Here 7 cylinders intersect following the corners of a rhombododecahedra.

With radius 1 the surface of the intersection is

and its volume is a third of the surface.

We have only two different not flat corners. The symmetric one has a solid angle of 36.02% and the other one a solid angle of 39.69%. The symmetric corner is identical with the 3 edge corner of the intersection of four cylinders.