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'A New Kind of Science'
Wolfram Goes from Simple Rules to Complex Forms

By Rich McManus

Photos by Ernie Branson

Sometimes, buzz alone can fill Masur Auditorium. Such was at least partially the case Sept. 17 when Dr. Stephen Wolfram — famous for hatching, out of more than a decade of relative solitude, what he calls "A New Kind of Science" — visited to explain his new book of that title, his new web site and his views on a mathematically contrived model system that appears to mimic, and perhaps underlie, forms found in nature.

"So, why did you come to this talk?" asked one guest, sitting in the fourth row, to his neighbor before the lecture began. "I don't know. Why did you come?"

"Because he's a famous guy."

So, in at least one neighborhood in the hall, folks had come simply to see what all the fuss was about. In a way, the rows of guests, some of whom communicated with neighbors to the side and rear, formed a kind of pattern of association, the sum of which could be represented by a series of black boxes and open boxes. Let black boxes indicate those who knew of Wolfram's work directly and open boxes represent those who knew him but barely, or only by reputation (Wolfram, a native of London, owes at least part of his fame to having been named a professor at Cal Tech at age 21, and for having won a MacArthur Foundation "genius" award early in his career; he is now CEO of Wolfram Research, Inc.). If you went row by row, all the way to the rear of the hall, applying the simple rules of "degree of familiarity with Wolfram," then the resulting matrix would either be interesting or not, depending on whether it was A) aesthetically pleasing in its own right, say, a nice sort of quilt pattern, or B) potentially meaningful to biology, because it resembles a form found in nature.

Dr. Stephen Wolfram, CEO of Wolfram Research, Inc., brought his "New Kind of Science" to NIH on Sept. 17.

That's a simplified way of presenting Wolfram's thesis, which he illustrated with an elementary example: from a simple row of 7 boxes (meant to resemble a line of cells, though it could be of any length) — the center one black and the three on either side open — one can generate successive rows by applying easy rules governing the color of neighboring boxes. For example, one rule might be that if there's a black box in the line above, there must be a black box below it. Or, if the box above is open, but adjacent to a black box, the box below must be black. Wolfram has elucidated some 256 "rules" based on eight available options governing the color of boxes; the options themselves rely on simple if/then rules of coloration based on proximity. Each rule spawns so-called "cellular automata," or successive generations that obey the rule in each iteration (see box).

One rule that Wolfram demonstrated produces, quite reliably, a pyramid shape. But once you get up around Rule 30, fascinating forms result — highly irregular and utterly random to the naive eye. Other rules create odd structures that proliferate awhile then die out, say after 3,000 steps or so. Fascinatingly, some of Wolfram's models are dead ringers for such natural forms as the variation seen in mollusk shell pigmentation patterns, and the forms taken by snowflakes and tree leaves. What so tantalizes Wolfram is that his models, which require such simple rules to generate, can result in such rich complexity; the math seems a good metaphor for rules embedded in nature. Or, in his words, "We've put so little in, but we've gotten so much out. It seems to violate our prejudices — that incredibly simple rules can produce incredibly complex phenomena."

Wolfram said he spent most of the past decade working on a "big intellectual structure," that has resulted in his new 1,200-page book, only 59 of whose pages deal specifically with biology. "During the past 300 years, mathematics and equations have been used in a serious way in science," he said, citing particularly apt applications in physics, such as determining the orbits of planets. "But (math) hasn't worked out so well in other areas...traditional mathematics has not been well used in biology. We might not be using the right building blocks for our models or descriptions of things."

The dearth of good math-based models prompted Wolfram to spend the past 15 years building the cellular automata concept so that "mathematics can be used like a microscope pointed at various objects — various flora and fauna. After all, look at all the stuff that's happening from one black box (or cell)."

That cellular automata can mimic forms in nature is evidence of "a very robust phenomena," Wolfram said. "Something very basic and fundamental is at work."

He pointed to the sequence of prime numbers, or the digits flowing forth from calculations of pi as something science has heretofore regarded as "a nuisance, or a distraction or a bug of some sort — not an important basic phenomenon." But the apparent randomness of these numbers has at least one satisfying aspect: the more complicated things look, the more we are likely to ascribe "naturalness" to them.

"We want (model) systems whose behavior we can readily predict and see," Wolfram said, "but nature operates under no such constraint." What models of natural systems can mathematics potentially evoke, he wondered? "I happen to think all of the universe and physics, but that's another lecture."

Wolfram addresses crowd.
Wolfram was careful to acknowledge the limits of mathematical modeling. "Modeling is always a difficult business," he said. "It takes a lot of knowledge of specific characteristics and essential features." His snowflake models were successful in that they "captured the basic morphology," but to get a closer approximation of the "thing itself," one needs to consider more factors that add detail and complexity. Biology is another order of magnitude harder to model than a two-dimensional crystal, he said. Yet if one looks at morphological structures and how they form in biology, there is "lots of regularity on a microscopic scale...growth processes in biology may actually be very simply described."

The most convincing evidence of his thesis were comparisons of mollusk shell pigmentation patterns, and even shell shapes, with patterns and shapes generated by cellular automata. The shells, it could be seen, "seem to be sampling simple cellular automata rules randomly...All cases get sampled in nature," Wolfram asserted.

Leaf shapes, too, in their huge diversity, can be mimicked by cellular automata. By dissecting leaf pods, Wolfram constructed models based on their branching, size and angles, then applied cellular automata to approximate a virtual forest of recognizable leaf types.

Wolfram claims no more than insight into how nature makes its choices, and leaves further exploration of how cellular automata may benefit biology to interested biologists. But he did offer some consolation: those provoked by the power of his models are invited to visit his web site ( to tinker with a program launched there just recently — A New Kind of Science: Explorer. "It's really best to learn on one's own," he counseled.

Build Your Own Cellular Automaton

If Dr. Stephen Wolfram's "New Kind of Science" intrigues you, then the following brief primer in building cellular automata can be a place to start deeper investigation. While his example, given in his talk Sept. 17 at NIH, involved a simple 7-cell system, things needn't be that elementary.

According to Todd Rowland of

1.) The row of boxes can be of any length.

2.) The cellular automaton rules dictate how to go from one row to the next.

3.) The color of a box on the next row depends on the color of the box on the previous row and the color of its two immediate neighbors.

4.) The cellular automaton rule tells what the next color will be based on the color of three cells, those being the previous cell and its two neighbors.

5.) A cellular automaton rule needs to dictate a color in 8 possible cases, covering all variations for three ordered cells (2x2x2=8 possible neighborhoods): {0,0,0}, {0,0,1}, {0,1,0}, {0,1,1}, {1,0,0}, {1,0,1}, {1,1,0}, {1,1,1}.

6.) Since there are 8 cases for a neighborhood, there are 2 to the eighth power=2x2x2x2x2x2x2x2=256 possible elementary cellular automata.

Rowland adds, "In case you were wondering: There are several ways to deal with the boxes on the edge of the rows, each of which is missing one of its neighbors. In most of the pictures shown (in Wolfram's lecture), the idea is to consider the missing neighbor as a white cell. Alternatively, one can make the row wrap around, like on a cylinder, but making the ends neighbors. This is more of a technical detail since the sizes of the pictures were chosen so that the interesting behavior would not depend on the choice of convention.

"This notion can be generalized," he continued, "by adding more colors, or using larger neighborhoods, in which case it would be called a one dimensional cellular automaton." For more basic information, visit

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