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One other mindblowing fact is that if you project the mass of the sphere onto a line that goes through the origin, you get a Gaussian distribution as the dimens
by ot 10y ago
One other mindblowing fact is that if you project the mass of the sphere onto a line that goes through the origin, you get a Gaussian distribution as the dimension goes to infinity.
If you're interested in more technical details about the weird geometry of balls, I can't recommend enough the first few pages of "An Elementary Introduction to Modern Convex Geometry" [1]. Ironically, the author's last name is Ball.
[1] http://library.msri.org/books/Book31/files/ball.pdf http://library.msri.org/books/Book31/files/ball.pdf
- repsilat 10y ago> you get a Gaussian distribution as the dimension goes to infinity. Wow, that's really cool. I wonder, I'm sure you could prove this in more direct ways, but is it also a consequence of the central limit theorem?
- effie 10y agoThat's doubtful, as no random variables are apparently involved in the questions of geometry. The situation with the Gauss function is perhaps similar to that of pi, which appeared originally in geometry of circles and only later was discovered in formulae for probability distributions or sums of infinite series. Neither Gauss function nor pi seem to be naturally connected to questions of probability; but applications of the probability theory to special situations lead to expressions that use these most basic mathematical concepts.
- rcthompson 10y agoIt should be clarified that this fact refers to a ball of volume 1 in N dimensions. Since the volume is held constant, as the dimension goes to infinity, so does the radius.
- ot 10y agoCorrect, this is different from the OP, which keeps the radius constant.
- TeMPOraL 10y agoWait, isn't it the other way around? I.e. if you want to keep a volume of the ball constant as dimensions rise, don't you have to reduce the radius? (I'm a noob at this whole dimensions thing.)
- legolas2412 10y agoVolume of a hyper-sphere of radius R reduces with increasing dimensions! https://en.wikipedia.org/wiki/Volume_of_an_n-ball https://en.wikipedia.org/wiki/Volume_of_an_n-ball To visualize, take a hyper-cube of length 1. It's volume is 1 unit for any n-dimensional hyper-cube. Scribe a hyper-sphere touching all the sides. How about its volume with increasing dimension? That decreases! (with respect to the cube). To visualize, think about the distance of the cube vertex from the centre, it increases with increasing dimension. In other words, Most of the cube's volume is closer to its vertex, not in the ball in the centre!
- TeMPOraL 10y agoThanks! I'll need to spend some time trying to visualize it though, I'm nowhere near grokking it.
- rcthompson 10y agoYou can start by working out the ratio of volumes between a sphere of radius 1 and the enclosing cube in 1, 2, and 3 dimensions (i.e. a line segment, a circle, and a sphere, respectively). You'll see that the ratio starts at 1 and decreases as the dimension increases.
- JadeNB 10y ago> You can start by working out the ratio of volumes between a sphere of radius 1 and the enclosing cube in 1, 2, and 3 dimensions (i.e. a line segment, a circle, and a sphere, respectively). You'll see that the ratio starts at 1 and decreases as the dimension increases. Be careful using low-dimensional phenomena as a guide to the asymptotic behaviour! If you take a sphere of radius 2, rather than radius 1, then the analogous ratio increases for a while (until n = 5), but then decreases to 0.