(The kind of complexity discussed here has nothing directly to do with complex numbers such as √ -1 introduced into mathematics since the 1600s.)

But eventually it was realized that quasi-Monte Carlo methods based on simple sequences could normally do better than ones based on pure randomness (see page 1085 ).

(Note however that I do not believe that such σ could ever actually be constructed in any explicit way by any real computational system—or in fact by any system in our universe.)

The Second Law was originally formulated in terms of the fact that heat does not spontaneously flow from a colder body to a hotter.

The Einstein–Hilbert action—and the Einstein equations—can be viewed as having the simplest forms that do not ultimately depend on the choice of coordinates.

The continued fractions for Exp[2/k] and Tan[1/2k] have simple forms (as discussed by Leonhard Euler in the mid-1700s); other rational powers of E and tangents do not appear to.

And the strong interactions responsible for holding nuclei together (and associated for example with exchange of pions and other mesons) seemed too strong for it to make sense to do an expansion with larger numbers of individual interactions treated as less important.

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.

The simplest known way of doing this (see note below ) involves a degree 8 equation with 60 variables: a  b c ↔ α [d, 4 + b e, 1 + z] ∧ α [f, e, 1 + z] ∧ a  Quotient[d, f] ∧ α [g, 4 + b, 1 + z] ∧ e  16 g(1 + z) λ [a_, b_, c_] := Module[{x}, 2 a + x 1  c ∧ (Mod[b - a, c]  0 ∨ Mod[b + a, c]  0)] α [a_, b_, c_] := Module[{x}, x 1 2 - b x 1 x 2 + x 2 2  1 ∧ x 3 2 - b x 3 x 4 + x 4 2  1 ∧ 1 + x 4 + x 5  x 3 ∧ Mod[x 3 , x 1 2 ]  0 ∧ 2x 4 + x 7  b x 3 ∧ Mod[-b + x 8 , x 7 ]  0 ∧ Mod[-2 + x 8 , x 1 ]  0 ∧ x 8 - x 11  3 ∧ x 12 2 - x 8 x 12 x 13 + x 13 2  1 ∧ 1 + 2 a + x 14  x 1 ∧ λ [a, x 12 , x 7 ] ∧ λ [c, x 12 , x 1 ]] (This roughly uses the idea that solutions to Pell equations grow exponentially, so that for example x 2  2y 2 + 1 has solutions With[{u = 3 + 2 √ 2 }, (u n + u -n )/2] .)

The result of this is that points in space can always be specified by lists of coordinates—although historically one of the objectives of differential geometry has been to find ways to define properties like curvature so that they do not depend on the choice of such coordinates.