Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Medium level:

Cantor Set

Figure 1
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!

SUM1 to infinity (2n-1/3n) = 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>.

Share

References:
    Any text on real analysis.

Keywords:    geometric series, measure theory
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Joshua Sabloff
Suggestions? Use this form.
4.14
 
current
rating 		Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!