Problem involving simple fractals (Koch snowflake problem)

[Stochastic13]

Homework Statement

Construction begins with an equilateral triangle with sides of length one unit. In the first iteration triangles with length one third are added to each side. Next, triangles of length 1/9 are added to all sides, etc., etc.

Is it possible for a bounded region to have a finite area and infinite perimeter? Explain.

Homework Equations

The Attempt at a Solution

Yes, If each time that iterations are increased the ratio of segment number to length is more than one, then by the geometric series test the series diverges and thus has infinite parameter. Also, if ratio of area is less than 1 as number of iterations goes to infinity, then the area converges by the geometric series test. Does that sound like I answered the question? What can you recommend for a better answer? Thanks.

 

[Chantry]

Uggggh, I just posted a response but the forum system logged me out before it sent.

I think your professor wants you to use the geometric series to solve this.

Note that the original triangle has area 1/2 and each iteration adds 3*4^(n-1) triangles.
Then, you can write this as the summation of terms (not including the 1/2) in the following format(I don't know how to use latex, so this will look ugly):

summation (from n = 0 --> infinite): a(r)^n

If r is < 1, which it will be,
this series converges to a/(1-r).

Considering it converges, you can find an exact area for the snowflake, which is finite even though there is an infinite perimeter.

You must also figure out that the perimeter is not finite by making a series that has r >= 1.

 

[Stochastic13]

OK, thanks very much.