Given the unit sphere it's simple to find the volume of the largest cube that can fit inside it. In the spherical cap above the cube we inscribe, again, the largest cube possible, and we keep going inscribing the largest cube inside the spherical cap above the previous cube. We'll form an infinite tower of cubes: is it possible to find a closed form for the infinite sum of the volumes of all these cubes?
I found the values for the sides of the first cubes:
$ s_0 = \frac {2} {\sqrt 3} \\ s_1 = \frac {2 \sqrt 3} {9} \\ s_2 = \frac {2 (2 \sqrt 7 - 5)} {9 \sqrt 3} \\ s_3 = \frac {-2 (5 + 4 \sqrt 7 - 2 \sqrt{74 - 5 \sqrt 7})} {27 \sqrt 3} \\ $
