Moore, Escher, Penrose: A Conformal Golden Braid

2026-10-01

AI Summary

This study addresses the breakdown of recursive structures and the failure of geometric constraints when diffusion models generate self-referential images. To overcome these limitations, we propose an interleaved sampling framework based on generalized inverse transformations. Specifically, we construct a generalized inverse operator that deeply couples denoising steps with geometric transformations, enabling the synchronous evolution of scenes and distortions. Furthermore, we design a Penrose identity projection mechanism to achieve joint denoising across both the source and transformed spaces. Notably, this approach requires no fine-tuning; it directly leverages frozen text-to-image diffusion models combined with conformal power mappings. Consequently, our method successfully generates complex self-referential compositions reminiscent of Escher’s *Print Gallery* while strictly preserving geometric connectivity.

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.

Problem

• self-referential image generation
• conformal transformation
• diffusion models
• recursive geometry
• out-of-distribution denoising

Innovation

• conformal transformation
• diffusion model
• generalized inverse
• self-referential image generation
• Penrose identity