How (and why) to take a logarithm of an image
The video explores how M.C. Escher's 1956 lithograph 'Print Gallery' can be mathematically analyzed through complex functions, specifically using logarithms to transform self-similar images into warped loops. The analysis reveals deep connections between Escher's intuitive artistic process and mathematical concepts like conformal mapping and elliptic functions.
Summary
The video begins by examining M.C. Escher's 1956 lithograph 'Print Gallery,' which creates a mind-bending self-contained loop where a young man looks at a print that features himself. The piece features a mysterious blank circle in the middle, creating ambiguity about the viewer's location within the scene.
The creator explains Escher's three-step process: starting with a straightened 'Droste effect' image (where a picture contains itself), creating a warped grid that distributes zoom factors across corners, and using mesh warping to transfer the image. Escher's grid maintains a crucial property - tiny squares remain approximately square, which is called conformal mapping in mathematics.
The video then transitions to complex analysis, explaining how complex functions naturally preserve shape at small scales due to their derivative properties. This leads to an exploration of the complex exponential function e^z and its inverse, the natural logarithm. When applied to images, the logarithm creates bizarre, doubly periodic patterns that repeat both vertically (due to rotation periodicity) and horizontally (for self-similar Droste images).
The mathematical recreation of Escher's effect involves three steps: taking the logarithm of the Droste image to create a periodic tiling pattern, rotating and scaling this pattern appropriately, and then applying the exponential function to unwrap it into the final loop. This entire process can be simplified to raising the input to a complex power.
The analysis reveals that Escher's intuitive artistic choices align with deep mathematical structures, particularly elliptic functions used in modern number theory. The video concludes by reflecting on how both artists and mathematicians can be drawn to the same universal structures for completely different reasons.
About this episode
Escher's Print Gallery, and the tour of complex analysis it invites. Check out our virtual career fair: 3b1b.co/talent Join channel supporters to see videos early: 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 Gerhson, Mitchell Zemil, and Anna Fedczuk. 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
Key Insights
- Escher called the Print Gallery 'the most peculiar thing I have ever done' in a letter to his son, describing the creation of a young man looking at a print that features himself
- The mathematicians De Smit and Lenstra reverse engineered Escher's straightened Droste image by taking the logarithm of the original piece
- Escher intentionally chose serial types of objects like rows of prints and blocks of houses because without cyclic elements, it would be more difficult to convey his meaning to viewers
- In Escher's final warped grid, tiny squares remain squares with lines intersecting at right angles, making the image transfer process easier and the final result more natural
- Complex functions have the special property that shape is preserved at small enough scales - tiny squares remain approximately square even after transformation, which is called conformal mapping
- The exponential function e^z is special because as the imaginary input increases at 1 unit per second, the output walks around a circle at exactly 1 radian per second
- The logarithm of a Droste image creates a doubly periodic pattern - periodic vertically because rotation is periodic, and periodic horizontally because the Droste image repeats as you zoom in
- The doubly periodic functions used in recreating Escher's effect are called elliptic functions, which play a prominent role in modern number theory and provide bridges to other parts of mathematics
Topics
Transcript
[0:00] Whenever I'm making one of these videos, there's sometimes a special moment where the act of animating involves solving a whole bunch of little technical puzzlers, and then the underlying math I'm trying to explain clicks for me in a way that it hadn't before once I see it alive on screen. The best versions of those moments often tell me when a video is going to be one of my favorites, and putting together the end of this piece right here was one such time when I got that feeling. Our story doesn't actually start in the math classroom today. [0:30] We begin in the art room. Imagine standing in a gallery, looking at a picture of a…
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