I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set the radius $ = 1 $ the volumes are respectively
$ V_2 = \frac{4}{3}\cdot4 $
$ V_3 = \frac{4}{3}\cdot6(2-\sqrt{2}) $
A natural question is to find the volume of the shape created by the intersection of more cylinders whose axes intersect all at a single point: Moreton Moore wrote an article where he calculates the volume of the intersection of $ 4 $ and $ 6 $ cylinders with the axes passing through the center of the opposite faces of the octahedron and dodecahedron respectively
$ V_4=\frac{4}{3}\cdot9(2\sqrt{2}-\sqrt{6}) $
$ V_6=\frac{4}{3}\cdot4(3+2\sqrt{3}-4\sqrt{2}) $
In another post was given the answer for the $ 10 $ cylinders case:
$ V_{10} = \frac{4}{3}\cdot\frac{15}{4}(24 + 24 \sqrt{2} + \sqrt{3} - 4\sqrt{6} - 7\sqrt{15} - 4\sqrt{30}) $
It's immediate to note that the limiting volume of infinite intersecting cylinders will be the unit sphere
$ V_\infty = \frac{4}{3}\cdot\pi $
So my question is if a formula is known for the general volume $ V_n $. The question was already asked here, but no response was given. Writing $V_n = \frac{4}{3}\cdot\sum_{i=1}^N a_i $ how can I calculate the algebraic coefficients $ a_i$ without calculating the integrals for each case? The corresponding serie would be $ \sum_{i=1}^\infty a_i = \pi$