What's the name of this fractal?

Question

I created it with some simple chaos game restrictions:

1. You can always move towards the point you moved to on the last step
2. When on the top left corner, you can go to the right corners
3. When on the top right corner, you can go to the left corners
4. When on the bottom corners, you can only go to the corner that is opposite

Answer 1

Your figure can be constructed by a graph directed iterated function system. An iterated function system typically has no restrictions on which transforms may follow each other. A graph-directed IFS has restrictions like the one you have imposed: there is a directed graph in which the transformations correspond to edges. In your example the nodes of this directed graph would correspond to the corners of your figure, like this:

For a reference, with fractal dimension calculations, see:

"Hausdorff dimension in graph directed constructions" R. Daniel Mauldin and S. C. Williams (Trans. Amer. Math. Soc. 309 (1988), 811-829) http://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961615-4/

We introduce the notion of geometric constructions in $R^m$ governed by a directed graph $G$ and by similarity ratios which are labelled with the edges of this graph.