# $\pi$ and 1-dimensional beings

The constant $\pi$ is commonly explained in terms of the relationship between the radius and perimeter of a circle, which is a 2-D object. It can also be explained in terms of some infinite series etc. For humans, as we are 3-D beings, the value of $\pi$ is quite critical in our physics. But for an imaginary one-dimensional being, does $\pi$ make any sense other than the sum of some fancy number series?

(In particular, I am curious about physical meanings rather than mathematical.)

One-dimensional creatures can take an object of mass $m$, attach it to a spring $k$ and they will find out the period of oscillations if this system is proportional to $\sqrt{m/k}$. The coefficient would be some strange number approximately equal to $6.28$, but not an integer or a rational (actually it's $2*\pi$).
Then one day some advanced one-dimensional mathematician will try to calculate how many pairs of integer numbers exists such that $x*x + y*y < R*R$. How fast does this number grow when $R$ grows? He would make some experiments and find out that this number seems to be proportional to $R^2$ and the coefficient is about $3.14$. Looks like half of that strange number which has to do something with oscillations, but come on, that simply can not be.
• R² looks two-dimensional, but we also do fancy calculations with hypercubes in for example 6-dimensional spaces. Mathematically, the dimensions that you calculate stuff with are not bound to how high dimensional your world is. You could still argue that the example gives a mathematical meaning rather than a physical one, but as I understood the answer, the story with the mathematician only shows how 1dimensional people could "find" $\pi$, it doesn't show physical meaning of it. The physical meaning is shown with the 1dimensional spring system. – Quantumwhisp Oct 31 '16 at 13:18
• @Caridorc We take R = 1000. Calculate $R^2 = 1,000,000$. Now check some pair of integers, let's say 5 and -3. 5*5 + (-3)*(-3) = 34 < 1,000,000. This pair is good, count it. (800, 800) is not good. And so on. Number of good pairs would be about 3,140,000, nothing 2-dimentional here. – lesnik Oct 31 '16 at 13:23