Can you have an infinite circumference?

[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?

 

[arildno]

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?

 

[Lwats80]

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.

 

[arildno]

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.

 

[Lwats80]

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.

 

[arildno]

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?

 

[Lwats80]

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.

 

[rtisbute]

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

 

[DaveC426913]

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.

 

[Hurkyl]

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)

 

[John Creighto]

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

 

[dodo]

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'.)

 

[arildno]

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.

 

[JasonRox]

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.