Generating self-referential images with conformal geometric transformations
Moore, Escher, Penrose: A Conformal Golden Braid
Computer Vision and Pattern Recognition
Summary
Some artworks show strange images that seem to look back at themselves in impossible ways. The authors studied how a famous image by M.C. Escher, called Print Gallery, uses a special kind of twisting geometry called a conformal map. They then used AI image models combined with these geometric rules to create new images that develop both the scene and its distortion together. This approach lets them produce complex, self-referential pictures that match the peculiar structure of the original artwork.
What this means in practice
- •For digital artists: Create novel self-referential artworks by combining AI image generation with conformal geometry constraints.
- •For computer graphics developers: Develop new image synthesis tools that maintain geometric consistency in recursive or twisted compositions using integrated transformation operators.
Authors
Sophia Feldman, Assaf Shocher
Abstract
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.'' So wrote M.C. Escher about his 1956 lithograph Print Gallery. Nearly half a century later, a mathematical analysis related its geometry to an untwisted source image through a conformal power map $z \mapsto z^α$, $α\in \mathbb{C}$. Building on this construction, we use a frozen text-to-image diffusion model to generate new self-referential scenes. Prompting alone does not enforce the recursion, while a post-hoc transformation can leave structures poorly connected. Applying the transformation during sampling is also insufficient: the denoiser may "repair" the intended distortion or drift out of the prescribed geometry. We construct a generalized inverse $T^\dagger$ of the non-invertible image transformation $T$, adapted to its recursive constraint. In the idealized formulation, the Penrose identity $TT^\dagger T = T$ makes $TT^\dagger$ an idempotent projection onto geometrically admissible images. Yet denoising only the transformed image remains an out-of-distribution task, even with projection. We therefore braid denoising steps with $T$ and $T^\dagger$: source-space steps develop the untwisted scene, while transformed-space steps refine its appearance and connections in the final geometry. We generate Print Gallery-like compositions and explore further transformations. Rather than distorting a finished image, we let the scene and its distortion develop together.