The Paradox of the Infinite Circle


from the work of 15th century philosopher,
mathematician, astronomer and Church Cardinal
Nicolaus Cusanus

Tags: paradoxes


If something that can't exist did, it would get pretty freaky. Wow. That is truely impressive. Really.
Posted by Impressed on 3/11/2009 4:59:45 PM
And so a circle of infinitesimally small diameter would for all practical sense, as viewed from this earthly realm, have no diameter and only point.
Posted by Zen Bonobo on 3/11/2009 6:19:08 PM
Wouldn't that also mean that any straight, infinite line would have to be a infinitely large circle?
Posted by Matt on 3/11/2009 7:01:21 PM
This is a very interesting paradox. Perhaps this is insight into the nature/shape of the universe. When a circle is "infinite" in size, the geometrical curvature is essentially zero for very small segments. However, over long/large distances, a curvature is created by the shape of the universe. Just a thought...
Posted by Rex on 3/11/2009 7:49:23 PM
This is not a paradox at all. Where's the self-contradiction? This is only the effect of a variable (radius) going to infinity.
Posted by glen on 3/11/2009 8:44:49 PM
The curvature approaches a line without getting there. any 'segment' of a circle is not a line. He is rounding off numbers that are immpreceivable to him. The argument is convincing because the same quanity is imipreceivable to all of us. excuse the spelling
Posted by icecow on 3/11/2009 9:36:16 PM
The only problem I see with this is something I learned in calculus. Infinity can never be reached, it can only be approached. This was discovered in modern calculus circa 1700's. Nicolaus Cusanus would not have known this. Thus the answer would be that the circle is approaching being a straight line, but is not a straight line.
Posted by Brad on 3/11/2009 10:36:14 PM
This is not a paradox. This is the concept of a mathematical limit. This is interesting in the same way as the concept of zero or using negative numbers is interesting. It's just a new way to think about something. Some mathmatical struictures have Real limits; for example, consider (1+1/x) as x increases. The limit of this as x becomes infinite is 1; although it will technically never get there if x is a real number. But if we introduce the concept of infinity, 1/infinity = 0; and 1 + 1/infinity = 1. So now we have a function that gives us the curvature of a circle in relation to its radius. I forget exactly what it is, but the line will always have some curvature, no matter how small, as long as the radius is real. However, once you introduce the concept of infinity, the curvature dissapears, in much the same way that it did when you considered 1/infinity.
Posted by jed on 3/11/2009 10:46:58 PM
It is not so interesting as it might seem. Consider a line, now move perpendicularly away from the line an infinite distance. The shortest distance from where you are to the line is now infinite. Now consider some other path from you to the line, its distance will be infinite as well. Are the distances equal? Well yes, because it would not make sense for there to be a path longer than the infinite one. I.e. there are no numbers greater than infinity, but neither is it a shorter distance. (at least not in this imprecise treatment)
Posted by ZeroBomb on 3/11/2009 11:43:21 PM
This actually happens in compactified models of R^n. See for example Mobius geometry where the point at infinity is included within the model. Constructing a sphere passing through this point is equivalent to making a plane. Here is a movie which demonstrates this in action:
Posted by Mik on 3/12/2009 2:13:17 AM
This is an example of being on the surface of the earth, we assume we are going in a straight line (and people believed the earth was flat for many a year) even though it is actually curved.
Posted by sponsoredwalk on 7/26/2009 10:04:54 AM
Another interpretation can be as follows: A straight line is a circle with a radius at infinity.
Posted by Frank on 11/30/2009 11:38:20 AM
I have been trying to wrap my head around this. A circle wouldn't be a circle if it had a straight line. Now, imagine the point of a circle. From that point, all the dimensions of the circle reach out to find its perimiter, but the perimiter would be ever expanding, so non-existant. Therefore, an infinite circle would really be just a point, wouldn't it? A point with theoretical dimesions reaching out?
Posted by Polybius on 5/25/2011 11:34:58 AM
A circle that is infinite is a plane. A circle has to have a defined edge in order to be a circle. Only a plane extends in 2 dimensions infinitely.
Posted by whatever on 6/24/2011 2:10:26 PM
@glen has it. This is not a paradox to anyone who's taken a basic Calculus class. As the diameter of a circle approaches infinity, the curvature of the circumference approaches zero. Limits. Duh.
Posted by mad_max on 7/21/2011 2:16:07 AM
congratulations, you've discovered limits....
Posted by anonymous on 7/27/2011 4:51:24 AM
oh the limits thats a tangent line :B
Posted by alex on 8/1/2011 1:45:31 AM
Just a nutcase thought or two! Imagine our universe as a sphere of infinitely large radius, or for that matter large diameter. Would that result in a straight line edged sphere? This is unlikely, unless it ended up as a cube. To do that it would have to have finite small radius corners making them in effect sharp corners, therefore making our universe a perfect cube. Now just imagine that there are countless other universes (an infinite number stretching through infinite space). All the universes fitting together like building blocks for ever and ever. Just a conjecture but other nutcases are welcome to comment on my thoughts. RemRaf
Posted by RemRaf on 8/20/2012 3:23:08 AM
There is actually a mathematical proof that when to objects went in opposite directions forming a straight line, the two points will meet at infinity. So its no longer a paradox.
Posted by Niknok on 6/13/2013 10:19:26 AM
In a simple demonstration, a straight line is the shortest distance between two points A & B, now what shall we call the shortest distance if am to go on a straight line back to point A through point B. That straight line is called a circle.
Posted by Larry Spence on 9/17/2013 9:19:25 PM
The only way a straight line can meet is in a circle. the paradox here is that we view things in terms of similarities and differences (synonyms & antonyms) and dwell less on functionality & relativity.
Posted by Larry Spence on 9/17/2013 10:00:15 PM

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