Greg Egan on Nostr: The volume of a ball of radius r in n dimensions is B(n,r) = π^{n/2}/Γ(n/2+1) r^n ...
The volume of a ball of radius r in n dimensions is
B(n,r) = π^{n/2}/Γ(n/2+1) r^n
If you look at the volumes of the balls (blue circles) inscribed inside hypercubes with edge length 1 and volume 1, they go to zero as n gets larger:
lim n→∞ B(n,½) = 0
If you look at the volumes of the balls (red circles) that circumscribe each hypercube (of diagonal √n), they go to infinity:
lim n→∞ B(n,½√n) = ∞
But what if you always choose the radius of an n-ball so that its volume is exactly 1 (green circles)?
r₁(n) = Γ(n/2+1)^{1/n} / √π
This gives us B(n,r₁(n)) = 1
It turns out that:
lim n→∞ (r₁(n) - √[n/(2eπ)]) = 0
In other words, r₁(n) is asymptotic to a multiple of √n, so it approaches a fixed ratio with the length of the diagonal of the hypercube with the same volume.
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"content":"The volume of a ball of radius r in n dimensions is\n\nB(n,r) = π^{n/2}/Γ(n/2+1) r^n\n\nIf you look at the volumes of the balls (blue circles) inscribed inside hypercubes with edge length 1 and volume 1, they go to zero as n gets larger:\n\nlim n→∞ B(n,½) = 0\n\nIf you look at the volumes of the balls (red circles) that circumscribe each hypercube (of diagonal √n), they go to infinity:\n\nlim n→∞ B(n,½√n) = ∞\n\nBut what if you always choose the radius of an n-ball so that its volume is exactly 1 (green circles)?\n\nr₁(n) = Γ(n/2+1)^{1/n} / √π\n\nThis gives us B(n,r₁(n)) = 1\n\nIt turns out that:\n\nlim n→∞ (r₁(n) - √[n/(2eπ)]) = 0\n\nIn other words, r₁(n) is asymptotic to a multiple of √n, so it approaches a fixed ratio with the length of the diagonal of the hypercube with the same volume.\n\nhttps://media.mathstodon.xyz/media_attachments/files/111/452/398/135/656/733/original/00bbe26ae60909af.mp4",
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