Moore, Escher, Penrose: A Conformal Golden Braid

"I don't think I have ever done anything as peculiar in my life. Among other things, it shows a young man looking with interest at a print on the wall of an exhibition that features himself. How can this be? Perhaps I am not far removed from Einstein's curved universe." This is how M.C. Escher described his 1956 litograph *Print Gallery*. This work of art has drawn mathematicians as much as artists. The image is a conformal power map of an ordinary picture. An ordinary generative model fails when prompted for such abnormality. This paper generates this form of art with off-the-shelf pretrained diffusion models. Applying the conformal transformation $T$ inside the sampling loop is not enough on its own: the model either "fixes” the twist as if it were damage, or drifts out of it. We therefore use the Moore–Penrose pseudo-inverse of the non-invertible map. The pair $T{T}^{†}$ is idempotent and projects any input onto the image of $T$. Even so, the denoiser would only ever see an out-of-distribution twisted picture. To solve this we interweave the diffusion steps with both directions of the transform: steps between $T^{†}$ and $T$ advance the underlying straight image, and steps between $T$ and $T^{†}$ realistically stitch the details of the transformed one. The Penrose identity $T{T}^{†}T=T$ guarantees that the result is a transformed image. We extend the construction to other conformal transformations, and obtain a tool for artistic, deliberately unrealistic image generation.