by Rucker, Rudy
What makes Mathematica even more useful is that everything you enter in a given session becomes integrated into a single document, called a notebook. A Mathematica notebook can include text, graphics, and mathematical expressions. You can save it, and if you open it again, all of the formulae are “live”—you can highlight a formula, change some of its numbers or symbols, and see the related parts of the whole notebook change accordingly, just like a spreadsheet. A Mathematica notebook fully embodies a once-futuristic concept that the physicist Richard Feynman longingly called “Magic Paper”—an intelligent writing medium, in which you can ask the paper to do your calculations for you.
Thanks to Mathematica’s notebook feature, you can watch what happens if the numbers in an equation change, or try out wild and crazy problems that ordinarily would be way too difficult to solve. Problem solving becomes a dynamic, experimental process.
The first time I saw Mathematica—this was Version 1, nine years ago—I used it to draw the kind of three-dimensional “Lissajous curves” you get if you had an object oscillating at different rates in each of three mutually perpendicular directions.
A 3-D Lissajous curve.
I’d seen two-dimensional versions in science museums and as drawing toys—a pencil or perhaps a slowly leaking container of sand hangs from a pendulum which is linked to a second, perpendicular pendulum. I’d always wondered what a three-dimensional Lissajous would look like. With Mathematica it was surprisingly easy. I typed a few lines of code and saw them.
A “baseball stitch” curve.
Before long, I’d exhausted the novelty of 3-D Lissajous curves, so then I imagined a new kind of curve I called a kappa-tau curve. These curves are defined in terms of their curvature (kappa) and their tendency to twist like a helix (tau). To my mathematical satisfaction, I soon got wonderful gnarly curves, some of them looking like the stitching seam on a baseball, as shown above. Yaaar!
But when I started wanting to look at lots and lots of my kappa-tau curves, and to set them to rotating in space, Mathematica became too slow. The very fact that it is a general-purpose system means that it is not going to be able to run some specific calculation over and over at the best speed. As I discuss in my essay, “How Flies Fly,” I ended up writing a stand-alone Windows program to show my kappa-tau curves. But I never would have gotten around to investigating these curves if I’d had to do it from scratch. You can find a my program and a Mathematica notebook for these curves on my website.
Mathematica makes research easy—well, easier. That’s one reason why it has sold a million copies to labs and offices around the world, at prices now around $1,000 a copy retail (but much cheaper for students).
The Mathematica software is the product of a company founded by Stephen Wolfram. Wolfram is a remarkable figure who helped invent the modern concept of complexity theory. Born in 1959, he got his Ph.D. in physics from Caltech at age 20 and won a MacArthur genius grant at the record-breaking age of 22. The first release of Mathematica came out in 1988, the second in 1991, and now, in 1996, Wolfram is out and about promoting Version 3.
What took so long? Wolfram offers two reasons. The first is what you’d expect to hear from any earnest software pitchman: Version 3 is so much better than Version 2 that developing it took a long time. The second reason is more intriguing. From 1991 to 1995, Wolfram was busy doing basic science research, concentrating on his book-in-progress, a monumental tome that may finally come out in 1999.
A secretive man, Wolfram is reluctant to give out details on his work, but asserts that, “I have in my sights a way to get a new fundamental theory for physics. My ideas are based on some insights into what it is that simple computer programs typically do.” Intriguingly, he says that he wouldn’t have gotten this far if he hadn’t had Mathematica to help him. Wolfram has enough intellectual credibility that one is half-tempted to think of Isaac Newton, who invented calculus and then used his new tool to unravel the secrets of celestial mechanics. It would be nice.
When pressed for more information, Wolfram says something like the following: “For the last three hundred years, people have been trying to use mathematics to model the natural world, but that this doesn’t work well for things like biology and complex systems. Equations are human constructs, but maybe nature follows more general rules than that. Maybe we have to go beyond human mathematics and look at how general computing systems work.”
Wolfram feels that there really is a simple fundamental theory and that we’ve been looking for it in the wrong way. He thinks that for the first time in about 50 years, somebody has a real chance to find it. Who? “I’m an ambitious guy, all right? My interest is to find the fundamental theory of the universe.”
The ultimate prize would be a simple computer program type thing that is the universe—not a model of the universe but the thing itself. One of the famous, if minor, successes of the cellular automata programs which Wolfram studied in the 1980s is that they’re good for modeling the intricate patterns which appear on the South Pacific seashells known as cone shells. But these cellular automata models are just models. Wolfram says, “If you can get a truly fundamental theory it’s not a model anymore, it’s the thing itself. It’s it.”
Talking about how useful Mathematica is to him, Wolfram says, “I can do my science because I use Mathematica so much. I can start typing in complicated things and they work. I can experiment. Programs are always smarter than you are. You have an idea for some kind of computation and you think this will never do anything interesting, but then you experiment and some version of it is interesting. Experiment is necessary. You have to look at lots of things. If it takes a minute to experiment, you’ll do it. If it takes an hour you won’t.”
Comparing his work on Mathematica to his scientific research, Wolfram says that his skill lies in finding the essence. “When I do science, I’m asking what is the essence of what goes on in nature. In designing Mathematica I ask what is the common essence of what people do when they do math. Every person just has one real skill. My skill is finding the essence of mechanisms of things.”
But would anyone outside a university ever need Magic Paper? One out of twelve copies of Mathematica are sold in Wall Street, where the software is used to build trading systems. And there are many engineering uses. When researching this review, I looked at a page of the Mathematica Web site with a lot of information about applications such as skateboards, shampoo, playground equipment, plastic surgery, and bicycle racetrack design.
Dale Hughes and the engineering consultant Chris Nadovich used Mathematica to design the bicycle velodrome track used in the 1996 Olympics. “Our design was in steel, which made the accuracy issue more demanding,” says Nadovich. “In wood you can always cut things to fit as you nail the boards together. Here the whole thing was manufactured in another place, shipped to Atlanta, and yes, everything fit. Well, it was off a quarter of an inch after a quarter-kilometer because the thickness of the paint wasn’t accounted for. The way they used to do bridges was to have two hundred guys calculating every piece. I just had me and a desktop PC and a couple of weeks. I couldn’t use AutoCAD [the popular computer-aided design program] because you don’t do 20,000 different parts by sitting there clicking and dragging. Mathematica has a tremendous amount of symbolic math capability. Initially I didn’t even know how many sections the track would have. I just solved the problem symbolically. Mathematica produced coordinate lists that I could load into AutoCAD to render the blueprints.”
Nadovich goes on to describe one of the formulae involved in the design. “We used a special curve called the Cornu spiral for the turns. It dates back to the 1800s when people were designing railroads for what they considered high speed. You can’t just go from a straight-away with zero curvature into a circular arc with a non-zero curvature. What you need is to have the curvature increase linearly as you go into the turn and decrease linearly as you come out. The centrifugal force is a linear function of the curvature, so if you change curvature
in a linear way it make sit easier for the riders to hold their position. They can concentrate on racing and not on steering.”
How did the riders like it? “They hated it! They didn’t like how the track felt. They said it was slow, it was bumpy, it was an ugly color, and they didn’t like the texture. Everybody was afraid of it. It was very depressing. But the riders that won did like it. In fact the track really was fast. They set two world records and twenty-one Olympic records on it. It felt different because it was a different material, most tracks are wood or concrete, just banked concrete roads.”
The most devoted user of a computer algebra program I know is Bill Gosper, an old-time hacker who was involved with the original Macsyma project. In fact he still uses Macsyma; not the modernized retail Windows version, but a massively customized, or one might say Gosperized, version of the program that lives on a file-cabinet-sized old Symbolics computer in his basement.
Going to visit Gosper has been a touchstone experience for me ever since moving to Silicon Valley eleven years ago. I find him grayer than before, sitting in a dusty, autumnal room of antique beige plastic artifacts. An ellipsoidal electric pencil sharpener. A stack of Symbolics computer monitors. Danish Modern chairs.
Gosper uses Macsyma to find weird algebraic equations. In high school algebra many of us may learned a few simple algebraic identities like
(x + y)2 = x2 + 2xy + y2.
Gosper interests himself in identities like this, only much gnarlier. The all-time champion of gnarly identities was the legendary Indian mathematician Sri Ramanujan. Gosper, who speaks in arcane hacker language, modestly rates the gnarliness of his own equations in milliRamanujans, and he gets his machine to show me one of his best, which he rates at a full 800 milliRamanunjans.
The left side of Gosper’s gnarly identity is a product of terms involving the geometrical constant pi divided by the trigonometric arctangent function. The right side of the identity is the pith root of 4. Not the cube root, mind you, the pith root. “Genius does what it must,” says Gosper of Ramanujan, “Talent does what it can. When I’m doing this stuff, I find something surprising and then try to make more surprising. I go for sensationalism.”
He begins rapidly keyboarding so as to show me more wonders, talking all the while. He is an artist, a symbolic acrobat without a care in the world for real world applications.
“The computer algebra field supports itself on one percent of what it can do. A key thing about computer algebra is that you have infinite precision. No roundoff. I’ll invert the same matrix twice and show how limited precision can screw it. Let’s set mumble to mumble.” He uses “mumble” as an ordinary word, as shorthand for expressions too complicated or dull to actually say. “Now we invert it. Oh my God, how long is this going to take. Twenty seconds, thirty seconds, whew. I’m worried if I’ve even got a patch in here to make this feasible. We can check while this is running. Says here it should work. This is a little discouraging. Christ on a crutch. Ah here it is, it’s done. Now we’ll set mumble and do it again. Ooh! It’s not converging. What the hell’s going on? 572??!!! It’s supposed to be 570! God help us. No, it’s still batshit. No no no it is 572. Oh, this should be a Taylor series, right? I have to stun it, I have to neutralize it. Now we can crank up the value. Now this is the right answer.” Gosper pauses and gives me a sly smile. “Now let’s see if I can earn my nerd merit badge.” “How?” “By typing in this number, which is the nineteen digits of two to the sixty-fourth power.”
Why does Gosper still use Macsyma instead of switching to Mathematica? That’s a little like asking why Steve Jobs doesn’t use Microsoft Windows. Gosper wrote a lot of the Macsyma code and it’s what he’s used to. He knows it inside out. If he were to switch to Mathematica, he’d be at the mercy of design decisions made by Wolfram’s team, and not all of these decisions are things Gosper would agree with. If you ask around in the symbolic mathematics community, you can find a real diversity of opinions about the best way to proceed.
Seeking further enlightenment, I talk to Bruce Smith, a programmer who’s done some third-party work for Wolfram Research. He talks about plans to extend the language you use to enter problems into the Mathematica. The dream would be to evolve a mathematical language in which one can readily write down expressions for everyday phenomena—including not only physics, but also biology and sociology. “Having an expression for something is a powerful concept that will become more popular in programming,” says Smith. “It’s not a coincidence that the word ‘expression’ is related to expressiveness.” Smith recalls a comment he heard in a 1970s talk by John Walker, one of the original developers of AutoCAD. Speaking about the future of computer-aided design, Walker said, “In the future, every manufactured object in the world will be modeled in a computer.” Smith feels that, in thinking about the future of the Mathematica language, we might extend this: “For every object we think about, we will want an expression for it so that a computer can think about it with us.”
Might there some day be a futuristic super programming language with expressions like Live, Think, Truth and Beauty? Realistically, it seems doubtful that anyone will ever take a comfortable human-scale problem like whom to date or where to go to lunch and say “let us calculate.” As G. K. Chesterton once put it, “Man knows that there are in the soul tints more bewildering, more numberless, and more nameless than the colors of an autumn forest.”
But within the domain of readily scientifically quantifiable problems, symbolic mathematics works great and is well worth the trouble of dealing with computers. The entire range of what is considered to be a reasonable solution to a problem is something that will expand greatly. Instead of just putting a vague bend where your track goes into a turn, you put in a Cornu spiral. Instead of guessing how a plastic-surgery patient’s appearance is going to change, you simulate the tectonics of the facial landscape. Instead of making dozens of cardboard models of your jungle-gym, you build it from equations in Virtual Reality.
How will all this affect the future of mathematics? The important thing about mathematics is that it acts as a concise, almost hieroglyphic, language for describing forms. The Mathematica program is an immense help for the tedious rote-work involved manipulating math’s hieroglyphs and converting them into visual images. Mathematica makes math more valuable than ever—for it takes a well-trained mathematical mind to know which kind of “hieroglyph” to use to model some particular situation. And it takes a mathematical genius to come up with a really good new hieroglyph.
New mathematics is developed as a result of a feedback loop involving theory and experiment. The great thing about programs like Mathematica is how much they accelerate the process. Thanks to computers, mathematics is at the dawn of a new golden age.
* * *
Note on “Mathematica: A New Golden Age of Calculation”
Written Fall, 1997.
Appeared in Seek!, 1999
I started working with Mathematica in 1988, when I got a job as a software engineer at the software company Autodesk in Sausalito. Mathematica was new then, and my employers gave me a nice big Mac computer with Mathematica on it. The two things I worked on were kappa-tau curves and three-dimensional Mandelbrot sets—which evolved into the Mandelbulb of the 2000s.
Over the years I’d gotten to know Wolfram, and it was easy for me to interview him. For his part, he was eager to promote his latest version of Mathematica. This article was supposed to appear in Wired magazine, but they chose not to run it.
The next essay details my work on the kappa-tau curves.
How Flies Fly: Kappa Tau Curves
It’s interesting to watch flies buzz around. They trace out curves in space that are marvelously three-dimensional. Birds fly along space curves too, but their airy swoops are not nearly so bent and twisted as are the paths of flies.
Is there a mathematical language for talking about the shapes of curves in space? Sure there is. Math is the science of form, and mathematicians are always studying n
ature for new forms to talk about.
Historically, space curves were first discussed by the mathematician Alexis-Claude Clairaut in a paper called “Recherche sur les Courbes a Double Courbure,” published in 1731 when Clairaut was eighteen. Clairaut is said to have been an attractive, engaging man; he was a popular figure in eighteenth-century Paris society.
In speaking of “double curvature,” Clairaut meant that a path through three-dimensional space can warp itself in two independent ways; he thought of a curve in terms of its shadow projections onto, say, the floor and a wall. In discussing the bending of the planar, “shadow” curves, Clairaut drew on then recent work by the incomparable Isaac Newton.
Newton’s mathematical curvature measures a curve’s tendency to bend away from being a straight line. The more the curve bends, the greater is the absolute value of its curvature. From the viewpoint of a point moving along the curve, the curvature is said to be positive when the curve bends to the left, and negative when the curve bends to the right. The size of the curvature is determined by the principle that a circle of radius R is defined to have a curvature of 1/R. The smaller the radius, the greater the curvature. The figure below shows some examples of circular arcs, with each arc drawn to be the same length.
Curvature along circular arcs in the plane.
We often represent a curve in the plane by an equation involving x and y coordinates. Most calculus students remember a brief, nasty encounter with Newton’s formula for the curvature of a curve; the formula uses fractional powers and the first and second derivatives of y with respect to x. Fortunately, there is no necessity for us to trundle out this cruel, ancient idol. Instead we think of curvature as a primitive notion and express the curve in a more natural way.