I'm trying to wrap my head around how higher dimensions work. Not over some academic pursuit, merely curiosity(which started with me discovering this neat recurrence relation). Given that the surface area of a sphere is an object of 1 dimension lower. So on a 3-sphere, the surface area is a plane and is measured in say m2. Then for a 4-sphere, the surface area is a 3-dimensional sphere with a volume measured in m3. Then I thought a bit about volumes of higher dimensional spheres before realizing hypervolume doesn't mean the same as normal cubic volume, throwing any intuition as what it is(let alone how big it is) out the window. However, in a 4-sphere, one can imagine taking slices at equal steps and obtaining a multitude of concentric 4-spheres, each of which has a "surface area" with a regular 3-dimensional volume. So if you define a step, using this method you get a meaningful answer for "What is a the 3d volume of a 4-sphere with radius r?" Say, S(step) = 0.2u (as in units) V4 = A4(u) + A4(0.8u) + A4(0.6u) + A4(0.4u) + A4(0.2u) Or ~41.22u3 if my calculations are correct. Does this even make sense? If no, why not?