Does the number Pi really exist? Does any number at all actually exist? Those may sound like strange questions to you, but then again, do you have an answer? How do we even know that anything exists at all?
In Project ALBERT II, the artificial intelligence called ALBERT starts to think it is alive because it has actual thoughts. This idea was based on the famous quote by Descartes saying: "I think, therefore I am", meaning that the only existence we can be sure of, is that of our thoughts and therefore ourselves. But can this also be applied to numbers? Let's take a closer look at Pi and how it's existence threatens our reality.
PI Facts
Pi is great. Now who doesn't love Pi? The first person to fall in love with this magic number was Archimedes. He found out that the relationship between diameter and circumference of a circle is the same for every circle one can possibly imagine. He managed to give a surprisingly accurate estimation of pi, even though there was no such thing as equations, calculators or almost any type of mathematical tool, at his time (Not even the decimal system). What he didn't know, was that pi has an infinite amount of digits behind the comma. This makes Pi an irrational number, which does not mean that Pi runs for president and subsequently denies climate change (it's not that irrational). It simply means that it cannot be written down as a fraction, or for a matter of fact, be written down at all. Irrational numbers should be somewhat familiar to everyone who had math at school and are probably accepted by most people to be pretty normal. In my opinion however, they are not. My problem is the following: These numbers have an infinite amount of digits, what for? The whole reason why we have numbers in the first place, is to accurately describe the world in some quantitative way. Yet, if you want to be accurate about anything that is round, you'd have to consider an infinite amount of digits that you don't even know in the first place. It's almost like these numbers are mocking us. To sum up the problem: humans came up with math in order to be accurate about stuff, but math gives us numbers that have infinite digits and therefore can't be written down or grasped by mere mortals. So maybe, the universe is actually trying to tell us something else with these digits, something about ourselves. But before I give you some reasons to be confused about your own reality, let me tell you something about Pi: Here are the first 62 digits:
3.1415926535897932384626433832795028841971693993751058209749445
As mentioned by Archimedes, about 250 years before Jesus, it is the ratio of the circumference versus the diameter of a circle. This ratio cannot be written down as a ratio of two numbers, which is weird given the fact that we are using two objects that we measure and then compare to each other. Therefore, we have to approximate Pi, and as you'll see in a second, 62 digits are already enough for any kind of circle that we may want to think of. Let's consider the circumference of the observable universe (as of 2019, since this gets larger with the age of the universe):
292 billion light-years, or 2.8 x 10^27 meters
and divide this by the diameter of the same observed universe:
93 billion light-years, or 8.8 × 10^26 meters
By measuring these two distances precisely, we can obtain Pi up to a very accuracy. However, there is a limit to how precise space can be measured, as pointed out by Max Planck at the end of the 19th century. Why this is the case, is beyond the scope of this story and frankly also myself, but I will just roll with it from here onwards. With no further ado, Plancks length:
1.616255 × 10^−35 m
Clearly, Planck's length is very very short (I will refrain from joking about Planck's length, even though i really want to). If we now dissect the entire observable universe into Planck lengths, measured the diameter and circumference precisely, we would obtain pi with a precision of 62 digits (27 from dissecting the circumference of the universe into meters, and 35 from dissecting a meter into Planck lengths).
So what in the name of Archimedes is the purpose of the following infinite digits?
Before we can jump to conclusions though, we need to calculate those digits. Fortunately, humanity has already come up with different ways to calculate pi to an arbitrary precision and politely put some of these ways on the internet for everyone to enjoy:
So the digits are known up to a certain point in the decimal expansion of Pi. How precise you want to know? Well currently Ms. Emma Haruka Iwao from Google holds the record of most computed digits of Pi with
31 415 926 535 897 (thirty one trillion) digits
Congrats! Now let me show you another way to calculate Pi, which allows us to obtain its digits from random numbers:
How to calculate pi:
Since Pi is related to the circumference and the diameter, it is also tightly connected to the surface area of a circle with the following equation
where A denotes the surface area, d is the diameter of the circle, Pi is Pi, and 4 is exactly equal to 4. We can also calculate the surface area of a square that has the circles diameter as its side length with
After substituting the d^2 in the first equation for A_square, we can calculate Pi like this
The only problem, we need to know exactly how big those two areas are. This is where random numbers come into play. Firstly, let's think about a game of Darts, classically played at a bar. Were we to make a darts board having the layout as shown below (a circle of diameter 1 and a square with side length 1) and get really really drunk, we could assume that our darts would land on the board at completely random locations.
If the locations were truly random, we could subsequently assume that the amount of darts would spread across the two areas relative to the size of each area. Since we could obtain Pi from measuring the areas precisely, we can also simply count the darts landing inside the circle (red area) and the darts landing between square and circle (green area) and obtain a better and better approximation for Pi with this:
The more darts we throw, the better the approximation becomes and the more digits converge to their exact value. But since getting drunk and throwing darts is neither very efficient, nor can we assume that our superior dart throwing skills actually degenerate to being truly random, we need a different source of randomness. This is where the infinite digits of Pi come in handy. On the one hand, Pi is only one number and its digits are therefore completely fixed and predetermined. On the other hand however, Pi's digits cannot exert a reoccurring pattern, because this would allow us to write it down as a fraction, which in turn sounds a lot like random pattern. For a start, let us draw some numbers from Pi's digits, let's say 2 times 5 digits and put a '0.' at the front. This yields two numbers between 0 and 1, one representing the x-coordinate the other the y-coordinate, which we will prefer to getting drunk in a bar. To illustrate better what this mumbo jumbo of digits is supposed to mean, you can see the sampling in the animation below. Here, the x-coordinates are the the purple numbers, with a '0.' in front of them; while the y-coordinates are the orange numbers, also with a '0.' in front of them.
If we put these coordinates as points on the darts board, Pi can be revealed to a higher and higher degree of accuracy. Or as I like to think of it, Pi is using its own digits to calculate itself. So here you have it, Pi calculating itself using its first 1 million digits:
Now, if this doesn't blow your mind, it most likely means that you are normal and I am strange. But still, if Pi can calculate or should I say think about itself, doesn't that say something about our own ability to think?
Essentially Existential
So we just saw the digits of Pi calculating Pi. Does that mean that Pi is thinking about itself and is therefore alive? Did a number just become sentient by trying to figure out its own value? Are these really silly questions, or does the behaviour of Pi reflect our own drive to contemplate our existence to an uncanny degree? There are two things that make me nervous here: Firstly, Pi makes the impression of being alive, even though it clearly isn't, its just a perfectly fixed number. Is this true also for us? Are we just very complicated machines that execute a completely predetermined sequence? And secondly, Pi needs to sample a lot of numbers from itself, just to find a couple of starting digits. This means that Pi will never be able to find its own value, even if we would let it think about itself until the heat death of the universe or even infinity. Is the universe trying to tell us that the same holds true for us? That we will never be able to find our own true value: the meaning of life?
Inspiration
I found inspiration for this short article from two books and two YouTube channels that I would like to credit here:
Infinite powers, by Steven Strogatz
Alex's adventures in Numberland, by Alex Bellos
Also the universe deserves some credit for being so damn amazing.
If you want the Python code for sampling the numbers and creating the animation, here is a GitHub link :)