5 ms·
It 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 doe
by rcthompson 10y ago
It 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.