Re: Mystery of High Dimensions [NOT OT]

On Thu, 02 Jan 2025 16:55:28 -0600, Physfitfreak wrote:
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Sounds like something one gets tempted to ask AI about it.
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There's no need for AI (even it could give an answer).
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The answer is in the "corners."
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A hypercube of n dimensions has 2^n corners, and it is these corners
that contain most of the volume.
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An inscribed hypersphere will always touch each face of the cube but
it always curves away from the corners.
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Furthermore, the distance from the center of the hypercube (and its
inscribed hypersphere) to each corner is sqrt(n) * r, where n is the
number of dimensions and r is the radius of the hypersphere and also
the length of 1/2 side of the hypercube.
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Thus as n increases, so too does the distance to each corner from
the center of the hypersphere.
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N-D spaces are fascinating places.
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For n=100, the cube has 2^100 corners with each corner located
10 units from the center of a hypersphere of radius = 1 unit.
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That's LOT of hypervolume outside of the hypersphere but still
inside the hypercube.