One can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedron, cube, and octahedron all admit very intuitive constructions. The cube is a cube, the octahedron is its dual, the tetrahedron has as vertices four pairwise non-adjacent corners of a cube. One can check that that everything you want holds on a single piece of paper.
Does anyone know a correspondingly elementary proof that the dodecahedron or icosahedron exists?
Best Answer
One of my favorite dodecahedron constructions goes something like this: Begin with two regular pentagons joined along an edge. Cut off the "far" triangles, leaving identical trapezoids joined along the edge. Finally, glue the severed triangles into to fold to create a "pup-tent".
The pup-tent has a perfectly-square base, and placing one such tent on each face of a cube causes trapezoidal faces of the tents to combine with the triangular faces of other tents to (re-)form the pentagonal faces of the dodecahedron.