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In the article the unit cubes have side length 1. So isn't the volume always 1, whatever the dimension?

Yes, the volume of the unit cube is 1 in any dimension, but the volume of the unit sphere goes to 0 as the dimension increases.

I believe thats because volume is a measurement of the ratio between "amount of stuff" in the hyper-object vs the amount of stuff in a hyper-cube

The amount of stuff increases with higher dimensions in a sphere, but not as fast as with a cube, hence the ratio between the two converge to zero