Euler’s Gem—The Polyhedron Formula and the Birth of Topology

They all missed it.’’ Richeson’s book begins with a strong and clear motivation for one of his key points on the nature and the historical development of mathematics. 

‘‘It’’ is ‘‘Euler’s Gem,’’ Euler’s polyhedron formula, one of the most beautiful formulas of mathematics (in fact, the author informs us, a survey of mathematicians found its beauty to be second only to $e^{pi} + 1 = 0$, also Euler’s). 

‘‘They’’ refers to all of Euler’s predecessors who, though active in the field of geometry, failed to come across this elegant and, to our eyes, even obvious relationship. Euler’s polyhedron formula is elegant and simple: In a polyhedron, the number of vertices (V), edges (E) and faces (F) always satisfy the equality $V–E + F = 2$. 

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