Last month, I wrote about the Cantor set, a mathematical space that is an interesting mix of small and large. It’s small in the sense that its length is 0. But it’s large in the sense that it’s uncountable. Once a mathematician get their hands on an object, one of their first instincts is to tweak it and see what happens. That’s how we end up with fat Cantor sets.

If someone talks about the Cantor set, they’re referring to the standard middle thirds set I wrote about earlier. It’s created by starting with the [0,1] interval and removing the middle third of it, so we’re left with the intervals [0,1/3] and [2/3,1]. Then the middle third is removed from each remaining interval, and that process is repeated forever. Surprisingly, there’s some stuff left at the end, but there’s no length. The total length removed sums up to 1, the length of the original interval.

Seven steps in the construction of the relatively svelte standard middle thirds Cantor set. Image: Public domain, via Wikimedia Commons.

The first logical tweak to make to the Cantor set is to change the middle third to some other fraction. What happens if we just remove the middle fourth of each interval instead? So we start with [0,1]. In the first step, we’re left with the intervals [0,3/8] and [5/8,1]. Then we keep removing the middle fourth (a segment of length 3/32) from each interval. You might think that because we remove less at each step, there would be more left at the end, but that’s not true. If you sum up the total length of all the intervals we remove, we still end up with 1. This Cantor set isn’t really any more interesting than the last one. We didn’t just get unlucky, either. If we keep the ratio the same each time, we’ll always end up removing intervals that sum up to length 1. There are ways of distinguishing between the middle thirds, middle fourths, or middle anythings Cantor sets, but for now we’ll try fiddling with the Cantor set in a different way.

The next thing to try is varying the proportion of the intervals we remove at each step. We’ll start by removing the middle fourth again, so we’re left with [0,3/8] and [5/8,1]. But for the next step, we’ll only remove an interval of length 1/16 from each remaining interval. Now things are changing a bit. Before, our second step was removing a segment of length 3/32, which is just a smidgeon longer than 1/16. In our new construction, we’ll keep shrinking the proportion of the remaining intervals we remove at each step.

At step 1, we remove an interval of length 1/4 from the one interval we start with.
At step 2, we remove an interval of length 1/16 from each of the two intervals remaining for a grand total of 1/8 of the original length removed.
At step 3, we remove an interval of length 1/64 from each of the four intervals remaining for a grand total of 1/16 removed.
We continue this pattern. At step n, we remove intervals with a total length of 1/2n+1.

The total length we remove if we do this infinitely many times is 1/4+1/8+1/16+…, which adds up to 1/2. Now we’re onto something!

This construction is called the Smith-Volterra-Cantor set or a fat Cantor set. The Cantor set had (one-dimensional) measure 0 because we removed all the length from out starting interval, but the fat Cantor set has a little meat on its bones–specifically half of the meat of the full interval [0,1]. But where is the meat? By design, the fat Cantor set has no solid intervals. We demanded that every time we saw an interval, we’d remove part of it. Somehow, there’s some length hanging around, but it’s not hanging around in a form we’re familiar with. If we try to put our hands on it, we’re only grabbing at dust.

The Cantor set challenges my intuition about small and large things. The fat Cantor set just blows my intuition out of the water. How can an object have an appreciable length when it has no little line segments sitting in it? Well, that’s not entirely fair. The set of all irrational numbers in the interval [0,1] has one-dimensional measure 1, so it’s as “long” as the entire interval, and that doesn’t seem terribly counterintuitive. The irrationals are all over the place. You can’t shake a stick without hitting one. Mathematically, we say they’re dense in the interval, or in the set of all real numbers, meaning every sliver of an interval we choose, no matter how small, will contain irrational numbers. Density, or the lack thereof, is what makes the fat Cantor sets even stranger. Fat Cantor sets aren’t dense in the [0,1] interval, and they’re not even dense in any smaller intervals inside it. No matter how much you zoom in, you’ll be able to find whole intervals that don’t have any points from the fat Cantor set. We call sets like that nowhere dense.

There’s nothing special about the fact that the fat Cantor set we constructed has length 1/2. In fact, by varying the sizes of intervals removed at each step, we can end up with Cantor sets that are as thin or as fat as we want, within reason. We can’t get a fat Cantor set that actually gets all the way up to length 1, but we can get as close as we care to. No matter how big we make our fat Cantor sets, they won’t have any whole intervals in them and they’ll be nowhere dense. There have stuff in them, but where is it?

One of the first places I saw the Cantor set was in the construction of the Cantor function, which I wrote about last month. The Cantor function, or Devil’s staircase, shows us some of the limits of the fundamental theorem of calculus that relates differentiation and integration. Fat Cantor sets can do that too. Specifically, Italian mathematician Vito Volterra (1860-1940) used one of these sets to cook up a function that’s differentiable but whose derivative is not integrable.

I’ll just pique your interest a bit with this picture of the Volterra function. It’s pretty wild, and you can learn more about it from Wikipedia or these slides from a lecture by David Bressoud. It just might break your brain.