Eighty-eight Thousand, Four Hundred and Eighteen (More) Ways to Fill Space

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Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, 2008 12:28 p.m. norton.

1 Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, :28 p.m. norton.tex page 108 Eighty-eight Thousand, Four Hundred and Eighteen (More) Ways to Fill Space Anderson Norton Anderson Norton teaches mathematics and researches mathematics education at Virginia Tech. His interests include fractal geometry, students mathematical conjectures, and philosophy of mathematics. He seeks to engage mathematicians (especially math graduate students) in mathematics education research. As the name implies, space-filling curves possess the fascinating property that they fill regions of two-dimensional (or even higher dimensional) space with the continuous image of a line segment. Until the end of the nineteenth century, mathematicians considered such a feat impossible. How could a continuous function possibly transform a one-dimensional object into a two-dimensional object? In this paper we present Hilbert s space-filling curve and generalize it to a new class of space-filling curves by establishing a connection with open-faced rook s tours, which we define as a path that begins in one corner of an m m grid (such as a chess board) and, traveling vertically and horizontally only, ends in an adjacent corner, having hit each square exactly once. We will see that every open-faced rook s tour on an m m chessboard defines a unique space-filling curve, up to symmetry of the square. Thus finding open-faced rooks tours is equivalent to inventing new space-filling curves. Hilbert s construction Hilbert s curve is the limit of a sequence of curves of increasing length. The first few stages of Hilbert s construction are shown in Figure 1. Note how the first curve traverses the (side length 1/2) quadrants of the unit square in a certain order, moving clockwise from the lower-left quadrant (quadrant 1). The second curve moves through the (side-length 1/4) sub-quadrants of these quadrants using a scaled down version of the first curve. Overall, however, the second curve traverses the larger (side-length 1/2) quadrants in the same order as the first curve. Similarly, each successive curve follows the preceding curve around the unit square, traversing quadrants, sub-quadrants, sub-sub-quadrants, etc., in a manner that respects the order used by all the preceding curves in the sequence. We refer to this property as the nested squares criterion. In general, the nth curve in Hilbert s sequence implicitly partitions the unit interval into 4 n sub-intervals and the unit square into 4 n sub-squares. The nth curve maps the ith sub-interval [(i 1) 4 n, i 4 n ] into the ith sub-square, while respecting the mappings from intervals to squares defined by the previous curves. Figure 2 represents the mappings by using quaternary expansions for the end points of the sub-intervals, and by indicating the destination of those sub-intervals by labeling the sub-squares with the right end points of the respective sub-intervals. The nested-squares criterion was not stated explicitly by Hilbert. We formalize it here in order to generalize his construction. 108 c THE MATHEMATICAL ASSOCIATION OF AMERICA

2 Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, :28 p.m. norton.tex page 109 Figure 1. The first four curves in Hilbert s sequence. Figure 2. Hilbert s mapping of sub-intervals to sub-squares. VOL. 40, NO. 2, MARCH 2009 THE COLLEGE MATHEMATICS JOURNAL 109

3 Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, :28 p.m. norton.tex page 110 Theorem 1 (The nested-squares criterion). Let { f n } be a sequence of continuous functions from the unit interval to the unit square satisfying the following condition: for all integers n 1 and all ordered pairs of integers ( j, k), 1 j, k 2 n, there exists a unique integer 1 t 4 n such that f m ([(i 1) 4 n, i 4 n ]) [( j 1) 4 n, j 4 n ] [(k 1) 4 n, k 4 n ] whenever m n. Then the limit of the sequence { f n } exists and is a continuous, spacefilling curve. Proof. We first prove that the sequence { f n } is uniformly Cauchy. Given ɛ> 0, we simply choose N such that 2 2 N <ɛ. This guarantees that, for all x, f n (x) f m (x) <ɛwhenever m, n N because, according to the nested-squares criterion, f n and f m map every x to the same sub-square with diagonal length 2 2 N. Hence { f n } is uniformly Cauchy, and from this fact it follows that the limit function exists and is continuous. We next prove that f is onto. The nested squares criterion guarantees that f n will hit every sub-square [( j 1) 4 n, j 4 n ] [(k 1) 4 n, k 4 n ]. This means that it will get within 2 2 n of every point in the unit square. Because f is the limit of { f n },theimageof f gets arbitrarily close to every point in the unit square. Because f is a continuous mapping from a compact set (namely, the unit interval), its image is closed in the unit square, and a closed subset that gets arbitrarily close to every point in the set actually contains the entire set [4]. Therefore, the image of the unit interval under f is the unit square. Generalizing Hilbert s construction Hilbert s curve relies on creating 4 n new sub-squares at the nth stage in the sequence. However, we could choose to partition the unit square into 9 n sub-squares, or m 2 subsquares, i.e., any perfect-square number of sub-squares. If we then choose an openfaced rook tour of an m m grid and use it as the first function in a sequence of functions satisfying a suitable generalization of the nested squares criterion, we will have a new space-filling curve. Figure 3 gives two examples. In both examples, the first curve in the sequence establishes a pattern and order for traversing the nested squares. Each successive curve respects the order established by previous curves in the sequence, while following a transformation of that pattern within the new sub-squares. In fact, these transformations are symmetries of the square (elements of D 4 ). For example, consider the second sequence illustrated in Figure 3. The second curve in the sequence hits each of the 25 sub-squares in the order established by the first curve, while following some transformation of the same 12 pattern within each of 25 2 new squares. Note that there is only one way to accomplish this because the starting and ending corners in the new squares are determined by the first curve and those corners uniquely determine the appropriate symmetry transformation of the 12 pattern. In other words, the pattern established by the first curve in the sequence defines a unique way to satisfy the nested squares criterion, thus defining a unique space-filling curve. Enumerating rook s tour space-filling curves The reader can now generate at least as many space-filling curves as there are openfaced rook s tours. But how many such rook s tours exist within an m m grid. The 110 c THE MATHEMATICAL ASSOCIATION OF AMERICA

Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, 2008 12:28 p.m. norton.tex page 111 Figure 3.

4 Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, :28 p.m. norton.tex page 111 Figure 3. Sequences of curves using 4 4(toprow)and5 5 (bottom row) nested squares. first curve in Hilbert s sequence is the one and only open-faced tour for m = 2. Nick Loehr [2], a colleague at Virginia Tech, has written a computer program that enumerates all of these tours. It turns out that m = 3 yields 2 tours; m = 4 yields 8 tours; m = 5 yields 86 tours; m = 6 yields 1,770 tours; and m = 7 yields 88,418 tours. Searching this sequence in the On-Line Encyclopedia of Integer Sequences [7], it turns out that in 1865, Möbius [3] identified this pattern as the number of walks from the NW corner of an n n array to the SW corner. Thus, Möbius unwittingly enumerated the entire class of the space-filling curves described here, three decades before Peano constructed the first space-filling curve! Closing remarks The discovery of space-filling curves raised topological questions that puzzled mathematicians for centuries. For example, if curves can fill area, should they be considered one-dimensional or two-dimensional? Giuseppe Peano [5] started the trouble in 1890 with an arithmetic construction involving the decimal expansions of points in the unit interval and points in the unit square. Hilbert followed with his geometric construction a few years later. Hans Sagan [6] has elaborated on both constructions and many others in his excellent book, Space-Filling Curves. This book also demonstrates how curves can be created to fill space in higher dimensions. Beyond providing for counter-intuitive fascinations, space-filling curves actually contribute to technological development. For example, new and improved halftoning techniques in digital imaging rely upon space-filling curves [8]. Digital gray-scale images (such as those produced by a laser printer) consist of an array of pixels in one of two states, on or off (black or white). In order to produce the illusion of shades of gray, halftoning algorithms arrange pixels so their average darkness fits the intended image within each small region of the plane. Arranging the pixels along a space-filling curve provides an efficient means of clustering the pixels to achieve appropriate darkness VOL. 40, NO. 2, MARCH 2009 THE COLLEGE MATHEMATICS JOURNAL 111

Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, 2008 12:28 p.m. norton.tex page 112 without altering the apparent contours of the original image.

5 Integre Technical Publishing Co., Inc. College Mathematics Journal 40:2 December 13, :28 p.m. norton.tex page 112 without altering the apparent contours of the original image. Moreover, a space-filling curve can accommodate any specified resolution. Acknowledgments. The space-filling curves illustrated in Figures 1 and 3 were generated using Mathematica [10] and an adaptation of Stan Wagon s Turtle Graphics code [9]. Thanks also to Ted Shifrin, Ed Azoff, Nick Loehr, Ezra Brown, Bill Floyd, Michael Henle, and the anonymous reviewers for their contributions to this article. References 1. D. Hilbert, Über die stetige abbildung einer linie auf ein flächenstück, Math. Ann. 38 (1891) N. Loehr, Rook s Walks: a computer program for generating numbers of rook s walks, Virginia Tech, A. F. Möbius, Gesammelte Werke, University of Michigan Library, Ann Arbor, J. Munkres, Topology: A First Course, Prentice Hall, Englewood Cliffs NJ, G. Peano, Sur une curbe qui remplit toute une aire plane, Math. Ann. 36 (1890) H. Sagan, Space-Filling Curves, Springer-Verlag, New York, N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences; available at com/~njas/sequences/a000532, L. Velho, and J. Gomes, Digital halftoning with space filling curves, Computer Graphics 25 (1991) S. Wagon, Mathematica in Action, 2nd ed., Springer-TELOS, New York, S. Wolfram, Mathematica TM ; available at c THE MATHEMATICAL ASSOCIATION OF AMERICA

Section 44. A Space-Filling Curve

44. A Space-Filling Curve 1 Section 44. A Space-Filling Curve Note. In this section, we give the surprising result that there is a continuous function from the interval [0,1] onto the unit square [0,1]