Moore, Escher, Penrose: A Conformal Golden Braid — Generating Self-Referential Scenes with Diffusion Models

Forum topic · 小凯 · 2026-10-03

Summary

This arXiv paper (2610.02210) by Sophia Feldman and Assaf Shocher, presented on zhichai.net, uses a frozen text-to-image diffusion model to generate self-referential scenes inspired by M.C. Escher's 1956 lithograph Print Gallery. Escher's famous recursive artwork is mathematically described by a conformal power map z -> z^alpha applied to an untwisted source image. The authors show that prompting alone cannot enforce recursive geometry, post-hoc warping yields poorly connected structures, and applying transformations during sampling lets the denoiser repair or deviate from the intended distortion. Their solution constructs a generalized inverse T^dagger for non-invertible image transformations T, so that T*T^dagger acts as an idempotent projection onto geometrically feasible images via the Penrose identity. Because denoising transformed images remains an out-of-distribution task, they interleave denoising steps between source space and transformed space, building the untwisted scene while refining its appearance in the final geometry. The result is new Print Gallery-like compositions where scenes grow together with their distortions.

Overview

Field: Computer Vision Authors: Sophia Feldman, Assaf Shocher Published: 2026-10-01 arXiv: 2610.02210

Abstract (translated)

M.C. Escher described his 1956 lithograph *Print Gallery*: "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." Nearly half a century later, a mathematical analysis related its geometry to an untwisted source image through a conformal power map $z \mapsto z^\alpha$ , $\alpha \in \mathbb{C}$ .

Building on this construction, the authors use a frozen text-to-image diffusion model to generate new self-referential scenes. Key challenges:

Prompting alone does not enforce the recursion.
Post-hoc transformation can leave structures poorly connected.
Applying the transformation during sampling is also insufficient — the denoiser may "repair" the intended distortion or deviate from the prescribed geometry.

Method

The authors construct a generalized inverse $T^\dagger$ tailored to the recursive constraints of a non-invertible image transformation $$T$$ . In an idealized formulation, the Penrose identity $T^* T^\dagger T = T$ makes $T^* T^\dagger$ an idempotent projection onto the set of geometrically feasible images.

However, even with this projection, denoising transformed images remains an out-of-distribution task. The solution is to braid the denoising steps with $$T$$ and $T^\dagger$ : source-space steps build the untwisted scene, while transformed-space steps refine its appearance and connectivity in the final geometry.

Results

The method generates *Print Gallery*-like compositions and explores additional transformations — rather than warping a finished image, the scene grows together with its distortion.

Original 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 $z \mapsto z^\alpha$ , $\alpha \in \mathbb{C}$ . 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 insuffici... (truncated)

*Auto-collected on 2026-10-03.*