Tilings and ESCHER

SH: Lastly, I would like you to speak about your "tiling," and perhaps also about M.C. ESCHER and aesthetic issues.

RP: Well, I should explain first of all that my father's father was a professional artist. He was a portrait painter, very representational, and came from a strict Quaker family. My father was one of four brothers, and they were all very capable, artistically; he used to do pen-and-ink drawings, and painted in oils. One of his brothers became quite a well-known artist; he was a surrealist painter and knew people like PICASSO and Max ERNST. So I suppose there is something of an artistic background in my family.

But as for the tilings, I used to doodle just for fun, designing patterns which would repeat-complicated things, different shapes that would make formal repeating patterns. One needed many of them before they would repeat. I was also interested in hierarchical structures, in which the pattern would appear at a larger scale. This was just playing around-there wasn't a feeling that this was science or anything.

One thing that must have been influential, although I didn't know it at the time, was a book my father had of KEPLER's works. Among these was a picture with many tiling patterns, some of them involving pentagons. I hadn't been thinking about them particularly, but it must have created the feeling that maybe pentagons were things that one could use for interesting designs. So I suppose that had some influence on me.

Then somebody wrote me a letter from a university in London, which has as a logo a pentagon subdivided into six other pentagons-one in the middle and five around. And I wondered what happens if one iterates that many times. One gets gaps and one has to think of ways to fill in the gaps. There's also a Japanese artist....

SH: ANNO Mitsumasa?

RP: Yes, he did something very similar-deciding how to fill the gaps. One can produce pentagons in a kind of hierarchical scheme, but there are always spaces between them. And when these spaces get big, one has to think about what to put in the spaces and make a choice whether to do it one way around or the other way. He did it the other way around, which was unfortunate, as that doesn't work so well. The other way, which I was doing, actually enables one to develop these things.

The story gets somewhat complicated, but I produced a non-repeating pattern with pentagons, and then somewhat later realized one could force these patterns into a kind of jigsaw puzzle. If one modified the shapes of the pieces a little, then one could assemble them in this way. And this led to six different shapes, which would force one into a non-repeating pattern.

Simon KOCHEN, who was visiting the Mathematical Institute in Oxford from Princeton, reminded me of Raphael ROBINSON, who had a set of six tiles that would tile a plane in a non-periodic way. He also mentioned that ROBINSON tried to keep his numbers down to a minimum. He had this non-periodic tiling based on squares-with modifications, but basically squares-and he had six different tiles to force non-periodicity. When I saw this again, I knew I could do better: my tiles were also six, but there was a redundancy; one could glue two pieces together and get it down to five. This was something of an improvement over what he had, but then I started thinking about it even more. And I realized one could get it down to two. But it's hard to give a simple explanation for why or where they came from.

The connection with ESCHER was different. Unfortunately, ESCHER never saw these tiles, because he died too soon. I'm sure he would have done wonderful things with them. When I was in my first year as a research student in Cambridge, I went to the International Congress of Mathematicians in Amsterdam and I saw a brochure of something that looked very strange to me. It was the catalog of an exhibition of ESCHER's work in a museum in Amsterdam, and I went to see it. I had never heard of ESCHER before. I became fascinated and tried to develop paradoxical designs.

Eventually I produced the "tribar"-which this triangle ["three-worlds triangle"] on the front of this book, The Large, the Small and the Human Mind, [Cambridge University Press, 1997] is based on-and I showed it to my father, who then started to produce all sorts other impossible buildings and things. He eventually produced this staircase, which goes around and around. We then wrote this into a paper and sent a copy to ESCHER, because he had really started us off thinking about these things. Yet the specific things that we had, he didn't-he hadn't seen before-after which he developed the staircase into "Ascending and Descending", one of his most famous works. And , which was based on our triangle.

On a later occasion, I actually visited ESCHER and showed him some of my tilings, which were not the non-periodic ones but other ones that were still quite complicated. Subsequently, the very last picture he ever produced, as far as I know, was based on this type of arrangement I had shown him. So that was an independent connection with ESCHER: impossible objects and those particular tiles. It's rather sad that he didn't live a bit longer to see the non-periodic ones.