The Cantor Ternary Set

This is the Cantor ternary set.

As you can see, it is formed from a line segment. (It is slightly thickened here so you can see it better.) It has no length. But it has infinitely many points.

Let me explain. This is how you make the picture above: Take a line segment.

Remove the middle third.

Repeat.

And again.

And again.

Keep doing this forever.

Geometrically, what you’re doing in the first step is taking a line segment of length, say, 1, and removing a third of it. You are thus multiplying its length by \frac{2}{3}. You are doing the same thing in all the other steps. So in the end, you are multiplying the length of the line segment by (\frac{2}{3})^\infty.

Yes. If you keep multiplying \frac{2}{3} by itself, it will become zero. In formal terms:

\lim\limits_{n\to\infty}(\frac{2}{3})^n=0

So, what we have in effect done is to multiply 1 by 0, which is 0. So the Cantor set has zero length.

Okay, let’s change perspectives. Instead of looking at this geometrically, we’ll look at it numerically. You know binary?

Good. Ternary is base 3.

So, take all the numbers between 0 and 1 on the number line, and write them out in ternary.

Chuck out all the numbers with a ‘1’ in the tenths place. Leave in 0.1—it can be expressed as ‘0.0\bar{2}’. ¹If you’re confused as to why this is possible, look at the Wikipedia article for 0.999…, the arguments there apply for base 3 too.

Now, take the remaining numbers. They all begin with ‘0.0’ or ‘0.2’. Remove all the numbers with a ‘1’ in the hundredths place. (Leave in 0.01 and 0.21, because they can be expressed as 0.00\bar{2} and 0.20\bar{2}.²See note 1.)

Keep repeating this. Now look at our number line.

It looks remarkably like the Cantor set, doesn’t it? Well that’s because it is. (This is also why this specific construction is called the Cantor ternary set.) But look at the set for a while.

We only removed numbers that had a 1 in them. In other words, if a number’s ternary representation consists of only 0s and 2s, it is in the set. There are obviously infinitely many such numbers. (For example, 0.2, 0.02, 0.002, etc.) So, I repeat what I said at the beginning:

The Cantor set has no length, but an infinite number of points

-me

This enormously interested Georg Cantor, who didn’t discover it, but popularized it. (If you’ve known math for any length of time, you’ll know that things never get named after who discovered them.)

Anyway, that’s the Cantor set. See you soon, or as Cantor himself would say,

Sources/Footnotes

• 1
  If you’re confused as to why this is possible, look at the Wikipedia article for 0.999…, the arguments there apply for base 3 too.
• 2
  See note 1.