Long ago, after having spent months in Switzerland, it stroke me just what that place reminded me of: “Welt Am Draht”. The movie was filmed to look “like no other place” and therefore mixes film footage from Munich, Cologne and Paris freely, to create a city that feels familiar but that you doesn’t quite seem to recognize.
This is exactly what Switzerland looks like. It features a mixture of old styles reminiscent of Germany, Italy and France simultaneously, a lot of frills, green idyllic pastures, cows… as well as water buffalo, ostriches, llamas, plus naked, ugly, grey concrete —lots and lots of it— sometimes hastily sprayed on by someone who would like at least SOME color somewhere. Sometimes it is freshly cleaned to return it to its original (did I mention UGLY?) concrete state, but fortunately it is often elaborately decorated by a sprayer group who obviously had a lot of practice. Miles-long practice.
Why do I mention that? Well, “Welt Am Draht” portrays its world like that because — spoiler — their world is fake. So being in Switzerland makes you wonder constantly, if you are living in a real country or a weirdly mixed-up computer simulation.
It got me thinking. The good news for our Reality is, there is Pi. Everything that is a circle has this number in it: it is the ratio between the circumference and the diameter of the circle. And since all the matter in this universe is made of atoms, it ultimately all comes down to spheres and other funnily-shaped figures with spherical parts that are the electrical fields of all the particles in the universe.
What does that mean for Reality? Well, since Pi is an irrational number, it automatically means that either the diameter (or radius), or the circumference of that circle has to be an irrational number as well. (You can check the details of this in the PS.)
It also means that a computer, which is a Finite State Machine, no matter how large and powerful, can not accurately represent the number.
Why? Because irrational numbers have an infinite amount of decimal points, and can not be represented any shorter.
So if the universe would be a simulation, the computer it is running on would have to regularly represent infinitely long numbers — for every particle, every field, every wave. Which, if the computer itself is finite, is actually impossible, because representing infinitely long numbers requires infinite amounts of memory — for each (!) irrational number in the universe.
Ergo, we just built a rudimentary mathematical proof that the universe is not a simulation running on any kind of finite state machine.
Neat, eh? Congratulations are in order. Now we can rest assured that this Reality of ours is the Real deal.
Until the next part of the Infinity series — where I show why it probably is not.
PS: Why either the circumference or the radius of a sphere has to be an irrational number:
- the circumference of a circle, ie the length of the line around the circle (or a sphere), is 2 times Pi times the radius of the circle.
- the definition of a rational number is, “any number that can be written as a fraction (a ratio) of two whole numbers”. (Incidentally, since 1 is a whole number — an integer — and 1/1 = 1, it also makes all the whole numbers rational as well — 2/1=2, etc.)
- So from any rational number you can always go back to a fraction of two whole numbers. e.g. 0.5 = 1/2. That means:
- Multiplying or dividing two rational numbers will always yield a rational number:
- you can convert both numbers into fractions, and multiply the resulting numerators and denominators according to your operation (either “n1*n2/d1*d2" for multiplication, or “n1*d2/d1*n2” for division). Since all the numbers involved in the multiplication are integers, you end up with one fraction of two integers —again, a rational number.
- Since irrational numbers are defined as “all the numbers that are not rational”, this trick does not work with them — they can not be expressed as a fraction of integers. Therefore, multiplying a fraction or dividing a fraction by an irrational number, it will yield irrational numbers either in the numerator, or in the denominator, rendering the final result irrational as well.
- ergo, since circumference and radius can be derived from each other by multiplication or division involving an irrational number, Pi, either the one or the other has to be irrational.