# What will be the shape of circle if it has no pi (π) [closed]

I not so good with mathematics

I like to know if there is no pi (π) existed in this world what will happen to a Circle. What will be the shape of it

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What do you mean by "no pi"? –  Qiaochu Yuan Jun 18 '12 at 13:40
We would rename it something else? Perhaps I should pose a question to you: what would a square look like if there was no length? What would be the shape of it? (What I'm alluding to is that this doesn't make any sense as written) –  mixedmath Jun 18 '12 at 13:41
Duplicate of this question? Or this one? –  Zev Chonoles Jun 18 '12 at 13:42
All known worlds without a pi have circles that are almost but not quite exactly isosceles, polka-dotted triangles. –  Henning Makholm Jun 18 '12 at 13:43
What do you mean by "no length"? Doesn't a line also have length? I can't make any sense of what you're asking. –  Qiaochu Yuan Jun 18 '12 at 14:17
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## closed as not a real question by MJD, azarel, The Chaz 2.0, Hans Lundmark, Zev ChonolesJun 18 '12 at 14:27

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## 3 Answers

$\pi$ is by definition the ratio of a circle's circumference to its diameter. If $\pi$ didn't exist but 'circles' did (I'm reaching a bit here), then that would imply that this ratio wasn't constant for all circles. This is something you can do in curved spaces (see here), but that doesn't work in flat space.

So I guess (to the point that this even makes any kind of sense), if $\pi$ didn't exist then many other aspects of how we treat space would also have to be changed. You couldn't just remove $\pi$ by itself from our normal ideas of geometry and 'see what happens'. You have to come up with a whole new set of rules since that fact is based on other, more basic ideas that have other implications.

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The number $\pi$ is defined as the ratio of a Euclidean circle's circumference to its diameter. It doesn't make sense to ask, "what happens to circles if there's no $\pi$?" Euclidean circles came before $\pi$, not vice-versa.

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On further thought: Let $F$ be the field of compass-and-ruler constructible real numbers. Then geometry in $F^2$ could reasonably be said to be a world where $\pi$ does not exist. The circles in that world look perfectly normal and circular -- it's just that their perimeter does not have the same length as any straight line.

That universe is rather substandard once we move past circles to more complex curves, though. For example, $F^2$ has curves that appear to cross each other but don't actually have any point in common. (Such as the $x$ axis versus the curve defined by $y=x^3-2$).

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It's not clear to me that there's even a notion of perimeter in the context of curves in $F^2$. –  Zev Chonoles Jun 18 '12 at 13:55
@Zev: I'm assuming that one can still cobble something together, such as generalizing a "curve" to mean a uniformly continuous map from a dense subset of the ordinary real unit interval to a metric space. –  Henning Makholm Jun 18 '12 at 13:57
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