Here's my drawing:



Finally, we pick a forth point p_4, which is not in the p_1, p_2, p_3 plane, and looking in the p_1, p_4, q plane we find the intersection of the line through q and the line between p_1 and p_4. This gives us a unique center point, and having already picked p_1, etc., we also know the length of the radius of the sphere.

Therefore, it takes 4 points, not coplaner (i.e., not picked from the same plane), no 3 of which are colinear, no 2 of which are the same, to define a unique sphere which goes through them.

Getting back to our polyhedron problem, we note that a regular polyhedron will have a center which is the same distance to every one of its corners (this should seem reasonable, even if it would take a little work to prove). Then every corner intersects the surface of the sphere (previously described) which goes through them,. Since every corner resembles the other corners, they must all look like tops of pyramids (with 3 or 4 or more edges all meeting at a corner) all coming together at the point in the sphere. It follows that non can be what is called a saddle point corner. Here is a picture of a saddle point corner:



It should reasonably follow that regular polygons are convex, and that in fact they are obtained by beginning with a solid sphere and carving off pieces to reveal polygonal faces.

Stop and try to name or describe some regular polygons.