Lifelikeness of fractals

It's well-known that fractal-type objects can be quite lifelike and easy to generate. I've been scripting Minecraft with Python, in preparation for teaching this to gifted middle- and high-schoolers this summer, and wrote a simple 3D turtle graphics class with pitch/yaw/roll support. Like many kids of my generation, I did 2D turtle graphics programming with LOGO in school, but a 3D turtle just has a load of new possibilities. In particular, the 3D turtle allows for nice 3D fractal generation.

Instructions on how to do this stuff in Minecraft are in my Python coding for Minecraft instructable.

It was very easy to generate the following fairly lifelike tree with a simple bit of recursive code and some randomness.

An L-system does a pretty lifelike job even without randomness (using rules from geeky.blogger):

There is something glorious about a world where structures are mirrored on multiple levels. It makes the different parts and levels of the world be like a work of art, with themes and intertextuality.

Trees and limits of probabilistic reasoning

Suppose you're one of the nodes of this infinite tree, cut off for the purposes of the diagram, but you have no information whatsoever on which node you are.  Region A is exactly like region B.  And region B is exactly like region C.

So that you're in D must be at least as likely as that you're in B or C, but that you're in B is just as likely as that you're in A, and ditto for being in C.  Hence that you're in D is at least twice as likely as that you're in A.  But that you're in D obviously has the same probability as that you're in A.

Thus, P(A)=P(D)≥2P(A).  Hence P(A)=0.  But of course the whole tree is equal to three copies of A, plus the point 0.  So if you can assign probabilities, then you're certain to be at 0.  Which is absurd, especially since you can recenter the graph at another point and run the argument again.

Philosophically, this is a nice illustration of the severe limits to probabilistic reasoning.  My eight-year-old son is looking at what I'm posting and says: "It's just probabilities and I'm not going to use probabilities on this tree.  It's obvious."  He wonders why I am posting such obvious things.

Mathematically, all that this displays is that of course there is a paradoxical decomposition of a regular tree, and hence that there is no finitely-additive symmetry-invariant probability measure on a regular tree.