Tales of Escher and Tiling Hyperbolic Space
Year: 2024 Authors: Grant Sanderson
Core claim
Escher’s encounter with Coxeter’s hyperbolic tiling on the Poincaré disk helped catalyze the circle limit works.
Topics
tiling patterns, hyperbolic space, wallpaper symmetries, Escher circle limits
Domains
Euclidean tilings, hyperbolic geometry, symmetry groups, Poincaré disk, M. C. Escher, Moorish architecture, graphic art, pattern design
Methods
historical narrative, visual comparison, mathematical correspondence
Media
images, sketches, Circle Limit IV
Paper text
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Sanderson
Tales of Escher and Tiling Hyperbolic Space
Grant Sanderson
3Blue1Brown, Wisconsin, USA; grant@3blue1brown.com
This talk describes the evolution of M. C. Escher’s interest in tiling patterns, originally inspired by Moorish architecture, and later evolving into a fruitful set of correspondences with several mathematicians. Pólya’s description of the 17 wallpaper symmetries [2] inspired more intricate Euclidean tilings in Escher’s work, and via his exhibit at the 1954 International Congress of Mathematicians this led to further correspondences with Coxeter.
Figure 1 displays an image from Coxeter’s 1957 symposium “Crystal Symmetry and its Generalizations” [1] depicting a hyperbolic tiling on the Poincaré disk, with Escher’s sketches on top trying to reverse engineer how the figure was created. Escher recorded that he was given “quite the shock” upon first seeing this image, which inspired a series of works that culminated in his circle limit series.
Figure 1: Illustration of hyperbolic tiling from Coxeter with Escher’s sketches on top.
Figure 2: M. C. Escher’s Circle Limit IV.
References
[1] H.S.M. Coxeter. “Crystal symmetry and its generalizations.” Transactions of the Royal Society of Canada, Series III, Vol. 51, Section 3, 1957, pp. 1-13. [2] George Pólya. “Über die Analogie der Kristallsymmetrie in der Ebene.” Zeitschrift für Kristallographie, vol. 60, 1924, pp. 278–282.