Defining Euler's number via elementary euclidean geometry (and a dimension limit) 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/387746/pi-from-the-unit-circle-sqrt-2-from-the-unit-square-but-what-about-e
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
 A: Four ants are on the corners of a square, each facing its neighbor in the counterclockwise direction. At the same time each begins marching towards its neighbor, all at the same speed. After moving through one radian of angular measure about the center of the square, they are each closer to the center of the square than they were at the beginning by a factor of $e$. 
Analytically, the ants trace out logarithmic spirals, describable as curves $t \mapsto \exp((-1+i)t) z_0$ where $z_0$ is the starting position. 
A: Carving Euler’s number in a carrot.
As noted by others, $e = 2.718...$ is transcedental, so we can’t construct this length with 'compass and straightedge'. So let's try it with 'carrot and knife'!
I will do this by carving the exponential function $f(\theta) = e^\theta$ on the mantle of a cylinder (a carrot).
1) Place the carrot under the $y-$axis, and position a knife on top of the line $y = x + 1$

2) Fixate the start of the blade at the intercept with the $x-$axis, such that this serves as a rotation point. With the start orientation of $45^{\circ}$, the knife initially cuts the mantle at $(x,y) = (0,+1)$:

3) Now start rotating the carrot in the counter clockwise direction. This makes that the knife will create a continuous carve on the carrot's mantle. As a result, the intersection point between knife and cylinder, will move away over the $y-$axis. 
Claim:
The resulting carve forms the exponential function $f(\theta) = e^{\theta}$.

Explanation
Note that the knife always cuts the carrot at the $y-$axis. This makes that for the tangent of the knife we have:
$$\frac{\Delta y}{\Delta x} = \frac{y-0}{0 - (-1)} = y$$
This yields a differential equation in the cylindrical coordinates of the carrot:
$$\frac{dy}{d \theta} = y(\theta)$$
Combined with the initial condition $y(0) = 1$, the solution of the differential equation $y’= y$ is indeed $y(\theta) = e^\theta$
The finish
Our cut $y=e^\theta$ intersects the line $\theta = 1$ at $(\theta, y) = (1, e)$. Connecting this point with $(\theta, y) = (1,0)$ creates a vertical segment with length $e = 2.71828...$

$\blacksquare$
PS: Rotating the carrot $180^{\circ}$, we get a length of $=e^{\pi}$ (Gelfond’s constant).
A: As far as I know, $e$ was encountered the first time as the answer to the question, what would happen, if the intervals, in which interest is paid and reinvested, tend to zero or, for short: $$ e := \lim_{n\to+\infty}\left(1+\frac{1}{n}\right)^n$$
The geometric interpretation is straight forward: 

$e$ is the limit-volume of an $n$-dimensional cube with sides of length $1+\frac{1}{n}$ 

As hyper cubes of the desired side-length can be constructed with straight-edge and compass, and because hyper cubes are of elementary geometry, that should answer the question
A: $e$ is the unique number $>1$ such that the area in the plane bounded between the three lines $y=0$, $x=1$, $x=e$ and the hyperbola $y=1/x$ is equal to $1$.

A: Speaking of hypercubes (drawing on the comment by Manfred Weis), there is another perspective that is geometric--but also combinatorial.
Here is Euler's Number expressed a series of hypercubes each divided by n!, where n is the dimension of the hypercube.
$$e^{x} = \sum_{n=0}^{\infty} \frac{x^{k}}{k!}   =  \frac{x^{0}}{0!} + \frac{x^{1}}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \cdots $$
You'll notice a similarity to elementary problems in combinatorics: from an arbitrary set {a,b,c,d,e} where each element is in [0,x], how many ways can you form coordinates in $\mathbb{R}$3? It's just a combination with repetition problem. There are
$$\binom{n + 1 - r}{r}$$
such ways. Similarly, each term can be thought of the set of combinations of coordinates with repetition in each hypercube space.
