Koch Flake

The length of the border of the flake is infinite. At each iteration, a border of length 1 become 4/3.

Starting from a straight line segment divided by 3, a broken line of 4 segments os obtained: the length is therefore increased by 4/3 (increase of 33%).

Example: After 2 iterations, a line of initial length $ l $ has a new length of $ l \times \frac43 \times \frac43 = l \times \frac{16}{9} $.

If the number of iterations is infinite, the length is infinitely times increased by 4/3. The total length of this fractal curve is infinite.

$$ \lim\limits_{n \to +\infty} \left( \frac43 \right)^n = +\infty $$

The area of the flake is finite and equals $ 8/5 $ of the area of the initial triangle.

For a side of the triangle $ a $, the final area of the flake is $ \frac{2a^2\sqrt{3}}{5} $

The algorithm is as follows:

0 - Draw an isosceles triangle and for each side (segment)

1 - Calculate the points at 1/3 and 2/3 of the segment

2 - Draw the isosceles triangle based on the segment formed with the 2 points found

3 - Remove the base of this new triangle

4 - Repeat from step 1 for each segment of the new figure

Mathematically speaking, the final drawing is called the Koch curve, and its base is a set of Cantor.