K., and J. J. Bartholdi III. 1989. Spacefilling curves and the planar travelling salesman problem. Journal of the Association for Computing Machinery 36:719–737.
• Sagan, H. 1991. Some reflections on the emergence of space-filling curves: The way it could have happened and should have happened, but did not happen. Journal of the Franklin Institute 328:419–430.
• Sagan, H. 1994. Space-Filling Curves. New York: Springer-Verlag.
• Sierpinski, W. 1912. Sur une nouvelle courbe continue qui remplit toute une aire plane. Bulletin de l’Académie des Sciences de Cracovie, Série A, 462–478.
• Velho, L., and J. de Miranda Gomes. 1991. Digital halftoning with space filling curves. Computer Graphics 25(4):81–90.

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