# Can you have an infinite circumference?

• Lwats80
In summary, the conversation discusses the concept of infinity in relation to the diameter and circumference of a circle. It is concluded that if the diameter is considered infinite, then the circumference would also be infinite. The idea of a shape with an infinite perimeter but finite area, such as a fractal, is also mentioned. Examples of shapes with infinite perimeters and finite areas are given, such as the Riemann sphere and a series of rectangles. The discussion also touches on the concept of dimension and how it relates to shapes with infinite perimeters.
Lwats80
Although a circumference would be finite as it encloses a finite space, if you said that the diameter was infinite (presuming you can say that) would this then make the circumference is infinite?

Another was to look at it, is if you take a circle of finite size and then keep enlarging it to infinity does this mean the circumference is infinite?

My head hurts, can anyone help?

Since the circumference in a circle equals (pi*diameter), the larger you make the diameter, the larger does the diameter become. If we allow for an "infinite" diameter, then of course the circumference of that circle is infinite as well.

But, more important than this triviality is your illogic premise:
"a circumference would be finite as it encloses a finite space"

This is totally wrong, why do you think it is right?

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Well, as you said the circunference of a circle does indeed equal pi*d so if we have a finite d we will also have a finite c. As for the space, the area of a circle=pi*dsq/4 as we all know, so if the area is finite the d will be finite and therefore using pi*d, the c will be finite.

Lwats80 said:
Well, as you said the circunference of a circle does indeed equal pi*d so if we have a finite d we will also have a finite c. As for the space, the area of a circle=pi*dsq/4 as we all know, so if the area is finite the d will be finite and therefore using pi*d, the c will be finite.

Sure enough, but are circles the only types of figures?
Some people use the words "circumference" and "perimeter" interchangeably; perhaps you don't, and your question was merely about circles, in which case your issue is resolved.

But there exists many figures with infinite perimeters, yet finite area.

As you can probably tell I am no physicist. I don't understand how you can have an infinite area with an infinite perimeter. Can you give me an example of such a shape and I will go and look it up.

Lwats80 said:
As you can probably tell I am no physicist.
What does that have to do with anything?
I don't understand how you can have an infinite area with an infinite perimeter.
Okay, you DID mean IN-finite in both parts here, did you?

Well I didn't originally but you have got me thinking. I have just been looking at the Koch snowflake and I think I am starting to get it. Thanks for the mental workout.

shpere

What about the Riemann sphere over the complex number field. You can carve out a circle where the diameter would include infinity. I would check it out!

rtisbtue

arildno said:
But there exists many figures with infinite perimeters, yet finite area.
Sounds like fractals. A hypothetical continent with an arbitrarily craggy coastline could have an arbitrarily long coastline, theoretically approaching infinity, while the continent's area remains finite.

A coastilne approaching inifinity would be a line but have a dimension much larger than 1; it have a dimension of 1.99999999999... i.e. =2.

DaveC426913 said:
Sounds like fractals. A hypothetical continent with an arbitrarily craggy coastline could have an arbitrarily long coastline, theoretically approaching infinity, while the continent's area remains finite.

A coastilne approaching inifinity would be a line but have a dimension much larger than 1; it have a dimension of 1.99999999999... i.e. =2.
If a shape has (Hausdorff) dimension bigger than 1, then it has infinite length. It doesn't matter how much bigger.

If a shape has infinite length, that doesn't imply its dimension is greater than 1. (I'm not sure what happens if we also assume the shape is bounded)

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rtisbute said:
What about the Riemann sphere over the complex number field. You can carve out a circle where the diameter would include infinity. I would check it out!

rtisbtue

I was thinking of that example. I wish I could remember some stuff about Nyquist plots. I seem to recall something about considering how loops at infinity cycle the point -1 in the complex plane but I forget the exact application.

http://en.wikipedia.org/wiki/Nyquist_stability_criterion

arildno said:
But there exists many figures with infinite perimeters, yet finite area.
And not necessarily a fractal. I believe some integrals of asymptotic functions provide an example. The bell-shaped normal Gaussian extends infinitely to left and right, yet the area between itself and the x-axis is 1. As Hurkyl points out, if the curve is bounded, that is another issue.

(Last minute edition: maybe this example doesn't count; if the curve never touches the x-axis, maybe we can't speak of a 'perimeter'. Yet, funnily enough, we do speak about an 'area'.)

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Another simple example:
Let us place alongside each other:
A unit square, a rectangle of length 1 and height 1/2, a rectangle with length 1 and height 1/4 and so on.

Now, for n such rectangles placed alongside each other, its perimeter is greater than 2*n, so with infinitely many such, the perimeter is infinite.

But the total area is just 1+1/2+1/4+1/8+++=2, that is, bounded.

arildno said:
Another simple example:
Let us place alongside each other:
A unit square, a rectangle of length 1 and height 1/2, a rectangle with length 1 and height 1/4 and so on.

Now, for n such rectangles placed alongside each other, its perimeter is greater than 2*n, so with infinitely many such, the perimeter is infinite.

But the total area is just 1+1/2+1/4+1/8+++=2, that is, bounded.

Exactly what I was thinking.

Good posts by Hurkyl earlier too.

## 1. Can a circle have an infinite circumference?

No, a circle cannot have an infinite circumference. The circumference of a circle is determined by its radius, and no matter how large the radius is, it will always have a finite circumference. This is because the formula for calculating circumference (2πr) involves multiplying a finite number (π) by a finite number (the radius).

## 2. Is it possible for a circle to have a never-ending circumference?

No, a circle cannot have a never-ending circumference. As explained before, the circumference of a circle is always finite, no matter how large the radius is. In mathematical terms, this means that the circumference is a bounded quantity and cannot be infinite or never-ending.

## 3. What about a spiral or a fractal shape? Can they have an infinite circumference?

No, even spiral or fractal shapes cannot have an infinite circumference. While they may appear to have a never-ending perimeter, they are still bounded by a finite number. Moreover, the concept of circumference only applies to perfectly round shapes, not irregular or complex shapes.

## 4. Can the circumference of a circle be infinitely small?

No, the circumference of a circle cannot be infinitely small. It can approach zero as the radius approaches infinity, but it will never be exactly zero. This is because, in mathematics, infinity is not a number, but a concept that represents something that is unbounded or limitless.

## 5. Is there a shape that can have an infinite circumference?

No, there is no shape that can have an infinite circumference. As discussed earlier, the concept of circumference only applies to perfectly round shapes, and all round shapes have a finite circumference. Other shapes, such as lines or curves, do not have a circumference at all.

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