How (and why) to take a logarithm of an image

M.C. Escher’s “Print Gallery” (1956) works like a visual paradox: a viewer can walk a continuous loop while the scene “zooms” deeper and deeper, yet the local geometry stays recognizable. The core finding behind the piece’s construction is that the loop can be recreated by translating the artwork into complex-number geometry—specifically by taking a logarithm to convert a self-similar zoom into a doubly periodic tiling, then rotating/scaling that tiling, and finally applying an exponential to map it back into a warped, conformal scene.

The ambiguity at the center of Escher’s lithograph—where a blank circular region forces the viewer to decide what “space” they’re in—turns out not to be a random artistic flourish. It’s the natural consequence of how the nested “Drosta effect” zoom is encoded. Escher’s scene contains a picture inside a picture inside a picture, repeating at a fixed zoom factor; in the Print Gallery, the self-similar copy is 256 times smaller. The viewer’s gaze around the loop effectively performs that zoom implicitly, so the “where am I?” question collapses into a single geometric transition.

To unpack how this happens, the analysis referenced in the video traces Escher’s process through three steps. First comes a straightened, self-similar reference image: a man looking at a print that contains the same man looking again, ad infinitum. Second comes a warped grid—an engineered mesh warp—that tells the artist how to copy tiny square regions from the reference image into their new positions. The grid is chosen so that, locally, the bounded regions remain approximately squares rather than becoming general parallelograms. That local “square-preserving” behavior is the hallmark of conformal maps, a special class of transformations in complex analysis.

The mathematical bridge is that complex functions with derivatives behave like locally rigid motions plus scaling: at sufficiently small scales, they preserve angles and keep tiny shapes approximately square. This is why polynomials like z² and z³ look warped globally but conformal locally. The video then narrows in on two functions that do the heavy lifting for Escher’s effect: the complex exponential e^z and the complex logarithm log(z). Exponentials turn vertical lines into circles because increasing the imaginary part by 2π completes a full rotation. Logarithms reverse that unwrapping: circles become vertical lines, and because exponentials are many-to-one, the logarithm becomes naturally multi-valued—handled by periodic tiling.

For the Print Gallery’s nested zoom, the logarithm converts the Drosta zoom into a pattern that repeats both vertically (from rotational periodicity) and horizontally (from the zoom factor, since multiplying by 16 becomes shifting by log 16). After that, a carefully chosen rotation and scaling realigns a diagonal path in log-space so that, once exponentiated, it closes into Escher’s loop while matching the required 2π periodicity. The final exponential “unwarps” the tiling into the warped scene, and the conformal property ensures local recognizability even as the global geometry spirals inward.

In the end, the piece isn’t just a clever optical trick. It’s a meeting point between artistic constraints—Escher’s insistence on a rigid local rule like “little squares stay little squares”—and deep mathematics, where doubly periodic structures connect to elliptic functions and modern number theory. The result feels like a puzzle that fits perfectly, not because the solution was obvious, but because the underlying structures were universal enough to attract both an artist and a mathematician.