A Zig-Zag Definition of Pseudo-arc:
An interview with John C. Mayer
Dermot Wilson Can you tell us when you first studied pseudo-arc? Whose work interested you in the beginning?
John Mayer The paper that I worked through first was Bing’s on the pseudo-arc [1] where he proves that it is homogeneous; you can take any two points of the pseudo-arc and move one to the other by homeo-morphism. So that everywhere the pseudo-arc looks the same. Which is a very surprising and counter-intuitive sort of thing if you look at a standard way to build it, which is say Moise’s construction [2]. Based on Moise’s construction you might not think the pseudo-arc is homogenous, that is, it’s the same everywhere like the circle. In Bing’s paper that was made clear. And Bing actually used patterns of Moise’s construction and saw in it that you could basically get any pattern out of it. This means that his pattern must have a lot of going back and forth. There’s a universality in the pattern. And you can build really any pattern out of it.
It took me a little while to understand that’s really what Bing was saying. Even though this pattern might at first might look very rigid in fact in some sense it already contains all patterns if you sort of squint at it differently.
I think one thing is to see that the pattern out of which the pseudo-arc is built is itself very simple. But it’s complicated in that it has layers and layers of simple things which you can glue together to get other simple things as opposed to one big complex thing that you can’t fathom because everything is tangled up. The pseudo-arc can be thought of as the intersection of strips in the plane that are long and thin and getting longer and thinner and wiggly inside so this is sort of an alternative to thinking of it as chains. But then you have to say something about how the strips sit inside each other and you have to say something equivalent to chains. So one can then think about what are the crooked patterns of different layers of complexity. The other day we were talking about the basic patterns for the pseudo-arc. It must have something that goes almost to the end of something and then almost all the way back and then finally all the way to the end and this is really the simplest pattern fore the pseudo-arc. This almost complete zig-zag.
DW So you can attach another one of those to the almost complete zig-zag and that is the pseudo-arc pattern?
JM Yes, except that we can think of this as maybe a basic pattern. If we say this does not have enough layers to even be the next stage of the pseudo arc. Because to get from here to here you have to have more layers and so what would something that has more layers look like? That would be a pattern that would be part of building a pseudo-arc. Well, you have to look at each of these and say to yourself its got to go from here to here but to get there it’s got to get almost there, go almost all the way back and get to the start again.
DW But you still proceed?
JM Yes, no matter how finely you are focusing down on the pseudo-arc you’ll always see that pattern in the large and in the small. In other words if you zoom out you see that basic pattern, if you’re looking just at one stage. If you zoom into that stage into even a small part of it, you’ll see that pattern again. But then if you say, “well I’m not going to look at that small part, I’m going to look at that small part AND the small part next to it.” Well, it’s not two zigzags one stuck to another. That doesn’t happen! That’s exactly what doesn’t happen. When you look at this small part and then you decide you’re going to look further there’s still got to be a zigzag that goes through BOTH of those parts, almost all the way, almost all the way back, almost all the way forward. So…one of the classical mistakes about the pseudo-arc, which if you pick up a textbook whose name I guess I won’t mention and look at the picture of the pseudo arc in it, you will see this classical mistake that the pseudo-arc is just a succession of little zigzags like this. That’s not what it is. It’s that in each zig there’s a zigzag.
You can think of this pattern as just having three layers. But suppose we wanted one of the basic patterns out of which you build the pseudo-arc that had more layers. It would be the next one? And I like thinking of this one in this way. If it’s going almost all the way, then it might go one, two , three but it wanted to go four. It can’t. It goes one, two three and it doesn’t quite make it. So it doesn’t do the fourth one. Then it has to go back one two and then finally it can go forward one two three. So you can think of this as a three, advance three steps, go back two steps, advance three steps. You’ve finally made four steps. That’s a four, but it consists of three two three. Three steps forward, two steps back, three steps forward. The next most complex building block would be a five. But it has to be four steps forward, three steps back, four steps forward. But you can’t take four steps forward, this is one of the rules of a pseudo-arc. Anytime you try to take four steps forward, you must take three steps forward, two steps back and three forward.
So the way you can think of a five as emerging from a three is you can first think of this as splitting them into two that are exactly alike. So you have one of them just like it and then you have another a copy of it, but this copy is shifted one ahead. Because remember you can do the first four and then you have to go back and then you can do the last four. So you want another copy of it but shifted ahead. And now this is two copies of the four. But it must go back before it can go ahead. So its going to go four three four. So we have to insert a three linking the two.
DW Threes are straight. So three is safe. You don’t have to make it any more crooked. We talked about the relational thing. There’s a sort of a marriage clause in there, the three linking the two fours. It links the two?
JM Yes. There’s the one in between that has to appear in between the ones that match. So when you think of a five as being defined from the four by applying this principle of go forward, go back, go forward. Then you’re going to have two forwards. You’re going have the go forward four, then you go back, then you have to go forward again to get to the very end. So you have your two fours which are married by the three that goes back. Well, now this is where something has to flip. Because if you want to go up to the six, it’s going to have to be the pattern five four five. It’s going to have to have one of these going forward. It’s going to have a copy of that on the other end. Also going forward though. But these two have to be married by a four. But that fours got to go backward. Because it’s got to go from the end of the first five to the beginning of the second five.
DW And that four of course will look like that.
JM It’s just like this but it will be flipped. So it’s a mirror reflection.
DW The five four five. You’re saying they are arithmetic as you go up.
JM Well, those are completely arithmetic in that you’re getting one pattern from the next in an arithmetic way. But if you’re building the pseudo-arc in stages and putting one strip inside the other then after you have your first strip, maybe you can put just one, the next strip makes just one zigzag. But that strip is now so long that you can no longer put the next generation in it. You have to put in something that is several generations ahead. Because you have to make that strip so much thinner and you have to make sure that you are crooked in the whole length of the second strip. In every segment of it. So, here’s where the mistake is. Okay you have this strip, right? And say you put just one zigzag in it. You have to keep in mind that each of the three parts of that zigzag is nearly as long as the whole thing. And it’s not enough to have a zigzag in each of those three parts. Right? You can’t go one third of the way to the end and back and one third of the way and think you are finished with that part. To truly be crooked, to truly be pseudo-arc, you have a zigzag and in that zigzag, you have to go from the start almost all the way to the end, but not quite all the way and then all the way back and then all the way forward again. But in each of those runs you have to do the same thing. Until you get down to a segment that is as short as the part we originally drew here.
ED So what you have there is the first stage. You are building something on a certain scale. You want to go from one side to the other side at a certain scale. And then you want to go inside of that from the beginning to the end at a finer scale.
JM At a much finer scale. But not only is the scale finer, but also the length you have to traverse in that scale. Because the first zigzag is very much longer. And that makes the next stage you have to put inside this one, much more complex with many more layers.
Murat Tuncali Moise's picture is that right? The other pictures that I have seen. They aren’t exactly like Moise's picture. This is one of the pictures that Judy Kennedy used.
JM Moise only suggests the third stage. He draws the second stage just the way I suggested. Almost all the way to the end of the four, almost all the way back, then all the way to the end finally. But then you change scales and on the new scale… It used to be that on this scale this was a pixel or a section, this was a section, this was a section, this was a section, this was a section, and to get through these five sections you had to go almost all the way to the end, almost all the way back, almost all the way to the end. But now you decide that in this strip your sections are now going to be smaller and the strip is going to be made of lots of them. And now it is a huge number, right? Because just here there are four or five. And now you have to be crooked in each section. And so that means to get from here, from this end to this end, you have to be crooked in every little sub-collection here. And not just in this one and in this one and in this one but in this one and in this one and in this one. As they are lengthening. So that means to get from here to here you have to get from here all the way to almost there. Right there, the next to the last one. But to get to that next to the last one, you have to get from here to the next to the next to the last one and to the next to the next to the last one. And so that’s where you’re going to get all these layers and layers and layers. And that’s what he is trying to show in here. You have these layers are starting in different, that look like they are starting in different lengths. And so you have been looking at something that has to go through, well not even as many sections as there are here.
DW And then on the next page?
JM Ah, yes, that’s the pattern for nine. And that would be if you only had nine sections to get through. Nine rooms of your house to get through and it’s going almost all to the end and going all the way back. But the trouble is the first stage had maybe four but maybe the next stage, because you had to make things smaller all the way along its length, it doesn’t have nine, it may have thirty. Then you have so many more layers. Because basically if you had to get through thirty. You’d have to go zig zig zig, that’s three, now you’re down to 29. Then in each of those you have to go zig zig zig and you’re 29 and you’re 28 and you’re 29. Now you have nine layers. And most of those layers extend across 29 of your rooms. And then in each of those you have to put zig zig zig and they extend across 28 of your rooms. And so if you are looking at a central room by this point you have 27 layers going through that room. And you are only nibbling a little bit at the ends. So by the time you get through ninety. You have something like three to the eighty-eighth layers.
DW It is defined though that you do get there.
JM Yes, it is finite. Three to the eighty-eighth is still only a finite number of layers.
ED You can even say how many steps you have to take. But the number of steps is very large. Very large, very fast. So if you actually want to draw it, unless you are using a very fine pen, all you will get will be a mess. Just a smudge because of the ink running into the other layers. So you have to somehow construct it in your mind. A picture is not very helpful. There’s just too much detail there.
JM I think that very quickly the number of layers exceeds the number of atoms in the universe. Assuming the universe is finite.
DW If we get to fifty links that’s it, we’re done.
JM It’s just that no physical pseudo-arc exists. Well. Of course a physical arc doesn’t exist either. After all, any physical arc has to have… it can’t be a line which has only length. A line segment has only length it has no breadth. So even thought the patterns are simply built up from just this idea of going almost all the way forward, almost all the way back and then finally all the way forward. Following that instruction and then following the instruction that to take four or more steps forward you must always be crooked you can figure out what the pattern would have to be for ninety or for a hundred. You can get a number and you can even say it has this many layers. If you went down the middle and counted off layers. You’d have a count of how many layers you have.
DW You know what Murat and I are trying to accomplish with our Pseudo-arc for the People Project. What do you think we might find?
JM I hope that you can give someone a visual impression of the complexity of the pseudo-arc, while they can still see how regular it is. Because after all, I started this by saying that maybe the first real research paper that I read was Bing’s paper proving that the pseudo-arc was homogeneous. But what he proved was that the pseudo-arc, like the circle, has the property that given any two points on it you can move that point to this point by just moving the circle rigidly. Now the motion on the pseudo-arc isn’t rigid in the same geometric way but it has the same property of being like one to one, each point goes to one other point. It doesn’t do any folding of the object. It just somehow slides things along in a nice way. The only way for that to happen is that something has to be somehow the same at every point. And the pseudo-arc does not look the same at every point. If you look at a point at the end, that’s exactly the same as a point somewhere in the middle. Intuitively that doesn’t look right. So it’s only when you descend through all these infinitely many steps in the construction that this actually occurs.
DW As you dig in layer upon layer and you see the same stuff there. It seems like a spring in a way. You can stretch it in a way and compress it.
JM Yes, it is like a spring that you can stretch but then you can also fold over onto itself. And that’s not obvious, right, that you not only can stretch it but you can fold it over onto itself and it is still the same thing. You have to look at the complex pattern and see, well if I squinted a little, I could think of these layers as all being glued together and forming just one layer. And then maybe the pattern, instead of going like a zigzag, would go like a U. If I don’t really separate the top two layers, if I think of them as together then it really goes from here like that. So, in here, there’s something that goes… you move these two together and you put this part together and it makes a U instead of a Z. Now what price do you pay for doing that? Well you are sort of considering these two layers you’re not right now considering them as separate stages. You are thinking of these together. And you think of them as going from here and turning here and going back and making a U and so whenever you have a zig zag you also have a U. And you make your U by, the term I think in Bing’s paper I think is “consolidating the chains”, you consolidate these two layers into one. Later you’ll separate them. The next stage down they’ll be separated. But right now, you think of, say, if you had bricks lined up here. You think of these as big bricks. So you have a big brick here and another big brick and another big brick and another big brick and then you have one two three four bricks this way and five bricks this way. And your brick at this end is maybe a bit big and the rest are a normal size. So the price you pay is you made your building blocks a bit bigger. If you think of your strip as divided into sections that we were looking at a moment ago. You’ve made your sections a little bigger this way in order to put these two layers together.
ED And this is what you do when we say that this pattern contains all other patterns. So that by doing this kind of consolidation you can build any pattern that you like. And so this says that this is somehow a universal object of a certain sort. So that you may form another object by following a pattern very much like this but instead of using five steps you might use seven at the next stage. But if you keep this up in the end you will have objects that are exactly the same. So that first you think there is a great deal of rigidity in the construction, but if you do this thing of going almost all the way forward, almost all the way back and then forward again. There’s a lot of slack in there with what you can do. But still, if you keep doing that same recipe, you wind up with the same objects at the end with exactly the same properties.
References
[1] Bing, R. H. , A homogeneous indecomposable plane continuum. Duke Math. J. 15, (1948). 729--742.
[2] Moise, Edwin E. An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua. Trans. Amer. Math. Soc. 63, (1948). 581--594.
Other Related References
Bing, R. H. Concerning hereditarily indecomposable continua. Pacific J. Math. 1, (1951). 43--51.
Bing, R. H. Snake-like continua. Duke Math. J. 18, (1951). 653--663.
Bing, R. H. Each homogeneous nondegenerate chainable continuum is a pseudo-arc. Proc. Amer. Math. Soc. 10 1959 345--346.
Kennedy, Judy, How indecomposable continua arise in dynamical systems. Papers on general topology and applications (Madison, WI, 1991), 180--201, Ann. New York Acad. Sci., 704, New York Acad. Sci., New York, 1993.
Kennedy, Judy A. A brief history of indecomposable continua. Continua (Cincinnati, OH, 1994), 103--126, Lecture Notes in Pure and Appl. Math., 170, Dekker, New York, 1995.
