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 s1 , 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.