Download Cantor Set - Missouri State University

Math Team Lesson
Field of Study: Fractals and Chaos
Topic: Cantor Set, Triadic Numbers, Prisoner Set
Source(s): Fractals for the Classroom; Peitgen, Jurgens, Saupe, et.al.; Springer-Verlag;
1992; pp. 79 - 90
Info: The Cantor Set is a self-similar set of real numbers which is an uncountable set.
In a geometric sense, it can be viewed as the segment from 0 to 1 on the number line
where the middle third of the segment is removed. Of the two remaining thirds, the
process is repeated (like a feedback system) where the middle third of each segment is
removed. This process is continued infinitely many times. (NOTE: When the middle
third is removed, the open interval is removed, leaving the endpoints.)
0
1
0
0
1/3
1/9
2/9
1/3
2/3
2/3
1
7/9
8/9
1
In a numeric sense, the Cantor Set is the set of all real numbers in the interval [0,1] that
has an equivalent notation in the triadic (base three) system which is made up of a finite
or infinite string of 0’s and 2’s. There cannot be any numbers in the set which have a 1 in
their triadic equivalency. However, there is some tricky play going on! Recall that in the
decimal system .999…. = 1. Likewise, in the triadic system .02222….. = .1. Since all of
the numbers in this interval are less than or equal to one, converting between the decimal
system and the triadic system requires working with negative exponents. For example,
the triadic number .1 would equal 1x3-1 which would equal ⅓. As it turns out, the
endpoints of the segments remaining in the set include: 0, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 2/27,
and so on. (NOTE: 13  0.13  0.2 3 )
 
10
Start your own Cantor Set on the segment below. List the endpoints of the segments that
remain in the set. Try to give the triadic (ternary) representation of those numbers as
well.
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
EXTENSION 1:
In a feedback (iterative) system, some initial points will set off a sequence of outputs that
will lead to positive or negative infinity. These initial points are called escaping points.
If an initial point does not lead to infinity, but arrives at an output from which it cannot
change, that initial point is called a prisoner point.
Starting with the feedback system given below, try initial points <0, >1 and then between
1 and 0. Determine where escaping points and prisoner points can be found.
x
3x
if x ≤ 0.5
-3x + 3 if x > 0.5
Describe the set of all prisoner points.
EXTENSION 2:
You have probably heard of the Sierpinski Triangle, which is analogous to the Cantor
Set. Now try to make a “Reverse Sierpinski” Triangle! Keep the middle ¼ of the
triangle and discard the outer ¾. Set up a series for the sum of the area of the fractal
design that is formed. Find that sum. Surprising??