I'm a non-mathematician and am trying to understand intuitively what happens to the volume and surface area of an n-ball as the size and dimensionalty increases. I've tried looking at similar topics in this forum but nothing answers my question (at least not in a way that I can understand).
What is puzzling me is that the volume of an $n$-ball peaks at $n = 5$ and then declines again (which seems odd in itself) but only for radius $= 1.$ For radius $2$ it's $n=24,$ for $3$ it's $n=56...$ the figure shows a graph of peak against radius for dimensions $1-350$ and it has a rather odd shape.
What I'm struggling with is why the radius should matter - it seems like the peak volume should be a property of the shape and shouldn't depend on whether the radius is in cm or m etc. Can anyone explain in non-mathematical terms? Many thanks!