Archimedes Revenge Solution

Congratulations to Runze Li, who solved the Archimedes Revenge problem. There are obviously 8 pieces which are the same. Each of these pieces can be split into a tetrahedron of volume 1/2 and 3 pieces which each have volume given by the integral I of [(z2+1) (π/2-2 arctan(z)) + z2-1 ]/2 from 0 to 1. (The third picture shows one of those regions). To solve the integral, note that the anti-derivative of arctan(z) is x arctan(x) - log(1+x^2)/2 (integration by parts). The answer of 1/2 + 3 I, which is log(2). Since there are 8 pieces, the volume of our beautiful solid is 8 log(2) = log(256).