Why is there no equation for the perimeter of an ellipse‽

See here for more on that: ▶ this video the info on calculating the actual value starts at 14 minutes (it's an infinite series).

Something smaller than may help you grasp this idea: ▶ this video There are things we can characterize to any desired degree of accuracy but that e don't get cute little equations out of... and some of those things are so simple they've been staring you in the face since middle school and you just didn't ever notice their absence from your formula sheets. This is not the exact same situation being described but it's a similar thing. Being able to put a complex system into a computer and arbitrarily manipulate it still doesn't mean we can extract some simple explanation. On the other end of the scale, see all the AIs coming out. They're 100% computer artifacts with theoretically no mystery in them whatsoever... but they're just tables of billions of opaque numbers and doing anything with the numbers beyond just running them is amazingly difficult.

There are neat formulas, but those are just approximations, as an exact (closed form) is provably not possible. link ▶ this video