Jeff's other projects are quite cool as well.

I was under the impression you could define any regular polygon with just three points.

You know, for kids!

Eideteker: Yea, but unlike those dirty polygons, circles are clean and perfect.

Hah! They're infinitely-sided polygons, man! The bigger you blow 'em up, the uglier they get.

If they're infinitely-sided, then you cannot possibly blow them up big enough to make them look straight-edged and ugly.

I was under the impression you could define any regular polygon with just three points.
Not uniquely (I can draw lots of squares that pass through (0,0), (0,1), and (1,1)) unless you insist that the points be vertices of the polygon. But if you insist on that, then most triples of points can't be vertices of a regular n-gon except maybe for a few special n's (try to draw me a square - or any regular polyhedron - with vertices at A=(0,0), B=(0,1), and C=(2,1). If a regular polygon had vertices at those points then of course the polygon would lie on the circle through those points. You'll have some difficulty finding any number of evenly-spaced points on the circle that include all three of the givens).

My geometry is rusty, but I was thinking center, vertex, and altitude/base (basically the midpoint on one side) could define a polygon of n sides (including circles). In other words, you describe a right triangle whose base is half of a side. Feel free to elaborate because I really am curious and am not just being too cool for school.

Well, OK, (you could also use the center and two consecutive vertices - why mess around with the half-of-a-side? - or three consecutive vertices - or just any three points with the stipulation that they have to be vertices of the polygon) but the issue is still that for any three points you plop down there usually won't be any n-gon that fits your points (again, except for maybe some few special n's). Just tweaking the previous example a little bit, try to form some n-gon with center at (2,1); a vertex at (0,0) and the midpoint of a side at (0,1).

Now, if I'd chosen points that came from an n-gon (and I told you what n was), you could recreate it uniquely. But the points I gave, for most n's (probably all, but to say for sure I'd need to know the answer to a difficult irrationality question) you won't be able to construct a suitable n-gon. I mean, for instance, trivially you can't be able to get a triangle or a square that does the job, right? For a circle, there's always one and only one.

At some level, the difference is a difference of dimension; the "space of all circles in the plane" is (real) 3-dimensional, and in fact circles in the plane are in a continuous, bijective correspondence with points of 3-dimensional Euclidean space.

Even once you commit to a particular n, describing an n-gon requires you to transmit four numbers. Say, two for the center; a third for the distance from center to vertex; and a fourth one from the interval [0,2Pi/n) to describe the orientation. Moreover there's some nontrivial topology there; since turning through 2Pi/n makes the n-gon coincide with itself again, there's at least one nontrivial element in the fundamental group in the space of planar n-gons. That space is, in fact, like the cartesian product of R^3 with a circle. As you let n get large, you can visualize the circle part of the product shrinking down (because 2Pi/n goes to 0) until it becomes a point, and you have R^3 cross a point - back to the space of all circles as a limiting case of the space of n-gons as n gets large.

Ok, but if I take graph paper and a pencil, and you give me three points that do not themselves make a single line, I can make you a regular polygon using those three points, simply by adding right triangles. Yes, you need to tell me how many sides, but if you say circle, you are just specifying that number to be infinity. So it's a cheat if you say you're not providing that information. I wish I could understand some of the more complex concepts at play here, but I tried it with pencil and paper and made at least the simple 3-6 sided polygons that I drew work. The site said, "Just like triangles, you only need three points to create [a circle]." I'm just saying that unless there's more of an explanation or a more rigorous definition, that's a trivial claim for all regular 2-d shapes. It sounds to me like you're specifying a whole lot of requirements here that the site did not (e.g., which point has to be the vertex, uniqueness).

Well, I did give you three points - (2,1), (0,0), and (0,1). And your description of your construction is that (2,1) should be the center, (0,0) should be a vertex, and (0,1) should be halfway along a side. And I think you're telling me you can do the construction for any number of sides I ask, as long as I tell you how many I want.

So tell me (or show me) what triangle, and what square you constructed by your method. (I mean that literally, not rhetorically, because I want to see what you've got.)

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The uniqueness is important - your original statement was "...you can define a regular polygon by three points." (People also typically say three points determine a circle, or two points determine a line) At least to a mathematician, that means once you give me the three points and tell me how many sides, there is one and only one polygon I can draw to do the job.

If we don't put some restrictions on where the given points lie (ie, they have to be vertices, or one of them is at the center of the polygon, or something), then you don't get uniqueness. You give me some points, tell me you want a square, and I can draw infinitely many different squares that contain all three of them.

If we do put restrictions on them (like my suggestion that all three should be vertices; or your center-vertex-midpoint description) then - unlike the case with circles - whatever system you've decided on for "how three points determine a polygon" there will be sets of three points that don't determine any regular polygon at all.

The linked site might not convey it clearly, but those TWO facts are the content of the great theorem about circles: (1) For ANY three points, there is a circle passing through all three of them, and (2) That circle is unique.

There is no corresponding theorem for regular polygons; not for triangles, not for squares, etc. You either lose uniqueness or, in adding extra conditions to get uniqueness, you lose the ability to carry out the construction for EVERY set of three points. But there's no point in my saying more unless I can see what you've done with the set of three points I gave.

The linked site might not convey it clearly, but those TWO facts are the content of the great theorem about circles: (1) For ANY three points, there is a circle passing through all three of them, and (2) That circle is unique.

Ok. In a roundabout fashion, that's (specifically 2) what I wanted to know. Thanks! I was wondering if determine/define was being used in a strict mathematical sense here or not. In the case that it is, you've got no argument from me.