In this chapter we give a construction of the icosahedron that is both self-evident and rigorous. This is a surprisingly neglected topic. Pictures of icosahedra are abundant in books on geometry, and they create in the reader the false impression that a beautiful picture proves the existence of the object.
Moreover, we separate clearly the issues of existence of the icosahedron and its uniqueness. As we shall see, the uniqueness is intimately related to the fact that the icosahedron has a rich set of symmetries that only a polytope can admit.
KeywordsSymmetry Group North Pole Convex Polytopes Continuity Principle Full Symmetry Group
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