Writing square root of two as a continuous fraction leads to a periodic sequence of coefficients. More generally, a continuous fraction is periodic if and only

Writing square root of two as a continuous fraction leads to a periodic sequence of coefficients. More generally, a continuous fraction is periodic if and only if the number it represents is a quadratic number, that is, a linear combination of a rational number and the square root of a rational number [1]. It does not exactly answer your question, yet in the sense of continuous fraction at least, square root of two is more structured than pi, and less random. Edit: the golden ratio is also a quadratic number, so this intuition is wrong in the end! [1]: https://en.m.wikipedia.org/wiki/Periodic_continued_fraction#Relation_to_quadratic_irrationals https://en.m.wikipedia.org/wiki/Periodic_continued_fraction#...