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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,
by rcthompson 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.
- 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.