In essentially all cases, however, the emphasis remained on trying to find some aspect of complex behavior that could be summarized by a single number or a traditional mathematical equation. … But despite all this, no major new scientific developments were forthcoming—not least because there was a tremendous tendency to ignore the idea of simple underlying rules and of what I had discovered in cellular automata, and instead to set up computer simulations with rules far too complicated to allow them to be used in studying fundamental questions.

The correspondence between multiplication rules and additive cellular automata can be seen even more directly if one represents all states by integers and computes h in terms of base k digits.

In a sense, however, the presence of such correlations is just a reflection of the idealized way in which the vacuum state is set up—with each field mode determined all at once for the whole system.

He argued that if the time evolution of a single state were to visit all other states in the ensemble—the so-called ergodic hypothesis—then averaged over a sufficiently long time a single state would behave in a way that was typical of the ensemble.

(All so-called Hurwitz numbers have continued fractions that consist of interleaved polynomial sequences—a property left unchanged by x  (a x + b)/(c x + d) .)

In all I recall nearly three hundred people who have helped me in these kinds of ways in the past twenty years (this does not include people—especially from the physics community—with whom my main interactions were before 1981, or those with whom my interactions have mostly been about Mathematica or the business of Wolfram Research): Ralph Abraham, Victor Adamchik, Ron Adrian, Guenther Ahlers, Berni Alder, Jan Ambjörn, John Baez, Jim Bailey, Igor Bakshee, Mary Barsony, Andrej Bauer, George Beck, Charles Bennett, Michael Berry, Philippe Binder, Lenore Blum, Manuel Blum, Bruce Boghosian, Enrico Bombieri, Phil Boyland, William Bricken, Bruno Buchberger, Art Burks, David Campbell, John Campbell, Chris Carlson, Pete Carruthers, Forrest Carter, Elise Cawley, Greg Chaitin, Steve Christensen, David Chudnovsky, Gregory Chudnovsky, John Conway, Barbara Cooper, Jack Cowan, Richard Crandall, Jim Crutchfield, Karel Culik, Predrag Cvitanovič, Gautam Dasgupta, Roger Dashen, Martin Davis, Richard Dawkins, David Deutsch, Kee Dewdney, Persi Diaconis, Whitfield Diffie, Freeman Dyson, Paul Erdős, Benson Farb, Doyne Farmer, Mitchell Feigenbaum, Carl Feynman, Richard Feynman, David Finkelstein, Michael Fisher, Mike Foale, Joseph Ford, John Franks, Ed Fredkin, Harvey Friedman, Uriel Frisch, Peter Gacs, Jill Gardner, Laurie Gay, Todd Gayley, Richard Gaylord, Murray Gell-Mann, Roger Germundsson, Etienne Ghys, Don Glaser, Nigel Goldenfeld, Shafi Goldwasser, Beatrice Golomb, Solomon Golomb, Bill Gosper, Peter Grassberger, Alfred Gray, Jeremy Gray, John Gray, Theodore Gray, David Griffeath, Misha Gromov, David Gross, John Guckenheimer, Charlie Gunn, Howard Gutowitz, Hyman Hartman, Jeff Harvey, Brosl Hasslacher, David Hawkins, Gustav Hedlund, Danny Hillis, Pierre Hohenberg, John Holland, John Hopfield, Bernardo Huberman, Alfred Hübler, Dominique d'Humières, Lyman Hurd, Ken Iverson, Raymond Jeanloz, Erica Jen, Leo Kadanoff, Dave Kammeyer, Kuni Kaneko, Stuart Kauffman, Karen Kavanagh, Jerry Keiper, Evelyn Fox Keller, Veikko Keränen, Scott Kirkpatrick, Sergiu Klainerman, Rob Knapp, Don Knuth, Rocky Kolb, John Koza, Bob Kraichnan, Yoshi Kuramoto, Jeff Lagarias, Rolf Landauer, Jim Langer, Chris Langton, Joel Lebowitz, David Levermore, Leonid Levin, Silvio Levy, Steven Levy, Debra Lewis, Wentian Li, Albert Libchaber, David Librik, Dan Lichtblau, Doug Lind, Aristid Lindenmayer, Kristian Lindgren, Chris Lindsey, Ed Lorenz, Saunders Mac Lane, Roman Mäder, Janice Malouf, Benoit Mandelbrot, Norman Margolus, Oleg Marichev, Olivier Martin, Yuri Matiyasevich, John Maynard Smith, Curt McMullen, Hans Meinhardt, Michel Mendès France, Nick Metropolis, John Miller, John Milnor, Marvin Minsky, Don Mitchell, Kim Molvig, John Moussouris, Walter Munk, Jim Murray, Lee Neuwirth, Alan Newell, Mats Nordahl, John Novak, Andrew Odlyzko, Steve Orszag, George Oster, Peter Overmann, Norman Packard, Heinz Pagels, Leonard Parker, Roger Payne, Holly Peck, Hans-Otto Peitgen, Roger Penrose, Alan Perelson, Malcolm Perry, Charlie Peskin, David Pines, Simon Plouffe, Yves Pomeau, Bjorn Poonen, Marian Pour-El, Kendall Preston, Lutz Priese, Ilya Prigogine, Itamar Procaccia, Charles Radin, Tom Ray, Jim Reeds, John Reif, David Reiss, Stanley Reiter, Ken Ribet, Jane Richardson, Ron Rivest, Igor Rivin, Terry Robb, Julia Robinson, Raphael Robinson, Robert Rosen, Gian-Carlo Rota, Lee Rubel, Rudy Rucker, David Ruelle, Jim Salem, Len Sander, Dana Scott, Terry Sejnowski, Rob Shaw, Tim Shaw, Steve Shenker, Bev Sher, Tsutomu Shimomura, Peter Shor, Brian Silverman, Karl Sims, Steven Skiena, Steve Smale, Caroline Small, Alvy Ray Smith, Bruce Smith, Lee Smolin, Mark Sofroniou, Gene Stanley, Ken Steiglitz, Dan Stein, Paul Steinhardt, Adam Strzebonski, Pat Suppes, Gerry Sussman, Klaus Sutner, Noel Swerdlow, Harry Swinney, Bart Taub, David Terr, René Thom, Bill Thurston, Tom Toffoli, Alar Toomre, Russell Towle, Amos Tversky, Stan Ulam, Leslie Valiant, Léon van Hove, Ilan Vardi, Hal Varian, Geerat Vermeij, Gerard Vichniac, Stan Wagon, Bob Wainwright, Bruce Walker, Denis Weaire, Eric Weisstein, Paul Wellin, Caroline Wickham-Jones, Tom Wickham-Jones, Amie Wilkinson, Stephen Willson, Jack Wisdom, Rob Wolff, Alexander Wolfram, Conrad Wolfram, Sybil Wolfram, Lewis Wolpert, Michael Woodford, Larry Wos, Larry Yaffe, Victor Yakhot, Jim Yorke, John Zerolis, Richard Zippel, George Zweig, Helio Zwi.

To go the other way, one uses the result that for all Church numerals x and y , Nest[s, k, n][x][y] is also a Church numeral—as can be seen recursively by noting its equality to Nest[s, k, n - 1][y][x[y]] , where as above x[y] is power[y][x] .

Similarly, assuming as in the rest of this note that all variables are non-negative, b  a + c + 1 has solutions that are exactly those integers that satisfy a < b , with c having some allowed value.

For almost all textbooks tend to discuss only those cases that happen to come out this way.

. • Mid-1970s: Tommaso Toffoli simulates all 4096 2D cellular automata of the simplest type, but studies mainly just their stabilization from random initial conditions. • Late 1970s: Douglas Hofstadter studies a recursive sequence with complicated behavior (see page 907 ), but does not take it far enough to conclude much. • 1979: Benoit Mandelbrot discovers the Mandelbrot set (see page 934 ) but concentrates on its nested structure, not its overall complexity. • 1981: I begin to study 1D cellular automata, and generate a small picture analogous to the one of rule 30 on page 27 , but fail to study it. • 1984: I make a detailed study of rule 30, and begin to understand the significance of it and systems like it.