Figure 1

Start with the interval [0,1].
Remove the (open) middle third of it, i.e. get (1/3, 2/3).
Now remove the middle thirds of each of the remaining
intervals, i.e. get (1/9, 2/9) and (7/9, 8/9).
Continue this process ad infinitum.
The points left over form a fractal called the
standard Cantor Set.
It is an infinite set since a lot of points, including the
endpoints of the removed intervals, are never
removed. Can you list the endpoints?
Now let's think about lengths.
The length of the original interval is 1.
Now how much "length" do we remove during the process?
At the first step, we remove an interval
of length 1/3.
At the second step, we remove two
intervals of length 1/9.
At the third step, we remove 4 intervals
with total length 1/27, etc.
What is the total length removed during the entire process?
A geometric series!
SUM_{1 to infinity} (2^{n1}/3^{n}) = 1.
Wait a minute ... you mean we have an
infinite set left over with 0 length??
Yes, and it's worse than that:
the set is uncountable!
Thus it has as many points as interval that we started with!
Presentation Suggestions:
Use this Fun Fact after students learn to sum
geometric series.
You may wish to assign the sum of the lengths "removed" as an
exercise before presenting this Fun Fact.
(Of course, you can also do the computation in terms
of how much remains after each stage and get a limiting
sequence instead.) Follow with the Fun Fact Devil's Staircase.
The Math Behind the Fact:
The real numbers are far stranger than any
of your students might have suspected!
The standard Cantor Set is quite interesting:
it is an uncountable, totally disconnected,
perfect (every point is a limit point)
set of "Lebesgue measure zero".
Another way to describe the standard Cantor Set is
the set of all real numbers in [0,1] expressible
without 1's in its base 3 expansion!
Though first constructed
as a pathological example, it arises naturally
as a fractal
in the study of dynamical systems.
How to Cite this Page:
Su, Francis E., et al. "Cantor Set."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
