Let $B_n$ be a closed ball in euclidean space $\mathbb{R}^n$, and consider the largest cube $Q_n$ contained in $B_n$. Then, let $C_n$ be a cube of maximal size that is contained in $B_n$ and disjoint from the interior of $Q_n$. Set $$r_n:=\frac{\text{vol }Q_n}{\text{vol }C_n}.$$ This ratio is independent of the choices of $B_n$ and $C_n$. It depends only on the dimension $n$. Now, if I did not completely screw up my calculations, it holds $$\lim_{n\to \infty} r_n=e^4,$$ where $e$ is Euler's number $2,718...$. I find this remarkable as it allows us, in principle, to define $e$ using only certain natural intrinsic geometric ratios. However, in contrast to the definition of $\pi$ as the intrinsic ratio between circumference and diameter of a circle, we need to pass to the dimension limit here, which arguably makes the approach to $e$ above much less elementary from a conceptual point of view.

In view of the situation above, I would like to know:

**Which other elementary geometric approaches to Euler's number are there?**

**Is there a textbook or an essay that deals with different elementary geometric approaches to $e$ and includes the one above?**

Edit: Maybe my question is too vague, but I hope that the spirit in which I asked it comes through. In view of the first answer I got, I would like to emphasize that I am aware of the "usual" definitions of $e$ that are found in standard textbooks and the fact that there are many definite or indefinite integrals of basic functions that could be used to define Euler's number. They represent an area under some curve, which allows one to view them as "geometric" objects. However, this is not what I mean, even though of course all volume and length measurements could be written as some integral, and all statements about intrinsic ratios could be formulated as statements involving some quantity being $1$. Thus, by *elementary euclidean geometry*, I mean the geometry of lines, circles, balls, cubes, cones, polygons, hyperbolas, parabolas, etc., not in a particular coordinate system but rather in an invariant fashion.

Edit 2: As pointed out by Dan Romik, there are related questions on stackexchange. However, all related questions at stackexchange I can find are of the open form "is there some geometric approach to $e$", and the best answers are very similar to Dan's. Yet I feel that the example above is conceptually somewhat different and I am concretely interested in approaches of a similar form, in contrast to asking a completely open question. Therefore, I would say that this question is not a duplicate of the related questions at stackexchange:

https://math.stackexchange.com/questions/382833/does-e-have-a-geometric-representation

https://math.stackexchange.com/questions/159707/is-there-any-geometric-way-to-characterize-e

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