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Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says...

A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.

A geometric plane would need to have an infinite number of points in order to represent a circle, whereas, say, a square could actually be represented with a finite number of points, in which case any geometric calculations involving circles would involve similarly infinitely precise numbers(pi, for example).

So when someone speaks of a circle as something other than a theory, are they really talking about a [ really big number ]-sided equilateral parallelogram? Or is there some way that they fit an infinite number of points on their geometric plane?

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    A circle is a completely theoretical object, just like a square is. In mathematics, we talk about the idea, but in the real world we usually mean objects which are close approximations to the shape.2011-06-12
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    All mathematical objects are theoretical. See [here](http://math.stackexchange.com/questions/19933/does-a-negative-number-really-exist) and [here](http://math.stackexchange.com/questions/154/do-complex-numbers-really-exist) and [here](http://math.stackexchange.com/questions/15571/do-infinity-and-zero-really-exist) for other questions of the form "does X really exist?".2011-06-12
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    @Eric: I disagree. If ask you to consider "all points at a given distance from the earth's center of mass and at a certain angle from the earth's axis of rotation", I have asked you to consider a perfect circle that exists in the universe. You can't touch it or perfectly describe it with numbers, but it exists. A circle is not theoretical in the way a 29-dimensional hypervolume is. It's a point of philosophical contention, really, but I think that a circle does exist.2011-06-12
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    Does anyone except me truly exist?2011-06-12
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    @JTL: Your example makes many assumptions about the status of "points" and "distance". I would contend they "exist" only as mental constructs, not "in the universe". A "point in space" is not a physical object. If I asked you to consider all points having a rational distance away from the center of the earth, do you really think that that "exists" in the same way the earth itself does?2011-06-12
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    @Zev: I guess that's a fair point of contention. I guess I just don't see anything wrong with taking the "natural attitude" toward things being "out there" in some sense. I would argue that a perfect circle is much more concrete than, say, an electron, for various reasons that don't fit here. Let's agree to disagree, for there's no answer =)2011-06-12
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    @Uronym: I think you're using "parallelogram" to mean "polygon". Every parallelogram has exactly four sides.2011-06-12
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    @Qiaochu: Can you offer me a proof that you do, indeed, exist? ;-)2011-06-12
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    @Zev: I agree about the existence of points and distance and circle and such as mental constructs...but I also challenge you to clarify what it means for something "to exist in the Universe." We have access to only that information which is filtered through and/or constructed by our perceptual/conceptual schemes (i.e. as mental constructs)...I'll let this go...but it is fun to pursue...!2011-06-12
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    This may entertain some people: http://en.wikipedia.org/wiki/List_of_circle_topics2011-06-12
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    @amWhy: Yes, this is of course getting into philosophy. I am not going to accept your challenge, since I think any such thing is impossible :) Ultimately "the riddle does not exist", as Wittgenstein said, but nonetheless it is fun to think about sometimes.2011-06-12
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    @Zev: understandable (the challenge was sort of rhetorical...); alas, Wittgenstein...one of my heroes! (I pursued graduate study in Philosophy before math...one degree down, one on the way!) Yes, it is fun to think about such matters, at times!2011-06-12
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    You may find this interesting: http://www.wired.com/wiredscience/2011/05/electrons-are-near-perfect-spheres/ "the electron differs from being perfectly round by less than 0.000000000000000000000000001 cm. To put that in context; if an electron was the size of the solar system, it would be out from being perfectly round by less than the width of a human hair."2011-06-12
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    @Qiaochu: perhaps; http://en.wikipedia.org/wiki/On_the_Internet,_nobody_knows_you're_a_dog :-)2011-06-12
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    @QiaochuYuan Everything *except* you exists. I can prove it.2011-09-29

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