Quadratic irrational numbers are perhaps the most basic examples of irrational numbers that arise as basic physical ratios: think of $\sqrt{2}$ as the distance between the corners of a square to the length of a side for example.

Similarly, the number $\pi$ is the ratio of the circumference of a circle to its diameter. Even $e$ can be realized in this sense as the distance $a$ such that $\int_{1}^{a}\frac{1}{x}dx=1$. Are there similar understandings for other irrationals?

>Can Liouville's number be expressed as the ratio of two measurements of a geometrical object?