the fact that higher-dimensional spheres are pointy is one of the weirdest things to think about
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@Felthry wait what, how.
@starkatt think about a square with four circles in it, and then add a fifth in the center, tangent to the other four
consider the ratio of that circle's area to the area of the other four
now think of a cube with eight spheres, and add a ninth, tangent to the other eight, in the middle. The ratio of that sphere's volume to the other spheres' is larger than the circle ratio
this ratio grows--without bound--as you increase the number of dimensions
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@starkatt in fact, as early as four dimensions the center hypersphere is also tangent to all eight sides of the hypercube!
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@starkatt separately, you also have that the ratio of the n-measure of a unit n-hypersphere to the n-measure of a 2-unit n-hypercube approaches zero as n approaches infinity, but the n-hypersphere remains tangent to the n-hypercube at all unit lattice points (i.e. (1,0,0,0,....), (0,1,0,0,...), (0,0,1,0,...) etc.)
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