The Droste-Effect and the Exponential Transform
Inhaltsverzeichnis
Reference
Bart de Smit: The Droste-Effect and the Exponential Transform. In: Bridges 2005. Pages 169–178
DOI
Abstract
In this note we give a correspondence between three kinds of pictures. First, we consider pictures with the “Droste-effect”: a scaling symmetry. These are drawn on a plane with a special point, which is the center of the scaling symmetry. Then we use the complex exponential map to transform these pictures to doubly periodic pictures, commonly known as wallpaper pictures, which are drawn on the entire plane. By rolling up the plane according to the periods, we get pictures drawn on a compact Riemann surfaces of genus 1: donut surfaces. As an application, we show the how the notion of a “Dehn twist” on a donut gives rise to a continuous interpolation between the straight world and the curved world of Escher’s 1956 lithograph Print Gallery.
Extended Abstract
Bibtex
Used References
[1] B. de Smit, Escher and the Droste-effect (website), Mathematisch Instituut, Universiteit Leiden, 2002; http://escherdroste.math.leidenuniv.nl
[2] B. de Smit and H. W. Lenstra, The mathematical structure of Escher’s Print Gallery, Notices of the Amer. Math. Soc. 50 no. 4 (2003), 446–451.
[3] B. Ernst, De toverspiegel van M. C. Escher, Meulenhoff, Amsterdam, 1976; English translation by John E. Brigham: The magic mirror of M. C. Escher, Ballantine Books, New York, 1976.
[4] D. Schattschneider, M. C. Escher: visions of symmetry, W. H. Freeman, New york, 1990.
[5] E. Th ́e, The magic of M. C. Escher, Harry N. Abrams, New York & London, 2000.
[6] J. H. van Dale, Groot woordenboek der Nederlandse taal, 12th printing, Van Dale Lexicografie, Utrecht, 1992.
Links
Full Text
http://archive.bridgesmathart.org/2005/bridges2005-169.pdf
