If you consider solutions to one of the simplest non-trival second-order differential equation, f''(x) = -f(x), then you will find that the solutions all have period 2π, no geometry needed.
I think we ought really see `R` as an algebraic stand-in for talking about geometry.
If you've got addition and multiplication then you can get the reals by just adding additive inverse, multiplicative inverses and completeness, none of which are inherently geometric properties. So any geometry must come from the distributive property, because there's not much else it could be.
In some sense, the algebraic numbers are indeed important in algebra as being the 'basic' field with infinite characteristic.
You get to `R` by demanding completeness (for others reading, the demand that every converging sequence has an existing limit). I would say that the idea of limits and completeness are fundamentally topological / analytical. Which is closer to geometry than to algebra.
I think the main conclusion here is that there is good reason that the field "algebraic geometry" exists.
Said differently: periodic things in some sense express motions on a circle.
Edit: natural frequency ω = Sqrt(k/m). ω comes out in rad/s, to convert to more natural units (cycles per second) π is required.
Now instead of space being Euclidean, you're saying space-time is? And effectively if space-time were no longer Euclidean, then the conversion to cycles/sec would use the "alternate pi" of this non-euclidean space time wouldn't it?
Likewise, it's hard to imagine what would happen to that very famous equation e^iπ + 1 = 0 if π were something different.
However, there is indeed a connection to circles. One way of seeing this by looking at the solutions to this differential equation over the complex "plane" and looking at the exp(i x) and exp(-i x) solutions, and seeing how these functions wrap the real number line around in a circle.
The complex plane fundamentally has a Euclidean geometry to it. So much so that if you grew up in a world with a strong non-euclidean geometry, or even grew up as some sort of digital being with no real notion of space at all, so long as you are able perform moderately sophisticated mathematics you are going to wind up discovering the complex numbers, and things like their absolute value and multiplication and the exponential function and how they map values around. Perhaps you never mentally arrange the complex numbers into a "plane", but none the less, all your basic geometric concepts are embedded into the algebraic operations on these complex numbers, and you will wind up effectively doing Euclidean geometry, even if you never identify it as such.
The self-similarity in these two DEs is why Euler's formula works. Are there any other such DEs?
f'(t) = M f(t)
f(t) = e^Mt f(0)
Something like f''(t) = -f(t) can be split up as follows:
f(t) = h'(t)
h'(t) = -f(t)
M = [[ 0 1]
[-1 0]]