Incommensurability between two magnitudes refers to the fact that their ratio cannot be expressed by a rational number and that, consequently, rational numbers are insufficient to describe reality. For example, it is known that the ratio between the circumference of a circle and its diameter is an irrational number, conventionally denoted by \(\pi\). The first mathematical proofs of existence of incommensurable magnitudes can be traced to Ancient Greece, namely the proofs of incommensurability between the diagonal and the side of a square and a regular pentagon.
For regular polygons with more than \(5\) sides, the diagonals (line segments joining non-adjacent vertices) do not all have the same length. We may, however, consider just the shortest diagonals (those with smallest length) in those polygons. Indeed, in the case of the regular hexagon there also exists a geometric proof, similar to those known for the square and the regular pentagon, showing the incommensurability between its shortest diagonal and its side. For regular polygons with more than \(6\) sides, however, it is not possible to establish this incommensurability using a geometric proof similar to the previous ones, despite the magnitudes involved also being incommensurable.
Looking at other diagonals, the situation is similar. For example, for every regular polygon with more than \(6\) sides, the second shortest diagonal and the side are always incommensurable, although there exist geometric proofs of this only for the cases of the regular octagon, decagon and dodecagon. As for the longest diagonal (the one with largest length), it is always incommensurable with the side for every regular polygon with more than \(6\) sides, but this can only be proved geometrically in the case of the decagon. Why is that?
This page aims at presenting the geometric proofs mentioned above, as well as the algebraic arguments which not only show the incommensurability for other cases but also allow us to understand why, in those cases, there is no geometric proof analogous to the previous ones.
We shall also see how it is possible to interpret, in terms of dynamical systems, the process of construction of new regular polygons used in the geometric proofs of incommensurability.
(*) This work was carried out under the guidance of Professor Maria Carvalho from the Universidade of Porto, under a grant by the Calouste Gulbenkian Foundation to develop a project for the promotion of Mathematics in Atractor.
Difficulty level: University