This interview was conducted
by a small newspaper in the New York City area. The *Metropolitan
Mathematician’s Journal* writes about advances in
mathematics and how they effect the world. They also interview
cultural icons who use mathematics in their lives. This interview
was conducted with M. C. Escher in 1948, at the height of his
success and popularity.

Q. Mr. Escher, you are famous for your graphic work which has surprised the art world. You do not conform your work to any modern movement and you do not seem to be trying to start your own movement. How would you describe what you do?

A. You are correct, I am not interested in starting or being a part of and art movement. I do my art because I think it is beautiful. That is what I believe art should be, a thing of beauty.

Q. Your art seems to have several recurrent themes to it, ideas or styles that you try to express. What are these?

A. Oh, there are far too many for me to tell you about now, but there are four that are my most popular. Firstly, I have always been interested in portraying the illusion of three dimensionality onto a two dimensional surface, in other words, making something on a sheet of paper look round. Secondly, I am very famous for another kind of illusion. I make images (again using the illusion of three dimensionality) that can not really exist. They can only be drawn on a flat surface, this is because a flat surface is not really three dimensional and does not have to follow the laws of reality. Thirdly, I express transformations in reality. I often do this by having a mirror in my picture reflect a scene that does not fit at all. Sometimes I make subtle changes to my picture that by the other side makes it completely different, a flock of birds changes into a city scape. In this style I often use tessellating figures gradually shifted to become something else. This brings me to my fourth and by far favorite style, regular divisions of the plane or tessellations. This is when a plane (or a section) of one represented by a sheet of paper is divided over its entire area by a regularly recurring series of lines. These must divide the plane into discernable shapes that repeat throughout its entirety. Or to put it in simple terms, a giant jigsaw puzzle in which all of the pieces are the same. Ever since I began experimenting with regular divisions of the plane, it has been my favorite thing to draw.

Q. Mr. Escher, are you aware of the amount of math involved in much of your work, and does that side of it interest you?

A. Yes, I am very conscious of the amount of math involved in my art, I find that to be one of the most fascinating aspects, much like my pictures where two distant worlds become one, art and math become one in much of my art, especially my tessellations. In fact, I consider myself more of a mathematician than an artist.

Q. Excellent! It is nice to hear about interest in mathematics in a wide variety of fields. Could you explain to us some of the basic mathematical principals that you use?

A. Certainly, although I am not proficient in advanced geometry. I use the basic principles of transformation geometry to make basic shapes that tessellate. To tessellate different kinds of shapes you must use different kinds of transformations. To tessellate a simple shape like a square, all you need is to slide it over or reflect it to tessellate, but for a slightly more complex shape like an equilateral triangle you must rotate it and then slide it in order to tessellate it infinitely over a plane. The 4 basic types of transformations are: Slide (translation) where you move a shape but keep its orientation, the reflection, where you take the reverse image of the shape and transfer it a certain distance over a line, the slide reflection, where you reflect a shape and then slide it, and the rotation where you change the orientation of the shape according to a certain point.

Q. Interesting, but your pictures do not involve only simple shapes-- you tessellate things like lizards and birds, how do you achieve that?

A. Interesting you should
ask--I developed a technique for that myself. To begin with you
take a shape that on its own will tessellate. Proceed by
‘cutting’ out pieces of the shape and
‘putting’ them around the outside in a specific way.
The easiest is in a square or rectangle where you can put the
piece on the same place on the other side of the shape, but, for
shapes that involve rotations to tessellate, it is much more
complex. In a hexagon for example you must place the piece
removed on the opposite part of an adjacent side. If you
‘cut’ a piece out of the side that you just
‘put’ a piece on, you must move that piece to the side
that the other piece came from. The artistic part in this is to
make that pieces you ‘cut’ out and ‘put’ on
look like something. Below is an example (*actually drawn by
Escher!!*) of how a complex tesselation starting from a
hexagon might come out.

This is how I used that tesselation to make a finished piece of art.

Q. Wow. That is really fascinating. Have you been doing this all of your life?

A. No, I first started doing graphic work in 1913, when I was 15 years old, but then I did simple black and white wood cuts. I gained recognition and was encouraged by my parents. I had gained a lot of fame and popularity for my complex lithographs (a complex system of printing) depicting worlds merging together by the time I began experimenting with regular divisions of the plane. I can date this back to around 1936. My first tessellations were based on old Arabic decorations, but I soon began to develop my own style. But it wasn’t until the mid 1940's that my technique had been perfected. It was a very gradual process, developed mostly by experimentation, because the complex mathematics of it escape me. I first mastered this technique with squares and rectangles, later moving on to triangles and finally hexagons and parallelograms. I am currently experimenting with irregular shapes (shapes with both positive and negative angles) that tessellate.

Q. That’s quite a history. I wish you luck in your further mathematical and artistic experimentation. Thank you for joining me here today Mr. Escher.

A. It was my pleasure, it is nice to be noticed for my mathematical work too.