Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
The short answer is that circles are simply everywhere. And thus pi.
Earth? A giant _3D “circle”_. Everything trigonometry? Sine, cosine and tangent are just aspects of triangles _in a circle_. Anything periodic? As sin and cos are the archetypal periodic things, lots of pi ensues. And so on and on.
So, why are circles omnipresent? Because they are the most “perfect” shape in 2D; similar with spheres in more dimensions:
They look the very same everywhere and from every direction. They are those points of exactly a fixed distance from the center. They optimize area/volume for a given circumference, related to why water forms little drops. They also optimize for other things, for example this is why large bodies in space are spherical.
There are so many things circles/spheres do, most of them as a result of their symmetry. And as a consequence, they are all over nature, from stars, planets, water drops, down to atoms and nucleons. If something in physics/reality does not change under rotation, such as all the laws we know, it will make spheres.
Saying that pi is ‘only’ related to the circle is downplaying things a bit, because circles are very important and simple mathematical objects that turn up *everywhere*. Any kind of repeating sequence, a *cycle* can be described with a circle somehow and pi will sneak in there, which is why it comes up a lot in physics, where waves are cyclical and waves are also *everywhere*.
Anything that rotates necessary pulls pi in because rotation happens in circular arcs.
The idea of a circle also generalises to spheres in higher dimensions, which gives pi opportunities to sneak in there too.
Sometimes pi sneaks in to seemingly unrelated places because a problem is converted into one that is solved using circles.
Can you describe a place that pi turns up where you can’t see the connection to the circle?
I will preface this by saying that I’m not a mathematician or a physicist, but I understand it as that pi applies to rhythms and cycles. These are to time what circles are to geometry, essentially functioning as waves. So pi then applies to equations where there are geometric waves (sin curves for example), but also temporal waves.
Easiest answer, given your own analogy… how many circles do you see in everyday life? Especially compared to straight lines?
I mean… the stars? The sun? Planets? Orbits? Even the grains of sand, and circles of the day itself. All of them… *described by Pi*.
If we see Pi in some equations, it pretty much always means somewhere in whatever is being calculated there is a circle. It’s not always obvious how the circle is related, but there is going to be one hiding somewhere. If you have an example of a man equation you’re thinking about, people may be able to point out where the circle is.