Mathematics is about logic and imagination. In a way, it’s an answer to questions like “if we imagine some rules, whatever they are, how far we can go with them by applying only logic?” One may think that mathematics is about numbers and calculations, but actually, mathematical ideas are about creativity, strict logic and lots of fantasy.
M.C. Escher (Maurits Cornelis Escher, 1898-1972) knew that well. Escher was a graphic artist, and his formal mathematical educations ended at high school. He believed he had no mathematical ability, and yet he painted math in a way that inspires mathematicians, scientists, engineers and many others to this day. Unfortunately, his works weren’t popular for most of his life, and only at the age of 70 he started to see a growing interest in them.
His works include impossible objects, perception illusions, symmetry, tessellation, recursion (when a thing is defined in terms of itself or of its type), self-reference, hyperbolic geometry and more. We surely will cover his life and his works in future articles, but in this one we want to talk about one specific picture, the “Print Gallery”.
The Print Gallery
In “Print Gallery” (Dutch: Prentententoonstelling), if you look at the bottom-right corner, you see a gallery. Going left (moving clockwise), you see more and more pictures in a bit distorted form, until you see a person in the bottom-left corner of the image. The person looks into a picture with a boat and a city at the top-left corner of the image. The city has many cubical buildings, spreading to the top-right corner of the image, and one of them is placed just above a print galley, that is located, well, at the bottom-right corner of the image. We are just where we started. Makes no sense, but still interesting, curious and beautiful.
A small weirdness in the image is its center. In the center there is a white circle, with Escher’s signature in it. Why is that? Same question raised in Dr. Hendrik Lenstra’s mind, a mathematician, when he flew to the Netherlands, and read a magazine with this picture in it.
He decided that he has to uncover the truth behind the image, and discover what should be in the middle of it – what hides beyond the white circle. Luckily, his profession and character gave him the right tools to do so. Eventually, he published his study, “The Mathematical Structure of Escher’s Print Gallery”, at Artful Mathematics: The Heritage of M. C. Escher, Celebrating Mathematics Awareness Month, Volume 50, Number 4 (http://www.ams.org/notices/200304/fea-escher.pdf).
It appears that the white circle can be filled, and what’s in it is quite amazing. Apparently, citing from the study, “the whole lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers and deduce that an idealized version of the picture repeats itself in the middle. More precisely, it contains a copy of itself, rotated clockwise by 157.6255960832… degrees and scaled down by a factor of 22.5836845286….”
To understand it better, see a video that is screening at Escher Museum at Den Haag, Netherlands.
Math and Art
How this incredible image was drawn? To dive into the secrets of this art-piece, the researches read The Magic Mirror of M. C. Escher by Bruno Ernst. Quoting The Magic Mirror and The Mathematical Structure of Escher’s Print Gallery, Escher started “from the idea that it must…be possible to make an annular bulge,” “a cyclic expansion…without beginning or end.” he “tried to put his idea into practice using straight lines, but then he intuitively adopted the curved lines. In this way the original small squares could better retain their square appearance.”
Escher knew that going with straight lines will create a distorted image. One that will be hard to understand, in which objects like pictures, windows and buildings will be almost unrecognizable, so he tried a form of elliptic curve (not to mistake with ellipses. Elliptic curve is a curve that is defined by a specific equation).
The cyclic expansion created by Escher evolved over time, until he came up with something like that:
The only thing that is left was to create a normal drawing, and adapt it to his new geometrically curved world. Pay attention that in his world, looking closely, each cell is square-like, and only by zooming-out one can see that the world is curved. That is exactly what Escher could not reach with straight lines, and it’s too has a mathematical principal, named “Conformal map” (angle-preserving, a transformation from one plane to another that preserves local angles).
This mathematical idea, conformal mapping, is exactly the reason why the image could not be made with straight lines, that do not preserve the angles of each square, like Escher’s curve did. Escher filled the squares, and performed the transformation.
For just another moment, let’s research this new curved world and its bizarre properties. At the illustration below, the left image shows a 5X5 square that starts at point A (follow the dots), finishes a closed loop, and ends, as expected, at point A. It does not look much like a square, but in this world, it is. On the other hand, if the square was a bit bigger, 7X7, as demonstrated at the right image, the situation would be completely different. Remember, that this curved world is infinite and curves into itself. So, a bigger square that starts at point A, would be “sucked” into the middle of the world, that contains a copy of itself, and end up at A’ – an analogues point to A in the inner part of the world. Now it really does not look much like a square, and still, in a way, it is!
Quoting The Magic Mirror and The Mathematical Structure of Escher’s Print Gallery, the realization of the idea of “a cyclic expansion…without beginning or end,” caused Escher “some almighty headaches.”
Escher conducted his own research, in his own way and with his own methods, and while his way is non-standard, his results amaze mathematicians, and develop curiosity in all people. Escher’s works not only help to explain some fundamental mathematical ideas, but also helped to develop new ideas and new fields of mathematical research. After the work is done, the only thing left to do is to learned, be amazed, and enjoy.
Featured image: Hand with Reflecting Sphere by M. C. Escher. Lithograph, 1935, also known as Self-Portrait in Spherical Mirror. Escher’s self-portrait.