It’s perfectly simple, Kathryn, and the kind of math anyone can follow.

Start with a triangle. There are all sorts of triangles, of course: tall

thin ones, short squat ones, no-particular-shape ones. The nicest one is

the one whose three sides are all equal length (& so its 3 angles all 60

degrees). That is a regular triangle. Same with 4-sided figures: the

square is the regular one–sides & angles all equal. There’s also a

regular pentagon, a regular hexagon, and so on–along, of course, with lots

of irregular ones in each case. So there is an infinite family of regular polygons. Now: what would be the equivalent in 3 dimensions? The 3-D

equivalent of “polygon” (2-D figure with straight sides) is “polyhedron”

(3-D figure with flat faces). The most regularity you can get is when (A)

all the faces are regular polygons, and (B) they are all the SAME regular

polygon. The ancient Greeks showed that there are just five convex

polyhedra satisfying both (A) and (B), the so-called Five Platonic Solids.

They are the top five in the top photograph on Jeff’s site. If you drop condition (B)–i.e. the faces must be regular polygons, but need not be all the same–you get the semiregular

polyhedra. There are 13 convex possibilities. If you drop the “convexity”

condition & allow, e.g., not only pentagons but also pentacles, you are in

the world of stellated polyhedra. Oh, in case you are wondering, there

are six regular convex figures in 4-D; but in any number of dimensions

higher than 4, there are only 3.