Here 21 cylinders intersect. 15 axis passe through the center of the edges of the dodecahedron
and 6 through the centers of the edges of the cube inserted in the dodecahedron. Added 2.5.2009.

With radius 1 the surface of the intersection
is

12.5991654

and its volume is a third of the surface.

The intersection has 384 facets.