A Platonic solid is a 3-dimensional shape built from identical flat figures called regular polygons, with the additional requirement that the same number of polygons should meet at each corner of the solid. A polygon is a closed shape in the plane made up of line segments. It is said to be regular if all the line segments have the same length and all the corner angles are equal. Thus, a 3-sided regular polygon is an equilateral triangle; a 4-sided regular polygon is a square; a 5-sided regular polygons is a pentagon; and so forth.

There are only 5 ways to assemble regular polygons into Platonic solids. When we build a Platonic solid, we must have at least 3 polygons meeting at each corner--two polygons are not sufficient to build a solid angle. Let's start by considering what Platonic solids can be built out of equilateral triangles. If...