We describe an algorithmic method to transform a Euclidean wallpaper pattern into a Circle Limit-style picture à la Escher. The design goals for the method are to be mathematically sound, aesthetically pleasing and fast to compute. It turns out that a certain class of conformal maps is particularly well-suited for the problem. Moreover, in our specific application, a very simple method, sometimes jokingly called the "Neandertal Method" for its almost brutal simplicity, proves to be highly efficient, as it can easily be parallelized to be run on the GPU, unlike many other approaches.