The intersection of three cylinders (and six half-cylinders) are interesting objects and make for fun random desk objects. They are also fun to build out of wood (takes some time) or 3D print (mostly trivial).
The intersection of three cylinder at a right angle results in a Steinmetz solid. Calculating the volume of the intersection is a classic math puzzle.(image courtesy of Van Helsing)
I made a virtual 3d model that you can interact with in openjscad.
The intersection of six cylinders results in a cube, which can seem counter intuitive at first given that cylinders are curved objects.
You can also play with this case in openjscad.