A Survey of Mathematics Related Topics: Pg. 45, Exercise 79

This activity allows the user to step through the process of building Cantor's Comb.

The Cantor Comb is a way of visualizing the famous Cantor Set. Start with a line segment like:

Then remove the middle third:

Next, remove the middle thirds of each remaining line segment. The next stage would be:

Now repeat this process indefinitely. The Cantor Comb is the stuff left after removing the middle thirds of the line segments "infinitely many times."

Other Cantor Combs can be created by removing different sized middle pieces. For example, we can remove the middle quarter (1/4) or the middle half (1/2) to get different Cantor Combs. To distinguish between these we sometimes call the original version the "Cantor middle-thirds comb" using a similar name for the others.

Georg Cantor (1845-1918) was very interested in infinite sets, especially ones with unusual properties. He built what is now called the Cantor set to illustrate an unusual infinite set. This set has only points left, no line segments - so no length. If you stop the process at any stage, there are line segments; very short ones. The idea that at any stage you have length, but in the limit you don't, makes the Cantor Comb a very curious set.