Trapezoidal Koch

This is the trapezoidal Koch curve. We've already explored the regular and the square Koch, this one is also achieved by a different first iteration.

As expected, the shape of the first iteration significantly affects the final fractal.

The length of each straight line is still 1, so the trapezoidal variant is actually longer in terms of wingspan due to the extra length in the middle section. The angle of inclination is still 60°, maintaining similarities with the regular Koch.

Here's how the various iterations look like:

As usual, after infinite iterations, it looks like the area of the trapezoidal Koch doesn't exceed a certain threshold. Let's explore the area of the trapezoidal Koch.

First, we need an expression for the area of a single trapezium.

Once we have this, we can start calculating the area of each iteration and see what it tends to.

Similar to the square Koch, each iteration adds 5 new trapeziums, but they are 3 times smaller in terms of length.

This equation allows to calculate the area of any iteration. Replacing k with infinity, we can find the area as the number of iterations tends to infinity.