On the non-existence of a perfect map from the 2-sphere to the Euclidean plane

Abstract

We present a proof of a theorem of Euler asserting that there does not exist any perfect map from the 2-sphere to the Euclidean plane. The meaning of the word „perfect" is explained. Euler's result is part of his broad work on cartography, whose aim was to find representations on the Euclidean plane, with the least possible distorsion, of regions of the sphere. The word „distorsion," in the context we consider, is explained. The ideas in the proof presented here are applied to the study of other similar problems.