Here 9 cylinders intersect. 6 axis passe through the middle of the dodecahedron faces and 3 through the middle of 6 edges arranged as an octahedron. Added 18.9.2008.

 

 

With radius 1 the surface of the intersection is

12.75636935

and its volume is a third of the surface.

 

The intersection has 96 facets.