600 
GEOMETRIC NOTIONS 
It was not known whether & would remain discrete if the con 
dition of limited variation was removed from both coordinates, 
until Osgood* exhibited a Jordan curve which is not discrete. 
This we will now discuss. 
582. 1. Osgood's Curve. We start with a unit segment 
T = (0, 1) on the t axis, and a unit square S in the xy plane. 
We divide Tin to 17 equal parts 
1 2 3 4 5 
1G 17 
1 
Si § 
m 
m m 
1**1 
1 1 
I * 
'VËËËËÊËËËËËiA 
m 
S u § 
I s t i 
m v/. 
I ^ 
&13 ^ 
m 
1 s » 1 
1 I 
I 
m 
1 
14 
intervals T n , T, 
^1’ ^2’ '** ^17’ (1 
and the square S into 9 equal 
squares 
#1, tf 8 , #5 ••• S 17 , (2 
by drawing 4 bands B 1 which 
are shaded in the figure. On 
these bands we take 8 segments, 
S 2’ S 4’ S 6 **' S 16’ 
marked heavy in the figure. 
Then as t is ranging from left 
to right over the even or black 
2’ J -4’ 
T, 
16 
marked heavy in the figure, the point a;, y 
on Osgood’s curve, call it £), shall range univariantly over the 
segments 3). 
While t is ranging over the odd or white intervals T x , T 3 ••• T 17 
the point xy on D shall range over the squares 2) as determined 
below. 
Each of the odd intervals 1) we will now divide unto 17 equal 
intervals T tj and in each of the squares 2) we will construct 
horizontal and vertical bands as we did in the original square 
S. Thus each square 2) gives rise to 8 new segments on D 
corresponding to the new black intervals in T, and 9 new squares 
corresponding to the white intervals. In this way we may 
continue indefinitely. 
The points which finally get in a black interval call /3, the 
others are limit points of the /3’s and we call them X. The point 
* Trans. Am. Math. Soc., vol. 4 (1903), p. 107.