Where have you seen the most famous M. C. Escher prints? Many, such as Relativity and Reptiles, show up on math and science textbooks and are used in school lectures. However, most are surprised to know that Escher had no formal training in mathematics.
Escher only received a high school education in mathematics and after graduating he enrolled in architecture school. A week into his education, one of his teachers who happened to be the successful graphic artist Jessurun de Mesquita was impressed by a few of Escher’s lithocuts. He asked Escher, “Wouldn’t you like to be a graphic artist instead of an architect?” Considering that Escher did not particularly like architecture as a career choice, he agreed to study graphic design and never took another math course.
Escher had never been interested in the math theories and equations hidden within his work. He was an avid observer of the world, and found patterns, symmetry and perspectives in daily life that he could recreate in his visual paradoxes. Although he did not use formal math in his designs, he understood how his designs were linked to math. He once said:
By keenly confronting the enigmas that surround us, and by considering and analyzing 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.”
The mathematician Hendrik Lenstra recognized the math in Escher’s work as he had always been fascinated by the numerical themes he saw within the pictures. Lenstra is a professor of mathematics at the University of Leiden in the Netherlands. In 2000, he started a two year project analyzing Escher’s Print Gallery in order to find an exponential function to Escher’s design process. Lenstra then generated a reproduction of the image using the exact formulation Escher discovered and refined the image. For a person who did not have a deep background in mathematics, Escher created what was exceptionally close to a perfect mathematical image.
Lenstra was particularly intrigued by the Droste Effect he saw in Print Gallery. The Droste Effect is when there is an infinite reproduction of an image within an image. As you can see within Print Gallery, through a row of windows a man is looking at a painting on the wall. The picture continuously expands as the eye moves clockwise from the central, blank circle. Lenstra was convinced that a mathematical transformation must have been used to create the image. He set out to prove that Escher was creating a continuous circular expansion.
Escher made grids of his own before forming the image, on which he could distort figures and make the spirals he desired. Take a look here at his distorted grid for Print Gallery. He imposed a straight grid on the scene and then transferred the picture one square at a time to the distorted grid.
Through high level math, Lenstra generated grids similar to Escher’s and eventually was able to reproduce the image using complex exponential functions. For a look at the math involved click here. Compared to the mathematically produced Print Gallery, Escher’s original was not far off from being perfect. Look here at the grid generated by the mathematical formula of Lenstra’s transformation. Compare it to Escher’s grid.
So the question remains. Was Escher a closet-mathematician? Did he understand what he was doing? Or did his inspiration from designs and repetition lead him to math?
If you are interested in seeing many of Escher’s prints and drawings, the New Britain Museum of American Art is opening M.C. Escher: Impossible Reality today! (Friday, July 16th 2010) Come see his works up close, so you can witness Escher’s unbelievable talent for creating “impossible images”.