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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.

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    $\begingroup$ Oscillation has one spatial and one temporal dimension - so, arguably, two dimensions. $\endgroup$
    – anon
    Mar 8, 2021 at 14:16
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    $\begingroup$ $\pi$ comes up in many more contexts than circles. You need it to understand the bell curve in statistics. $\endgroup$ Mar 8, 2021 at 14:27
  • $\begingroup$ @EthanBolker As 3blue1brown would put it; "If you see pi, there is a circle hiding somewhere in the background" :) $\endgroup$ Mar 8, 2021 at 14:36

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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.

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    $\begingroup$ I don't quite agree with your statement on the quaternions. Quaternions are the vector space $\Bbb R^4$ endowed with a funny product (just like $\Bbb C$ is $\Bbb R^2$ with a funny product), and I don't see you encoding it in $\Bbb R^4$ (well, you could, but ultimately you'd just give funny names to the elements of $\Bbb R^4$: you'd still be able to find a base of $4$ elements for some action of $\Bbb R$). $\endgroup$
    – user239203
    Mar 8, 2021 at 14:44
  • $\begingroup$ The quaternions are isomorphic to the vector space $\mathbb{R}^4$, yes - but my statement reads, the quaternion relations can be described in a one dimensional sense. That is $i^2 = j^2 = k^2 = ijk = -1$. Once again, the basis elements $i, j, k$ are in some sense the fundamental constants of the quaternions. $\endgroup$ Mar 8, 2021 at 14:47
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    $\begingroup$ I like this answer. I guess it was unnecessary to specify the dimensions as being spatial, since fundamentally spatial dimensions are dimensions nevertheless, and are not necessarily of greater importance. $\endgroup$
    – Aidan Daly
    Mar 8, 2021 at 15:06
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    $\begingroup$ @AidanDaly That is definitely true, but the reason I included "spatial dimension", is to bring home the fact that I was really using the Euclidian/Vector space definition of dimension. The emphasis is crucial because there are other notions of dimension of objects that do not contain $\pi$ or $e$ or other constants. For example take the set of all polynomials with coefficients from $\mathbb{Z}$. This set forms what is called a ring, and has an associated dimension called the Krull dimension. This set with a well-defined dimension, does not contain any of our favorite constants. $\endgroup$ Mar 8, 2021 at 15:19

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