A Geometrization of Lebesgue's Space-Filling Curve

Hans Sagan

1. Introductory Remarks

G. Cantor demonstrated in 1878 that any two finite- dimensional smooth manifolds, no matter what their re- spective dimensions, have the same cardinality [1]. This is true, in particular, for the interval I = [0, 1] and the square Q -- [0, 1] • [0, 1], meaning that there exists a bijective map from I onto Q.

The question arose almost immediately whether or not such a mapping can possibly be continuous. E. Netto put an end to such speculation by showing in 1879 that such a bijective mapping is, by necessity, discontinuous [8]. Is it then, it was asked, still possible to obtain a contin- uous surjective mapping if the condition of bijectivity is dropped? G. Peano settled this question once and for all in 1890 by constructing the first "space-filling curve" [10]. (A "space-filling curve" is a continuous map from I to E n (n > 2) whose image has positive n-dimensional Jordan content [11].) Other examples followed: [3, 6, 14].

However, this was not the end of it because the fol- lowing related question arose: Whereas the interval I cannot be mapped continuously and bijectively onto an n-dimensional set of positive Jordan content (such as Q in E2), can it be done with an image set of positive outer measure? In other words, are there Jordan curves (continuous injective maps from I into E n) with positive n-dimensional outer measure? There are indeed, as W. F. Osgood showed in 1903 [9] when he constructed a one- parameter family of such curves. In fact, Osgood's curves are Lebesgue measurable. It is reasonable to assume that word of the Lebesgue measure had not reached Harvard by Thanksgiving of 1902 when Osgood submitted his paper for publication, because Lebesgue's pivotal thesis only appeared that year in the Annali di Matematica pura et applicata and there was no airmail. One does wonder, however, why Osgood did not make use of the Borel measure, which would have given him a stronger result. The limiting arc of Osgood's family is Peano's space- filling curve. This is not a coincidence. Peano's ingenious result undoubtedly inspired Osgood's construction.

Jordan curves with positive Lebesgue measure are now called Osgood curves.

In 1917, K. Knopp constructed another family of Os- good curves with Sierpifiski's space-filling curve as its limit [4, 13, 14].

Apparently unaware of this earlier w o r k T. Lance and E. Thomas, in a recent note, developed the same idea, albeit by a different approach [5]. It is the purpose of this note to put Osgood's and Knopp's approach in jux- taposition to the one by Lance and Thomas and to point out how the latter narrowly missed obtaining Lebesgue's space-filling curve as the limit of their family of Osgood curves and therewith the opportunity to put the first ge- ometric generating process of Lebesgue's space-filling

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~)1993 Springer-Verlag New York 37

curve on record. Geometric generations of the space- filling curves by Peano,Hilbert, and Sierpifiski have been known as long as the curves themselves in the cases of Hilbert's and Sierpifiski's curves and almost as long in the case of Peano's curve [3, 7, 12, 13]. Besides being of interest by themselves, such geometric generations lead to uniformly convergent sequences of approximat- ing polygons, which, in turn, lead to simple proofs of the continuity of the map. In the case of the Lebesgue curve, in particular, such a proof turns out to be much simpler than the conventional proof that is based on the structure of the Cantor set (such as the one in [2]).

2. The Curves by Osgood and Knopp

Osgood's construction of a family of Jordan curves with positive Lebesgue measure consists in the successive re- moval of grate-shaped regions from squares, starting out with the unit square and proceeding as indicated in Fig-

ure 1 for the first two steps. The shaded squares are what is left after each step. Starting with square $1, they are then connected by "joins" as indicated by the bold line segments in Figure 1. The dimensions of the grate- shaped regions can be chosen so that the sum of their areas tends to some positive ,k < 1. If An denotes the point set consisting of the 9 n squares and 9 '~- 1 joins that are obtained at the nth step and d(An) its Jordan content, then the set C = N~_lAn that is obtained af- ter infinitely many steps has Lebesgue measure #(C) = l i m n ~ d(An) = 1 - ,~ > 0. It represents a Jordan curve [9] that may be parametrized as follows: Dividing the interval I into 17 congruent subintervals and excluding the even-numbered subintervals without beginning and endpoint, and repeating the process for each of the re- maining 9 closed subintervals, and then again for each of the remaining 81 closed subintervals, and then again and again, ad infinitum, generates a Cantor-type dis- continuum F17. At the first step, the remaining 9 closed

Figure 1. Osgood's construction.

Figure 2. Construction of Peano's space-filling curve.

38 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO, 4,1993

Figure 3. Knopp's construction of Osgood curves.

intervals are mapped into the squares S1, 82 . . . . . $9 and the excluded 8 open intervals are mapped linearly onto the joins (without beginning points and endpoints) from Si to $2, $2 to $3 . . . . . $8 to $9, and the process is re- peated ad infinitum. The complement of P17 is mapped bijectively onto the set of all joins, and r17 is mapped onto N~=I Qn, where Qn represents the 9 n squares of the nth it- eration. (Osgood mapped the excluded even-numbered closed subintervals onto the joins with beginning points and endpoints and the remaining odd-numbered open intervals into the squares. We have deviated from his construction with a view toward what we are going to do in Section 3.)

In the limit with & ~ 0, Peano's space-filling curve is obtained. The first two steps in the construction of Peano's curve are illustrated in Figure 2 where the bold polygonal lines indicate the order in which the squares have to be lined up. Joining the beginning points of these polygonal lines to (0, 0) and their endpoints to (1, 1) by straight line segments yields the approximating poly- gons we mentioned in Section 1. (Another notion of ap- proximating polygons may be found in [7, 12, 13].) Com- pare Figure 2 with Figure 1. Since consecutive squares now have an edge in common, the injectivity of the map- ping is lost. This is to be expected because manifolds of different dimensions cannot be homeomorphic.

Knopp's construction of a family of Osgood curves consists in the successive removal of triangular regions

from an initial triangle as indicated for the first four steps in Figure 3, where the shaded triangles are the ones that are left after each step. (The initial triangle T need not be a right isosceles triangle.)

At the first step, we remove a triangle of area rim(T), where re(T) is the area of the initial triangle T and where rl C (0~ 1), to be left with two triangles To, T1 of a com- bined area re(T)(1 - rl). From To and T1, we remove triangles of area r2m(To) and r2ra(T1) for some r2 E (0, 1) to be left with four triangles Too, To1, Tlo, and Tu of a combined area re(T)(1 - rl)(1 - r2), etc. In the limit, we obtain a point set C of Lebesgue measure #(C) = re(T) II~_ 1 (1 - rk). If we choose the rk such that E~~ rk converges, then #(C) is positive. If, at each step, the trian- gles that are to be removed are placed judiciously, then all dimensions of the remaining triangles tend to zero and the remaining triangles shrink into points. If the interval [0, �89 (all numbers 020(22(23...) is mapped into To and [�89 1] (all numbers 021(22(23...) into T, so that [0, �88 (all numbers 0200(23(24...) is mapped into Too, [�88 �89 (all num- bers 0201(23(24 . . .) into To1, [�89 3] (all numbers 0210(23(24...) into T1 o, and [ 3, 1 ] (all numbers 0~ 11 (23 a4..-) into T11, etc., with the points common to two adjacent intervals being mapped into the vertices common to the corresponding adjacent triangles, we see that the mapping from [0, 1] to C is bijective and continuous and C is an Osgood curve of Lebesgue measure #(C) > 0.

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993 39

If we choose, for example, as initial triangle a right isosceles triangle of base 2 [and hence, m(T) = 1] and rk = 1/4k 2, k = 1,2, 3 , . . . , we obtain from Weierstrass's factorization theorem that the corresponding Osgood curve has Lebesgue measure 2/Ir. Setting rk = r / k 2 in- stead and taking the limit as r --* 0, a space-filling curve, namely, the Sierpifiski curve, is obtained because the combined area of the removed triangles shrinks to zero (see also [13]). Since adjoining triangles are then not only joined at vertices but also along edges, the injectivity of the mapping is lost. (See also Fig. 4, where the bold poly- gons indicate the order in which the triangles have to be taken, and compare Figures 3 and 4. These polygo- nal lines, when extended by line segments to entry point and exit point, represent approximating polygons that converge uniformly to the Sierpifiski curve.)

3. The Lance-Thomas Curve and the Lebesgue Curve

Instead of removing grates from squares or triangles, Lance and Thomas, by contrast, remove cross-shaped regions from squares as we have indicated for the first two steps in Figure 5 [5]. They connect the remaining squares by joins as indicated in Figure 5 by bold line segments. As in the preceding cases, the dimensions of the regions that are to be removed may be chosen so that the sum of their areas tends to some positive ,~ < 1. An Osgood curve of Lebesgue measure 1 - ,~ is obtained.

Its parametrization may be accomplished as in the case of Osgood's original example, using instead of a Cantor- type discontinuum the Cantor set itself, mapping the ex- cluded (open) middle thirds linearly onto the joins and the remaining closed intervals into the squares. Specifi- cally, at the first step, the closed intervals [0, 1/9], [2/9, 1/3[, [2/3, 7/9], [8/9, 1] are mapped into the squares S1 to $4 with the endpoints going into the appropriate cor- ners and the open intervals (1/9, 2/9), (1/3, 2/3), (7/9, 8/9) linearly onto the joins (without beginning points and endpoints) from S1 to $2, 82 to $3, and 83 to 84. The process is continued ad i n f i n i t u m to obtain a continuous bijective map from I onto n~= 1 An, where An denotes the set consisting of the 4 n squares and 4 n - 1 joins that are obtained at the nth iteration. As in the case of Osgood's example, Nn~ An is Lebesgue measurable with measure limn--,~ J ( A n ) = 1 - )~ > O.

While every part of Knopp's Osgood curve is again an Osgood curve, this is not the case for the examples by Osgood and Lance and Thomas because of the presence of joins. Because of this, Knopp leveled in [4], p. 109, footnote 2, some justifiable criticism at Osgood's con- struction. (In the same footnote, he dismisses an attempt by Sierpi~ski in [15] to construct a curve without this shortcoming as too complicated.) This criticism applies to the construction by Lance and Thomas as well, and it could be viewed as a throwback to Osgood's original attempt, were it not for the fact that a slight modifica- tion of their construction yields Lebesgue's space-filling curve as the limit.

Figure 4. Generating the Sierpifiski curve.

40 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993

Figure 5. Generating the Lance-Thomas curve.

Lebesgue's space-filling curve is defined on the Cantor set

F = { 0 3 2 a 1 2 a 2 2 a 3 . . . : a j = 0 or 1}

by (1)

x = O 2 a l a 3 a 5 . . . , Y = O 2 a 2 a 4 a 6 . . . .

and on the complement F c = [0,1]\F of the Cantor set by linear interpolation (see [6] or [11]). If one re- places the joins in Figure 5 by the ones in Figure 6 and starts with the lower left corner, one obtains Lebesgue's space-filling curve as the limiting arc as ,~ --* 0. This may be seen as follows: By our construction, the in- terval [0, 1/9] (all numbers of the form 0 3 0 0 a 3 a 4 a s . . . )

is mapped into $1 in Figure 7 and the points of $1 have coordinates (020b25354. . . ~ 020c2c3c4...). The inter- val [2/9, 1/3] (all numbers 0302a3a4...), is mapped into $2 with points (0~0b2b364..., 021c2c3c4...), the inter- val [2/3, 7/9] (all numbers 0 3 2 0 a 3 a 4 a 5 . . . ) into $3 with ( O ~ 1 6 2 6 3 6 4 . . . , 0 2 0 c 2 c 3 c 4 . . . ) and [8/9, 1] (all numbers 0 3 2 2 a 3 a 4 a 5 . . .) into S4 with ( 0 2 1 b 2 b 3 b 4 . . . , 0 2 1 r 1 6 2 1 6 2 . . ) .

The process is to be repeated with the intervals [0, 1/81], [2/81, 1/27] . . . . . [80/81, 1] (all numbers of the type 0 3 0 0 0 0 a 5 a s a 7 . . . ~ 0 3 0 0 0 2 a s a 6 a 7 . . . , . . . , 0 3 2 2 2 2 a 5 a s a 7 . . . )

and the squares S~j within each of the squares Si (i, d = 1,2,3,4) with points (0~00b364b5 . . . . 0200c3c4c5...), ( 0200b3b4b5 . . . , 0 2 0 1 c 3 c 4 c 5 . . . ) , . . . , (0211b3b4b5..., 0 ~ 1 1 c 3 c 4 c 5 . . . ) , etc. This process, continued a d i n f i n i t u m ,

demonstrates that the mapping satisfies (1). Because = 1/3 = 0~02 is mapped by (1) into the point (1/2, 1)

and t = 2/3 = 032 into (1/2, 0), the limiting position of the join from the exit point of the second square in Figure

Figure 6. Recursion operator leading to Lebesgue's curve.

6 to the third square, as & ~ 0, represents the linear inter- polation on (1/3, 2/3) C FC; because t = 1/9 = 03002 is mapped into (1/2, 1/2) and t = 2/9 = 0302 into (0, 1/2), the limiting position of the join from the first square to the second square represents the linear interpolation on (1/9, 2/9) c FC; because t = 7/9 = 03202 has the image (1, 1/2) and t = 8/9 = 0322 the image (1/2, 1/2), the limiting position of the join from the third square to the fourth square represents the linear interpolation on (7/9, 8/9) c pc, etc. (See also Fig. 6.) Repeating this argument a d i n f i n i t u m reveals the limiting positions of the joins to

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993 41

represent the linear interpolation on F c as called for by Lebesgue's definition.

We can now utilize this geometric generation of the Lebesgue curve to construct approximating polygons as follows: In each square, we join entry point and exit point by a straight line (diagonal) and leave the joins as they are as we have indicated in Figure 8 for the first two steps. (In our illustration, we have rounded off some corners to prevent the polygon from bumping into itself and obscuring its progression.) These polygons are ap- proximating polygons in the conventional sense because within each square, the distance from the polygon to the Lebesgue curve is bounded above by the length of the diagonal of the square, namely, by 21/2-n, and the poly- gons coincide with the Lebesgue curve along the joins.

Hence, they form a sequence that converges uniformly to the Lebesgue curve, the continuity of which is thus established.

In conclusion, let us note that Osgood's limiting curve, namely, the Peano curve, and Knopp's limiting curve, namely, the Sierpifiski curve, are nowhere differentiable, whereas the limiting curve of the Lance-Thomas family is, just as the Lebesgue curve, differentiable a.e.

Acknowledgment

Dionne R. Wilson prepared the illustrations for this article by scanning and enhancing the author's hand- drawn sketches, using Adobe Illustrator on an Apple Macintosh.

82

81

84

83

~22

S 21

8 24

~23

S 14

S 42

8 41

~ 2

8 44

34

Figure 7. Image of the Cantor set.

//// ////

Figure 8. Approximating Polygons for Lebesgue's space-filling curve.

42 THE MATHEMATICAL [NTELLIGENCER VOL. 15, NO. 4, 1993

References

1. G. Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Crelle J. 84 (1878), 242-258.

2. A. Devinatz, Advanced Calculus, New York: Holt, Rinehart, Winston (1968), 253.

3. D. Hilbert, Ueber die stetige Abbildung einer Linie auf ein Flaechenstueck, Math. Ann. 38 (1891), 459-460.

4. K. Knopp, Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch, Arch. Math. Phys. 26 (1917), 103-115.

5. T. Lance and E. Thomas, Arcs with positive measure and a space-filling curve, Amer. Math. Monthly 98 (1991), 124- 127.

6. H. Lebesgue, Lecons sur l'Intdgration et la Recherche des Fonc- tions Primitives, Paris: Gauthier-Villars (1904), 44-45.

7. E. H. Moore, On certain crinkly curves, Trans. Amer. Math. Soc. 1 (1900), 72-90.

8. E. Netto, Beitrag zur Mannigfaltigkeitslehre, Crelle J. 86 (1879), 263-268.

9. W. E Osgood, A Jordan curve of positive area, Trans. Amer. Math. Soc. 4 (1903), 107-112.

10. G. Peano, Sur une courbe qui remplit toute une aire plane, Math. Ann. 36 (1890), 157-160.

11. H. Sagan, Some reflections on the emergence of space- filling curves, Franklin ]. 328 (1991), 419-430.

12. H. Sagan, On the geometrization of the Peano curve and the arithmetization of the Hilbert curve, Int. J. Math. Educ. Sci. Technol. 23 (1992), 403-411.

13. H. Sagan, Approximating polygons for the Sierpifiski- Knopp space-filling curve, Bull. Acad. Sci. Polon. 40 (1992), 19-29.

14. W. Sierpifiski, Sur une nouveUe courbe continue qui rem- plit tout une aire plane, Bull. Acad. Cracovie, (Sci. Mat. Nat. Serie A) (1912), 462-478.

15. W. Sierpifiski, Sur une courbe non quarrable, Bull. Acad. Cracovie (Sci. Mat. Nat. Serie A) (1913), 254-263.

Department of Mathematics North Carolina State University Raleigh, NC 27695-8205 USA

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