NemoClaw Knowledge Wiki

M.C. Escher's Print Gallery: Complex Analysis and Conformal Mapping

5 min read

M.C. Escher’s Print Gallery: Complex Analysis and Conformal Mapping

Generated: 2026-06-17 · API: Gemini 2.5 Flash · Modes: Summary

Clip title: How (and why) to take a logarithm of an image Author / channel: 3Blue1Brown URL: https://www.youtube.com/watch?v=ldxFjLJ3rVY

Summary

This video provides a deep dive into the mathematical underpinnings of M.C. Escher’s iconic 1956 lithograph, “The Print Gallery.” The main topic is how complex analysis, specifically conformal mapping, can explain and even recreate the paradoxical, self-referential loop depicted in Escher’s artwork. The presenter begins by illustrating the artwork’s unique quality: a man looking at a print that features himself, forming an infinite, inward-spiraling scene, which Escher himself called “the most peculiar thing I have ever done.” Mathematicians Smits and Lenstra’s 2003 analysis of this piece is highlighted, promising to not only explain the mind-bending self-contained loop but also reveal what should fill the seemingly ambiguous blank space in the center.

The video first offers an intuitive, three-step breakdown of how Escher likely constructed “The Print Gallery.” Step one involves starting with a “straightened-out” version of the self-similar image (known as the Droste Effect), where a picture of a scene contains a smaller version of the same picture, endlessly. Step two introduces a “warped grid,” which Escher meticulously drew, implicitly encoding the desired scaling and rotation needed for the artistic effect. Finally, step three utilizes a “mesh warp” technique, where the content of the straightened image is copied piece-by-piece onto the warped grid, distorting it to create the final, circular, and spiraling scene. A key insight here is that Escher’s grid maintains local “squareness” despite global distortion, a property that is mathematically significant.

The core of the mathematical explanation then moves into complex analysis. The video elaborates on complex numbers and functions, introducing the concept of a “conformal map” – a function that preserves tiny shapes (like squares) at a local scale, even as it globally distorts the space. Key functions explored include the complex exponential (e^z), which maps vertical lines in the input space to concentric circles in the output space, and its inverse, the natural logarithm (ln(z)), which unwraps circles back into lines. Crucially, the complex logarithm of the self-similar Droste image creates a pattern that is “doubly periodic,” repeating both vertically and horizontally. By combining these functions—taking the logarithm, applying rotation and scaling, and then exponentiating—the video demonstrates how Escher’s effect can be mathematically reproduced, effectively showing how the central “hole” is filled by a continuous, infinitely spiraling version of the image itself (represented by w^c, where ‘w’ is the input and ‘c’ is a complex constant).

In conclusion, the video bridges the gap between Escher’s artistic genius and deep mathematical principles. The mathematical model not only replicates “The Print Gallery” but also reinforces the artistic choices Escher made, such as preserving local shapes despite global distortion. This adherence to mathematical rigor (even without formal training by Escher) results in the elegant and satisfying fit of the puzzle. The video suggests that the structures Escher intuitively gravitated towards—such as doubly periodic patterns and conformal mappings—hide profound mathematical realities, even connecting to advanced concepts like elliptic functions in number theory. This shared fascination between artists and mathematicians for these underlying structures highlights a universal appeal in the intricate patterns of logic and aesthetics.

Video Description & Links

Description

Escher’s Print Gallery, and the tour of complex analysis it invites. Explore our virtual career fair: https://3b1b.co/talent Join channel supporters to see videos early: https://3b1b.co/support An equally valuable form of support is to share the videos. Home page: https://www.3blue1brown.com

Original paper by de Smit and Lenstra: https://pub.math.leidenuniv.nl/~smitbde/papers/2003-de_smit-lenstra-escher.pdf

The book I was showing is “Magic of MC Escher” by J. L. Locher https://amzn.to/4d7zXTT

If you want to play with this concept interactively, Jürgen Richter-Gebert put together a nice page: https://mathvisuals.org/PrintGallery/

This piece was co-written by Paul Dancstep, who handled many of the animations in the art section, including the delightful mesh warp scene.

Aaron Gostein helped with the manim animations in the section introducing complex functions.

Artwork provided by Talia Gershon, Mitchell Zemil, and Anna Fedczuk. https://sites.google.com/view/taliagershon https://mitchellzemil.com/ https://anna-fedczuk.framer.website/about

Music by Vincent Rubinetti

Timestamps:

0:00 - The print gallery 13:04 - Conformal maps from complex analysis 21:41 - The complex exponential 25:56 - The complex logarithm 32:32 - 3b1b Talent 33:14 - Constructing the key function 40:16 - The deeper math behind Escher

These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim

3blue1brown is a channel about animating math, in all senses of the word animate. If you’re reading the bottom of a video description, I’m guessing you’re more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly.

Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

Tags

Mathematics, three blue one brown, 3 blue 1 brown, 3b1b, 3brown1blue, 3 brown 1 blue, three brown one blue

URLs

Graph View

Backlinks