590 
GEOMETRIC NOTIONS 
sponding to t' either lies in the same square of D n as the point Q 
corresponding to £, or in an adjacent square. But the diagonal 
of the squares = 0, as n = go. Thus 
Dist ((T Q') = 0 , as w = oo. 
^ 1US <£(£')— <£(£) , and f (O - f(0 
both = 0, as t’ = t. 
As t ranges over 51, the point x, y ranges over every point in the 
square 53. 
For let Q be a given point of 53. It lies in a sequence of 
squares as 3). If Q lies on a side or at a vertex of one of the g 
squares, there is more than one such sequence. But having taken 
such a sequence, the corresponding sequence 2) is uniquely de 
termined. Thus to each Q corresponds at least one P. A more 
careful analysis shows that to a given Q never more than four 
points P can correspond. 
2. The method we have used here may obviously be extended 
to space. By passing median planes through a unit cube we 
divide it into 2 3 equal cubes. Thus to get our correspondence 
each division _Z)„ should divide each interval and cube of the pre 
ceding division into 2 3 equal parts. The cubes of each divi 
sion should be numbered according to the 1° and 2° principles of 
enumeration mentioned in 1. 
By this process we define 
x — $i(0 > y=<\> 2 (0 » z=tf> 3 (0 
as one-valued continuous functions of t such that as t ranges over 
the unit interval (0, 1), the point a;, y, z ranges over the unit 
cube. 
574. 1. Hilbert's Curve. We wish now to study in detail the 
correspondence between the unit interval 51 and the unit square 
53 afforded by Hilbert’s curve defined in 573. A number of inter 
esting facts will reward our labor. We begin by seeking the 
points P in 51 which correspond to a given Q in 53- 
To this end let us note how P enters and leaves an g square. 
Let B be a square of B n . In the next division B falls into four