The Unavoidable Flatness of Pentagon

A pentagon in R^3 with all side lengths and all angles equal must be planar.
The theorem is not true for isogonal equilateral n-gons with n = 4 or n ≥ 6.
This statement was first mentioned (without proof) in J. Waser’s doctor thesis (1944), “in the course of an electron-diffraction study of gaseous arsenomethane (AsCH3)n”. J. Waser is a chemist, and this property of pentagon had not been recognized among mathematicians until the chemist J. D. Dunitz discussed the problem with van der Waerden and G. Polya in 1970. Later, van der Waerden found a proof of it. Another proof, which is more elementary, was found by S. Smakal using Gram’s determinant. It is rare that such a beautiful mathematical theorem was unnoticed by the Greeks, nor Euler or Gauss, and first appeared in one humble sentence buried in a chemistry thesis.