Mathematical impossibilities

September 10, 2020

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‘Nothing is impossible’ is one of the most used phrases to convince people that they can accomplish anything, no matter how hard you have to work for it. But to what extent is this phrase actually true? The word impossible is something that a lot of people already have big trouble with getting their head around. If you would ask me for an example of ‘what is impossible?’, the first thing that comes to my mind is teleportation. But the second after thinking that, a new thought comes up: ‘is it really impossible though?’. Ofcourse, you can come up with some super trivial impossibilities, such as you cannot pour a drink from an empty bottle or you cannot read a book without any letters in it. But what if we go a bit deeper, for example impossibilities in nature? Mathematicians like theorems, properties and facts. Theorems always hold (under certain given conditions), since every theorem has an extensive proof. But what if things occur in the world that contradict a proof or a calculation? In this article, we will look at a very interesting ‘paradox’ that, mathematically, contradicts something that happens everywhere in the world.

Introduction to the paradox

Here in the Netherlands we (unfortunately) experience approximately 130 days of rain per year. You probably still recall from primary school how rain arises: water evaporates and rises, clouds are formed and at a certain point, these clouds ‘drop’ rain. Seems reasonable, right? It’s probably not a surprise to you that raindrops are very interesting if we look at it from a mathematical and physical perspective. What probably will be a surprise to you, is that rain is mathematically impossible. How is that possible? It happens everywhere, right? This is one of these weird ‘impossibilities’ that we will look at.


Liquids hate surfaces, or, differently put, liquids like to bind together and be in an as-closed form as possible. That is why it is for example possible to put more water on the top of a glass or to put quite some amount of water on the top of a coin. This is because creating a surface costs more energy than creating a volume. Suppose you are evaporating water in saturated air from a gas to a liquid. Every cubic centimetre of water that is being made releases energy, but to make each squared centimetre of the surface of that water requires an input of energy. For water, the energy you obtain from the volume, which is approximately r^3, where r stands for the radius of the water, is more than enough to compensate for the energy loss of the surface area, which is approximately r^2.

The problem

As you know, cubing positive numbers usually makes the number bigger than squaring them. For example, 5^3 > 5^2. You can trust me on the fact that this holds for all numbers greater than 1.   Let us now consider the situation r^3 = r^2. This is equivalent to r(r^2 - r) =  r(r(r - 1)) = 0, which solutions are given by r = 0 and r = 1. Looking at the graphs of r^2 (blue) and r^3 (red), we see that between the point 0 and 1, r^3 < r^2.  Now, since raindrops are very small at the beginning, we are quite safe to say that the radius of a raindrop will typically be under 1 centimetre (and, trivially, more than 0 centimetre): problem!  There is not enough energy to form the raindrop and, eventually, r^3 will approach 0 as the size of the raindrop gets smaller and smaller. Rain can now not exist!

But how?

As you probably do agree with me, rain does exist. But how come we were just able to ‘prove’ that it goes against physics? That is because we considered raindrops to consist solely of water. In reality, it contains a lot of microscopic substances that fully mix up the composition of the raindrop. On top of that, we ‘skip’ a lot of different physical laws that ‘fight’ this. Still, it is very interesting to see that we could even set up such a proof. There are many more mathematical impossibilities that, in case you find these types of things interesting, I would recommend looking up and reading about. For now, let’s hope this proof does hold for the upcoming days, such that we can enjoy the good weather a little bit longer!

Dit artikel is geschreven door Lars Beute:

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