math & escher
Mauritis Cornelis Escher (1898-1972) was a Dutch graphic artist who is well known through out the world for his impossible constructions or illusions created by a type of printed drawing called a lithograph. His works have been especially inspiring to and admired by mathematicians. Examples of M.C. Escher's work in this style include stairwells and waterfalls that ascend or descend in an infinite strange loop depending on the direction the observer follows (Ascending and Descending, 1960; Waterfall, 1961), geometric patterns that morph into living organisms and vice versa (Liberation, 1955; Metamorphosis II, 1940), and a picture of a man looking at a picture which contains itself in another spiral strange loop (Print Gallery, 1956).
Escher had only a high school level education in mathematics. Although, it is believed by experts that he probably did not understand the more sophisticated mathematics underlying his creations, Escher was astutely aware of the intimate relationship his work had established to mathematics:
By keenly confronting the enigmas that surround us, and by considering and analysing the observations that I have made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists.
-M.C. Escher (from To Infinity and Beyond, Eli Maor)
Escher created "Print Gallery" by constructing a grid of a continuous circular expansion where cells on the grid increase by a factor of 256. Escher then transformed an image on an undistorted cell to consecutively distorted cells. In 2002, Professor Hendrik Lenstra who holds positions both at the University of California, Berkeley and Universiteit Leiden in the Nertherlands was perplexed by a seemingly unfinished hole in the center of Escher's "Print Gallery." His team was able to mathematically model the transformation function within "Print Gallery" and write a computer program that could complete the transformation within the blank central hole. What resulted was an inverted image of "Print Gallery" shrinking and repeating itself infinitely. It is believed Escher was not precisely aware of how the picture was to finish itself.
By keenly confronting the enigmas that surround us, and by considering and analysing the observations that I have made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists.
-M.C. Escher (from To Infinity and Beyond, Eli Maor)
Escher created "Print Gallery" by constructing a grid of a continuous circular expansion where cells on the grid increase by a factor of 256. Escher then transformed an image on an undistorted cell to consecutively distorted cells. In 2002, Professor Hendrik Lenstra who holds positions both at the University of California, Berkeley and Universiteit Leiden in the Nertherlands was perplexed by a seemingly unfinished hole in the center of Escher's "Print Gallery." His team was able to mathematically model the transformation function within "Print Gallery" and write a computer program that could complete the transformation within the blank central hole. What resulted was an inverted image of "Print Gallery" shrinking and repeating itself infinitely. It is believed Escher was not precisely aware of how the picture was to finish itself.
REFERENCES
B. de Smit & H. W. Lenstra Jr. (April 2003). The Mathematical Structure of Escher’s Print Gallery. Notice of the AMS, 50(4), 446-451.
Hofstadter, D. R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. New York, NY: Vintage Books.
Robinson, S. (October 2002) M.C. Escher: More Mathematics Than Meets the Eye. SIAM News, 35 (8) Retrieved from: http://www.siam.org/news/474.pdf
B. de Smit & H. W. Lenstra Jr. (April 2003). The Mathematical Structure of Escher’s Print Gallery. Notice of the AMS, 50(4), 446-451.
Hofstadter, D. R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. New York, NY: Vintage Books.
Robinson, S. (October 2002) M.C. Escher: More Mathematics Than Meets the Eye. SIAM News, 35 (8) Retrieved from: http://www.siam.org/news/474.pdf