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Why the ratio of a circle's circumference to its diameter is always the constant $\pi$? Moreover many of you know better about golden ratio $\phi$, Euler's number e, and many other constants. How explain this while everything seems to be in a complex system?

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closed as off topic by Waffle's Crazy Peanut, dmckee May 7 '13 at 16:53

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The fact you mentioned about circle is actually quite amazing. On the one hand you can draw a circle on your copy and check that ratio of circumference to the radius is constant. On the other hand you can begin with assumptions of set theory and develop real numbers, real plane and calculus from it purely on the base of logical reasoning; and then you find that mathematical circle is the same as physical circle that we observe in this world. Actually the similarity between the physical circle and the mathematical (flat) circle has to do with the fact that curvature of our space is quite small. – user10001 May 6 '13 at 18:23
I think this two questions on Math.SE would be interesting: Does π depends on the norm? and π in arbitrary metric spaces. – m0nhawk May 6 '13 at 18:35

$\pi, e,$ and $ \gamma$ are all mathematical constants. They are not dependent on the physical universe. For example,

$$\pi = 4\left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots\right)$$

but it's unlikely for you to ask why the numbers, 4, 1, 3, 5, 7 aren't changing in a chaotic universe.

Real things aren't circles; circles are just math. For some real things, though, measurements of them can be well-approximated by circles.

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If we want to create a circle, first we do some calculations based on math. Then the circle becomes an approximation to what were in our mind. Never the same. We could do math so we could build something that is very approximate to what we imagined. So, maybe the physical universe is an approximation to some ideas that we can find its footprints on constants. We define pi, but we can not change the ratio of a imaginary circle's circumference to its diameter to be constant. If we could define another language better than math, maybe we could define better the real world. – nikamed May 6 '13 at 15:30
Of course there are not constants in the real world, but there are approximations to them. This give the idea that there may be constants in somewhere imaginary. – nikamed May 6 '13 at 15:41
+1 for the natural numbers. But claiming that they are mathematical does not explain why they are like they are in the universe. That is the question. Do you mean that no other universes can exist without these numbers? – Val May 6 '13 at 16:56

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