Here 12 cylinders intersect. 6 axis passe through the middle of the dodecahedron faces and 6 through the middle of the edges of a cube inserted in the dodecahedron. Added 1.12.2008.

 

 

With radius 1 the surface of the intersection is

12.771001363

and its volume is a third of the surface.

 

The intersection has 192 facets.