sound of random sequence, 585 Historians and free will, 1135 Historical perspectives, 844-846 see also history items in specific entries History accidents of in universe, 505 as central to human condition, 846, 1196 of complexity research, 862 computations done in human, 837 of definition of sciences, 863 effect on math of, 792 as explaining axiom systems, 816 general comments about, 850 of life on Earth, 387, 825 of math and ease of concepts, 1177 recorded by physical objects, 1195 role of in biological evolution, 388 theory of and chaos, 971 uniqueness of for universe, 504 History register see Shift registers Hobbes, Thomas (England, 1588-1679) and free will, 1135 and nature of space, 1028 Hodgkin-Huxley equation as example of PDE, 161 Hofmeister, Wilhelm F. B. (Germany, 1824-1877) and phyllotaxis, 1007 Hofstadter, Douglas R. (USA, 1945- ) and recursive sequences, 880, 907 Hokusai, Katsushika (Japan, 1760-1849) and fluid turbulence, 997 Hölder exponents, 959 Holland, John H. (USA, 1929- ) and genetic algorithms, 985 in Preface, xiii Homeobox genes, 419, 1010 Homoclinic tangles, 972 Homogeneity origins of, 353 Homogeneous structures see Cellular automata Homomorphisms and generalized additivity, 952 Homoplectic processes (extrinsic randomness generation), 973 Homotopy and structures in networks, 1046 and topological defects, 1045 Homunculus hypothesis, 1010 Honda, Hisao (Japan, 1943- ) and branching in plants, 1005 Honeycomb form of, 1011 length minimization in, 988 Hooke, Robert (England, 1635-1703) and snowflakes, 992 Hopfield, John J. (USA, 1933- ) and neural networks, 1100, 1102 in Preface, xiii Hopi language and Sapir-Whorf hypothesis, 1181 Hopper crystals, 993 Horizon problem (in cosmology), 1055 Hormones in brains, 1099 in plants, 404 Horn shell growth of, 415 Horner form and evaluation of powers, 1093 Horns (animal) as artificial-looking, 828 compared to antlers, 1008 growth of, 413 Horse races as source of randomness, 968 Horton stream order (in tributaries), 1001 Hours-minutes-seconds origin of, 902 Hoyle, Fred (England, 1915-2001) and extraterrestrials, 1190 HP 700 and creation of this book, 854 HP calculators reverse Polish notation in, 896 Huffman, David A. (USA, 1925-1999) and data compression, 1069 Huffman coding, 564 adaptive, 1069 implementation of, 1071 and JPEG compression, 1074 Human behavior and computational irreducibility, 1132 free will in, 750-753, 1135 and game theory, 1104 laws of, 1014 Occam's razor for, 1025 Human condition defining features of, 1196 scientific approaches to, 845 Human form parametrizations of, 1010 Human history vs. biological evolution, 399 Human languages see Languages (human) Human relics in archeology, 1183 Human thinking, 620-631 and constructs in math, 1177 emulation by computers of, 733 and Gödel's Theorem, 1159 and Principle of Computational Equivalence, 733, 1130 Human will see Free will Humanities relativism in, 1131 summary of relations to, 9, 863 Humans artifacts of early, 1184 defining properties of intelligence of, 822 and future technology, 1195, 1196 genetic programs of, 383, 1002 height distributions of, 977, 1003 making randomness, 306, 970 mathematical power of, 1126 as only intelligence, 822 uniqueness of, 844, 1195 visual system of, 577 weight distributions of, 977 Humboldt, F. W. H. Alexander v. (Germany, 1769-1859) and branching in plants, 1004 Hume, David (Scotland, 1711-1776) and purposes in nature, 1185 Humidity and weather, 1177 Huntington, Edward V. (USA, 1874-1952) and axioms for logic, 1151 Huntington axioms for logic, 773, 1151 Hurwitz numbers, 914 Hydraulic jump, 377 Hydrocarbons considered interesting, 1194 Hydrodynamics see Fluid flow Hydrogen atom as exactly soluble, 1133 and history of quantum theory, 1056 Hydrology and forms of landscapes, 1001 Hyena pigmentation pattern of, 426 Hyperbolic equations, 923, 940 Hyperbolic space, 1049 Hypercube Boolean, 1095 and trivalent networks, 1029 Hypercycles and origin of life, 1179 Hyperelliptic Diophantine equations, 1164 Hyperelliptic functions in solution of spinning top, 1133 Hypergeometric functions asymptotic behavior of, 917 and continued fractions, 914 and curves from curvature, 1009 and nested patterns, 612 Hypergeometric1F1 in Calogero-Moser model, 1133 Hypergeometric2F1 and caps on hyperspheres, 1050 and quintic equations, 945 in Seiberg-Witten model, 1133 and special functions, 1092 and Thue-Morse sequence, 1092 and trinomial coefficients, 1091 HypergeometricPFQ and 6j symbols, 1055 difficulty of evaluating, 1134 from rational integral, 916 Hypergraphs, 1029 Hyperreals as generalizing numbers, 1168 Hypersonic flow see Supersonic flow Hypersphere volume of, 479, 1050 Hyphenation lack of in this book, 852 I-1 base, 932 I Ching and substitution systems, 893 IBM 370 computer my use of, 854 IBM 7090 computer and pi, 911 ICBMs radar used by, 970 radars for detecting, 1188 Ice hopper crystals in, 993 snowflake crystals of, 370 Ice-ray style of Chinese lattice, 874 Icosahedral group, 938 Icosahedral symmetry in quasicrystals, 994 Icosahedron clusters in sphere packing, 986 and GoldenRatio, 891 IDEA cryptographic system, 1086 Ideal forms and Platonism, 1176 Ideal gas CA idealization of, 446 Idealism and free will, 1135 and history of universality, 1109 and logic as foundation, 860 and models of math, 1149 and nature of space, 1028 and theories of communication, 1181 and universe as intelligence, 1196 Idealizations of mathematics, 776 in modelling, 364 and Principle of Computational Equivalence, 728 in traditional science, 736 and ultimate theory of physics, 465 Ideals (algebraic) as generalizing numbers, 1168 Idempotence of And, 817 in operator systems, 801 of Or, 817 Identity cellular automaton, 883 Identity function in combinators, 1122 IEEE floating-point standard and chaos experiments, 919 If (conditional) emulated by Gamma, 1129 Illative logic and symbolic systems, 102 ILLIAC IV and history of CAs, 877 Illinois, University of, xiii Illusions visual, 1076 Image compression lossless, 560 lossy, 572 Image processing by averaging, 1073 history of, 1077 and history of 2D CAs, 877, 928 and visual perception, 1076 and Walsh transforms, 1073 Imaginary time formalism, 1043 as example of model feature, 992 Imitation and bird songs, 1180 see also Emulation Immune system and Boolean networks, 936 and combinatorial chemistry, 1194 and growth of animals, 418 randomness in, 970, 1002 shape space in, 1184 Impact craters as landscape elements, 1001 Implementation see also Emulation see also entries for specific systems Implicants in Boolean formulas, 1095 Implicational calculus, 803, 1170 models in, 1171 possible models of, 805 theorems in, 813 Implicit rules constraints as, 210, 940 Implies axioms for, 803 axioms in terms of, 1151 in multivalued logic, 1175 non-confluence of, 1037 non-universality of, 1174 and structure of proofs, 1155 truth tables for, 802 Impossibilities mathematical, 1137 Impulse response for additive systems, 952 spectrum of, 1081 Incompleteness, 782 essential, 1159 in multiway systems, 783, 797 of Robinson arithmetic, 1169 see also Gödel's Theorem Inconsistency in axiom systems, 781 in multiway systems, 797 Increment function in combinators, 1122 in register machines, 97 in Turing machines, 1144 Incremental improvement and natural selection, 392 Incremental programming, 894 Independence from axioms, 1167 of axioms, 803 from more powerful axioms, 1166 of theorems in logic, 818 Independence results since Gödel's Theorem, 1163 Indeterminism and free will, 750, 1135 Index features of this, 852 names in this, 852 Indian studies of prosody, 875 Indigenous peoples and animism, 1195 Indo-European languages, 1103 Induction (mathematical) and automated proofs, 1158 in axioms of arithmetic, 1152 and recursive sequences, 907 in reduced arithmetic, 800 as scheme for proofs, 1177 statements unprovable without, 1169 Induction (scientific) and experimental math, 899 and ultimate theory of physics, 466 Inequivalence as name for Xor, 1173 Inertia and uniformity of space, 1028 Inertial motion, 521 Infeasible computations, 1143 Inference of cellular automaton rules, 1089 statistical, 589 Inference rules in proofs, 1151 Infinite acceleration in 5-body problem, 1130 Infinite evolution and undecidability, 755 Infinite impulse response cellular automata as, 1035 Infinite loops in Mathematica, 1137 and proving undecidability, 1137 Infinite objects and non-standard arithmetic, 1169 Infinite patterns generated in finite time, 732 Infinite trees and symbolic systems, 898 Infinitesimals and computational irreducibility, 1132 as generalizing numbers, 1168 and non-standard analysis, 1172 and PDEs, 161 Infinities in QED, 1057 Infinity and the continuum, 1127 forms of, 1162 and symbolic representation, 788 and transfinite numbers, 1162 and undecidability, 788 Inflationary universe, 1026, 1055 Information in axiom systems, 819 in block encoding, 1071 in cellular automata, 959 and definition of life, 1178 in multicolor encodings, 1111 and radiation in rule 37R, 455 in theorems of logic, 818 and thermodynamics, 1020 Information content algorithmic, 1067 vs. amount of computation, 1133 of biological organisms, 1002 entropy as, 960 and history of chaos theory, 971 in initial conditions, 920 and lower bounds, 1143 measures of, 1181 Information dimension, 959 Information-preserving data compression, 560 Information theory algorithmic, 1067 and cryptography, 1086 and data compression, 1069 and defining complexity, 1068 and defining randomness, 1068 and human languages, 1181 Information transmission and causal networks, 520 in cellular automata, 252 in class 4 systems, 281 and data compression, 1069 and error-correcting codes, 1101 in financial markets, 1015 limited by speed of light, 518 in PDEs, 923, 925 and quantum entanglement, 1065 and universality, 694 Initial conditions for cellular automaton fluids, 381 and complexity, 41 and computability, 1129 finding as NP problem, 1142 information content in, 920 as input for computations, 637 nested in rule 184, 272 NP completeness of finding, 769 and oracles, 1126, 1129 periodic, 266 persistent structures from, 283 and Principle of Computational Equivalence, 724 random, 223- 296 randomness from, 304-314 for rule 110 universality, 689 sensitive dependence on, 153 sensitivity to, 250-252 sets of for Turing machines, 1139 shape dependence on, 179 and thermodynamics, 443 and time travel, 1043 for Turing machines, 710 and undecidability, 756 for the universe, 1026 Initial value problems in general relativity, 1053 in PDEs, 940 Injectivity in cellular automata, 959 Ink and printing of this book, 852 spreading of as diffusion, 978 Inner and associative CAs, 956 InputForm for programs in notes, 854 Insanity and free will, 1136 Inscribed circles, 986 Insert (insert element) and random networks, 1038 Insertion sort, 1142 Instabilities in CA evolution, 950 and chaos theory, 153, 971, 972 in crowds of people, 1014 in financial systems, 430 of fluid vortex sheets, 997 in numerical PDEs, 924 in numerical turbulence, 997 in PDEs, 988 and quantum measurement, 542 in reaction-diffusion, 1013 and the Second Law, 1020 in self-gravitating systems, 1021 in splashes, 1000 Instantons in path integrals, 1057 Instincts and defining intelligence, 1178 Institute for Advanced Study, xiii Institutions and complexity research, 862 role of in science, 857 Instruments (musical) as nonlinear oscillators, 971 sounds produced by, 1079 Insurance and history of statistics, 1082 Integer equations, 790, 944 and undecidability, 787 see also Diophantine equations Integer factoring, 1090 see also FactorInteger Integer functions patterns generated by, 870 Integer linear programming NP completeness of, 1145 Integer sequences, 123, 128-131 IntegerDigits basic examples of, 854 and computational reducibility, 747 concatenation of, 913 implementation of, 901 and rule 90 pattern, 870 and substitution systems, 889, 891 and sums of three squares, 910 IntegerExponent and 3n+1 problem, 904 and additive CA attractors, 963 behavior of, 911 and binomial coefficients, 870 and cyclic multiplication, 950 and decimation systems, 909 and period doubling, 892 and a recursive sequence, 906 relation to DigitCount of, 902 and rule 150 pattern, 885 and spectra, 1081 and symbolic systems, 897 TM computation of, 1144 and TM enumeration, 1139 and Turing machine 600720, 1145 and Turing machine for increment, 759 IntegerQ (integer test) and fraction systems, 1115 Integers algorithmically simple, 916 as encodings of lists, 1120 and human experience, 1177 number of, 1127 as ordered set, 1152 represented by symbolic expressions, 1121 transfinite, 1162 Integrability of PDEs, 1023 in three-body problem, 972 Integrals as exact solutions, 1133 Integrate and distance in curved space, 1048 impossibilities in, 1137 numbers generated by, 916 systematic methods for, 1177 Integrated circuits see Chips Intelligence analysis using human, 620 and Anthropic Principle, 1026 attributed to universe, 1195 definition of, 822, 1178 in Drake equation, 1191 extraterrestrial, 635, 822 machine, 628, 1099 and Maxwell's demon, 1021 and philosophical implications, 1197 physics as showing, 1191 as special feature of humans, 844 testing of, 1104, 1178 in the universe, 822 see also Thinking Intelligent design and complexity, 861 and teleology, 1185 Intentionality see Purpose Interdependence and definition of life, 1178 Interestingness of chemicals, 1194 in mathematics, 793, 816 of theorems in general, 821, 1176 of theorems in logic, 817 Interfaces (boundaries) effective rules for in CA, 980 Interfaces (computer) and concept of halting, 1137 graphical vs. language, 631 history of, 1102 Interference and quantum computers, 1148 in quantum mechanics, 1059, 1062 of string updates, 503 Interferometer device like, 311 Intermediate degrees (in computation theory), 734, 1130 Intermediate growth groups, 938, 945 Intermediate steps in multilevel logic, 1096 in proofs, 811, 1155, 1157 in solving PCP, 1140 Intermittency and 1/f noise, 969 Internationalism in mathematical notation, 1182 Internet businesses and speculative stocks, 1015 Interpolation of CA patterns, 1092 and fitting models, 1084 Interpretability axiom system encoding, 816 Interpretation of quantum mechanics, 1058 Interpreted languages, 1109 Interrupts (computer) as source of randomness, 970 Intersection and finite set theory, 1171 Interstellar radio communication, 1189 Interstellar space molecules in, 1179 turbulence in, 1188 Interstices packings that use, 986 Interval, maps on the see Iterated maps Interval arithmetic and generalizing numbers, 1168 Intractability (computational), 758-771 and cryptography, 1089 Intrinsic curvature, 1049 Intrinsic randomness generation, 315-326 and continuum behavior, 333 early reactions to, 971 in early universe, 1056 experiments showing, 976 in financial systems, 432 in fluid flow, 382 in fluttering, 971 and free will, 752 in Ising models, 982 my first paper on, 882 in quantum systems, 543, 1063 in reversible conserving systems, 462 and Second Law, 450 see also Randomness generators Intrinsically defined curves, 1009 Introspection and free will, 1135 Intuition about objects in nature, 828 and computer experiments, 856 and continuous math, 925 development of my, 21 in human thinking, 627 mathematical and proofs, 1156 mechanism for in brain, 1136 need for new, 39-41 from practical computing, 716, 872 and Principle of Computational Equivalence, 726 and ultimate theory of physics, 468 Intuitionism as foundation of math, 1176 Intuitionistic logic and double negation, 1158 and Peano arithmetic, 1152 Invagination in animal development, 418, 1009 Invariances and cellular automata state transition graphs, 963 Invariant configurations in 1D cellular automata, 941, 954 in 2D cellular automata, 942, 954 as attractors, 276 in Game of Life, 964 and satisfying constraints, 348 undecidability of, 1138 Invariant entropy (spacetime entropy), 960, 961 Invariant interval (in relativity theory), 1042 Invariants and axiom system proofs, 1170 in dynamical systems, 961 for knots, 1046 see also Conserved quantities Inventions vs. discoveries in math, 1176 for randomness generation, 969 Inverse problems and NP completeness, 771 and perception, 551 Inverse square law for gravity, 536, 1047 InverseFunction and CA encodings, 1118 Invertible cellular automata see Reversible cellular automata Involute of circle, 418 Involutions cellular automata as, 1017 Ion traps for quantum gates, 1148 Ionosphere radio emissions from, 1187 IQ (intelligence quotient), 1178 tests, 1104 Iran ornamental art in, 874 Iraq ornamental art in, 873 Iris (eye) patterns randomness of, 1014 Iron Age ornamental art from, 873 Irrational numbers and 3D non-periodic tiling, 943 generating additive systems, 953 generating quantum gates, 1148 and idealized billiards, 1022 iterated multiplication by, 903 and musical chords, 1079 in Peano arithmetic, 1162 Irreducibility see Computational Irreducibility Irreducible representations (of groups) and isotropy, 980 and spin, 1046 Irregularities see Regularities Irreversibility in data compression, 572-576 and memory, 625 in physical systems, 441-457 and quantum measurement, 1063 Ising, Ernst (Germany/USA, 1900-1998) and Ising model, 981 Ising model, 981 exact solution of 2D, 1133 ground state of, 944 and history of CAs, 876 lack of exact 3D solution, 1133 and lattice gas models, 999 and my work on CAs, 880 and P completeness, 1149 phase transition in, 982 as precursor to my work, 879 as probabilistic model, 1083 and randomness tests, 1085 undecidability in, 1138 see also Spin systems Islam and ornamental art, 872 as rejecting animism, 1195 Islamic art, 43 Isocorrelation textures, 1078 Isomers (variant molecular structures), 1194 Isotropic tensor, 980 Isotropy in aggregation systems, 978 in biological growth, 1007, 1010 in code 746 CA, 334 of differences in 2D CAs, 950 in discrete systems, 980 in physics and CAs, 473 of random walks, 977 of sums of squares, 910 Iterated aliquot sums, 911 Iterated bitwise operations, 906 Iterated division and continued fractions, 143 Iterated function systems, 191 and pictures of ferns, 1005 Iterated maps, 149-155 algebraic iterates in, 1098 attractors in, 961 of bit operations, 921 from CA densities, 953 on the Cantor set, 869 complex, 933 and computation universality, 1129 and computer experiments, 899 emulating discrete systems, 1129 as evidence of determinism, 972 and financial markets, 1015 and fluid flow, 998 history of, 918 and history of chaos, 971 and history of complexity, 49 of integers, 122 in Newton's method, 1101 periodic points in, 955 as precursors to my work, 879 and QCD, 1061 and recurrence relations, 906 spectra in, 1080 and substitution systems, 921 two-dimensional, 921 Iterated morphisms (substitution systems), 893 Iterated radicals, 915 and GoldenRatio, 891 Iterated rules sizes of formulas for, 1096 Iterated run-length encoding, 905 Iteration as basis for algorithms, 1141 compared to repetition, 990 examples of in Mathematica, 853 Iteration theory and iterated maps, 918 $IterationLimit and avoiding undecidability, 1138 Iterative automata see Cellular automata Iterative improvement for engineering designs, 1193 in satisfying constraints, 344 IUPAC chemical nomenclature, 1194 J combinator, 1123 Jacobi, Carl G. J. (Germany, 1804-1851) and sums of four squares, 910 JacobiP (Jacobi polynomial) and trinomial coefficients, 1091 JacobiSN (Jacobi elliptic function) asymptotic oscillation of, 917 and background for PDE, 924 and logistic map formulas, 1098 in solution of spinning top, 1133 JacobiSymbol and Hadamard matrices, 1073 and quadratic residues, 1090 and spectra, 1081 Jacquard, Joseph-Marie (France, 1752-1834) and programmable looms, 1107 Jamming-resistant radio and shift registers, 878 Jansky, Karl G. (USA, 1905-1950) and radio astronomy, 1188 Japanese work on CAs, 877 Java (computer language) and chaos experiments, 919 and computer communication, 1182 as example of language, 1109 and register machines, 101 Jellies large alkanes as, 1194 Jen, Erica (USA, 1952- ) in Preface, xiii and rule 30, 871 Jet stream, 1001 Jets turbulence in, 377 Jevons, W. Stanley (England, 1835-1882) and logic machines, 1107 Job, Book of and understanding nature, 876 Johnson noise, 968 Join in lattice theory, 1153 Join (join lists) as associative function, 1094 basic example of, 853 evaluation chains based on, 1095 theorems about, 1168 Joint denial as name for Nor, 1173 Joints characteristic shapes of, 1183 Jones, James P. (USA/Canada, 1941- ) and Diophantine equations, 1161 Josephus problem, 909 Joule, James P. (England, 1818-1889) and nature of heat, 1019 JPEG compression, 573, 1074 Judaism and free will, 1135 and human uniqueness, 1195 as rejecting animism, 1195 Judgement as crucial in modelling, 365 in social sciences, 9 Julesz, Bela (Hungary/USA, 1928- ) and texture perception, 1076 Julia, Gaston M. (France, 1893-1978) and iterated maps, 879 Julia sets, 933, 934 analog of in tree space set, 1006 Junctional calculus, 803, 1170 possible models of, 805 theorems in, 813 Jupiter and extraterrestrial life, 1179 radio emissions from, 1187 Red Spot of, 377, 999 K combinator, 711, 1121 K^0 particle and time reversal violation, 1019 K_33 non-planar graph, 527 Kabala (universal object), 1127 Kaluza, Theodor F. E. (Germany, 1885-1954) and notions of space, 1028 Kampé de Fériet hypergeometric functions, 1009 Kaneko, Kunihiko (Japan, 1956- ) and continuous CAs, 922 in Preface, xiii Kansas grid of towns in, 1187 Kant, Immanuel (Germany, 1724-1804) and complexity in biology, 861 and free will, 1135 and purposes in nature, 1185 Karatsuba, Anatolii A. (Russia, 1937- ) and multiplication, 1142 Kardar-Parisi-Zhang (KPZ) equation, 925 Kari, Jarkko J. (Finland/USA, 1964- ) and undecidability in 2D cellular automata, 1138 Kármán, Theodore von (Hungary/Germany/USA, 1881-1963) and vortex streets, 998 Karnaugh maps, 1097 Kasiski, Friedrich W. (Poland, 1805-1881) and cryptanalysis, 1086 Kauffman, Stuart A. (USA, 1939- ) and Boolean networks, 936 in Preface, xiii Kazakhstan straight railroad in, 1187 KdV equation as exactly soluble, 1133 Kells Book of, 873 Kelvin (William Thomson) (Scotland, 1824-1907) and harmonic analyzer, 1107 and minimum area packings, 988 and models of space, 1027 and Monte Carlo methods, 1192 and thermodynamics, 1019 and vortices in ether, 1044 Kelvin-Helmholtz instabilities, 988 Kepler, Johannes (Germany, 1571-1630) and 2-body problem, 972 and computers, 1107 and Fibonacci numbers, 891 and inhabitants of Moon, 1190 and rules of nature, 860 and snowflakes, 992 and sphere packings, 986 Keratin repetitive structure of, 1003 Kernels in visual perception, 579 Kerr solution as exact solution, 1133 and time and computation, 1130 Key sequences in cryptography, 598 Keyboard characters for programs in notes, 854 Keystrokes number to create this book, xiv as source of randomness, 970 Khinchin (Khinchin's constant), 914 Kidney ducts branching in, 1008 Kinetic theory and diffusion equation, 1024 and thermodynamics, 1019 "Kingdom was lost" and chaos theory, 971 Kingdoms of organisms, 1003 Kingsnake pigmentation pattern of, 426 Kirby, Laurence A. S. (England/USA, 1952- ) and unprovable statements, 1163 Kirkpatrick, E. Scott (USA, 1941- ) in Preface, xiii and simulated annealing, 985 Kites characteristic shapes of, 1183 Kleene star (RepeatedNull), 957 Klein, Oskar (Sweden/Denmark/USA, 1894-1977) and notions of space, 1028 Klein-Gordon equation discretization of, 1060 vs. sine-Gordon, 922 Kleinian groups and circle packings, 986 KleinInvariantJ (modular elliptic function) and almost integers, 915 Knapping (of stone tools), 1183 Kneading randomness in, 306 Kneading sequences, 955 Knossos maze at, 873 Knots Alexander moves in, 1038 as complex artifacts, 1183 as constructed by rules, 875 in ether as atoms, 1028, 1044 model of atoms as, 1024 as not in 1D systems, 927 theory of, 1046 Knuth, Donald E. (USA, 1938- ) and dragon curves, 893 and Knuth-Bendix procedure, 1037 in Preface, xiii and tests of randomness, 1084 Knuth-Bendix procedure, 1037 Koch, N. F. Helge von (Sweden, 1870-1924) and nested curves, 934 Koch curve, 191, 933 history of, 934 Kolakoski, William G. (USA, 1944-1997) and cyclic tag systems, 895 Kollectiv (collective) and defining randomness, 1068 Kolmogorov, Andrei N. (Russia, 1903-1987) and algorithmic randomness, 1068 and fluid turbulence, 997, 997' Kolmogorov complexity see Algorithmic information Kondo model, 1057 Koran calligraphy from the, 874 Korec, Ivan (Slovakia, 1943-1998) and encodings of arithmetic, 1163 and register machines, 1121 KPZ (Kardar-Parisi-Zhang) equation, 925 Krohn-Rhodes decompositions, 1069 Kronecker product, 1073 KroneckerDelta and orthogonal bases, 1072 Kufi calligraphy, 874 Kummer, Ernst E. (Germany, 1810-1893) and Fermat's Last Theorem, 1166 Kuramoto-Sivashinsky equation, 925 Kuratowski's theorem, 1045 Kurdyumov, Georgii L. (Russia, 1953- ) and transitions in CAs, 981 L-shaped tiles, 932 L systems, 82-87 2D, 187-189 and biological models, 1004 and plant branching, 1005 see also Substitution systems Labyrinth pattern of, 873 Labyrinthine patterns from 2D cellular automata, 427 in code 746, 334 Lag systems, 894 Lagrange, J. Louis (La Grange) (Italy/Germany/France, 1736-1813) and continued fractions, 915 Lagrange points simple geometry of, 1187 in three-body problems, 972 Lagrangian density in general relativity, 1053 for PDEs, 923 in quantum field theory, 1061 LaguerreL (Laguerre polynomials) in hydrogen atom, 1133 Lambda calculus, 1121 and Church's Thesis, 1125 and combinators, 1122 confluence in, 1036 and history of universality, 1110 as idealization of math, 1150 undecidability in, 1136 Lamé equations, 995 Laminar fluid flow, 376 in the atmosphere, 1001 Laminated sphere packings, 986 Land grants and regular grid patterns, 1187 Land use effects on weather of, 1177 Landau, Lev D. (Russia, 1908-1968) and fluid turbulence, 997 Landau-Ginzburg equation, 925 Landscapes (abstract) constraint satisfaction on, 346 for memory, 1101 Landscapes (geological), 1001 Langton, Christopher G. (USA, 1948- ) and 2D Turing machines, 930 and CA classes, 948 in Preface, xiii Langton's ant (2D Turing machine), 931 Language games, 1181 Language of mathematics and science, 859 Languages (abstract) as attractors in TMs, 961 context-free, 939, 1091, 1103 context-sensitive, 939 regular, 939 universal and Leibniz, 1149 unrestricted, 939 see also Formal languages Languages (computer), 1103 as examples of software, 1108 functional and combinators, 898 and history of computing, 1108 intuition from existence of, 872 logic programming, 1158 Mathematica, 853 meaning in, 1182 notion of types in, 898 for parallel computing, 1035 thinking in, 1104 universals in, 1182 see also Computer languages Languages (human), 1103 as analogy in atomism, 876 animals trained to use, 826 center in brain, 1099 conventional aspect of, 1181 and cryptanalysis, 599 and defining intelligence, 822, 1178 frequency of words in, 1014 and history of universality, 1109 influence of on thinking, 1181 information content in, 1181 logic in everyday, 806, 1173 and math thinking, 1177 reasons for features of, 1181 recognizing meaning in, 826 redundancy of, 1086, 1181 relativity hypothesis in, 1181 rules for and programs, 860 rules in grammar for, 875 and theories of communication, 1181 and thinking, 630 translation of and AI, 1100 translation of and universality, 643 understanding dead, 1184 and Zipf's law, 1014 Laplace, Pierre-Simon (France, 1749-1827) and Central Limit Theorem, 977 and determinism, 1135 Laplace equation as constraint, 940 and diffusion-limited aggregation, 994 and flatness of growing tissue, 1007 and network dimensions, 1031 Laplacian in curved space, 1050 LARC computer and gas simulation, 864 Large cardinal axioms, 1167 Lasers as technology components, 1195 Latencies in circuits, 1095 Latent heat and crystal growth, 370, 372 Lateral inhibition and pigmentation patterns, 427 in plant phyllotaxis, 409 in snowflake growth, 370 Latin logic operations in, 1173 Latin squares and quasigroups, 1171 as satisfying constraints, 945 Lattice dynamical systems, 155-160, 922 see also Continuous cellular automata Lattice gases, 999 enumeration in, 959 see also Cellular automaton fluids see also Spin systems Lattice gauge theories, 1057, 1061 and computational irreducibility, 1133 and discrete space, 1027 and quantum gravity, 1054 Lattice models see also Cellular automata see also Spin systems Lattice packings, 987 Lattice points numbers inside spheres, 910 Lattice theory axioms for, 1153 and posets, 1041 Lattice vectors and linear congruential generators, 974 LatticeReduce and digit sequence cryptanalysis, 1089 Lattices isotropic behavior on, 980 lines at slopes on, 916 possible, 929 random walks on, 329, 977, 978 Law of large numbers, 976 Laws (human) and defining randomness, 1068 and free will, 1135 and regularities in radio spectrum, 1188 and responsibility, 1136 Laws of Form and Nand, 1173 Laws of physics vs. mathematics, 772 vs. programs in biology, 397 and ultimate theory, 470 Layout of networks, 193 NP completeness of, 1145 LCDs see Liquid crystals LCGs see Linear congruential generators LCM (lowest common multiple) and Lissajous figures, 917 and Riemann Hypothesis, 918 Leading digits, 914 and number compression, 1070 of powers, 903 Leaf shapes see Leaves LeafCount (expression size) and algebraic expressions, 945 and combinator expressions, 1122 and number of recursive functions, 908 and symbolic systems, 897 for universal Diophantine equation, 1161 Learning of bird songs, 1180 of computer languages, 1104 and defining intelligence, 823, 1178 human, 626 models of, 1099, 1102 the science in this book, 855 as trick in evolution, 1002 Learning curves, 1014, 1102 Least squares fitting, 1082, 1083 Leaves arrangements of (phyllotaxis), 408-412, 1006 evolutionary theories of, 1005 flatness in growth of, 1008 growth of, 401, 412, 1005 shapes of, 401-407, 1005 LED segments and textons, 1076 Leech lattice isotropy of, 980 and sphere packings, 987 Leftmost innermost evaluation, 906 Leg (animal) development of, 419 Legendre, Adrien-Marie (France, 1752-1833) and sums of three squares, 910 LegendreP (Legendre polynomials) and isotropy measures, 980 in solution of Mie scattering, 1133 Lehmer, Derrick H. (USA, 1905-1991) and linear congruential generators, 974 Lehr, Eduard (Germany, ~1932) and parametric curves, 1009 Leibniz, Gottfried W. v. (Germany, 1646-1716) and binary numbers, 902 and calculators, 1107 and free will, 1135 and I Ching, 893 and idealizations of math, 1149 and logic, 1099 and logic as basis for science, 860 and math notation, 1182 and the nature of space, 1028 and origins of universality, 1109 and symbolic logic, 1151 and universal language, 1181 Lemmas defining characteristics of, 1176 and lengths of proofs, 1175 in proof of axiom for logic, 811 in proof searching, 1157 in structure of proofs, 1155 Lempel, Abraham (Israel, 1936- ) and data compression, 1069 Lempel-Ziv encoding, 565 Lemur pigmentation pattern on, 426 Length basic example of, 853 Lenses characteristic shapes of, 1183 Lenticular clouds, 1183 Lenz, Wilhelm (Germany, 1888-1957) and Ising model, 981 Leonardo da Vinci see da Vinci, Leonardo Leopard pigmentation pattern of, 426 Leptons as types of particle, 1043 Less (<) encoded as integer equation, 1160 Letters frequencies of in English, 1069 visual perception of, 1076 Lettuce ammonite septa like, 1008 growth of, 413 Leucippus (Greece, ~480 - ~420 BC) and atomism, 876 Levin, Leonid A. (Russia/USA, 1948- ) and NP completeness, 1143 in Preface, xiii and transitions in CAs, 981 Lévy, Paul P. (France, 1886-1971) and nested C curve, 893, 934 Lévy flights, 969 lex and regular languages, 958 Lexical analysis and regular languages, 958 LFSRs (linear feedback shift registers) see Shift registers LHASA (chemical synthesis software), 1194 Li, Wentian (USA, 1959- ) and CA classes, 948 in Preface, xiii Lichens on Mars, 1180 Lichtenberg figures, 995 Lie groups and additive systems, 953 and sphere packings, 987 Life definition of, 557, 823, 1178 and definition of complexity, 1069 on Earth, 825 and history of complexity, 50 on Mars, 1179 origin of, 1179 Life, Game of see Game of Life Life force, 1178 Life insurance and history of statistics, 1082 Lifeforms alien, 825 LIFO (last-in first-out) stacks and context-free languages, 939 Light caustics in, 984 perception of, 577 and quantum history, 1056 Light bulb filaments, 842 Light cones, 519 for cellular automata, 949 differential geometry of, 1051 and random causal networks, 1052 Light levels and object recognition, 1076 Lightning and origin of life, 1179 pattern of from space, 1187 randomness in, 995 as source of radio emissions, 1187 Lights of cities patterns of from space, 1187 Likelihood maximum, 589, 1083 Lily pads, 401 Limit cycles for differential equations, 961 in iterated maps, 961 see also Repetitive behavior Limit ordinals, 1162 Limitations of science, 6, 748, 846 and ultimate theory of physics, 471 Limited size systems, 255-260, 267 Limits of cellular automaton rules, 948 and fractal dimensions, 933 of networks as space, 1030 undecidability of, 1138 see also Continuum limits Lincoln cathedral, 873 Lincos language, 1189 Lindenmayer, Aristid (USA/Netherlands, 1925-1989) and L systems, 893 and plant branching, 1005 in Preface, xiii Lindgren, Kristian (Sweden, 1960- ) in Preface, xiii and universal CAs, 1115 Line defects, 1045 Line-printer output of my early cellular automata, 19 Linear algebra and CA invariances, 1022, 1023 and dimensions of networks, 1031 and models of memory, 1101 Linear cellular automata see also Additive cellular automata see also Elementary cellular automata Linear codes, 1101 Linear congruential generators, 318, 974 based on powers of 3, 903 and chaos on calculators, 920 cryptanalysis of, 1089 historical use of, 968 and randomized algorithms, 1085 as shift register analogs, 975 Linear differential equations and time series, 1083 Linear differential operators, 953 Linear Diophantine equations, 944, 1164 and polynomial time, 1146 Linear equations, 940 as defining rationals, 916 and free field theories, 1061 Linear feedback shift registers see Shift registers Linear filters in sound compression, 1080 and spectra, 1082 in visual perception, 1076 Linear models and computational irreducibility, 1132 in statistics, 589 Linear predictive coding in sound compression, 1080 Linear programming NP completeness of integer, 1145 for optimizing designs, 1193 Linear recurrences, 891, 906 and additive CAs, 1018 and history of CAs, 878 and time series, 1083 see also Linear congruential generators Linear stability analysis, 988 Linear systems vs. nonlinear, 940 Linearity compared to additivity, 952 in quantum mechanics, 541, 1060 LinearSolve iterative algorithm for, 1141 and LFSR cryptanalysis, 1087 and linear systems, 940 and NC computations, 1149 Linguistics see Languages (human) Linial (Gulden), Samuel (USA, 1927- ) and underivability of logic axioms, 1170 Linkages in chaos toys, 1183 characteristic shapes of, 1183 and constructible reals, 1129 for randomness generation, 969 Linked lists for Game of Life, 949 Linker for register machines, 1114 Linnik's problem, 910 Linux and creation of this book, 854 Liquid crystals as components in technology, 1195 phase transitions in, 981 topological defects in, 1045 Liquid-gas phase transitions, 981 Liquids flow of, 376 see also Fluid flow Lisp (computer language) grammar of, 1104 and network systems, 936 representing expressions in, 897 theorems about, 1158, 1168 Lissajous figures, 917 List manipulation examples of in Mathematica, 853 Listable and implementing CAs, 866 ListConvolve and aggregation systems, 978 and autocorrelation, 1080 basic example of, 853 and CellularAutomaton, 886 and elementary CAs, 865 and fast evolution rules, 1094 and general CAs, 886 and hexagonal CAs, 992 and image processing, 1077 and integer multiplication, 1093 and Ising model, 981 and multidimensional CAs, 927 recursive algorithm for, 1142 in sandpile models, 989 and Sierpinski pattern, 931 and visual perception, 1076 ListCorrelate and additive CAs, 870, 952 and visual feature extraction, 1077 ListPlay and sound output, 1080 Lists encoded as integers, 1120, 1127 see also Sequences Lithic analysis, 1183 Little group (spacetime symmetries), 1046 Littlewood, John E. (England, 1885-1977) and chaos theory, 971 and Goldbach's Conjecture, 911 and iterated maps, 919 and numbers of primes, 910 Livermore (Lawrence Radiation Laboratory), 864 Lizards pigmentation patterns of, 426 Local conservation laws, 1023 see also Conservation laws Local minima, 346 Localization in class 1 and 2, 252 as origin of repetition, 355 Localization (of eigenstates) and randomness tests, 1085 Localized structures and 3n+1 problem, 904 in class 4 systems, 281-296 in code 20 CA, 285 in code 357 CA, 286 in code 1329 CA, 287 in continuous CAs, 160, 244 as engineering components, 678 finding in 2D CAs, 965 and long-range communication, 252 in network evolution, 525 origins of, 990 and particles in physics, 525 in PDEs, 164, 1129 in presence of randomness, 526 in QCD, 1061 from random initial conditions, 228 in reversible CAs, 440 in rule 30 CA, 700 in rule 37R CA, 455 in rule 45 CA, 701 and rule 110 axiom system, 1168 in rule 110 CA, 31, 292, 964 same in different 2D rules, 949 in three dimensions, 949 in turbulent fluids, 997 in Turing machines, 888 and universality, 692 see also Gliders Locomotion in animals, 1011 Log (logarithm) algebraic independence of, 1164 computation of nth digits in, 912 and computational reducibility, 744, 747 continued fractions for, 144 digits of, 141 leading digits in, 914 randomness in digits of, 967 as source of randomness, 968 uniform distribution mod 1, 904 and Weber's law, 1014 Log-periodic antennas, 1183 Logarithmic growth of number representations, 1070 Logarithmic law for leading digits, 914 Logarithmic series, 1134 Logarithmic spiral, 418 Logging and messages to Moon, 1189 regular patterns of, 1187 Logic as applied to math, 780 axioms for, 773, 808 and combinators, 711, 898 and communication, 1181 and computational irreducibility, 1196 and defining intelligence, 1178 difficult notation in, 1104 and discreteness in programs, 976 as example of lattice theory, 1153 as example of rules, 875 and finite set theory, 1171 as foundation of math, 1176 history of, 1097, 1099, 1151 history of notation for, 1182 and history of universality, 1109 in human languages, 1173 in human thinking, 627 implications for, 1197 interesting theorems in, 816 lengths of proofs in, 1175 machines for doing, 1107 multilayer, 1096 multivalued, 814, 1175 and natural science, 860 non-universality of basic, 784 as not mainstream math, 859 possible models of, 805 in practical computers, 1108 and quantum mechanics, 1058 reversible, 1097 shortest And, Or axioms for, 1174 shortest Nand axioms for, 808 ternary, 814 and Thue on substitution systems, 893 see also Predicate logic Logic circuits emulated by CAs, 662, 1112 emulated by Game of Life, 1117 and universality, 1119 Logic expressions see Boolean expressions Logic minimization, 1095 compared to searching, 1193 Logic programming languages, 1158 Logic Theory Machine, 1157 Logical depth and defining complexity, 1069 for Turing machines, 1144 Logical functions see Boolean functions Logical positivism and theories of communication, 1181 Logical rules see Rules Logicism as foundation of math, 1176 LogIntegral (logarithmic integral li) and distribution of primes, 133, 909 and Zeta, 918 Logistic differential equation and iterated maps, 918 Logistic map, 920 closed forms for iterates of, 1098 and continuous CAs, 922 and history of chaos theory, 971 see also Iterated maps Logistics and causal networks, 1033 and complex systems, 862 Lognormal distribution for human weights, 977, 1003 and multiplicative processes, 977 Logo (computer language) and 2D TMs, 930, 931 Long division, 139 Long-term memory, 1102 Long-time tails in molecular dynamics, 999 and shaking, 969 Longest paths and NP completeness, 1146 Looms and history of computer, 1107 Loop general relativity, 1048 Loop quantum gravity, 1054 and spin networks, 1055 Loop switching method and network layout, 1031 Lorentz, Hendrik A. (Netherlands, 1853-1928) and relativity theory, 1041 Lorentz contraction, 1041 Lorentz gas, 1022 Lorentz transformations, 1041, 1042 Lorentzian spaces, 1051 Lorenz, Edward N. (USA, 1917- ) and chaos theory, 971 and complex ODE, 879 and experimental math, 899 and fluid turbulence, 998 in Preface, xiii Lorenz equations as giving strange attractor, 922 and history of chaos theory, 971 and Lissajous figures, 917 and weather prediction, 1178 Los Alamos and history of CAs, 877 and my work on CAs, xiii, 880 Lossless data compression, 560 Lossy data compression, 572 Lotka, Alfred J. (USA, 1880-1949) and oscillatory chemical reactions, 1013 Lotteries, 969 Lovelace, A. Ada (Byron) K. (England, 1815-1852) and universality, 1110 Low-level languages and practical computing, 1109 and register machines, 101 Löwenheim-Skolem theorem, 1172 Lower bounds on computations, 761, 1143 on parity, 1096 on sorting algorithms, 1142 LPC in sound compression, 1080 Lucas, F. Edouard A. (France, 1842-1891) and binomials mod k, 870 Lucas-Lehmer test for primes, 911 Lucas numbers, 891 Lucian (Greece, 120 - ~180 AD) and inhabitants of Moon, 1190 Lucretius, Titus Carus (Italy, ~95 - ~55 BC) and atomism, 876 and models based on rules, 860 Lukasiewicz, Jan (Poland, 1878-1956) and axioms for logic, 1151 and multivalued logic, 1175 Lukasiewicz representation, 896 Lunes and ornamental art, 872 theory of, 873 Lungs branching in, 1008 Lyapunov exponents in cellular automata, 950 and entropies, 960 in iterated maps, 921 probabilistic estimates of, 953 LZW compression, 565, 1069, 1072 MacArthur Foundation, xiii Mach, Ernst (Austria/Czech., 1838-1916) and Mach's Principle, 1047, 1053 and origins of relativity, 1042 Mach numbers, 997 Machiavelli, Niccolò (Italy, 1469-1527) and rules for military drill, 875 Machine 1447 function computed by, 761 and longest halting times, 1144 Machine 1507 as complicated TM, 1144 function computed by, 761 Machine 596440 as candidate for universality, 709 properties of, 1120 Machine 600720 behavior of, 763 and longest halting times, 1144 properties of, 1145 Machine code and history of computing, 1109 and implementation of CAs, 866 of programs by Alan Turing, 1013 and register machines, 97 Machine integers and linear congruential generators, 974 Machine intelligence, 628, 1099 Machine numbers for Turing machines, 888 Machines see also Register machines see also Turing machines Machining plans and Voronoi diagrams, 987 Macintosh and computer interfaces, 1102 Magic Eye random dot stereograms, 1076 Magic squares as defined by constraints, 945 for SETI, 1190 Magnetic chaos toys, 1183 Magnetic field compared to gravitational, 530 and gauge invariance, 1045 Magnetic moments and size of particles, 1044 Magnetic monopoles, 1046 Magnetization in Ising model, 981 phase transition in, 981 Magnetosphere radio signals from, 827, 835, 1187 Magnus, Wilhelm (Germany/USA, 1907-1990) and word problem for groups, 1141 [...]... Mathematica Book, 853 writing style in, 849 Mathematical constants, 13 6-1 44 Mathematical equations as foundation for science, 1 in history of science, 44 see also Equations Mathematical formulas and computational irreducibility, 737 and engineering, 829 as results of models, 368 and science education, 1133 Mathematical functions, 14 5-1 48 as computable, 1128 difficulty of evaluating, 1134 evaluated by CAs,... Mars, 1180 Newton, Isaac (England, 164 2-1 727) and 2-body problem, 972 and calculus, 901 and color vision, 1075 and computational irreducibility, 1132 and invariance of mechanical laws, 1041 and law of gravity, 1047 and math in science, 859, 860 and nature of space, 1028 in quote by Kant, 861 and theology, 861 Newtonian physics and determinism, 967 Newton's Laws character of as principle, 1126 Newton's... vacuum fluctuations, 1062 Pairing functions (sigma), 1127 Pairwise sorting algorithms, 1142 Palatine Chapel, 873 Paleolithic art, 43, 873 Palermo, Sicily, 873 Paley, Raymond E A C (England/USA, 190 7-1 933) and Walsh transforms, 1073 Paley, William (England, 174 3-1 805) and natural theology, 861 Paley family of Hadamard matrices, 1073 Paley order for Walsh functions, 1073 Palindrome systems, 12 5-1 27 Palindromes... trees, 40 5-4 07, 1006 and universal objects, 1127 Parametric differentiation as framework for integrals, 1177 Parametric surfaces in shell models, 1008 Paramodulation as step in proof, 1156 Parasitism and biological evolution, 1002 Parastichies (phyllotaxis spirals), 1007 Parentheses language of balanced, 939, 1103 pattern from balanced, 1091 pattern of balanced, 989 and single combinators, 1123 Paris,... speed of light, 1041 spin states of, 1046 Master equations in cellular automata, 953 Matching of networks, 1038 and paramodulation, 1156 of parentheses, 1091 of strings, 1038, 1101 MatchQ (test for matching) and constraint systems, 941 Material equivalence as name for Equal, 1173 Material implication as name for Implies, 1173 Materialism and free will, 1135 Materials fracture in, 994 search-based discoveries... problem), 1128 Non-Abelian groups CAs based on, 887, 956 and charge quantization, 1046 in gauge theories, 1045, 1057 see also Groups (mathematical) Non-Abelian plane waves, 1061 Non-Archimedean fields as generalizing numbers, 1168 Non-commutative geometry and general relativity, 1048 Non-compact groups and tachyon spin, 1046 Non-computability of entropies, 958 of fractal dimension, 933 see also Undecidability... 1142 Partition function and phase transitions, 981 PartitionsP (number of partitions) and number of commutative groups, 1172 Parton model as mechanical model, 1026 Pascal, Blaise (France, 162 3-1 662) and arbitrary number bases, 902 and his calculator, 1107 and Pascal's triangle, 870 Pascal's triangle and associative CAs, 956 for continuous moduli, 922 and encodings of arithmetic, 1164 history of, 870 and... 1085 and scale invariance, 955 undecidability of, 1138 and uniformity, 354 Phases and localized structures, 990 of matter as analogy for cellular automaton classes, 235 separation of in 2D CA, 336 Phelan, Robert J (USA, 193 3- ) and minimum area packings, 988 Pheromones alkanes as, 1194 and animal communication, 1180 phi see GoldenRatio Philippines as source of shells, 1011 Philosopher's stone and alchemy,... circle packings, 985 forms of, 385, 1011 skeletons as animal artifacts, 1184 Radix of numbers see Bases (number) Radó, Tibor (Hungary/USA, 189 5-1 965) and Busy Beaver Problem, 889 Railroads as speculative stocks, 1015 as visible from space, 1187 RAM (random-access memory) emulated by CA, 663, 1 112 Ramanujan, Srinivasa A (India/England, 188 7-1 920) and almost integers, 915 and divisor sequences, 910, 911 and... structure of proofs in, 1151 undecidability in, 1138 unfamiliarity of notation in, 853 see also Foundations of mathematics Mathematical models history of, 992 and Principle of Computational Equivalence, 728 Mathematical morphology, 1077 Mathematical notation, 1182 grammar of, 1104 influence on thinking of, 1181 for logic, 1173 vs Mathematica, 853 relations vs axioms in, 1169 Mathematical physics functions of, . Karatsuba, Anatolii A. (Russia, 193 7- ) and multiplication, 1142 Kardar-Parisi-Zhang (KPZ) equation, 925 Kari, Jarkko J. (Finland/USA, 196 4- ) and undecidability in 2D cellular automata, 1138. Invariances and cellular automata state transition graphs, 963 Invariant configurations in 1D cellular automata, 941, 954 in 2D cellular automata, 942, 954 as attractors, 276 in Game. machines, 930 and CA classes, 948 in Preface, xiii Langton's ant (2D Turing machine), 931 Language games, 1181 Language of mathematics and science, 859 Languages (abstract) as