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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. –  Eric Naslund Jun 12 '11 at 0:20
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All mathematical objects are theoretical. See here and here and here for other questions of the form "does X really exist?". –  Zev Chonoles Jun 12 '11 at 0:21
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Does anyone except me truly exist? –  Qiaochu Yuan Jun 12 '11 at 0:32
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@Uronym: I think you're using "parallelogram" to mean "polygon". Every parallelogram has exactly four sides. –  Jim Belk Jun 12 '11 at 0:36
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@Qiaochu: Can you offer me a proof that you do, indeed, exist? ;-) –  amWhy Jun 12 '11 at 0:42
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4 Answers

It may be interesting to note that the nature of the circle on the basis of the Euclidean axioms is somewhat less than one might think, and is consistent with some strange behavior. The reason is that Euclid had no formal continuity assumptions as a part of his axiomatic framework, and it turns out that the rational plane $\mathbb{Q}\times\mathbb{Q}$, consisting of points having only rational coordinates, satisfies all the Euclidean axioms.

This fact can be used to show that some of Euclid's arguments and constructions are not actually correct. For example, Euclid describes how to construct the perpendicular bisector of a line segment KM, by constructing circles P and R with radius KM and joining the intersection points A and Z as below. The line AZ is the perpendicular bisector of KM.

bisector

But the difficulty here is that Euclid never proved that circles with a common radius must intersect, and so he doesn't know that A and Z actually exist. The fact that A and Z exist is a hidden unstated continuity hypothesis in the system. Worse, it is consistent with Euclid's axioms that the circles do not actually intersect, in that there are no such points A and Z. For example, this is the case in the rational plane $\mathbb{Q}\times\mathbb{Q}$, since when K and M are rational, then A and Z are not. And so it is not possible to prove on the basis of Euclid's axioms that the circles do intersect in points A and Z. The circles may simply somehow pass through each other without touching. In the rational plane, these circles do not intersect, but instead pass through each other without meeting, and this construction does not succeed in building the perpendicular bisector.

The conclusion is that it is entirely consistent with Euclid's axioms that circles have these strange holes in them and that circles with a common radius may not intersect.

There are several other similar issues with the Euclidean axioms, and these led to various formal corrections to and axiomatizations of the Euclidean axioms in the early twentieth century.

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Interesting!${}$ –  joriki Jun 12 '11 at 8:41
    
In the rational plane there are circles, some of which intersect. There are not equilateral triangles. –  Henry Jun 12 '11 at 8:59
    
@Henry: What does that tell us? –  joriki Jun 12 '11 at 9:42
    
@joriki: at least two things: (a) circles are not the most difficult problem; and (b) as JDH says, Euclid missed some important axioms –  Henry Jun 12 '11 at 10:33
    
Note also that the midpoint of KM does exist in the rational plane, and there are rational points on the bisector outside the circle: so in the rational plane, one can form a line segment connecting a point inside a circle to a point outside the circle, without intersecting the circle. –  JDH Jun 12 '11 at 11:47
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A. B. Kempe's lecture on linkages How to Draw a Straight Line (1866/1867) suggested that a circle was a much easier construction than a straight line.

As regards the circle we encounter no difficulty....The apparatus I have just described is of course nothing but a simple form of a pair of compasses, and it is usual to say that the third Postulate postulates the compasses.

But the straight line, how are we going to describe that? Euclid defines it as “lying evenly between its extreme points.” This does not help us much. Our text-books say that the first and second Postulates postulate a ruler (2). But surely that is begging the question. If we are to draw a straight line with a ruler, the ruler must itself have a straight edge; and how are we going to make the edge straight? We come back to our starting-point.

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Ah, but in the modern age we have lasers. –  Qiaochu Yuan Jun 12 '11 at 0:50
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@Qiaochu Lasers which are subject to the force of gravity. –  Tauf Jun 12 '11 at 2:10
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@Tauf: Gravity which defines straight lines. –  joriki Jun 12 '11 at 8:42
    
@joriki The light will be [microscopically] refracted by the air. –  muntoo Sep 29 '11 at 3:07
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In the same sense as you think a circle is impossible, a square with truly perfect sides can never exist because the lines would have to have infinitesimal width, and we can never measure a perfect right angle, etc.

You say that you think a square is physically possible to represent with 4 points, though. In this case, a circle is possible - you only need one point and a defined length. Then all the points of that length from the initial point define the circle, whether we can accurately delineate them or not. In fact, in this sense, I think a circle is more naturally and precisely defined than a given polygon.

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@ JTL; On a grid, though, is not a circle impossible? A square could be accurately and exactly represented on a grid, but a circle would need an infinite number of points to represent each point on it, whereas a square could be represented with a 3x3 grid. I can't see how a circle could exist if it cannot be represented on a grid, could you clarify that? –  Uronym Jun 12 '11 at 0:26
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@Uronym: A square grid also can't represent an equilateral triangle. Does that mean equilateral triangles don't exist? –  Zev Chonoles Jun 12 '11 at 0:29
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@Uronym: Wouldn't you agree that a square has an infinite number of points on its perimeter, as well? The perimeter of a circle isn't really what defines it. It's defined by the location of the center and the length of the radius. Similarly, the square is defined by the length of the sides and the location of the corners, or the center if you prefer. I don't think that its possible to draw either one perfectly, not even the square, but we can certainly define both of them perfectly. –  barf Jun 12 '11 at 0:30
    
Would it not depend on how large the square/rectangle were on the grid? If a rectangle has a top left corner at 3, 5 and the bottom right corner at 4, 10, the square is 2, 5 in size, and the area is 10. The perimeter also fills 10 squares on the grid. That seems rather exact, whereas you would need an infinite number of squares on the grid to represent the circle. So perhaps a better question is whether a circle can truly exist on a grid? –  Uronym Jun 12 '11 at 0:43
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@Uronym: I cannot understand why you stick to grid-based thinkng, since a grid itself is also a theoretical setting. Conversely, if we admit grids, then there is no reason to prefer a rectilinear grid to a radial grid, too. A square is as realistic as a circle in Euclidean geometry. –  sos440 Jun 12 '11 at 1:10
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I think it quite depends on how you understand "exist" and how you define $circle$. For example, in terms of cartesian coordinates, a circle of radius $1$, centering at the origin of the $xOy$ plane, can be "defined" as the set $$\{(x,y)\in{\mathbb R}^2:x^2+y^2=1\}$$ Then when you ask whether such set exists or not, the answer is yes, according to the axioms for set theory.

The keyword in the excerpt you quote from wiki, I think, is "set"(the set of points in a plane that...). So when you are asking "can a circle exist" you are actually asking "can such set exist".

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protected by Qiaochu Yuan Jun 12 '11 at 0:14

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