After learning how to intersect two cylinders, let’s try it with more. The next simple case are three cylinders, and the most symmetric case has them aligned along the coordinate axes. The union is on the left, and the (rescaled) intersection on the right.
We obtain an inflated cube that has twelve cylindrical rhombi as faces, one for each edge of the cube. This is therefore a version of the rhombic dodecahedron. Using four cylinders, the most symmetric way is to align them with the diagonals of a cube. We see here (again) that each cylinder contributes a chain of cylindrical faces that touch at opposite vertices.
For six cylinders, there are two notable choices for the axes. The first is to use the six edges of a tetrahedron, recentered so that they all pass through the origin. Alternatively, we could use lines that connect midpoints of opposite edges of a cube.
The other, even more symmetric choice is to use the diagonals of an icosahedron. Using axes related to the platonic solids creates simple patterns that can all be understood in terms of finite reflection groups acting on the sphere.
By now, I have probably created the impression that the appearance of the chains of equally colored cylindrical faces has something to do with the symmetry of the chosen axes. Not so. The next image shows a random choice of 10 cylinders of equal radius through the origin, and their intersection to the right.
The black curves are the waists of the cylinders, i.e. their intersections with the sphere of equal radius centered at the origin. The waist of a cylinder must belong to the intersection, because any other cylinder will properly contain it. Thus our chains appear along these waists, and they are pinched whenever to waists meet.