Intersecting Spheres

If in 2-D we intersect two circles (called one-spheres by mathematicians), the intersection is either a point, two points, or a circle. In 3-D the intersection of two spheres (each called a two-sphere) will be either a point, a circle, or a sphere. What can the intersection of two three-spheres be? And three three-spheres?

Answer

Two identical three-spheres can intersect in a point, a circle, a sphere (two sphere), and a three-sphere. Now bring in a third identical three-sphere to intersect with the former two in appropriate combinations of points, circles, spheres, and a three-sphere, the latter when all three are coincident. With three intersecting identical three-spheres, a resulting single two sphere can be obtained only when the three three-spheres form a symmetrical configuration. If the leptons and quarks of the Standard Model of particle physics are physical manifestations of the finite rotational symmetries of the 3-D Platonic solids and their 4-D analogs as proposed in a model by F. Potter (see the reference below), then the intersections of three-spheres will become important in fundamental physics. A quark would be defined in a 4-D space, and its mathematical behavior would depend on the properties of three-spheres. The proton, for example, is a real particle composed of three quarks in our 3-D world that is, three 4-D entities according to the proposed model. So three three-spheres (representing the quarks) must intersect to form a two-sphere that “lives” in our 3-D space.