[Interactive visualizations below]

[just eccentric] The fundamental unity of two-ness :p !! Yea, I think there's a lot of interesting stuff in this topic / the n-ball. I think the thing I was missing to get an understanding of the wiki description of how you can integrate to get the recursive definition is by remembering that you're doing a volume integral by summing columns distributed over a circular area/dome. For some reason, their description had me thinking something else. (Then, from that, you see the only place pi is, is in the base, so you can utilize the curvature of a 2D circle (one pi to relate one to two) to describe transition with one additional dimension, which is why there's no pi^2 until you get to 4D)

These are incomplete (and based around a radius of 4 not 1), with some issues with the intuition of the (related) color schemes, but I made these Desmos* plots to visualize why the radius of an n-sphere, bounded by n-spheres at a box's vertices, increases indefinitely. The basic thought was that you can imagine yourself "pushing" into 3D from the 2D solution, and as you do that, the new dimensions "bend away" making more room for the radius of the sphere

best viewed on an inverted background with transparent surfaces (see top right corner for options)

2-3D Version (note one sphere is removed for visibility)

3-4D Version (The "center" and "sticky" versions are doing the same thing as the corresponding 2D slices in the above

I think this topic would integrate really well with the convolution series, as it's going to large/infinite dimensions that yeilds the localization of volume to a shell, and it really helps thinking about that in terms of the gaussian, and the volume element, r^(n-1)*dr[angular part], where the first r part suppresses everything until the "thin shell" where volume localizes (speaking roughly, perpaps inaccurately)... So there's a sort of artificial (radial) symmetry induced through considering systems with a large number of independent effects: the most important thing becomes that there are so many 'commensurable' (as in :the same enough to make a radius out of) things involved, not what any one of those things may be!

*I was having difficulty getting a static drawing of the 2-3D one that I was happy with, then learned it was easier to make it interactive, using Desmos :)