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?

I will add my inspiration for asking this question in the hopes of encouraging more discussion that can help me focus these ideas. While pondering the argument for the existence of two irrational numbers $a$ and $b$ such that $a^b$ is rational using $\sqrt{2}^\sqrt{2}$ and the "law of the excluded middle", I wondered at the linked wikipedia article's note that an "intuitionist" would not accept this argument as proof. I feel like the only way to reject this argument as proof would be to question the assumption that $\sqrt{2}^\sqrt{2}$ exists as a real number, and in fact must be either rational, or irrational. (Any intuitionists want to way in?)

This led me to ask myself, how do I know that $\sqrt{2}$ exists? I of course immediately answered that it was the ratio of the diagonal of a square to the length of a side! This led me to consider the "existence" of other irrational numbers as "some sort" of "physical ratio."

In considering this I would (for now) accept the description of $e$ above as being a "physical ratio."

Can anyone help me define what I mean by these "physical ratios?" I suppose I am looking for some argument for the "existence" of otherwise incomprehensible numbers, and whether or not every irrational could be explained in such a way.

I of course must note that this is incredibly vague and that the only one who can truly decide what I mean is myself, but any advice would be great!

physically. I realize that with a ruler and compass you can draw a curve whose length is $\pi$. What precisely is one allowed to do here in creating geometric objects? Are you admitting any curve length (or ratio of curve lengths) that can be constructed with a ruler and compass? – Benjamin Dickman Dec 19 '12 at 6:30