The composition of conformal maps is a conformal map. While the rotation, exponential and quadratic maps are a few basic conformal maps, by way of composition one can produce even more interesting maps.
By composing the exponential map with a scaling and rotation map and again with a logarithmic map, Escher was able to produce an interesting non-rectangular grid system which he used to create his well known etching Print Gallery.
Learn more at Escher and the Droste Effect.
The applet below shows a piece of Escher's transformation on a rectangular grid.
Yet Another Note on Branch Cuts:
Since the transformation involves composing an exponential map with a logarithmic map the inverse transformation will involve composing a logarithmic map with an exponential map. As with some of the previous applets, Arg z ∈ (-π, π] which means that the negative real axis is a branch cut in both the rectangular and Escher grids.
Please draw slowly over the branch cuts!
One can produce interesting paintings with this applet by drawing concentric shapes centered at the origin (i.e. Droste paintings).
Using the applet.