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Moore, Escher, Penrose: A Conformal Golden Braid

Sophia Feldman, Assaf Shocher

arXiv:2610.02210v1cs.CV

TL;DR

Prompting and post-processing do not fully address the challenge of generating scenes composed within Print Gallery’s self-referential geometry. The paper combines a matched generalized inverse with braided denoising across source and transformed representations; its results show geometry shaping composition throughout generation, though finite resolution and denoising limit exact recursion.

  • Problem

    The paper seeks to generate scenes composed within Print Gallery’s geometry, which prompting does not constrain and post-processing cannot let the model adapt to resulting connections.

    Source: §1 · p. 2

  • Method

    A matched generalized inverse and alternating denoising in source and transformed representations braid a frozen image model with the geometric maps.

    Source: §1 · p. 2, §3.2 · p. 6

  • Results

    The results show how geometry can shape a composition throughout generation, despite finite resolution and imperfect scene consistency.

    Source: §6 · p. 9

  • Takeaways & Limitations

    Recursive scenes can develop together with their distortion rather than applying the distortion only to a finished image.

    Source: §6 · p. 9

    Finite resolution and subsequent denoising limit geometric consistency, so continuous identities do not guarantee exact recursion in the final raster image.

    Source: §5 · p. 9

Abstract

from arXiv · show

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 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.

1 INTRODUCTION

The paper aims to generate scenes composed within the self-referential geometry of Print Gallery, where prompting, post-processing, and applying the transformation during sampling each leave a different gap. It introduces a matched generalized inverse and braided sampling, and distinguishes this approach from prior work on geometric reconstruction, multiscale generation, and constrained diffusion.

1 INTRODUCTION

  • The Print Gallery depicts a young man viewing a town print whose buildings lead back to the gallery in which he stands.Penrose and his father’s impossible triangle and endless staircase later inspired Escher’s Waterfall and Ascending and Descending.Source: §1 · p. 2
  • Prompting does not enforce exact relations between image regions, post-processing leaves connections unadapted, and denoising can undo intended distortion.The method instead uses a matched generalized inverse and braids denoising between source and transformed representations.Source: §1 · p. 2

1.1 RELATED WORK

  • de Smit & Lenstra (2003) formalized Print Gallery’s geometry and reconstructed its missing centre, while this work generates new scenes and updates both representations during sampling.Cipolina-Kun et al. (2022) compared inpainting approaches after straightening the artwork; Generative Escher Meshes synthesizes tileable shapes and textures.Source: §1.1 · p. 2
  • Unlike multiscale methods that allow content to change with scale, this construction repeats the same content.It differs from fractal image coding and Neural Collages in prescribing admissible images while drawing content from a frozen model.Source: §1.1 · pp. 2–3
  • Prior constrained-diffusion and coupled-view methods steer frozen priors or synchronize views, while this work uses matched operators for recursive geometry and alternates source and transformed representations.The forward operator discards content outside a selected source band, and its generalized inverse selects a Droste extension.Source: §1.1 · p. 3
  • Unlike learned idempotent generators or other nonlinear inverse constructions, this work specifies the geometric operator analytically and keeps the image prior frozen.These inverse formulations differ in how they select among algebraically consistent inverses.Source: §1.1 · p. 3

2 PRELIMINARIES

The preliminaries define scale-recursive source images and the conformal power map that produces Escher geometry. They then establish generalized-inverse properties and the paper’s chosen projection convention.

2.1 FROM SCALE REPETITION TO ESCHER’S GEOMETRY

  • The source image repeats under scaling, s(λz) = s(z), and the conformal power map introduces rotation to produce Escher geometry.The choice of exponent makes sampling independent of the logarithm branch; for λ = 256, the smaller copy scales by approximately 1/22.58 and rotates clockwise by 157.63°.Source: §2.1 · pp. 3–4, Figure 2 · p. 4
  • A source band R/λ < |z| ≤ R determines the scale-recursive image through repetition inward and outward, with output locations reading colour from corresponding source locations.Figure 2 shows the progression from a repeated tile in logarithmic coordinates to scale repetition and then the rotated Escher geometry.Source: §2.1 · p. 4, Figure 2 · p. 4

2.2 GENERALIZED INVERSES AND IDEMPOTENT PROJECTIONS

  • The Moore–Penrose inverse returns a minimum-norm least-squares solution; retaining only the first two Penrose identities gives reflexive generalized inverses whose round trips are idempotent but need not be orthogonal.For nonlinear maps, Gofer & Gilboa (2024) retain reflexive consistency and select first by minimizing output error, then input norm.Source: §2.2 · p. 4
  • Braided sampling interleaves denoising with untwisted repetition and switches between Droste and Escher representations using T and T †, ending with Escher-space refinement.Each switch uses 4× super-resolution before the map and resizes to working resolution afterward; the noise-level axis is schematic.Source: Figure 3 · p. 5
  • The paper’s T † is a chosen reflexive generalized inverse selected by scale-repeating Droste structure and matched fundamental bands, not by minimum norm or orthogonality.Its round trips restore recursive image structure, and its projection means idempotence without a nearest-point guarantee.Source: §2.2 · p. 5

3 METHOD

The method generates recursive scenes with a frozen text-to-image model by braiding denoising with geometric maps between untwisted and transformed representations. The maps and sampling schedule develop the scene and its transformed connections together.

3.1 MOVING BETWEEN REPRESENTATIONS

  • The matched T and T† maps preserve admissible images on round trips and restore recursion in arbitrary Escher images by extracting and extending a selected band.Their composition TT† is an idempotent projection onto the admissible Escher class in the continuous construction.Source: §3.1 · pp. 5–6

3.2 SAMPLING SCHEDULE

  • The schedule warms up scale repetition, then alternates source- and Escher-space denoising with T and T† on clean predictions, re-noising after maps; defaults switch at σhi = 0.87 and σlo = 0.5.Fresh noise avoids warping existing noise, whose covariance a linear resampler would change.Source: §3.2 · p. 6

3.3 RESOLUTION AND TRANSFORMATION CHOICES

  • Switches map 4× superresolved images before resizing, while explored geometries include conformal, multicenter, square, Möbius, and Rimrings arrangements.Rimrings omits inverse steps because inversion is numerically unstable; an optional final denoise can smooth residual seams.Source: §3.3 · p. 6, Figure 4 · p. 7

4 EXPERIMENTS

The experiments qualitatively inspect recursive layouts and smaller copies, using a frozen text-to-image backbone. They show how conformal and alternative geometries shape scenes, and how cumulative sampler components affect their nested details.

4 EXPERIMENTS

  • All experiments use a frozen text-to-image backbone and qualitatively inspect recursive layout and content within smaller copies.Baseline and ablation rows hold the base prompt and initial seed fixed, except for the stated prompt extension.Source: §4 · p. 7

4.1 CLASSICAL PRINT GALLERY TRANSFORMATION

  • In conformal examples, recursion follows architectural frames, mirrors, camera displays, and screens; the library prompt uses geometry to nest an ordinary reading room.Figure 5 presents the conformal construction with prompts or excerpts.Source: §4.1 · pp. 7–8, Figure 5 · p. 7

4.2 TWIST AND TRANSFORMATION CHOICES

  • Poles and Möbius repeat motifs around two centres, while Square produces roughly rectangular nesting; recognizable scene elements persist as arrangements change.Figure 6 varies twist and transformation family for three fixed prompt–seed pairs.Source: §4.2 · p. 8, Figure 6 · p. 8
  • Rimrings forms bands into tortilla layers, garden ledges, coils, and plate edges, retaining intentional divisions rather than merging them into a continuous scene.These examples omit the unstable inverse, and bands accumulate near an outer rim.Source: §4.2 · p. 8, Figure 7 · p. 8

4.3 BASELINES AND COMPONENT ABLATIONS

  • Prompt-only and extended-prompt baselines lack nested structure, while post-hoc T enforces repetition but carries over scene boundaries the denoiser cannot adapt.Figure 8 compares these baselines with cumulative sampler additions.Source: §4.3 · p. 8, Figure 8 · p. 9
  • The inverse/forward pair develops another nested scene, and in-loop SR clarifies small edges and details; optional time travel can amplify seams.The 8× crops show cumulative component effects; time travel is excluded from the main sampler.Source: §4.3 · p. 8, Figure 9 · p. 9, Figure 8 · p. 9

5 LIMITATIONS

Finite resolution and subsequent denoising limit geometric consistency, and a fixed recursion centre may miss the intended subject.

5 LIMITATIONS

  • Finite resolution and subsequent denoising prevent the continuous identities from guaranteeing exact recursion in the final raster image.A fixed recursion centre can place the nest on a window instead of an off-centre painter’s canvas; optional time travel can amplify seams.Source: §5 · p. 9

6 CONCLUSION

Braided sampling combines a frozen image model with matched geometric maps to develop recursive scenes in source and transformed representations. The results show how geometry can shape a composition throughout generation despite finite resolution and imperfect scene consistency.

6 CONCLUSION

  • Braided sampling combines a frozen image model with matched geometric maps to develop recursive scenes in source and transformed representations.The results show how geometry can shape a composition throughout generation despite finite resolution and imperfect scene consistency.Source: §6 · p. 9

A IMPLEMENTATION DETAILS

The baseline comparison fixes the base prompt and initial seed within each row, apart from the stated prompt extension.

A IMPLEMENTATION DETAILS

  • Each of the five cumulative ablation columns uses the same central 128 × 128-pixel crop from a 1024 × 1024 output, enlarged 8× with identical crop coordinates and no separate enhancement.The crop uses rows and columns 448 through 575, with zero-based coordinates.Source: A IMPLEMENTATION DETAILS · p. 13

B SAMPLER AND HYPER-PARAMETERS

The sampler alternates denoising with geometric warps, while the transformation definitions specify how output locations map back to recursive source images. The recipe also transfers to a pixel-space backbone with an adapted noise schedule.

B SAMPLER AND HYPER-PARAMETERS

  • The FLUX sampler uses 128 flow-matching Euler steps, with a zero-spin warm-up followed by three window warps in a T–T†–T schedule.Before each window warp, the clean prediction is upscaled 4× with Real-ESRGAN and downsampled; optional time travel is off in the main sampler.Source: B SAMPLER AND HYPER-PARAMETERS · p. 13

B.1 TRANSFER TO PIXELDIT

  • On PixelDiT, re-matching noise levels restores contrast and recursion while retaining the FLUX recipe; 1024^2 sampling takes about 20 seconds on one RTX 5090.The schedule changes warm-up from 0.95 to 0.96, σhi from 0.87 to 0.92, σlo from 0.5 to 0.69, and time travel from 0.75 to 0.82.Source: §B.1 · p. 14

C TRANSFORMATION DEFINITIONS

  • Conformal twists produce logarithmic spirals, poles create local recursion centres, and the Square chart replaces circular nesting with square-shaped nesting.The conformal map is locally conformal away from the origin; the square map is non-conformal and changes derivatives across corner rays.Source: C TRANSFORMATION DEFINITIONS · pp. 14–15
  • Möbius maps precompose the twist, while Rimrings add a radial term that crowds repeated bands near the outer circle; Rimrings are not inverted because inversion is numerically unstable.The visible divisions between bands are part of the desired appearance.Source: C TRANSFORMATION DEFINITIONS · p. 15
  • The renderer selects the largest-radius scale-equivalent source sample that stays inside the canvas, using shared coordinate rules for standalone and in-loop rendering.Standalone rendering uses bicubic sampling on a supersampled grid; in-loop warps use bilinear sampling.Source: C TRANSFORMATION DEFINITIONS · p. 15

C.1 THE DROSTE-SELECTED GENERALIZED INVERSE

The construction factors the forward transform and its matched generalized inverse into extraction and extension maps. The inverse reads a selected Escher band, untwists it, and extends it by scale repetition.

C.1 THE DROSTE-SELECTED GENERALIZED INVERSE

  • The representative index n(q) is fixed from coordinates; on a bounded canvas, the convention chooses the largest index whose representative remains inside the canvas.In the unbounded continuous construction, n(q) may be zero.Source: §C.1 · p. 16
  • T† folds an image into the source band and selects its matched representative; TT† regenerates an admissible Escher image by extending that band.For an admissible image, T† selects the unique source member mapped to it; for an arbitrary image, the selected band determines the repeated copies.Source: §C.1 · p. 16
  • The extraction and extension maps factor T and T†, yielding B1E1 = B2E2 = id and both reflexive identities.The construction defines T = E2B1 and T† = E1B2.Source: §C.1 · p. 16
  • The continuous construction excludes the recursion centre, and its exact identities do not cover finite sampling, interpolation, super-resolution, or subsequent denoising.The inverse is an algebraic selection, not a minimum-norm or nearest-point choice.Source: §C.1 · p. 16

D EMPIRICAL CHECK OF IDEMPOTENCE

The geometric round-trip check finds small rasterization residuals even though the continuous construction returns identical transformed images. The check is separate from denoising and super-resolution.

D EMPIRICAL CHECK OF IDEMPOTENCE

  • Rasterized round trips retain the recursive layout but show blur and edge differences; this check does not establish exact consistency of final denoised images.In the continuous construction, P(Tx) = P^2(Tx) = Tx for P = TT†.Source: D EMPIRICAL CHECK OF IDEMPOTENCE · p. 17
  • 0.011, 0.012, and 0.014 residuals for λ = 2, 4, and 16 sit at the bilinear-resampling floor, while the continuous transformed images coincide.The residuals are measured as |P^2(Tx) − P(Tx)|.Source: Figure 10 · p. 17, D EMPIRICAL CHECK OF IDEMPOTENCE · p. 17
  • For the PixelDiT transfer, the adapted settings change warm-up from 0.95 to 0.96, σhi from 0.87 to 0.92, σlo from 0.5 to 0.69, and time travel from 0.75 to 0.82.Per-cell tags retain the FLUX naming convention and do not denote the adapted noise levels.Source: Figure 11 · p. 18

E ADDITIONAL RESULTS

The additional results show that the recipe transfers to a pixel-space model and provide a larger gallery of FLUX.1-dev scenes and transformations. The gallery spans subjects, transform families, inset scales, and spiral orders.

E.1 TRANSFER TO A PIXEL-SPACE MODEL

  • The recursion still forms with PixelDiT-1300M after matching noise levels, supporting transfer across the latent-to-pixel change.The recipe keeps the same prompts, numerical seeds, transform, and scale as FLUX, with adapted intervention noise levels.Source: §E.1 · p. 18

E.2 CURATED FLUX GALLERY

  • The curated FLUX.1-dev gallery extends beyond the main-text examples across subjects, conformal, square, poles, and Möbius transforms, inset scales, and spiral orders.The gallery spans two pages.Source: §E.2 · p. 18, Figure 12 · pp. 19–20
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