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understanding of non-linearity, the Fermi-Pasta–Ulam theorem. Looking for problems that could be computed on the new MANIAC computer at Los Alamos, the scientists tried a dynamical system that was simply a vibrating string—a simple model “having, in addition, a physically correct small non-linear term.” They found patterns coalescing into an unexpected periodicity. As Ulam recounts it: “The results were entirely different qualitatively from what even Fermi, with his great knowledge of wave motions, had expected…. To our surprise the string started playing a game of musical chairs, …” Fermi considered the results unimportant, and they were not widely published, but a few mathematicians and physicists followed them up, and they became a particular part of the local lore at Los Alamos. Adventures, pp. 226–28.

“NON ELEPHANT ANIMALS” quoted in “Experimental Mathematics,” p. 374.

“THE FIRST MESSAGE” Yorke.

YORKE’S PAPER Written with his student Tien-Yien Li. “Period Three Implies Chaos,” American Mathematical Monthly 82 (1975), pp. 985–92.

MAY CAME TO BIOLOGY May.

“WHAT THE CHRIST” May; it was this seemingly unanswerable question that drove him from analytic methods to numerical experimentation, meant to provide intuition, at least.

STARTLING THOUGH IT WAS Yorke.

A. N. SARKOVSKII “Coexistence of Cycles of a Continuous Map of a Line into Itself,” Ukrainian Mathematics Journal 16 (1964), p. 61.

SOVIET MATHEMATICIANS AND PHYSICISTS Sinai, personal communication, 8 December 1986.

SOME WESTERN CHAOS EXPERTS e.g., Feigenbaum, Cvitanović.

TO SEE DEEPER Hoppensteadt, May.

THE FEELING OF ASTONISHMENT Hoppensteadt.

WITHIN ECOLOGY May.

NEW YORK CITY MEASLES William M. Schaffer and Mark Kot, “Nearly One-dimensional Dynamics in an Epidemic,” Journal of Theoretical Biology 112 (1985), pp. 403–27; Schaffer, “Stretching and Folding in Lynx Fur Returns: Evidence for a Strange Attractor in Nature,” The American Naturalist 124 (1984), pp. 798–820.

THE WORLD WOULD BE “Simple Mathematical Models,” p. 467.

“THE MATHEMATICAL INTUITION” Ibid.

A GEOMETRY OF NATURE

A PICTURE OF REALITY Mandelbrot, Gomory, Voss, Barnsley, Richter, Mumford, Hubbard, Shlesinger. The Benoit Mandelbrot bible is The Fractal Geometry of Nature (New York: Freeman, 1977). An interview by Anthony Barcellos appears in Mathematical People, ed. Donald J. Albers and G. L. Alexanderson (Boston: Birkhäuser, 1985). Two essays by Mandelbrot that are less well known and extremely interesting are “On Fractal Geometry and a Few of the Mathematical Questions It Has Raised,” Proceedings of the Inter national Congress of Mathematicians, 16–14 August 1983, Warsaw, pp. 1661–75; and “Towards a Second Stage of Indeterminism in Science,” preprint, IBM Thomas J. Watson Research Center, Yorktown Heights, New York. Review articles on applications of fractals have grown too common to list, but two useful examples are Leonard M. Sander, “Fractal Growth Processes,” Nature 322 (1986), pp. 789–93; Richard Voss, “Random Fractal Forgeries: From Mountains to Music,” in Science and Uncertainty, ed. Sara Nash (London: IBM United Kingdom, 1985).

CHARTED ON THE OLDER MAN’S BLACKBOARD Houthakker, Mandelbrot.

WASSILY LEONTIEF Quoted in Fractal Geometry, p. 423.

INTRODUCED FOR A LECTURE Woods Hole Oceanographic Institute, August 1985.

BORN IN WARSAW Mandelbrot.

BOURBAKI Mandelbrot, Richter. Little has been written about Bourbaki even now; one playful introduction is Paul R. Halmos, “Nicholas Bourbaki,” Scientific American 196 (1957), pp. 88–89.

MATHEMATICS SHOULD BE SOMETHING Smale.

THE FIELD DEVELOPS Peitgen.

PIONEER-BY–NECESSITY “Second Stage,” p. 5.

THIS HIGHLY ABSTRACT Mandelbrot; Fractal Geometry, p. 74; J. M. Berger and Benoit Mandelbrot, “A New Model for the Clustering of Errors on Telephone Circuits,” IBM Journal of Research and Development 7 (1963), pp. 224–36.

THE JOSEPH EFFECT Fractal Geometry, p. 248.

CLOUDS ARE NOT SPHERES Ibid., p. 1, for example.

WONDERING ABOUT COASTLINES Ibid., p. 27.

THE PROCESS OF ABSTRACTION Ibid., p. 17.

“THE NOTION” Ibid., p. 18.

ONE WINTRY AFTERNOON