re: Zippergons, High Fashion, and Really Big Numbers.

(In the interview, MM comes before questions asked by Math Munch and RES comes before Rich’s responses. Other names that precede questions belong to the Math Munch readers who submitted them.)

Andres: Why did you want to become a mathematician?

RES: First, I have loved shapes and numbers ever since I was about 3 years old. But, when I was a kid, I didn’t know that I COULD be a mathematician. Math seemed more like a game that I loved but would eventually have to give up when I had to choose a career.

By the time I got to high school, I realized that I could become a scientist and keep playing games. I was very good at chemistry, so for a while I thought that I would become a chemist. In hindsight, I can see that what I liked about chemistry were the mathematical parts of it.

Once I got to college, I realized that I was better at math than at any other subject, and also I could see mathematicians around me – my professors. I found math a fascinating and endless subject, nothing at all like what I had been learning in high school.

I loved the logical precision of math, and also the creative side. I enjoyed talking math with my friends, and we would challenge each other with all kinds of interesting puzzles and problems. Also, I liked how I kept making progress in math, learning new and powerful techniques all the time. By the time I got to the end of college I knew for sure that I wanted to be a mathematician.

MM: Is there a memory of your advisor Bill Thurston that you could share with us?

RES: The strongest memory I have of Bill was when, during my second year of graduate school at Princeton, I asked him to be my thesis advisor. You have to understand that the Princeton math department is a very intimidating place, and Bill was (from afar) one of the most intimidating people there.

When I finally met with Bill, he told me that he liked to work on simple things, from scratch. He said that he liked to work on the outskirts of subjects, on topics which other people didn’t really consider much. He was very modest and straightforward, and that was a big relief. I really liked his approach. It was very encouraging to find a great match in an advisor.

Most of my memories of Bill come from when I would meet him in his office during graduate school. I liked to work on all kinds of crazy topics, most of which didn’t work out. For the first five or ten minutes of every meeting, I would know more than Bill about whatever I was thinking about. It was fun being on something like an equal footing with him for a brief time. But then he would quickly get a bead on what I was thinking about, and after another five minutes he was completely on top of things. Then, the rest of the meeting was spent with him making numerous and half-comprehensible suggestions about whatever I was thinking about.

Bill had a mystical quality to him, because he often saw a clear mathematical landscape where most other people just saw fog. So, often he would describe things that he would see to people who didn’t see them at all. He would make cryptic statements like “this is like an infinite jungle gym”, or “this is like bubbles in a froth”, and then you’d go home for 6 months and think about it and eventually you would see that he was exactly right.

Liam: How long have you been a math professor and when did you take interest in writing about big numbers?

RES: Well, it depends what you mean by “professor”. I got my PhD in 1991, when I was 24 years old. But then I had about 4 years when I was a “postdoc”, which is sort of halfway between a graduate student and a professor. Then in 1996 I was an assistant professor for 1 year, and then in 1997 I got to be a tenured professor. So, I guess I’ve been a full-fledged professor for 17 years.

I think that I got an interest in writing about big numbers when I was in 11th grade, listening to my chemistry teacher explain how big Avogadro’s number is. (That is roughly the number of air molecules in a balloon full of air.) I didn’t really do much drawing until about 20 years ago, and then I didn’t actually start on the numbers book until about 3 years ago. So, really, 3 years ago was when I really started working on the book.

Shelby: What was the inspiration for writing your book?

RES: One inspiration was what I already said in answer to the last question: my chemistry teacher in high school talking about Avogadro’s number. Another inspiration came from a game I used to play with a friend in college. We would give ourselves one minute to write down the biggest number we could, and then we would try to figure out which of us had written a bigger number.

Audrey: How did you find out that all of these huge numbers exist?

RES: Since I was pretty young, I had learned that the numbers go on forever. Now that I am a professor, I can tell you that it is really just a human assumption (or axiom) that the numbers really do go on forever. The assumption is sometimes called the Axiom of Infinity. It is one of the axioms of what is called Zermelo-Fraenkel set theory. Zermelo-Fraenkel set theory is sort of a list of all the things that you need to assume in order to really do mathematics from the ground up, and one of the assumptions on the list is that the (counting) numbers go on forever.

Even though the numbers go on forever (at least if you believe the axioms in Zermelo-Fraenkel set theory) the names for them run out pretty quickly. So, I was completely sure that the numbers existed (because I believe in the Z.F. axioms) but the challenge was to come up with good names for the ones which didn’t already have them.

Alex: Did you count to try to reach those numbers?

RES: Yes and no. In a way I did count but I had to skip a lot of the numbers to reach the really big ones described in the book. If you just count 1,2,3,4,… you won’t really get very far. One of the things I wanted to tell people in the book is that if you want to reach really big numbers you have to devise good ways of counting through them, and that means skipping some.

MM: Your artwork has a distinctive visual style. Can you tell us about your background as an artist?

RES: When I was a kid, I thought that art was all about drawing pictures of scenery, like the ocean and the clouds. I was terrible at doing that, so I figured that I could never be an artist.

When I became an adult, I drew some comic books in which the pictures were like stick figures. I didn’t think that they were very good but my wife liked them a lot and she encouraged me to continue drawing. I still like to draw simple cartoons by hand but usually I just put them up on my refrigerator for my family to look at.

As a mathematician, I often use the drawing program called xfig to illustrate my articles. xfig lets you draw simple shapes like circles and triangles. After a while, I started fooling around with the program and using it to make cartoons. Eventually I got pretty good at xfig and started making some comic books using the program. Eventually I used xfig for my first published kid’s book, “You Can Count on Monsters“.

A few years ago, I switched from xfig to Inkscape, which is another drawing program. Inkscape is a much more sophisticated program, and I enjoyed being able to make more versatile and refined pictures. My artwork hasn’t changed too much since I switched, but I think that it is a bit better now. I used Inkscape to illustrate the Big Numbers book.

So, the short answer is that I am self-taught as an artist, and to some extent the drawing program I use dictates the style.

Audrey: How did you create your self-portrait?

RES: I took a picture of myself using the computer’s camera and then I loaded the picture into xfig. Then I traced over it using the drawing program. For part of the picture, like my glasses, I zoomed way into the picture and tried to approximate the patches of color I saw on the screen with colored polygons. When I zoomed back out, the picture looked pretty realistic, much to my surprise. After I had drawn myself, I used the program to draw in a monster and an alien-looking background.

Ronald: To achieve an understanding of large numbers, do you find that it helps to have an automatic recall of math facts and place value?

RES: That’s an interesting question. I would say that having an automatic recall of math facts is useful for understanding any kind of math, in the same way that being a good speller helps you write more quickly, or knowing how to work the shower and the sink in your bathroom helps you get ready in the morning. If you can recall something automatically, it means that you don’t have to pause to figure it out and that lets you work more quickly.

More specifically, if you can quickly recall addition, multiplication, and other math operations, you will better appreciate big numbers because you can see how they build up on themselves. It helps to have a good “arithmetic sense”, like why 10 times 10 is much smaller than 10 to the 10th power. I’m not sure if “place value” really comes into this. It is more a question of using the basic number operations in revealing ways.

Roark: What is the biggest number that can be named (other than infinity)?

RES: There is no biggest number (other than infinity) that can be named. As soon as you name one, you can always make up a new name which describes what happens when you add one to the number — unless, of course, you think of the name just an instant before you die. Then you won’t have time to think of the next name.

You might ask “What is the largest number that HAS been named?” I can’t answer that question.

Incidentally, there is more than one size of infinity. The smallest size of infinity is usually called “Aleph nought”. Aleph is the first letter in the Hebrew alphabet. The next smallest size of infinity is called “Aleph one”, and then the next is called “Aleph two”, and so on. There are many subtle questions about these different sizes of infinity.

Dawnae: I would really enjoy reading that book. What is the biggest number you know?

RES: I guess that the last number I wrote in my book is the largest one that I have thought about for more than a few seconds, but if I really think about any number I immediately think about a larger one. So, I can’t give you an answer to this.

Max: How many numbers are in a googolplex?

RES: That’s kind of a funny question. You might say that a googolplex is just a single number. Another answer is that there are a googolplex numbers in a googolplex. Let me explain…

First of all, you have to know something about sets. A set is just another name for a thing. Sets are made of other sets, and the things constituting a set are called the members of a set. That’s how Zermelo-Fraenkel set theory starts out.

One of the axioms says that “nothing exists”. The way it is phrased is that there is a set which has no member. It is called the empty set, and it is often written like this: {} (Two brackets with nothing in it.) The number 0 is often considered a name for the empty set.

Once you know that the empty set exists, another axiom says that there is a set which just has the empty as a member. It is written like this {0}. This set is often called 1.

Once you know that 0 and 1 exist, the axioms tell you that there is yet another set which has them as members. This set is written as {0,1}. The set {0,1} is often called 2. So, in summary

0 is another name for the empty set
1 is another name for {0}
2 is another name for {0,1}
3 is another name for {0,1,2},
and so on.

Each number, when considered a set, has exactly that number of members. So, a googolplex has a googolplex numbers in it.

MM: How has your relationship with math changed since you’ve had kids?

RES: My math research hasn’t changed that much, except that sometimes I would rather do things with my kids (or wife) than work on research. So, I am less single-minded about math than I was when I didn’t have kids. (But I am still pretty single-minded when I am really working on a project.)

I’ve probably done my best math research since having kids, but that might just be because I’ve had kids for most of my career as a mathematician.

My interest in elementary math is much higher since I’ve had kids, because I can see them learning things in school and I want to help out and explain things to them. I have to say, though, that usually they are so sick of their homework that they don’t want to hear much more math from me. It sometimes bothers me that the schools do not seem to teach them much that is interesting in math, but they keep them so busy that they don’t really want to learn anything interesting outside of school.

One reason I wrote “Really Big Numbers” is that I want to give kids something in math to think about besides their school work.

Shelby: What was it like creating a game? Was it hard and was it fun?

RES: The one game I’ve really made is called “Lucy and Lily“. I was just fooling around and then eventually I noticed that something kind of magical happened when you set the game up a certain way. So, the discovery was really fun, and then it was exciting to make the game work.

I’ve also enjoyed all the comments I’ve had about the game over the years. The game is not popular like Pac-Man or Tetris, but still a lot of math people have heard of it and I enjoy getting comments about it. The game has been the subject of a number of student projects (usually college students) and I enjoy hearing about that.

The game wasn’t hard to make. I’m a pretty experienced computer programmer, and so it wasn’t that much work for me to make it.

Genesis: What are you making next?

RES: After I wrote the Big Numbers book, I wrote another picture book. (I made the latest one this summer.) The new picture book is about alien life and alien ways of thinking. It is called “What are Aliens Like?” I’m not sure if this book will ever get published, but I really enjoyed making it.

Right now I am thinking about doing some computer experiments on something called Langton’s ant. Langton’s ant is a “mathematical ant” that moves around on an infinite checkerboard according to certain rules, and there are some famous conjectures about how it behaves.

I would (of course) like to solve these conjectures, and I have been trying to imagine what the ant might be “thinking about” as it moves around. I’ve got the idea that maybe I can combine some ideas from what is called “boolean algebra” with some ideas from what is called “integrable systems” to make sense of the question, “What is the ant thinking”? So, I might try to make a computerized ant farm next.

Julia: Can you try to stump us with a number challenge?

RES: OK. Suppose you have 3-cent coins and 5-cent coins. What is the most amount of money you can’t make exactly? This one is easy. you can’t make 7 cents, but you can make 8=3+5 and 9=3+3+3 and 10=5+5 and 11=3+3+5 … So, the answer is 7. Now suppose that you have 13-cent coins and 37-cent coins. What is the most amount of money you can’t make exactly? This one is hard, but not too hard. It is known as the Frobenius coin exchange problem.

Here’s a much harder one. Suppose that you do the following thing when you have a number. If the number is even, divide by 2. If it is odd, multiply by 3 and add 1. What happens when you do this over and over. Let’s try with 3:

3 –> 10 –> 5 –> 16 –> 8 –> 4 –> 2 –> 1 –> 4 –> 2 –> 1, etc.

Once you get down to 1, you get a cycle that goes 1,4,2,1,4,2,… Here’s the question. Is it true that, no matter which number you start with (like a googolplex), the number eventually gets down to 1? Nobody knows! This is a famous unsolved problem. Sometimes it is called “The 3N+1 problem” and sometimes it is called “The Collatz Conjecture“. You can have fun thinking about this one, but most people think that it is practically impossible to solve.

MM: It seems like using computer programs plays a big role in the way that you do math. Can you say something about your history with using computers to do math and how computers and paper-and-pencil math fit together in your workflow?

RES: I’m a self-taught computer programmer. I first learned how to program in Basic when I was in high school. The most interesting thing I did back then was write a computer program which helps the user solve puzzles like Rubik’s Cube and similar kinds of things.

I didn’t do too much programming in college, but then in graduate school I went back to it with a vengeance. At some point I taught myself how to program in C, and also I used Mathematica a lot. (It doesn’t take very long at all to learn how to use Mathematica.) Sometime after I got my PhD I learned Tcl, which is a language with good graphics capabilities. (Tcl is very much like Python.) For many years I wrote programs using a combination of C and Tcl.

Around 10 years ago, I switched from C/Tcl to Java. Java is almost as fast as C, and it has great graphics rolled into it. Also, it is known as an object oriented language, and these kinds of languages are very appealing to mathematicians. So, now I write most of my programs in Java, though occasionally I use Mathematica when I want to do something quickly.

My usual approach is to pick some problem that people don’t know much about and then devise some computer experiments to try to understand what is going on. In good cases, the experiments turn up something useful, and suggest more interesting and refined experiments. In great cases, this process continues, creating a feedback loop which gives me more and more knowledge of the problem as I make better and better experiments.

I never really think about solving a particular problem. Rather, I just think about getting a clearer picture of what is going on. If this feedback keeps going long enough, I get a clear picture and then the solution to the problem just sort of emerges. This doesn’t happen that often, but when it does, it is really exciting.

After I discover something on the computer I have the (mostly) unhappy task of converting it into a traditional math paper. Sometimes things clarify further as I try to express them in conventional language, but sometimes I get frustrated that I have to express ideas in a traditional style. I feel as if I was told that I had to draw all my pictures with crayons.

Many mathematicians won’t use a computer program, and many are skeptical that a rigorous mathematical proof could reside in a computer program. This is partly a cultural issue and partly a scientific one. The problem with computer programs is that they seem to rely on the particular computer that runs them, and they tend to become unusable after a few decades (or sooner). So, at least currently, written papers are much more durable. I keep hoping that this will change, but meanwhile I keep writing papers because I want my work to be accepted in the conventional way as valid mathematics.

MM: What are your most favorite and least favorite parts of math?

RES:

most favorite: discovering something new!
giving lectures on new things I have discovered.
making an interesting computer program
talking with my friends about math
teaching students who are really interested in math

least favorite: refereeing papers (to decide if they should be published.)
organizing math conferences
giving out grades to students
participating in national scientific panels
reading college admission or job applications

I’d say that generally I like the creative side of mathematics, but not the parts that have to do with administration or evaluation.

MM: Many of our Math Munch readers are young people. Are there any closing thoughts you’d like to share with them?

RES: One thing that my wife and I often talk about — she’s a very talented fiber artist — is that you shouldn’t discount the things you do well because they seem easy. Often you might have a special talent for something but it comes so easily to you that you think it is just plain stupid. But then it turns out that other people find it quite difficult. So, you should always take seriously what you do well and easily. At the same time, some things (like math) take a ton of work even if you have a talent for it. So, you might also want to take seriously things which don’t come so easily.