These are all regular (edges are all the same length) polyhedra (Greek: poly = many + hedra = seat) whose vertices (the sharp points) all touch the surface of an associated sphere called the circumscribed sphere (drawn as translucent red in this graphic). Each polyhedron thus sits precisely inside its circumscribed sphere. Another sphere (the inscribed sphere) sits precisely within each polyhedron and so glancingly touches each facet. The Platonic solids therefore lie sandwiched between their two associated spheres.