I think here is something that helps me understand intuitively how volume goes to zero in an n-sphere I think the counterintuitive part for me was I kinda misu

I think here is something that helps me understand intuitively how volume goes to zero in an n-sphere I think the counterintuitive part for me was I kinda misunderstood what volume really means So lets think instead in terms of how hyperdimensional volume relates to a basic concept of the "amount of stuff" contained within an n-dimensional object. The "amount of stuff" in a cube increases as you go from a 2d hypercube (i.e. a square), then to a 3d hypercube, and so on to higher dimensions, even though measurement of the volume (volume for 3d, area for 2d, etc...) stays the same Another way to think about this "amount of stuff" is to imagine some kind of infinite dimensional "fluid" with extremely small particles that covers a square with one layer of particles. Thats going to have X**2 number of particles right? Where X is the approximate number of particles along each dimension. Now if it covers a cube, that will then have X**3 number of particles. So the number of particles of that fluid thats covering an n-cube increases exponentially as you increase dimensions of the n-cube Similar thing happens with the number of particles blanketing first a circle, then a sphere, then to higher dimensions, the number of particles increases, but not as fast as the unit volume. I think this can be more easily seen where the diameter is 1, rather than the radius being 1, and comparing that to the unit cube So the volume is really kinda like the ratio between the number of those hyper particles in the object vs the number of those hyper particles in the sphere And this ratio converges to zero at higher dimensions, even though the amount of stuff increases (Perhaps to infinity as well? I suspect so but not entirely sure on that one. That would take a mathematical proof and I'm not quite cut out for that right now)