Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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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.
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