A Space Filling CurveAuthor(s): Liu WenSource: The American Mathematical Monthly, Vol. 90, No. 4 (Apr., 1983), p. 283Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2975763 .

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1983] NOTES 283

A SPACE FILLING CURVE

Liu WEN Department of Mathematics, Graduate Center of the City University of New York, New York, NY 10036

Hebei Institute of Technology, Tianjin, the People's Republic of China

The purpose of this paper is to give an example of a space filling curve which is simpler than previous examples (cf. [1], 455-458).

Divide the interval [0,1] into ten equal subintervals:

ak = [ k/lO,(k + 1)/10], k = 0,1., I9. Let f and g be continuous real valued functions defined on [0,1] which satisfy the following conditions:

(i) f(t) = (f l when te61 u( 3, (i) ~~~~~~~~when te5 U67;

0 when t E 1 U 85,

t-1 when t E3 U 87;

(ii) f(0) = f(l) and g(0) = g(l). Extendingf and g to (- oo, + oo) periodically, we get two continuous periodic functions which are still denoted f and g respectively.

Let 10 00

,(t) = E f(10k 1t); 4(t) = E kg(lok It). k=1 2 k=1 2

Obviously 4 and 4 are continuous. We will now easily show that

(X - O(t) 0 < t A< I

is a curve filling the unit square. In fact, let x and y be arbitrary real numbers in (0, 1) with base 2 binary expansions

x = .XIX2... Xk... ; eachxk = 0or1; y = YlY2 .Yk l ;eachyk = 0or1.

Let

1 if Xk = 0 and Yk = 0

t= 3 if Xk = 0 and Yk = 1

k 5 f Xk = 1 and Yk 0 7 if Xk = l and Yk =1*

Now let t be the number whose base 10 decimal expansion is t = .tlt2.. tk..--

It follows immediately from the definitions that x = +(t), y = At,

and so the curve passes through every point of the unit square.

Reference

1. E. W. Hobson, Theory of Functions of a Real Variable, vol. 1, Dover, New York, 1957.

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