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ReapReason
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Can a circle have an infinite diameter? I say no.
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MarkThompson
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No it can’t (in plane geometry, at least), although one can certainly talk about what happens to a circle as its diameter tends towards infinity.
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Spirithound
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Well, it’s unclear to me. If we ever were to encounter a circle with infinite diameter, we would never know it, since we would never see the edge of the circle.
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Alberti_Devoveo
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Only if its radius were half of infinity
. On a serious note: Spirithound is right; we could never know because we would never be able to see any of the boundaries.
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ReapReason
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This is part of the reason why I don’t think it is logical to say that a circle can be infinite, because that which defines the circle has no quantifiable state. If there is no edge that can be defined, then we lose that by which we understand a thing to be a circle. I think that any shape by definition is finite.
Its like saying that there is an infinitely small circle. That to is irrational because there is always a quantifiable diameter.
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ReapReason
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But surely an infinite diameter is meaningless since an infinity has no boundaries? I think we can know that it is irrational.Only if its radius were half of infinity
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ReapReason
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Why do you say no? Can you speak more about what you mean by “what happens to a circle as its diameter tends towards infinity”.
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Spirithound
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Ah, but an infinitely small circle is easy: it is but a single point.
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Alberti_Devoveo
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Our universe has no boundaries, is it too meaningless?
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ReapReason
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It depends on the context in which you say that a thing has no boundaries. Why do they say that the universe has no boundaries? I think you will find they mean something quite different from an infinite circle
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ReapReason
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A circle necessarily has a diameter, and therefore is quantifiable.
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MarkThompson
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In Euclidean geometry, a circle is the set of points equally distant from a given center. Suppose that (a, b) is the center; then a circle with radius r must contain the point (a+r, b). Therefore a circle with radius ∞ must contain the point (a+∞, b). But this is not a point, because it has no defined x-coordinate. You see, despite the fact that we can write it with the simple symbol “∞” we must remember that infinity is not a number and it is not a quantity. There is simply no such thing as “a point that’s infinity away from the center.”Why do you say no? Can you speak more about what you mean by “what happens to a circle as its diameter tends towards infinity”.
As to your second question, I mean that we can explore things such as certain limits as r → ∞. For instance, we can say that as the radius of a circle tends toward infinity, its circumference also tends toward infinity at 2π times the rate and its area at π times the square of the rate.
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Qoeleth
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Nicholas of Cusa demonstrates that an infinite circle is actually a straight line, since its curvature becomes nothing. He also demonstrates that an infinite triangle is a straight line.
I suppose it would not really be a circle, but the concept could be used, even though it can’t be imagined. Perhaps the concept of the concept of infinite circle exists. But not the thing itself.
I suppose it would not really be a circle, but the concept could be used, even though it can’t be imagined. Perhaps the concept of the concept of infinite circle exists. But not the thing itself.
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GEddie
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The curvature which is 1/R, would likewise go to zero.
However, not only would you never see it’s edge, you’d never find the center!
God Bless and ICXC NIKA
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ReapReason
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Then Nicholas of Cusa demonstrates that an infinite circle does not exist since a circle by its nature is not a line.
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thewanderer
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Seeing as the very definition of an infinite straight line is that it is unbounded, ie, has no end-points, it is impossible for a circle to have an infinite diameter. The diameter is the straight line which goes through the centre of a circle and is bounded by the circles circumference. This contradicts the very definition of an infinite straight line.
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MarkThompson
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Or let’s put it another way. Let’s say that (a, b) is the center of a circle C which is an “infinite circle.” Let (x, y) be an arbitrary point anywhere in the plane; thus by hypothesis a, b, x, y ∈ ℝ, i.e., they are real numbers. Then by the Pythagorean theorem, the distance d from (a, b) to (x, y) is d = √(x - a)² + (y - b)² ]. Now let us call a point outside a circle if the distance from the point to the center is greater than r, on the circle if (by the definition of a circle) the distance from the point to the center = r, and inside the circle if the distance is less than r. However, since a, b, x, and y are all real numbers, then the distance from (a, b) to (x, y) is likewise ∈ ℝ for all arbitrary points (x, y). But every real number is “less than infinity.” Therefore, for all points (x, y), d < r, and so all points are inside the “infinite circle.” Since all points are inside the circle, there can be no points on the circle. Therefore C = ∅. But a circle is defined as “a set of points that are equidistant from a center.” Since C is an empty set, it cannot be a circle.
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ReapReason
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Agreed.Seeing as the very definition of an infinite straight line is that it is unbounded, ie, has no end-points, it is impossible for a circle to have an infinite diameter. The diameter is the straight line which goes through the centre of a circle and is bounded by the circles circumference. This contradicts the very definition of an infinite straight line.
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InspiritCarol
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Something that complicated can not be reduced to these one-liners.
If you look at any point on a circle very closely, it may be taken as a “straight line” for mathematical purposes.
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ReapReason
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Assumptions aside, a straight line taken by itself evidently does not have the nature of a circle. You can use points to make what we understand to be a shape or a circle, but that’s irrelevant.
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