The "e" in apple

Question

I cut a thin slice from a spherical apple, cutting along a plane.

Then I realized, there is an "e" in apple.

Why is there an "e" in apple?

Hint:

It doesn't have to be an apple. It just has to be a spherical solid object from which a thin slice can be cut. An apple is a good choice because it has two colors (red and yellow), making the explanation a bit easier.

OK, time to drop another hint:

Consider the four quantities:

r = red surface area of the small piece
y = yellow surface area of the small piece
R = red surface area of the big piece
Y = yellow surface area of the big piece

Find a limit expression with these four quantities that equals Euler's number, e.

Answer 1

I had the following idea but it felt a bit vague. But after seeing the second hint, seems like this could be the intended answer.

The area of the red part of the thin slice is 2πrh and that of the yellow part is 2πrh(1-h/2r). Here r is the radius of the sphere and h is the thickness of the slice (i.e., r minus the distance of the plane from the centre of the sphere). The total surface area of the sphere is 4πr². The ratio of the first two areas is 1/(1-h/2r), whereas the ratio of the third and the first areas is 2r/h. Take the first ratio to the power of the second ratio and take the limit h/r to 0 (thinner and thinner slice). This yields e.

I was wondering if there’s a better (geometric or perhaps probabilistic) motivation for taking the power of ratios in the last step, but I don’t have one yet.

Answer 2

There is an "e" (or its Greek form "ε"). As you mentioned and reaffirmed in a comment, the slice is "super thin." In math, epsilon can refer to a small, arbitrary number. Thus, being a super thin slice, in a way, the "e" stands for "ε."

Answer 3

Here is my answer (essentially the same as @Pranay's answer).

The moral of the story is that e is hidden in every ball.

Taking OP's hint, we consider the four quantities:

$r=$ red surface area of the small piece
$y=$ yellow surface area of the small piece
$R=$ red surface area of the big piece
$Y=$ yellow surface area of the big piece

We will show that the following limit (note the elegant structure) equals $e$: $$L=\lim\limits_{r/R\to 0}\left(\frac{r}{y}\right)^\frac{R}{Y}$$
Assume the radius of the sphere is $1$.
Let $t$ be the thickness of the slice. Archimedes tells us $r=2\pi t$.
Let $a=\sqrt{1-(1-t)^2}=\sqrt{2t-t^2}$ be the radius of the yellow disks.

$L=\lim\limits_{r/R\to 0}\left(\frac{r}{y}\right)^\frac{R}{Y}$
$=\lim\limits_{t\to 0}\left(\frac{2\pi t}{\pi a^2}\right)^\frac{4\pi-r}{\pi a^2}$
$=\lim\limits_{t\to 0}\left(\frac{2}{2-t}\right)^{\frac{4-2t}{2t-t^2}}$
$=\lim\limits_{t\to 0}\left(\frac{2}{2-t}\right)^{\frac{2}{t}}$
$\overset{t=-\frac{2}{n}}{=}\lim\limits_{n\to \infty}\left(\frac{2}{2+\frac{2}{n}}\right)^{-n}$
$=\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$
$=\boxed{e}$

Answer 4

I doubt this is the answer that you're looking for, but here's my stab in the dark. It's a "topological" take on the question.

"e" stands for edge. The uncut apple is a sphere, a manifold without a boundary. After the cut, the surface of the apple now has an edge.