I guess no discussion of this formula held after 2011 would be complete without links to https://shreevatsa.wordpress.com/2011/04/12/how-does-tuppers-self-referential-formula-work/ (a.k.a. how the formula just encodes the picture inside the initial condition) and http://jtra.cz/stuff/essays/math-self-reference/index.html (a formula that includes all the parameters inside the plot).
The paper may be of interest. Lots more to it than just one demo. http://www.dgp.toronto.edu/people/mooncake/papers/SIGGRAPH2001_Tupper.pdf