Thank you for explaining this. Well written, easy to understand.

I understand the irrational term for pi, but I do see the rationality for the outcome of pi. I'll need to explain this in 3D, introducing the 4D system.

In this image, you can see the cube that we associate with 3D, but the focus is now on the 8 corners, establishing four lines that represent 4D (time is not involved in this system).

To see 4D more easily, envision a globe (the earth) inside the cube. Not the 6 planes, but the 8 corners stick out as the main features.

Now, if we were to organize earth with a specific northpole and southpole, then we need to pick one set of the 4D lines. In the following image, that is made visible with a Rubik's Cube. In top, the ‘northpole’.

What should be visible immediately, and with it we are finally coming to pi, is that the equator is not expressed as a single line, but rather that all 6 remaining corners are sticking above and below the equator line. As such, pi it that line found as the balancing act of these corners.

My conclusion is that the irrational nature of pi is quite rational. It is not based on itself, but is the resulting balancing act of the parts.