> But most important, the fractal possesses various counterintuitive mathematical properties. Continue to pluck out ever smaller pieces, and what started off as a cube becomes something else entirely. After infinitely many iterations, the shape’s volume dwindles to zero, while its surface area grows infinitely large.
I'm struggling to understand what is counterintuitive here. Am I missing something?
Also, it's still (always) going to be in the shape of a cube. And if we are going to argue otherwise, we can do that without invoking infinity—technically it's not a cube after even a single iteration.
This feels incredibly sloppy to me.
Think of a 3-dimensional object (unlike a surface, which is 2-dimensional, regardless of the shape), with the volume zero. That's not easy to wrap your head around.
> shape’s volume dwindles to zero, while its surface area grows infinitely large
I think it's easy to grok when you get it, but that's certainly counter-intuitive on the surface, no?
I won't say it isn't possible that someone might struggle with this—it's quite subjective, obviously—but I do think it's unlikely that anyone with a general understanding of both volume and surface area would struggle here.
Even just comparing two consecutive iterations, I feel confident that any child who has learned the basic concepts would be able to reliably tell you which has more enclosed volume or surface area.
I will happily concede that the part you quoted could be quite unintuitive without the context of the article or the animation included in it. :)
I think Gabriel's Horn is a great explanation of how this is counter-intuitive[1]. This is a shape which you could fill with a finite amount of water, say a gallon. Yet it would take an infinite amount of paint to paint the surface. Of course, part of the reason it's counter-intuitive is that there is no 0-thickness paint that exists.