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

2026-03-22 · Source: 3Blue1Brown · Field: Science & Research — Mathematics & Computational Sciences, Research Methodology & Innovation · Depth: Advanced, extended

Summary

M.C. Escher's 1956 lithograph, "The Print Gallery," a self-contained loop where a young man views a print featuring himself, is analyzed for its mathematical depth. Mathematicians De Smit and Lenstra provided a 2003 analysis, revealing the piece's underlying structure. Escher's intuitive three-step creation process involved starting with a straightened, self-similar "Drosta effect" image, creating a warped grid, and then using this grid to transfer the image. The self-similar copy in Escher's work is 256 times smaller than the original. The mathematical foundation for this warping is explained through conformal maps, particularly using complex numbers and functions like the complex exponential (e to the z) and the natural logarithm. These functions transform lines into circles and vice-versa, enabling the creation of the warped, self-similar effect.

Key takeaway

For Creative Technologists or Research Scientists exploring generative art or visual mathematics, understanding conformal maps and complex logarithms offers a powerful framework. This mathematical approach provides a precise method for creating self-similar, warped visual effects, moving beyond intuitive artistic methods. You should consider how these functions can transform visual elements, enabling the creation of complex, self-referential designs with inherent mathematical elegance.

Key insights

Escher's "Print Gallery" intuitively applies complex analysis principles, specifically conformal maps and complex logarithms, to create its self-referential loop.

Principles

• Conformal maps preserve local shapes (tiny squares remain square).
• Complex exponentials turn vertical lines into concentric circles.
• Complex logarithms unravel circles back into lines.

Method

Recreating Escher's effect involves taking the logarithm of a Drosta image, rotating and scaling the resulting doubly periodic pattern, and then applying the complex exponential function.

In practice

• Use mesh warp in graphic design for controlled image distortion.
• Explore complex functions for generating unique visual transformations.

Topics

• M.C. Escher
• Print Gallery
• Drosta Effect
• Conformal Maps
• Complex Logarithm

Best for: AI Student, Research Scientist, Creative Technologist

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Editorial summary, takeaway, and curation by AIssential. Original article published by 3Blue1Brown.