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.)
Best Answer
Sure it does.
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.