Koch's Snowflake & Antisnowflake: Find Finite Area Formula

[nofx]

I have a question to do with Koch's snowflake and anti snowflake..
I have no clue how to find the formula for the total area of each.
This is my table for snowflake
http://img485.imageshack.us/img485/8959/snowflake2an.jpg
& this is it for anti-snowflake
http://img356.imageshack.us/img356/2616/antisnowflake8vz.jpg

I've realized that the area for both are finite, with the snowflake being 81x(8/5) and antisnowflake being 81x(2/5)

 

[eok20]

since you know how much area is added, you can set up a recurrence relation. for the snowflake, A_n = A_n-1 + 27*(4/9)^(n-1). since A_n-1 = An-2 + 27*(4/9)^(n-2) and so forth, A_n = A_0 + 27*(4/9)^(1-1) + 27*(4/9)^(2-1) + ... + 27*(4/9)^(n-1). this is a geometric series with the ratio 4/9<1 so it will converge as n goes to infinity.

 

[Architeuthis Dux]

^(3n-2)

Great question! Koch's snowflake and anti-snowflake are fascinating fractal objects that have infinite perimeter but finite area. To find the finite area formula for each, we need to understand the process of how they are created.

Koch's snowflake is created by starting with an equilateral triangle and then replacing the middle third of each side with two equal segments that form an equilateral triangle. This process is then repeated infinitely, creating a snowflake with increasingly intricate patterns.

To find the area of the snowflake, we can use the formula for the area of a regular polygon, A = (s^2 * n) / (4*tan(π/n)), where s is the side length and n is the number of sides. In the case of the snowflake, n starts at 3 for the initial triangle and then increases by a factor of 4 with each iteration. So for the first iteration, n = 3 and for the second iteration, n = 12. We can continue this pattern to find the total number of sides after n iterations, which is 3 * 4^(n-1).

Now, we need to find the side length, s, for each iteration. In the first iteration, the side length is simply the length of one side of the original triangle. But for subsequent iterations, we need to take into account the fact that the length of each side is reduced by a factor of 3. So for the second iteration, the side length is s/3 and for the third iteration, it is (s/3)/3 = s/9. We can continue this pattern to find the side length for any given iteration, which is s/3^(n-1).

Putting all of this together, we can create a formula for the area of the snowflake after n iterations:

A = (s^2 * 3 * 4^(n-1)) / (4 * tan(π/3)) * (1/3)^(n-1)

Simplifying this, we get:

A = (s^2 * 3 * 4^(n-1)) / (12 * √3) * (1/3)^(n-1)

And since we know that the initial side length, s, is 81, we can plug that in to get the final formula:

A = (81^2 * 3 * 4^(