# Is there any geometric way to characterize ee?

Let me explain it better: after this question, I’ve been looking for a way to put famous constants in the real line in a geometrical way — just for fun. Putting $$√2\sqrt2$$ is really easy: constructing a $$45∘45^\circ$$$$90∘90^\circ$$$$45∘45^\circ$$ triangle with unitary sides will make me have an idea of what $$√2\sqrt2$$ is. Extending this to $$√5\sqrt5$$, $$√13\sqrt{13}$$, and other algebraic numbers is easy using Trigonometry; however, it turned difficult working with some transcendental constants. Constructing $$π\pi$$ is easy using circumferences; but I couldn’t figure out how I should work with $$ee$$. Looking at

made me realize that $$ee$$ is the point $$ω\omega$$ such that $$∫ω11xdx=1\displaystyle\int_1^{\omega}\frac{1}{x}dx = 1$$. However, I don’t have any other ideas. And I keep asking myself:

Is there any way to “see” $$ee$$ geometrically? And more: is it true that one can build any real number geometrically? Any help will be appreciated. Thanks.

For a certain definition of “geometrically,” the answer is that this is an open problem. You can construct $\pi$ geometrically in terms of the circumference of the unit circle. This is a certain integral of a “nice” function over a “nice” domain; formalizing this idea leads to the notion of a period in algebraic geometry. $\pi$, as well as any algebraic number, is a period.
It is an open problem whether $e$ is a period. According to Wikipedia, the answer is expected to be no.