Perimeter verus area

[the tree]

In one of Terry Pratchett's Discworld novels he describes the Proverbial Quantum Butterfly as having infinitely complex wings that had an infinite edge and therefore an infinite area.

A friend of mine disagree's with this and says that as Koch's snowflake fits can fit into a circle with a finite area, the snowflake itself must also have a finite area, although I agreed with that at first but then I figured that sometimes things are not that simple, as with Gabriel's Horn and the other stuff I don't understand.

So if something has an infinite perimeter, then does it have to have an infinite area?

[RyanJ]

So if something has an infinite perimeter' date=' then does it have to have an infinite area?[/quote']

 

By definition it should I guess but as always provingthings withinfinities inthem is always a challange! I hope someone can

 

The question of an area fitting inside a shape that has a finite area too me means that the shape must have a finite area, other wise it could not fit in any confined area at all...

 

Cheers,

 

Ryan Jones

[Connor]

nope.

 

consider an infinite integral with a converging sum

[the tree]

O.k. To be honest I don't understand integerals beyond the very basics (stuff like [math]\int{f'(x)}dx=f(x)[/math]), could you explain what an infinte integeral is?

[Sisyphus]

Ever hear of the paradox where, in order to go some distance, first you have to go half, and then half of the remaining half, and then half of the half of the half, and so on to infinitity, and thus you'll never actually get there? That's like an infinite sum with a finite total. You keep adding terms of 1/(2^n), and the sum as n approaches infinity is finite: one.

 

BTW, an analogous integral would just be the integral of f(x)=1/(2^x), from 0 to infinity.

[the tree]

Thanks, I guess I'd better learn definate integerals (I think that's the word) if I'm to carry on with abstract stuff like this. I looked at the MathWorld article but didn't get it, do you know of an easy tutorial for them?