Robert Brooks: The Wizard of Shape, part 3.

RW:  Dr. Brooks, could you explain or give examples of geometrical shapes derived from a sphere?

RB:  Let me give you an example of the torus, which resembles a doughnut.  This was part of a lecture I gave in my class, which dealt with the Classification of Surfaces.  This is one of the great theorems of topology.  To set up the statement of the theorem, take a sphere(see Figure 3).  Now take a torus(see Figure 4).  Then take a two-holed torus(see Figure 5).  And now down below you have a projective space(see Figure 6).  It’s not oriented, so you can’t draw it properly.  And then take the Klein bottle, which you also can’t draw(see Figure 7).

RW:  Right.  The Klein bottle is the surface that continues into itself without end, somewhat like a three-dimensional Mobius strip.

RB:  Yes.  Let’s return to the projective space(see Figure 8).  We’ll put a little handle on it(see Figure 9).  You can take other spaces, and put more handles on them.TOPO 4A

RW:  So you’re taking the basic sphere and adding holes.

RB:  Well, think of it as adding handles.  A torus is a sphere with one handle, and the two-holed torus is a sphere with two handles.  To understand what I mean by handle, just think of what’s attached to your luggage.  So, if you think of your luggage as a sphere, and attach a handle to it,  you get Figure 10a.  And then if you start shrinking the sphere down and pull the handle, you get Figure 10b.  Eventually, your shrinking and pulling will result in Figure 10c.TOPO 5A  And so the shape in Figure 11(a-c) is the same as pulling the two handles on the sphere. TOPO 6A And then you take the real projective plane(the space of all lines going through the origin.  It is obtained by identifying opposite points on a sphere.  It is also called the cross cap.) and the Klein bottle, and you put handles on them.  By taking a square, and folding the appropriate ends such that the arrows are lined up and pointing the same way, you can obtain the shapes we have been discussing:  torus, Klein bottle, sphere, and real projective plane(see Figures 12a-12d).TOPO 7A

First, I will define a surface as an object such that any sufficiently small piece of it can lie in the plane.  Second, I will assume that the surface has no boundary.  For instance, if you puncture a sphere, it goes out like a balloon, and you can imagine it continuing like an infinite plane.  The third thing that I’m going to insist on is that the surfaces are compact, which means whenever you have infinitely many points, they have to pile up.

RW:  So by compact we’re placing a limit?

RB:  Yes.  Then the question is, “What are all the possible surfaces, if you call two surfaces the same if you can obtain one from the other by deforming them through stretching?”(See Figures 13-19). TOPO 8A And the theorem states that these shapes form a complete list.  Okay, take the sphere, and add a number of handles on it.  Take the projective plane, add a number of handles on it.  Take the Klein bottle, and add a number of handles and that will form all possible surfaces.  Okay, in the terms that I specified in this theorem are all the two-dimensional shapes which don’t have any boundary and are compact.  This is it.  It’s beautiful, you know.  I don’t know if you count this as an abstract theorem.

RW:  Well, it’s a generalization, which is an abstraction of sorts; moving from a specific example to some kind of rule.

RB:  From that point it is an abstraction.  You go to the proof, and, in step one, you list all the parts that such a surface might be built out of.  There are seven parts.

RW:  It’s interesting when we look at these shapes that some of them look very familiar.  The upside-down version of the Y-shape looks like a pair of pants on a store dummy.

RB:  Actually, the technical term for this shape is “pairs of pants.”

RW:  So your point is that there’s something very concrete about all of this.  We can recognize some things as ordinary objects.  Yet, it’s also true that we are aiming for certain generalizations about these objects.  This seems to me to be somewhat abstract.

RB:  I think, in terms of the metaphor you used about looking into the soul of an object, that the soul needn’t be some abstract thing.  It could be made out of bits and pieces of these shapes.  On the one hand, it’s abstract in the sense that you’re looking into the essence of things.  On the other hand, it’s concrete;  you can sit down and play with these shapes with your children.

Robert Brooks: The Wizard Of Shape, Part 2.

“A man’s reach should exceed his grasp

or what’s a heaven for?”–Robert Browning

RW:  Perhaps you could tell us something about the origin of topology.

RB:  The founder of topology is considered to be Henri Poincare, who had a very strong philosophical outlook towards mathematics and science.  He was led to topology through differential equations, and, in particular, by the “N-Body Problem”, which depicts gravitation.  This situation raises the following questions:  What’s going to happen to these “N”(number) of bodies in the long run?  Are they going to settle into some definite pattern?  Are they going to collide with each other?  Are they going to fly off?  The prospect of trying to solve such a system of equations is just preposterous.

RW:  Is it because there are too many variables involved?

RB:  Yes.  Poincare also wrote a beautiful essay on probability in which he gave the following example:  Imagine you have some large number of molecules of gas A in a bottle, and you ask, “What’s going to happen to this gas in the bottle?”  Then, you make up some experiment in which you have two different gases in two different parts and you remove the partition.  Now some people believe(Albert Einstein:  “God does not play dice.”) if you could find the location of all the particles and work out the equations, you would know how the system evolved.  But the system is so complicated and the equations so difficult, all you can do is guess.  Poincare wanted to show that you get the right answer precisely because you don’t know all of the factors!  It’s the kind of paradox that he always seems to be looking for in mathematics.

On the one hand, you have a determinist point of view, “Well, if we can just sit down, and calculate all these things, we’ll know the right answer.”  On the other hand, Poincare wrote that just by stepping back from the situation, and relying on insight without getting distracted by all the little things, that you get a very clear picture of what’s going on.  And that’s what topology is all about.

RW:  Now in topology are we talking about the position of points, and the possible arrangement of shapes?

RB:  Well, I tend to think about topology as being certain mental habits;  certain ways of thinking about things.  Let me go to these shapes hanging on my door.TOPO 1ABoth shapes are built out of the same fundamental pieces, in this case a figure “H”.  So the H’s are the building blocks of these surfaces.TOPO 2ATOPO 3AThe blocks are put together in a fairly clever way, so that on the one hand,they’re the same, and on the other hand, they’re different.  They’re the same in that they make the same sounds, but they’re different enough to make different shapes;  one’s longer, and the other’s fatter.  So we put similar pieces together in different ways, and say. “What can happen so that they’re the same in some ways and different in other ways?”

RW:  But topology has this other characteristic that Michael Guillen points out in his book, Bridges to Infinity, that basically deals with the souls of geometric objects, those aspects of their geometry that don’t change under certain kinds of transformations.

RB:  Yes.  It goes back to the image of Poincare and the molecules of gas.  You step back from the millions and millions of little interactions that you have no control over and look into the soul of these objects.  That’s the beauty of topology.

Robert Brooks: The Wizard of Shape, part 1.

During my adult life, I became interested in discovering why forms were the way they were, the laws governing the formation of forms, the applicability of forms, and the inks between different forms.  I realized that for the purpose of my study I would have to come up with an appropriate definition of form.  I chose to define form as a perceived structure or concept represented by a definite pattern.  Applying my definition to the study of forms, I saw that forms would fit into three major categories:  NATURAL FORMSCREATED FORMS, and THEORETICALLY-DERIVED FORMS.  Natural forms pertain to those forms which exist in the physical world independent of human beings.  Cloud patterns and ocean waves are natural forms.  Created forms are those forms which arise from the human imagination.  These forms include poetry, architecture, and sculpture.  Theoretically-derived forms involve forms which arise through logic and reasoning.  Cycloids and lemniscate are examples of such forms.

Topology belongs to the last category.  It may be called the “aerobic” branch of mathematics, because it looks at the properties of shapes after they are twisted and stretched.  Topology is a qualitative form of mathematics, involving concrete shapes that ten-year-olds could play with.  Stephen Barr’s Experiments in Topology, offers numerous examples of topological fun.  One such, is twisting a two-sided strip so that it has one edge, resulting in the Mobius Strip.    For those that want to read about the creative, imaginative side of topology, I recommend Clifton Fadiman’s two collections of short stories and verse:  Fantasia Mathematica,  and The Mathematical Magpie.

When I started selecting interviewees for my book The Magicians of Form, The late Robert Brooks, then a Professor of Mathematics at the University of Southern California, was my first choice.  Dr. Brooks taught a course in topology, and had an ability to make the complex simple.  His warmth and enthusiasm put me at ease, and I found myself even more interested in the subject matter of topology.  What follows is an excerpt of an interview that took place in his office at USC.IMG_5961

                              “I think the thing that motivated me was the thought that ‘They’re holding something back from us’.”

RW:  Dr. Brooks, perhaps you could say something about your early interest in mathematics.

RB:  I think I wanted to be a mathematician since I was in the 4th or 5th grade.

RW:  Does that mean you had a natural aptitude in solving mathematical problems?

RB:  Well, I like to think I have a natural aptitude.  But let me tell you a story…  This was in the 1st grade, and we were doing primitive addition, and learning to add several digit numbers together.  Then we began to learn carrying, and it dawned on me that all the numbers we had been given to add up until that time, had been kind of “cooked up”, so you didn’t have to carry.  I was a little upset that no one had pointed that out to me; and I said to myself, “I wonder what else they’re holding back?”  And I must have spent about two weeks adding random numbers together.  Then I came to the conclusion that the only thing you had to know in adding two numbers together was carrying, and then you could any two numbers, no matter how many digits they contained.  But I felt I had to prove that.

This was the first problem I remember thinking seriously about.  I recall working on it for a long time, and I ended up giving up.

RW:  So you had the desire to go from the specific to a general rule, to an overall proof/

RB:  I think the thing that motivated me was the thought that “They’re holding something back from us”, and I wanted to be on top of what was going on.

RW:  You had a certain lack of trust in the whole procedure.

RB:  Absolutely!  And I think one thing that’s so appealing to me about mathematics is its real immediacy;  that you’re basically on your own with the materialand if there’s something thereyou’ve got to find it.

RW:  So you’re the pioneer?

RB:  You’re just about everyone in this business.  You’re the pioneer, you’re the explorer, you’re the critic.  In many cases, you’re the audience.

RW:  Then it’s really your world.  You’re immersed in this abstract universe that you’ve created.

RB:  That’s right….  But topologists have a certain disdain for abstraction.  Topologists want to show what’s there.