This was originally going to be a small section in a short series building up to a simulation of evolving trees, but it has turned out to be extremely interesting on its own. I have a few parts worth of interesting math to go through related to fractal trees, but in this part we’ll just go through what they are, some cool examples, and an interactive demo where you can mess with the parameters and see what comes out.
A fractal tree is part of a class of fractals that are composed of several copies of themselves. To build one start with a trunk of any length, then draw two branches coming out of the top of the trunk each slightly smaller than the trunk and rotated to point in a different direction. Then draw two more branches coming out each of the first two branches rotated and scaled in the same way and repeat forever. You can change four parameters: the scaling factor and angle for each branch every time a branch splits into two more branches. Messing with these parameters can get you some really interesting shapes.
Some of these examples look a fair bit like actual trees like this half of a christmas tree:
Or this tree with some oddly level leaves:
You can also get some more fractally shapes like this infinite series of self-similar loops on loops:
Or this angry-looking shark fin:
You can also get some more well-known shapes like the Dragon Curve (plus some scaffolding)
And the related Twin Dragon Curve
You can also get this space-filling fractal that looks like a sheet of A4 paper with a ratio
There are also some pretty normal but unexpected shapes like this octagon.
How does it work?
This demo draws the trees up to 100 iterations of branching. Given that the number of branches doubles each time, that’s about