LETTERS TO THE EDITORS
To the Editors:
In the May–June Computing Science column “Crinkly Curves” by Brian Hayes, I found the concept of constructing a curve that fills a space interesting in that the link between points on the space-filling curve and the points on a line is not easily recognizable and seems to be counterintuitive.
For the Hilbert curve, the basic unit is a square grid of four smaller squares. Since these units are to be sequenced to make larger collections of squares, it seems that the Hilbert curve should enter a unit on one side and leave it on another. Thus, the unit square for the Hilbert curve should have a line entering the bottom side of the bottom left square, proceeding through the centers of the four squares of the unit and exiting out the right side of the bottom right square. The basic unit, after rotation or reflection, could be tiled to form larger sequences, and there would be no need to add the connecting lines between the sequenced units. The physical size of a sequence would have to decrease as the number of sequences increases so that each sequence would fit into a unit square.
But actually, we can start with two smaller squares: one with a line passing straight through the center, entering and exiting on opposite sides; and another with a line making a turn at the center, entering and exiting on adjacent sides. The curves in these two squares could be assigned a direction; and then the squares could be rotated, reflected and finally sequenced to form the basic four-unit square for the starting Hilbert curve.
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Brian Hayes responds:
Dr. Csuha is correct that the lines linking subunits of the Hilbert curve can be eliminated, but only in the limiting case where the number of subunits becomes infinite and their size goes to zero. In that circumstance, the connecting lines have zero length, and every point of the curve is the vertex of a right angle. As I said in the column, the curve becomes “all elbows.”
Let me take this opportunity to correct an error in the column pointed out to me by Doug Robertson of the University of Colorado. I wrote that the Hilbert Curve “is not a fractal because its dimension is not a fraction.” But the defining property of a fractal is not that the dimension be a noninteger. A fractal is an object whose “fractal dimension” exceeds its “topological dimension.” The Hilbert curve has a fractal dimension of 2, because it fills an area of the plane; the curve’s topological dimension is 1, because it is a line segment.