Moore, Escher, Penrose:A Conformal Golden Braid

Sophia FeldmanAssaf Shocher

Technion — Israel Institute of Technology

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Top-left: M. C. Escher, Print Gallery (1956). Surrounding panels: generated recursive scenes. Right: a conditional image of a reader holding this paper, re-noised and refined with our braided sampler. The reader, like you, believes they hold the outermost copy.

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 ↦ zα, α ∈ 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† of the non-invertible image transformation T, adapted to its recursive constraint. In the idealized formulation, the Penrose identity TT†T = T makes TT† 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†: 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.

Scenes within scenes

Explore all results

Hover to follow the recursion. Click or tap to open a larger view and read the prompt.

Let the scene and its distortion
develop together

Our sampler moves between an untwisted recursive scene and its transformed representation. The same frozen diffusion model develops the content in one space and refines its connections in the other.

Braided sampling: an untwisted repetition warm-up, alternating denoising blocks linked by T and T dagger, and final refinement in the Escher representation.
Geometric switches act on predicted clean images, followed by fresh noise; ordinary denoising steps run between switches. Each switch uses super-resolution before mapping and resizing. The early blurred preview is illustrative.
  1. 01

    Establish repetition

    A zero-spin warm-up encourages the source scene to repeat across scales.

  2. 02

    Braid the representations

    Alternate denoising with the forward map and its matched generalized inverse.

  3. 03

    Refine the connections

    Finish in the transformed space, where the scene meets its smaller copies.

One prompt, several geometries

Changing the map changes how the scene repeats.

A photorealistic toy puppet theater on a theater stage; its miniature stage shows this same theater, the same audience, and the same puppet theater, and a marionette climbs over the footlights into the real stage; velvet curtains, warm footlights.

Rimrings uses a separate forward-only procedure; its unstable inverse is not used.

BibTeX

@misc{feldman2026moore,
  title  = {Moore, Escher, Penrose: A Conformal Golden Braid},
  author = {Feldman, Sophia and Shocher, Assaf},
  year   = {2026},
  url    = {https://assafshocher.github.io/escher/}
}