Another cool thing about lungs and trees is that they’re both fractals shaped by completely different natures but converge on these similar forms, requiring irrational constants like the golden angle.
From an information view, their fractal dimension exceeds the bounds of 3D space, defying purely rational models.
They accomplish this via iterative growth over time, compressing vast complexity into finite volume — and most bizarre of all, we can model this mathematically, but only with irrationality as a presupposition.
There’s even more complex fractal models, such as fluid dynamics and electromagnetic waves, that rely on imaginary numbers (sqrt(-2)) — an axiomatic impossibility, but used to successfully simulate things like airplane turbulence.
Nature is our witness to the limits of our own reason and rationality, and not to mention the epistemic miracle that we can actually know any of this.