Many, many years ago I came up with an idea about a fractal that would be based on right triangles and squares. People that had less mathematical experience than I, didn’t take me seriously, when I told about my idea. They thought that this was just some kind of nonsense.
I didn’t visualize my idea and forgot the whole thing… Later, after many years when I started to read e-books about fractals, I found sophisticated ideas how to make nice colored Pythagorean fractal trees. I think I’ve been dealing with wrong people in the past… 🙂
This article discusses only about simplest possible Pythagorean fractal tree, most symmetrical version of it.
We start with a right triangle with two 45 degrees angles and one 90 degrees angle and visually speaking turn the triangle so that hypotenuse is at the bottom:
Next we draw squares against the sides of the triangle so that the length of the side of the square is the length of the side of the triangle:
Next we imagine new right triangles to the farthest side of the ”branch” squares and do the same process as earlier: We draw squares to all the sides of the triangles.
This process is continued infinitely and we have a fractal! …that looks like a tree.
The length of the new hypotenuse c‘ = cos(45°) * c, where c is the length of the hypotenuse of previous iteration loop’s right triangle(s).
These methods make it quite easy to generate a fractal tree. When drawing the tree into screen, only the squares are drawn.