Very counterintuitive. I understand how the proportional space of an n-ball bounded in an n-cube can go to zero, but I don't understand how the volume of an n+1-ball can be of a smaller volume than an n-ball.
> Very counterintuitive. I understand how the proportional space of an n-ball bounded in an n-cube can go to zero, but I don't understand how the volume of an n+1-ball can be of a smaller volume than an n-ball.
It can't be, in the same sense that it's not true that π square meters (the area of a unit circle) is less than (4/3)π cubic meters (the volume of a unit sphere); they are simply not comparable quantities. What is true, as wging very nicely put it (https://news.ycombinator.com/item?id=31349002), is that the (n + 1)-ball (of radius 1) eventually takes up a smaller fraction of the (n + 1)-cube (of side 2) than the n-ball does of the n-cube.
I agree that this is still probably sufficiently unintuitive that the best one can do is to argue why one shouldn't disbelieve it, not to try to make it intuitive; but nonetheless I will share one of the closest things I've seen to an explanation, which is that the number of corners in a cube grows exponentially with the dimension of the cube, so that, sooner or later, the cube is "mostly corners"—and the ball does not poke into the corners.
> volume of an n+1-ball can be of a smaller volume than an n-ball
Volume is the ratio of the n-ball to the n-cube.
Saying that n+1-ball has “smaller volume” than the n-ball is equivalent to saying that the ratio of the n+1-ball to the n+1-cube is smaller than the ratio of the n-ball to the n-cube.
> I understand how the proportional space of an n-ball bounded in an n-cube can go to zero
The volume comment is saying that the ratio of the n-ball to its bounding n-cube goes to zero faster than (base-2) exponential:
A two-dimensional n-ball is a circle. If you extend it into 3 dimensions, by default it is a cylinder, of h=1 and r=1. That’s not a 3-dimensional n-ball however.
If you increase the dimensions of that 2-dimensional n-ball into 3 dimensions, it becomes a 3-dimensional n-ball—-a sphere of r=1.
And now you have a sphere that is smaller than the circle that was extended into 3 dimensions as a cylinder. The ball-ness of the n-ball formula chopped off the corners of the cylinder.
And that chopping off continues in higher dimensions.
I mean, we can directly calculate the volume of a cylinder by modeling it as a big stack of circles and adding up their areas. That's the whole idea of integral calculus.
If you have enough circles, they will form a cylinder.
