Sur l’ incommensurabilité dans les polygones réguliers

One of the most important concepts in mathematics is ‘incommensurability’. In this paper, we propose a didactic path referred to the commensurability among elements of regular polygons. Avoiding the method of ‘infinite descent’, we will show that ‘the only regular polygon whose side is commensurable with the circumradius is the hexagon’ and that ‘the only regular polygon whose side is commensurable with the inradius is the square’. A special analysis is devoted to the case of regular heptagons, where we prove that ‘two distinct elements among sides, diagonals, inradius, and circumradius are incommensurable’.