Is the Fractal Perimeter Infinite but Area Finite?

[Gib Z]

Hey if you guys look up Fractal on Wikipedia, you see the author states that the Koch Snowflake, a common and famous fractal, supposedly has an infinite perimeter yet finite area. It sed it would be infinite perimeter because it keeps on adding perimeter with each iteration. How ever, i thought since it keeps on adding Less with each iteration, it would remsemble a series that continually adds less. i haven't worked out the actual series yet, i will soon, but basically since it keeps adding less and less, it should eventually converge into a finite number eventually, right? sort of like if u kept on adding 10^0 + 10^-1 + 10^-2 + 10^-3 + 10^-4 so on so forth, sure u keep adding numbers, but the first one is 1, 2nd term 0.1, 3rd 0.01, 4th in 0.001, so on in that fashion, adding to 1.11111111111111111111111..., or 1 and 1/9. well yea, so this fractal i think really had finite area

 

[StatusX]

Not all series which have terms that go to zero converge. For example,

$$\sum_{n=1}^\infty \frac{1}{n} = \infty$$

To see this, note:

$$1+ \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4} \right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} +\frac{1}{8} \right) + ... > 1 + \frac{1}{2} + \left( \frac{1}{4} + \frac{1}{4}\right) + \left( \frac{1}{8} +\frac{1}{8} +\frac{1}{8} +\frac{1}{8} \right) + ... = 1 + \frac{1}{2} +\frac{1}{2} + ... = \infty$$

But this doesn't matter here, because in the Koch snowflake, at each step you increase the perimeter by 4/3, so the terms in the series actually increase, and so it obviously diverges.

 

[MeJennifer]

How ever, i thought since it keeps on adding Less with each iteration, it would remsemble a series that continually adds less.

It is true that each triangle part adds less per iteration but at the same time there are more of them per iteration.

 

[Gib Z]

yea that's for the help, i just realized wen i actually bothered to work it out this morning. wow that is quite extra ordinary, infinite perimeter inside a finite area. does anyone know the equations of any fractals, hopefully that i can plug into Graphamatica 2.0e?

 

[HallsofIvy]

Well, no, the perimeter is NOT "inside" the area!

 

[Gib Z]

excuse me? yes i think quite so. I was no referring to inside the finite area of the fractal, but say, a square around it.