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 FORMS, CREATED 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.

                              “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 material, and if there’s something there, you’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.