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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.)

Emilio Pisanty
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Sreehari S
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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.

SwiftsNamesake
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lesnik
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  • What does a 1-D spring look like? – Brondahl Oct 31 '16 at 11:57
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    @Brondahl Let it be a piece of rubber rope. Now we have a problem, how to tie an object to it. Can not make a knot. Need to use some glue. How to put a glue between a rubber and an object? I can see no easy way. – lesnik Oct 31 '16 at 12:07
  • @lensik I think we can make some kind of spring and put in their (1D beings') pathway. Then they will see it as an object coming forth and then going back and then repeating...Am I correct? – Sreehari S Oct 31 '16 at 12:47
  • I guess you are correct: they will see it as a moving "wall" and if you attach to it you would oscillate together with it: move forward and backward with some period. – lesnik Oct 31 '16 at 12:51
  • $R^2$ looks two-dimensional to me. – Caridorc Oct 31 '16 at 13:04
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    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
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    @Caridorc We take R = 1000. Calculate $R^2 = 1,000,000$. Now check some pair of integers, let's say 5 and -3. 55 + (-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
  • @Quantumwhisp yes. looks reasonable, I can write $x^4$ no problem, but visualizing it is a different story. – Caridorc Oct 31 '16 at 14:12
  • The one dimensional mathematicians would eventually realize the relationship between oscillation and the number-theoretic result once they hit on the idea of state variables and they wrote the harmonic oscillator equations as $p = \dot{x},, \dot{p} = -x$ and then understood that the motion conserves $p^2+x^2$, i.e. it is a constant arclength per unit time traversal of a 2D circle in $(x,,p)$ phase space. – Selene Routley Apr 28 '17 at 07:59