Area of the Koch snowflake as a recursive formula and as an explicit formula in terms of n, quick = best?

Question

This is a question on my math packet, and my brain is fried right now. I just can't seem to figure it out.

Details are appreciated immensely, and are clear explanations.

Side length--s--is 1, and I need the area for iterations 0 through 6 with the recursive formula and the explicit formula, as well as the finite area of the curve.

From Googling, I found the final area asfter an infinite amount of iterations to be either 8x/5 or (2 * sq. root 3)/5 *s^2.

I don't understand the discrepancy.

Thank you in advance.

Answer 1

The formula with the s in it gives the area when the starting triangle has sides of length s. The formula with the x in it gives the area based on the area of the original triangle.

The area of an equilateral triangle with sides of length s is A = s²√(3) / 4
Substitute this for x in the formula and you get the other formula.

To make your own iterative formula for the area, try thinking about pieces of the problems seperately. First, figure out how many new triangles you will be adding at step n. Then figure out what the area of one triangle will be at step n. The total area added will be the number of triangles times the area.

See the link I provided for more help on doing this.

Answer 2

Koch Curve Formula