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.
Figures & tables
Figure 1
Figure 2: The Droste effect in three representations. Left: a tile repeated in logarithmic coordinates, with horizontal period L=logλ and vertical period 2π . Middle: returning to image coordinates gives scale repetition, s(λz)=s(z) . Right: the conformal power map adds rotation, producing the Escher geometry with y(eαLw)=y(w) . All three panels use the same pattern, with λ=16 .
Figure 3: Braided sampling. Warm-up interleaves denoising with untwisted repetition T0 . The sampler switches between Droste and Escher representations using T and T† , with denoising blocks D(n) between switches. It ends with refinement in the Escher representation. At each switch, 4× super-resolution precedes the map, then resizing to working resolution. Dashed arrows indicate repetition; the noise-level axis is schematic. The early Droste preview is an illustrative blur of the later panel, not a recorded checkpoint.
Figure 4: Overview of generated results. Diverse scenes with conformal twists, square spirals, and multi-center maps. See the animated supplementary material .
Figure 5: Conformal results with input prompts. Longer prompts are excerpted. Animated views appear in the supplementary material .
Figure 6: Twist and transformation choices. Each row shares a prompt and initial seed. The conformal columns use ∣p∣=1,2 (negative p for the bookshop), followed by Poles, Möbius, Square, and Rimrings. Longer prompts are excerpted. Rimrings omits inverse steps. See the supplementary material for animated views.
Figure 7: Rimrings: repetition with visible boundaries. Bands accumulate near an outer rim; their divisions form material layers, ledges, coils, or plate edges. These examples omit the unstable inverse. Animated views appear in the supplementary material .
Figure 8: Baselines and cumulative ablation. Rows fix the base prompt and seed; the blue callout supplies the prompt extension. The first three columns show prompt-only, extended-prompt, and post-hoc T baselines. Starting from T at σ=0.87 , subsequent columns add warm-up, T† followed by a final T at σ=0.5 , in-loop SR (ours), and optional time travel. Identical central crops are enlarged 8× below.
Figure 9: Failure cases. Left: recursion anchors on the window rather than the canvas. Right: time travel amplifies the seam (without, then with).
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Component
Setting
Base model
FLUX.1-dev ( 12 B rectified-flow DiT), FluxPipeline , fp16
Guidance
distilled guidance embedding, scale 3.5 (no CFG pass)
Table 3: Operator schedule over the 128 steps ( σ:1→0 ).
Name
Symbol
Default
Swept
Inset scale
sin
1/4
1/8,1/16
Periods per turn
p
1
−1,2
Warm-up start
σw
0.95
off, 0.99
First op
σhi
0.87
0.80,0.92
Op gap
steps
9
5,13
Time travel
σtt
off
0.60,0.75
Appendix
Table 4: Swept hyper-parameters and their defaults.
Figure 10: Checking geometric round trips. From left to right: x , Tx , P(Tx) , P2(Tx) , and the residual ∣P2(Tx)−P(Tx)∣ , where P=TT† . Rows use deck ratios λ=2,4,16 . The three transformed images coincide in the continuous construction; the rasterized comparison shows blur and residual differences near checkerboard edges. The measured residual ∣P2(Tx)−P(Tx)∣ is 0.011 , 0.012 , 0.014 (top to bottom), each at the bilinear-resampling floor.
Figure 11: Transfer to a pixel-space backbone. The conformal T -cycle recipe applied to PixelDiT-1300M, with the same prompts, numerical seeds, transform, and scale as the FLUX configuration, and adapted intervention noise levels. The warm-up start changes from 0.95 to 0.96 , σhi from 0.87 to 0.92 , and σlo from 0.5 to 0.69 . Where time travel is used, its noise level changes from 0.75 to 0.82 . Per-cell tags retain the FLUX naming convention (e.g. tt75 ) and do not denote the adapted noise levels.
Figure 12: Additional Escher-recursion results (FLUX.1-dev). Curated gallery across subjects, transforms (conformal, square, poles, Möbius), inset scales and spiral orders; (Page 1 of 2.)
Glasgow College, University of Electronic Science and Technology of China · School of Mathematical Sciences, University of Electronic Science and Technology of China