I wasn't too sure if I should have posted this on the mathematics or physics stack exchange, however I figured perhaps since this is in reference to fundamental constants it would be better placed here.
I have recently been thinking of fundamental constants. Specifically, the minimum number of spatial dimensions you would need to understand them. If you take pi for example, I thought at first the minimum number of spatial dimensions you would need to understand (and therefore be able to discover) pi was two. My reasoning was that, in one dimension, circles don't exist and therefore there would be no reason to hold the irrational number pi in elevated regard. However, I figured that you can have oscillations in one dimension, I.e a point moving up and down from position $-y$ to $y$, and where oscillations exist you could inevitably discover the functions sin and cos and thus the number pi. So my question is this - are there any fundamental constants that require a number of spatial dimensions > 1 to understand? Or do all fundamental constants exist in all spatial dimensions?
Apologies that this is both vague and abstract. Ι also wasn't quite sure if the tags I used were relevant. I would love to hear any thoughts on the matter.
Depends really on what you mean by "understand". Many fundamental constants can be (in one way) defined in a 1-dimensional sense, in terms of their infinite sums. Sure this might not be the most intuitive or the easiest way to introduce said constants; but it can be defined this way nonetheless.
For example, $$e = \lim_{n\to \infty} \left(1+ \frac{1}{n}\right)^n = \sum_{i=1}^\infty \frac{1}{n!}$$ $$\pi = 4\sum_{i=1}^\infty \frac{(-1)^{k+1}}{2k-1}$$ and so on...
Even $$i^2 = -1$$ or the quaternion relations can all be described (and dare I say, understood) via summations in the real line.
In this vague sense, every fundamental constant needs only one dimension to be defined, and therefore understood via the definition. But by all means, a flavor of them exist in all spatial dimensions. Again, here the word dimension is very vague; and I am sort of interpreting it as the dimensionality of Euclidian/Vector spaces.