It's a representation of a construction of the cantor set.
I've known about the Cantor set for a while, but this part
of the Wikipedia page still freaks me out:
Wikipedia wrote:... a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and 2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points.
So far, so good. I mean, provided you accept that a strict subset can contain the same number of elements as its superset. But let's not be picky here.
Wikipedia wrote:It may appear that only the endpoints are left, ...
Yes. That must be it. It's obvious. Anything else would be ridiculous.
Wikipedia wrote:... but that is not the case either. The number 1/4, for example, is in the bottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottom third of that, and in the top third of that, and so on ad infinitum—alternating between top and bottom thirds. Since it is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middle third. The number 3/10 is also in the Cantor set and is not an endpoint.
Huh? How can it be? But these are special cases, though, right? Most of what's left is the endpoints?
Wikipedia wrote:In the sense of cardinality, most members of the Cantor set are not endpoints of deleted intervals.
So, @ysth: No, I don't think anyone can explain it. (Though many internets to whoever wrote that Wikipedia section.)