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Platonic solids are those forms in 3D that have all their edges and angles and faces identical to each other. But why are there only 5 of them?
Think about how many different ways there are to make a 3D corner out of 2D polygons. The combined total of degrees meeting at each vertex must be less than 360 or the 'corner' will be flat or even overlapping. Also, at least three polygons must meet to form a 3D corner. It is very much worth convincing yourself that the 5 possibilities below are the only ones.
If you know this... then can you figure out why there are 13 Archimedean solids where the requirement for identical faces and angles is relaxed, and all that is neccessary is that each vertex is identical to all the others?
