The Surprising Story Behind the Infinite Staircase Painting | Roger Penrose & Jordan Peterson

So now you also had some interactions, at least at arm's length, with Escher.

Oh yeah, yeah. So what I read was that you and your father had been interested in Escher's work, and you worked out with him the ever ascending staircase, which, by the way, seems to me quite similar, especially to the music in Bach's third Brandenburg concerto. I talked to a musician this week about how Bach managed to make this continual ascending spiral that never really goes up.

That's true; there is a thing like that.

Yeah, right. Yes, and then you sent the drawings of the staircase to Escher.

Well, the story was a little bit longer than that. I had been at this — I was a graduate student, I think in my second year; I can't quite remember. A colleague and I went to Amsterdam to go to the International Congress of Mathematicians, which happens every four years. At this congress, I happened to see one of my lecturers, and he had a catalogue that had one of these action pictures. I asked, "What on earth is that?" You see, and he said, "Well, this exhibition in the Van Gogh Museum by this artist M.C. Escher."

Never heard of him before. I went to see the exhibition; I was absolutely blown over by these pictures. One in particular, I think was called 'Relativity,' and I came away thinking, "Gosh, that's amazing! I wonder whether I could do something a little bit different that I hadn't actually seen in the exhibition.”

There, so I tried to make a construction with bridges and roads going into possible ways. I simplified it down to this thing that people refer to as a tri-bar. I've seen the tri-bar, and I showed my father. I mean, I didn't know that there's a Swedish artist called Oscar Reutersvärd who had done things very similar earlier, but Escher didn't know about him either.

But anyway, there are other artists who've done things like that. If you look carefully in the old, there's a piece which has a picture of gallows, and they're joined up differently at the top.

Yes, I've seen that. I've seen that picture.

So there are other people who had played with these ideas, but I hadn't quite seen it in Escher. And so my father and I wrote an article. He developed this; the staircase was his actually. He was designing buildings, and then he produced the staircase which went round and round.

We decided to write a paper on this; we had no idea what the subject was, what journal did we send it to? So my father said, "Well, I happen to know the editor of the British Journal of Psychology, so let’s call it psychology." So we sent it to them and they accepted it. He said he thought he could get the editor to accept it. They did, and this was — we gave reference to Escher's catalogue and the Escher’s exhibition.

Then my father had correspondence with Escher, with letters going backward and forward. Then I think I was driving in the Netherlands for some other conference, I think, and I was curious. When I was reasonably close to Escher, I phoned him up. I got the phone number from my father, and he was very nice. He invited me and my then-wife to tea.

I just had a chat with him, and he sat at one end of a long table; I was at the other end. He had two piles of prints, and he said, "Well, this pile I don't have many left, I'm afraid." Then he pushed the other pile to me and said, "Choose one." So I sort of went through these things, and I picked one out. It was pretty hard to choose one out of all that, and I chose one called 'Fish and Scales.'

He was actually rather pleased because he said, "Well, most people don't understand that one." So I felt a bit flattered by that. This I then gave him a set of little pieces of just one shape, and I gave him a set of them and said, "Well, can you tile with those?"

Then a little while later he wrote to me and said he'd seen how to do it, but he wanted to know what the underlying principle was. So, I did; I wouldn't say that was a very bad correspondence. It took me a little while before I got back to him, but I showed him what it was based on.

On the basis of that, he produced what I believe was his last watercolor, maybe his last picture, as far as I know. I think it's called 'Ghosts,' which is based on this. It's the only tiling, as far as I know, that he ever did which is what's called non-isohedral. You see, he usually did periodic ones, but they're periodic in the strong sense that if you find a shape, the next time you see it, it has the same relation to the pattern as a whole.

So you could move this one into that shape, and the whole pattern goes with it into itself. But the one I showed him was what's called non-isohedral, that you can have different instances of the shape.

So this one has a different relation to the pattern as a whole from that one. And so if I move this one into that, I can't bring the whole pattern along with it. So you have two different roles that the shape plays in the last one of his pictures showed this.

So, I'm curious too about two things now. I'm interested in why you're so fascinated by the relationship of a geometric shape that can be arrayed in a variety of different manners to this underlying problem of mapping. Like, so you're reducing, you're reducing or establishing a relationship between the problem of mapping a large terrain to the utilization of like very stringently defined, what would you call them, representational systems, or that's a geometric form.

What is the geometric form conceptually in relationship to the problem of mapping?

Well, you have a shape, and then you have certain rules about which pieces will fit next to it. But there's certain freedom in that rule. You could put this one that way or another way, you see? And, you know, if it's just — if it's a shape which very clearly has to fit that way next to it, then it just repeats.

You see, but if there's some freedom as to what the next one will do, then you might have to make that choice, and certain choices will run you into difficulties later, and other choices maybe will allow you to continue.

Is there a relationship between that and what composers do with music? Because, I mean, there's a certain — well, there's a certain repeating determinacy in music, but obviously, a composer just doesn't take a pattern and repeat it indefinitely. They take a pattern, and the pattern seems to allow for some choice in movement from that pattern forward.

Well, maybe; I don't know. I mean, what makes a piece of music into a good piece of music? I mean, I have no idea; that's a much deeper issue.

Well, we do know a bit about it. We know that if it's too simple and repetitive, your interest gets exhausted.

Yes, exactly; it gets stale very rapidly, and then as it moves towards purely unpredictable, it becomes indistinguishable from noise. So there's some place in between there, and you could probably move on that place where you get some ultimately harmonious relationship of predictable form and something like the play of novelty.

That seems to me to be analogous to that possibility of shifting the shapes in this tiling problem. I mean, I think music is tiling something.

Yeah, it's a representational form.

No, that's probably some connection. It's just that music has so much more freedom as to what you do. You see, with these tiling shapes, it's forced on you that either it fits or it doesn't. You see, with music, it's much more subtle.

Right, I would hate to make too much of a leap.

Yeah, yeah, yeah.

Yeah, fair enough. One more question along that line. Now that that triangle you made, yes. What's the relationship between those paradoxical forms and the tiling problem?

Not much, because they seem to be — I mean, there's a play of representation, an image there.

I mean, one of the things I've been wondering; I looked at all your diverse contributions, and I thought, "Wow, there are a lot of things happening in a lot of different places." But there must be something that’s not random; there’s something at work that’s kind of a uniting principle that might be — I don't know, it might be the problem that you're trying to solve in some deepest way that's uniting all these elements of exploration and interest.

I don't know; you're asking too hard a question. I don't know. I mean, sometimes I don't see any overriding principle. I mean, there's a sort of thing, you know something feels right. Now, why does it feel right? I mean, that could be something very subtle.

Yeah, yeah, maybe wrong too.

Or maybe they are wrong.

Yes, yes. It seems to me that that also is related in some important sense, psychologically, to that notion of understanding. You know, the feeling that it's right — it's interesting that it can be wrong, but it's also interesting that it can be a predictor of — I got a student; she was very creative, and she would come up with hypotheses that were damn good.

But she was more creative than the typical psychologist, and I don't say that in a denigrating way. I mean, she was more like an artist than a researcher. And then what she would do is spend like six months writing out the algorithmic pathway to that conclusion, even though that is not how she derived it.

But she had a pretty unerring ability to jump forward to the right place with her intuition. It's something like — I think it's something like a deep form of pattern recognition. You know, you don't need the full pattern to infer what the pattern might be.

You can have a sparse representation of it, leap to what might be analogous to a tiling solution, I suppose, and that seems to be something related to the accuracy of intuition.