Watermelon Paradox: A Line/Surface/Object/HyperObjext/... has infinite length/area/volume/hypervolume? [closed]

Question

Watermelon Paradox?

I discovered (I do not know if I'm the first one tho) that when cutting a watermelon in half, I added two surfaces while the volume of the two different pieces is the same.

(You gotta be silly with the name of the paradox or no one will remember it, but it need to contain some truth or it makes no sense)

So the surface area of the watermelon increased while the volume stays the same.

To simplify, I will just use a 2D square, the area acts as volume and the circumference of the square acts as the surface area.

Imagine a square with the length of 2:

|--2--|
 -----  
|     | 
|     | 
 -----  

(not a very accurate square)

$A=2^2=4$

$C=2*4=8$

Then, we split the square in two:

 --   --  
|  | |  | 
|  | |  | 
 --   -- 

Now, the long side, $a$ is 2, and the short side, $b$ is 1

$a=2$

$b=2/2=1$

$A_1=a*b=2*1=2$

$A_2=a*b=2*1=2$

$A_{total}=A_1+A_2=4$ (Same as the unsplitted square).

$C_1=(a+b)*2=6$

$C_2=(a+b)*2=6$

$C_{total}=C_1+C_2=12$ ($C_{total}-C=4$ because there is two more segments with length 2 added)

Then we split the two rectangles into 4 rectangles so that the circumference is

$C_4=(\frac{2}{4}+2)*4=20$

So, the formula for getting the total circumference of a square of 2x2 splitted into a Real Number of rectangles without changing the total area of the square is:

$C_x=(\frac{2}{x}+2)*x=2x+2=2(x+1)$

Funny thing is, it doesn't converge, meaning

$$\lim_{x \to \infty} 2(x+1) = \infty$$

If we want to see the total circumference of x (x is a Real Number) number of splits of a mxm square without changing the total area is:

$C_x = (\frac{m}{x}+m)*x=mx+m=m(x+1)$

Now, here's the catch:

If a rectangle's short segment is zero, it becomes a line.

in this formula: $$\lim_{x \to \infty} m(x+1) = \infty$$

It describes when a square of mxm size is cut into infinite pieces along the vertical axis. Therefore, the short length of the rectangle becomes infinitly close to zero.

Therefore, a line is infinitly long.

Although I counted a line as two segments, you can always divide the formula by 2, and a line is still infinitly long.

$$\lim_{x \to \infty} \frac{m(x+1)}{2} = \infty$$

It also works with 3 Dimensions (and N Dimensions?)

Question:

I know this is not quite right, since a line have a length.

Bonus:

I found that the larger the square, the smaller the ratio of $m(x+1)$ to the area.

$$\frac{m(x+1)}{m^2}=\frac{x+1}{m},m\ne0$$

Meaning, $\frac{d}{dx} \frac{x+1}{m}$ decreases when m gets bigger.

So

$$\lim_{m \to \infty} \lim_{x \to \infty} \frac{x+1}{m} = 0$$

More wierdly, how can $\frac{\infty}{\infty}=0$?

Can we conclude from the above that $$ m(x+1) = m^2$$?

My Works in 2D splitting: https://www.desmos.com/calculator/rlrcirxmnx

Answer 1

This is a nice example, and getting clear about what's going on here requires getting clear about how limits in two variables work. The resolution is that you are trying to commute two limits that do not commute.

Let's begin with a rectangle of length $\ell$ and height $h$, so initially its perimeter is $p(1, \ell, h) = 2 \ell + 2h$. If we split the rectangle into $n$ smaller rectangles lengthwise, we get $n$ rectangles of length $\frac{\ell}{n}$ and height $h$, whose total perimeter is

$$p(n, \ell, h) = n \left( 2 \frac{\ell}{n} + 2 h \right) = 2 \ell + 2nh.$$

Normally I would have just written $p_n$ but in this context we need to be clear that $p$ is a function of multiple variables. Now you've observed that:

1. If we take $n \to \infty$ then for any $h \neq 0$ (note this!), the total perimeter diverges: $\lim_{n \to \infty} p(n, \ell, h) = \infty$.

2. As $h \to 0$, the rectangles become lines and the total perimeter converges to the total length of the lines, which is constant: $\lim_{h \to 0} p(n, \ell, h) = 2 \ell$.

Both of these are true. But to conclude that "all lines are infinitely long" what you are trying to do (I think, I don't quite follow this part of your argument since you never make $h$ explicit) is to argue that

$$\begin{align*} \lim_{n \to \infty} \lim_{h \to 0} p(n, \ell, h) &= 2 \ell \\ &\color{red}{\stackrel{?}{=}} \lim_{h \to 0} \lim_{n \to \infty} p(n, \ell, h) \\ &= \infty \end{align*}$$

and that middle step where you try to commute the limit $n \to \infty$ and the limit $h \to 0$ simply doesn't work (and of course the fact that $2 \ell \neq \infty$ is the proof that it doesn't work).

If you never explicitly talk about limits but only talk about what happens for specific, finite values of $n$ and $h$ then there is no paradox and it's clear what happens: the total perimeter is just $2 \ell + 2 nh$, and the size of this number simply depends on the relative sizes of $n$ and $h$. If $n$ is very large, say $n = 10^{100}$, and $h$ is only kinda small but not that small, say $h = 10^{-10}$, then $nh = 10^{90}$ remains large. But if $n$ is only kinda large, say $n = 10^{10}$, and $h$ is very small, say $h = 10^{-100}$, then $nh = 10^{-90}$ is very small.

So there is no well-defined two-variable limit $\lim_{n \to \infty, h \to 0} p(n, \ell, h)$; the behavior of $p(n, \ell, h)$ for large $n$ and small $h$ depends on how quickly $n$ gets large compared to how quickly $h$ gets small, and that's all. Typically the first time you'd see an example of this is in multivariable calculus where it's relevant to questions like when partial derivatives commute, but the underlying phenomenon is pretty elementary if you think of it in terms of how big and small various things get.

---

Incidentally, this fact that the surface area of a watermelon (or anything else) gets larger when you chop it up into smaller bits is an important observation in several parts of science, for example it is related to why we chew our food (this increases the available surface area for enzymes to digest it!); see surface-area-to-volume ratio for more on this.