The video I made for Saturday's post made me wonder what the picture would look like if you didn't have a branch cut at all - if you make the logarithm function generate all of its multiple values (or equivalently, choose a sheet of the Riemann surface at random). Computers can't count up to infinity so the random method must be used. We can't make all the sheets equally likely as their number is unbounded so we need to pick some distribution. I chose a method that's rather simple to implement - I just roll a (virtual) dice until a 1 or a 6 comes up, and subtract the number of 2s and 3s from the number of 4s and 5s (a 1D random walk). That makes positive and negative numbers equally likely, but makes small (positive or negative) numbers more likely than large ones.
Here is the result:
The infinite tower is transformed into a fractal fern-like structure. I was surprised at how different this looked from the second image on this post - while the non-principal branches are not visited very often, they cover a lot of space so make a lot of difference to the image.