Why isn it obvious that a regular dodecahedron exists?
What is wrong with the following argument for the existence of a regular dodecahedron, an argument which is supposed to describe what one actually does when making one out of cardboard? Draw a regular pentagon in the plane, and surround it by five further regular pentagons of the same size, each one sharing a different edge with the original pentagon. Now fold these upwards, all by the same angle, until they just touch each other, so that you have a sort of cup. The top of this cup consists of ten edges, two for each of the five further pentagons, that zigzag round roughly in a circle. Let us label the original pentagon A and the five subsequent ones B,C,D,E and F, ordered cyclically. The upper corners of the zigzag make an angle of exactly 108, the angle of a regular pentagon, since each one is formed by two edges of one of B,C,D,E or F. I claim that the same is true for the lower corners. This can be seen as follows. Consider the corner of the zigzag at the top of the edge e shared b
