Pentagon, Hippasos

Consider the Pentagon ABCDE, the length of each side is s₀ and the length of the each diagonal (e.g. EC) d₀. Consider that all sides of CDED' are of equal length and all the four sides of AC'A'D also.

Hippasos used a geometric analog of Euclid's algorithm to show that the ratio d₀/s₀ is an irrational number. From the Pentagon ABCDE and the diagonals a smaller pentagon A'B'C'D'E' can be formed with diagonal d₁ and side s_{1 ,} and from this a smaller pentagon can be again constructed with a diagonal d₂ and side s₂ and so forth. One can find the following relations between the sides and the diagonals of these pentagons:

d₀-s₀ = d₁ < s₀, s₀ – d₁ = s₁ < d₁,

d₁-s₁ = d₂ < s₁, s1 – d₂ = s₂ < d₂,

d₂-s₂ = d₃ < s₂, s2 – d₃ = s₃ < d₃,

and so forth.

Hippasos showed that the Euclid algorithm will never stop! Therefore it is impossible to expressed the ratio d₀ /s₀ as a ratio of integers.