A mathematical curiosity: A fractal tree object with branches in every dimension, up to infinity

I stumbled upon a curious mathematical beast: a fractal with infinite dimensions. I will be producing 3 dimensional renderings to get an idea what this ‘looks’  like, but let’s start with the maths. Imagine a trunk of a 1-dimensional tree, i.e. a fat line. After a while it splits into two branches in a 2-dimensional…