5 ms·
Volume 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_
by legolas2412 10y ago
Volume 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.