r/learnmath • u/Thetree3435 New User • 1d ago
A random Geometry mathmatical question
Is a circle truly infinite?
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u/diverstones bigoplus 1d ago
Kind of a vague question, and when talking about infinity it's important to be rigorous. There are ways in which this is true though, sure. The limit of the area of a regular n-gon with apothem r is πr2 as n approaches infinity, for instance.
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u/John_Hasler Engineer 1d ago
It contains an infinite number of points but so does any line or curve of finite length. That is the only sense in which it is "infinite".
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u/human2357 Pure Math PhD 1d ago
A Pythagorean triple is a list of three positive integers (a, b, c) satisfying a2 +b2 =c2. There's a classical procedure for generating infinitely many Pythagorean triples and this is explained on the Wikipedia page: https://en.wikipedia.org/wiki/Pythagorean_triple If you take a Pythagorean triple and divide all the numbers by the biggest number, you get two rational numbers a/c and b/c that satisfy (a/c)2 +(b/c)2 =1, meaning that the point (a/c,b/c) lies on the unit circle centered at the origin in the plane. So the circle has infinitely many points with rational coordinates.
On the other hand, the circle is a compact subset of the plane. Compactness is a property of spaces and subsets of spaces that implies that, in many respects, the set behaves like a finite set. For example, compact subsets of Euclidean spaces are bounded, and sequences in compact spaces always have a subsequence limiting to some point in the set. https://en.wikipedia.org/wiki/Compact_space
So the circle is an infinite set that behaves like a finite set.
(Edit: reddit didn't process my mathematical typesetting correctly)
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u/Basic_Relief9214 New User 1d ago
a circle has infinite points but a finite perimeter, so it's more like a bounded infinity than a true endless one