Stephen Wolfram - A New Kind Of Science Episode 12 doc

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 Flu

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

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

reverse Polish notation in, 896

Huffman, David A (USA, 1925-1999)

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, 9863 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

radar used by, 970

radars for detecting, 1188

Ice

hopper crystals in, 993

snowflake crystals of, 370

Ice-ray style of Chinese lattice, 874

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

IEEE floating-point standard

and chaos experiments, 919

and history of CAs, 877

Illinois, University of, xiii

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 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

and symbolic representation, 788

and transfinite numbers, 1162

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

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

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 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

numbers generated by, 916

systematic methods for, 1177

Integrated circuits

see Chips

Intelligence

analysis using human, 620

and Anthropic Principle, 1026

and Maxwell's demon, 1021

and philosophical implications, 1197

Intermediate degrees (in computation theory), 734, 1130

Intermediate growth groups, 938, 945

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

in financial systems, 432

in fluid flow, 382

in fluttering, 971 and free will, 752

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

see Computational Irreducibility

Irreducible representations (of groups)

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

Iterated aliquot sums, 911

Iterated bitwise operations, 906

Iterated division

and continued fractions, 143

Iterated function systems, 191

and pictures of ferns, 1005

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

sizes of formulas for, 1096

Iterated run-length encoding, 905

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

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

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

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

"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

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

effects on weather of, 1177

Landau, Lev D (Russia, 1908-1968)

and fluid turbulence, 997

Langton, Christopher G (USA, 1948- )

and 2D Turing machines, 930

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

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

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

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 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

Lenz, Wilhelm (Germany, 1888-1957)

and Ising model, 981

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 additive systems, 953

and sphere packings, 987

see Game of Life

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

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

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

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

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

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 minima, 346 Localization

in class 1 and 2, 252

as origin of repetition, 355 Localization (of eigenstates) and randomness tests, 1085 Localized structures

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

shortest And, Or axioms for, 1174

shortest Nand axioms for, 808

ternary, 814

and Thue on substitution systems, 893

see also Predicate logic

Logic programming languages, 1158

Logic Theory Machine, 1157

Logical depth

and defining complexity, 1069

for Turing machines, 1144

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 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

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)

random generators on, 320, 974

Maintainability of software

and defining complexity, 1069

Majority cellular automaton

and P completeness, 1149

Majorization of functions (growth rates), 1162

Makanin, Gennadií S (Russia, 1938- )

and sequence equations, 1141

and undecidable word problems, 1141

Malacology (study of molluscs), 1008

and Boolean networks, 936

and defining complexity, 1069

and history of complexity, 862

Mandelbrot, Benoit B (France/USA, 1924- )

and branching models, 1005

and computer experiments, 899

and financial processes, 1014

and fractal landscapes, 1001

and fractals, 880, 934

and history of complexity, 862

and Mandelbrot set, 934

discrete approximations to, 1050

undecidability in equivalence of, 1051, 1138

Manneville, Paul (France, 1946- )

basic example of, 853

and causal networks, 1033

from linear congruences, 974

local and linear, 953

price fluctuations in, 429

Markov, Andrei A (Russia, 1903-1979)

and sequential substitution systems, 894

and statistics of language, 1181

as statistical models, 590 Markov random fields and texture discrimination, 1077 and visual perception, 584 Markov systems (sequential substitution systems), 894 Marr, David C (England/USA, 1945-1980)

and visual perception, 1076 Mars

question of life on, 1179, 1179', 1180 sending messages to, 1189

as source of radio signals, 1188 Martin, Olivier C (USA/France, 1958- ) and additive CAs, 951

as coauthor of paper, 881

in Preface, xii, xiii Martin-Löf, Per E R (Sweden, 1942- ) and randomness, 1068

Martingales (gambling systems), 1068 Marxen, Heiner (Germany, 1957- ) and Turing machines, 889 Masers, natural, 1187 Masking

of sounds, 587 visual with textures, 1077 Mass

of elementary particles, 1046 equality of inertial and gravitational, 1047

in relativity theory, 1041

in Schwarzschild solution, 1053

in ultimate theory of physics, 528 Massey, James L (USA/Switzerland, 1934- ) and Berlekamp-Massey algorithm, 1087 Massively parallel computers

and history of CAs, 877 Massless particles and 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 of, 1193 Mates

and bird songs, 1180

Mathematica

algorithms in discovered by searches, 1193 analysis of internal code of, 1003 and authoring system for book, 852 avoidance of data types in, 898, 1154 axiom systems in, 1150

as basis for discoveries, 46 cellular automata in, 867 chaos experiments with, 919 computation of pi with, 911 computation of PrimePi in, 909 computation of PrimeQ in, 909 and computer communication, 1182 and computer experiments, 113, 899 and concept of halting, 1137 confluence of rules in, 1037 creation of, 20

and creation of this book, xii and creation of this index, 852 curves of math functions in, 145 and definition of math, 859 design of for human use, 1177 diversity of universality in, 644 evaluation order in, 898, 906

as example of language, 1109 and examples in education, 1133 and experimental math, 899

generic inequality of symbols in, 1169

grammar of, 631

history of my work on, 864, 881

and human thinking, 627

as idealization of math, 1150

infinite loops in, 1137

and math insight vs proofs, 1156

mathematical functions in, 1092

mathematical notation in, 1104

minimal model for, 102

name of, 859

as necessary skill, 856, 857

as notation for computation, 1182

notations for logic in, 1173

NP completeness in, 1143

number of names in, 1103

number representation in, 1070

operator precedence in, 1150

patterns

see Patterns (in Mathematica)

patterns and network evolution, 1037

patterns and operator systems, 898

and pictures in this book, 851

proofs and design of, 1158

pure functions in, 1121

random programs in, 1183

randomness generation in, 317, 973

scoping of variables in, 1152

and sequential substitution systems, 894

size compared to genomes, 383

structure of, 1103

and structure of proofs, 1155

symbolic expressions in, 896

theorem proving and, 1158

and use of digit sequences, 902

use of hashing in, 1100

and history of universality, 1110

and intermediate degrees, 734

and neural networks, 1099

and problem of free will, 1135

structure of proofs in, 1151

undecidability in, 1138

unfamiliarity of notation in, 853

see also Foundations of mathematics

and Church's Thesis, 1126 Mathematics

analysis using, 606-620 applied to biological form, 1004 axioms in, 773

as central in education, 855 and continuity of space, 472 definition of, 859

in economic systems, 1015 and extraterrestrial intelligence, 837, 1189

as foundation for science, 1 and history of complexity, 862 history of in science, 5859 and human intelligence, 822 idealizations of, 795, 1149 implications for, 772-821 invention vs discovery in, 1176 and models of thinking, 629 new results in this book in, 863 notation in, 853

numbers in, 116 ordering of constructs in, 1177 and PDEs, 161

in physics, 433

vs physics, 821 self-limiting of, 792 set theory in education, 1154

in social sciences, 9 summary of relations to, 7863 and systems with constraints, 221 types of theorems in, 1176 use of rules in, 875 see also Foundations of mathematics MathieuC (Mathieu function)

asymptotic oscillation of, 917 MathLink

and CA programs, 865 and computer experiments, 899 and creation of this book, 854 Mating

and defining species, 1003 Matiyasevich, Yuri (Russia, 1947- ) and Diophantine equations, 1161

in Preface, xiii and word problem for semigroups, 1141 Matrices

and 2D substitution systems, 933 and cryptanalysis, 602

as generalizing numbers, 1168 Hadamard, 1073

mappings of, 921 and non-commutative rings, 1153 random, 977

satisfying constraints, 945 and solving linear systems, 940 Matrix mechanics, 1056 Matrix memories, 1101 MatrixPower and CA entropies, 958 and correlation functions, 1082 and Fibonacci, 890

and substitution systems, 890 and undecidability, 1138 Matter

as distinct from space, 1028

in Einstein equations, 1052

as generating curvature, 532, 536 Mauritania

Richat structure in, 1187 Max (maximum)

as associative, 1094 related to Or, 952 Maximal periods

of cellular automata, 950, 1088

of linear congruences, 974

of random generators, 318

of shift registers, 975 and symmetries, 950 Maximal principles variational principles, 1185 Maximum entropy method, 1083 Maximum likelihood, 589, 1083 Maximum-of-t test, 1085

and electromagnetism, 1041

and fluttering of paper, 971

and gauge invariance, 1045

and Maxwell's demon, 1021

and statistical mechanics, 1019

and stochastic models, 968

Maxwell's demon, 1021

Maxwell's equations

as examples of PDEs, 161

and relativistic invariance, 1041

as simple physical models, 1025

May, Robert M (Australia/USA/England, 1936- )

and chaos theory, 971

Mazoyer, Jacques (France, 1947- )

and firing squad problem, 1035

McCarthy, John (USA, 1927- )

and theorems in Lisp, 1168

McCulloch, Warren S (USA, 1898-1972)

and neural networks, 880, 1099

and universality, 1110

McCulloch-Pitts models, 1099

and finite automata, 958

McCune, William W (USA, 1953- )

and axioms for groups, 1153

and axioms for logic, 1151

Mean curvature, 1009

Mean field theory

for cellular automata, 953

and rate equations, 984

for self-avoiding walks, 978

Meander patterns (mazes), 873

Meanders (in rivers), 1001

in Preface, xiii and reaction-diffusion, 1013 Meiosis, 970

Melting

of ice, 372

as phase transition, 981 Melting points

of alkanes, 1194 difficulty of computing, 1194 MemberQ (list membership test)

in aggregation systems, 978 Memo functions, 869, 906 Memory (animal)

in lower organisms, 623 and shell patterns, 1012 Memory (computer) emulated by CA, 663 and history of computing, 1108

in practical computers, 1108 and PSPACE completeness, 1142 Memory (extraterrestrial)

in Black Cloud, 1190 Memory (human), 1102 and free will, 753, 1136 and future technology, 1195 and human intelligence, 823 and human thinking, 621 inanimate analogs of, 1178 visual, 623

and weather prediction, 1177 Memory-based encoding, 565 Menger, Karl (Austria/USA, 1902-1985) and nested patterns, 934

Mental functions, 1099 see also Brain Menu interfaces, 1102

vs languages, 631 Mercury

advance of perihelion of, 1047 Meredith, Carew A (Ireland, 1904-1976) and axioms for logic, 1175

Meredith axioms for logic, 808, 1175 Merging

in multiway systems, 937 Meristems (in plants), 409, 1004 Mersenne, Marin (France, 1588-1648) and primes, 909

Mersenne primes and perfect numbers, 911 Mertens Conjecture, 910 Mesh generation and discrete spaces, 1050 and Voronoi diagrams, 987 Mesons (in particle physics), 1057 spins of, 1046

Mesopotamian art, 43, 873 Mesoscopic systems and quantum theory, 1058 Metabolism

and definition of life, 1178 Metamathematics, 1150 empirical, 1176 and Gödel's Theorem, 1158 Metaphysics

implications for, 1196 Metastable states and phase transitions, 341 and quantum measurement, 1063 Meteorites

and origin of life, 1179 Meteorology

and history of chaos theory, 971 Meter

speed of light and defining, 1041 Meter in poetry

rules for, 875

for Lorentzian spaces, 1051

for numerical data, 1101

Riemann tensor expansion of, 1049

in space of CA rules, 948

in space of CA states, 869

in unified field theory, 1028

volume density from, 1050

Metric spaces

networks as, 1030

Metric tensors, 1048

Metrics

for complexity of software, 1069

Michelson, Albert A (USA, 1852-1931)

and experimental math, 899

and Michelson-Morley experiment, 1041

Microwave background radiation

see Cosmic microwave background

Middle A

frequency of, 1079

Middle Ages

animism in, 1195

concept of microcosm in, 1196

understanding of nature in, 876

use of randomness by, 1192

Millen, Jonathan K (USA, 1942- )

and code 20 CA, 877

Miller, Stanley L (USA, 1930- )

and origin of life, 1179

Miller-Rabin algorithm (for PrimeQ), 1192

Millipedes

locomotion patterns of, 1011

Mimesis, 1178

Mind

see also Brain

see also Thinking

computing forms of, 1009

and deformable packings, 988

vs Einstein equations, 1052

and general study of form, 967

Minimal systems for particular purposes, 1186 for particular sequences, 1186 see also Shortest programs Minimal theories, 1186 Minimality

and recognition of purpose, 831 Minimization

Minkowski space, 1051 and quantum field theory, 1061

as solving Einstein equations, 1053 Minors (in networks), 1045

Minotaur home of the, 873 Minsky, Marvin L (USA, 1927- ) and discreteness of space, 1027

in Preface, xiii and register machines, 896, 1115 and simple Turing machines, 879, 889, 1143 and tag systems, 1120

and universal Turing machine, 1115, 1119, 1119' Miracles

and ultimate theory of physics, 1025 Mirror

as amplifier, 310 Mises, Ludwig E von (Austria/USA, 1881-1973) and free will, 1135

Mises, Richard von (Austria/USA, 1883-1953) and defining randomness, 1068 Missiles

radar pulses used by, 970 MIT

and my work on CAs, 880 and Paterson worms, 930 and precursors to my work, 879 and work on nesting, 934 Mitotic cell division, 970 Mixed states (in quantum theory), 1062 Mixing

of initial conditions in rule 30, 976 Mobile automata, 71-77

2D, 931 2D and causal networks, 1033 active cell motion in, 887 causal-invariant systems for, 1033 causal networks from, 1033 emulated by CAs, 657, 1111 emulating CAs, 664, 1112 experiments on, 112 generalized, 76 implementation of, 887 implementation of generalized, 887

on networks, 1040 random initial conditions in, 949 reversible, 1018

and time in universe, 486 Möbius transformations and complex maps, 933 and continued fractions, 914 and elliptic functions, 1093 Mod (modulo)

and algebraic forms for CA rules, 885 basic example of, 854

and computational reducibility, 747 computed by rule 132, 638 and cyclic addition systems, 950 difficulty of evaluating, 1134 emulated by continuous functions, 1129 encoded as integer equation, 1160 and formulas for repetition, 607 and idealized billiards, 1022 and primitive recursion, 907

in rolling motion, 971 TMs to compute, 1144 Mod 1

uniform distribution, 904 see also FractionalPart Mod 2 rule, 25

see also Additive cellular automata

personal experiences of, 991, 992

and Principle of Computational Equivalence, 728

see also Ultimate theory of physics

see also Simulations

system based on, 255

see also Mod

Modus ponens, 1150, 1151

and equational axioms, 1151

and equivalential calculus, 1170

and implicational calculus, 1170

MoebiusMu

difficulty of evaluating, 1134

and distribution of primes, 910

and maximal periods, 950

and complexity in biology, 390

and recognition of purpose, 1184

Molecular chaos assumption

and H theorem, 1020

Molecular dynamics

and cellular automaton fluids, 378

computer simulations of, 999

fluid equations from, 996, 999

and fracture processes, 995

and inspirational book cover, 864

unpredictable properties of, 1194

vibrational modes of, 1081

Mollusc shells

see Shells

Molluscs

classified by shell patterns, 1012

pigmentation patterns of, 389

models for space with, 1028 Monasteries

ornamental art from, 873 Monocotyledons (plants) branching in, 1004 symmetries in, 1007 Monoids

axioms for, 773 enumeration of, 805 and generalized additivity, 952 and multiway systems, 938

vs semigroups, 1153 universality of, 1159 Monolith in 2001, 1183 Monopodial branching (in plants), 400 Monotheism

and rejection of animism, 1195 and teleology, 1185

Monster Group, 938 and sphere packings, 987 Monte Carlo methods aggregation systems as, 332

as application of randomness, 1192 for CA conservation laws, 1022 and computational irreducibility, 1132 and digit reversal, 905

in Ising models, 982

in lattice gauge theories, 1061 and probabilistic models, 1083 quasi-randomness in, 1085 for satisfying constraints, 344 and stochastic models, 968 Moon

circular craters on, 1187 circular shape of, 861, 875 computing orbit of, 313, 1107 sending signals to, 1189 supposed life on, 1180, 1190 and three-body problem, 972 times of rising of, 973 Moore, Cristopher (USA, 1968- ) and P completeness, 1149 Moore, Edward F (USA, 1925- ) and 2D cellular automata, 928 Moore neighborhood (for 2D cellular automata), 177, 928 Moorish (Islamic) art, 874

Moral responsibility, 1136 Moral theories

and free will, 1135 Moray eel

pigmentation pattern of, 426 Mordell conjecture, 1164 Mordell equation, 1165 Morning glory leaves, 1006 Morphisms

in category theory, 1154

of words, 82-87 see also Substitution systems Morphogenesis (in biology), 400-422, 1003 undecidability in models of, 1138 Morphogenesis (of landscapes), 1001 Morphogens, 1004

Morphology

of biological systems, 386 history of, 1004 mathematical, 1077 Morphometrics, 1010 Morrison, Philip (USA, 1915- ) and SETI, 1189

Morse, H C Marston (USA, 1892-1977) and substitution systems, 893 Morse code, 1069

Morse-Thue sequence, 83 see also Thue-Morse sequence Mortality problem for matrices, 1138 Mosaics, 873

Moseley, Henry (England, 1801-1872)

Mu-law sound encoding, 1080

mu operator (in general recursion), 907, 1121

Muchnik, Albert A (Russia, 1934- )

and intermediate degrees, 1130

Mud

and spontaneous generation, 1179

Mug (circular reflector)

caustics from, 984

Mules

as not self-reproducing, 1178

Mullins, William W (USA, 1927-2001)

and dendritic growth, 993

Multichannel analysis

in SETI, 1189

Multifractals

generalized dimensions for, 959

and history of fractals, 934

Multilayer logic, 1096

Multinomial (multinomial coefficients)

and additive CAs, 1092

patterns generated by, 870

recursive algorithm for, 1142

see also Times

Multiplication systems, 257

reversibility in, 1018

state transition graphs for, 961

Multiplication tables

CAs based on, 614, 886

consistent with axioms, 805

for groups, 945, 1172

see also Operator systems

Multiplicative digit sequences, 902

Multiplicative processes

and lognormal distribution, 977

MultiplicativeOrder

and additive CAs, 951, 1133

and cyclic multiplication, 950

difficulty of computing, 1147

and digits of rationals, 912

and linear congruence periods, 974

as period in doubling system, 257

and quantum computers, 1148

Multipole moments

and isotropy, 980

Multiregister machines, 1114

Multiresolution analysis, 1074

Multispin coding (bitwise optimization)

and cellular automata, 866

and branching of time, 504

canonical forms in, 1036

causal invariance in, 507

vs causal networks, 1037

and cell division in animals, 1009

and chemical synthesis, 1194

compared to sequential substitution systems, 204

confluence in, 1036 convergence in, 1036 emulated by predicate logic, 1159 emulated by TMs, 765

emulating multicolor, 1115 equivalences in, 1172 frequency of behavior in, 938 generating Euclidean spaces, 1036 and Gödel's Theorem, 782 halting problems for, 1137 history of, 938

limited size versions of, 939

as models of math, 776 multidimensional, 939 and NP completeness, 1145

vs operator systems, 1169 and quantum mechanics, 771 random initial conditions in, 949 randomness in, 205

reversible, 1018 sampling of as axiom systems, 795 searching for paths in, 1157 simulation as NP problem, 1142 spacetime networks from, 1036 states generated by, 207 and string matching, 1101 terminating and confluence, 1036

in terms of operators, 1172 undecidability in, 1136, 1137 word problems in, 1141 Multiway tag systems, 1141 and correspondence systems, 1140 Munching foos

as precursors to my work, 879 Munching squares program, 871 Municipal districting

and Voronoi diagrams, 987 Muon

mass of, 528, 1046 models of, 1044

as type of particle, 1043 Murray, James D (England/USA, 1931- )

in Preface, xiii and reaction-diffusion, 1013 Muscles

CA models for, 878 control of, 625, 1099

as excitable medium, 1013 patterns of cells in, 385, 1007 randomness in twitching of, 1011 Music

and auditory perception, 585

in bird and whale songs, 826, 1180 chords in, 917

and extraterrestrials, 1191 frequencies in, 1079 grammatical rules in, 875, 1181 and math thinking, 1177 scores from programs, 1080

on Voyager spacecraft, 1190 Musical chairs

randomness in, 968 Musical instruments and nonlinear oscillators, 971 shapes of, 1183

Mutations (biological) effects of, 390 Mutual information and defining complexity, 1069 Mycoplasma bacterium, 1179 Myhill, John R (USA, 1923-1987) and firing squad problem, 1035 Myhill-Nerode theorem (minimal finite automata), 957, 957' Mysticism

and combinatorics, 1082 and non-Western thinking, 1196 and universal objects, 1127 Mythology

as models, 992 and origin of complexity, 861 n-body problem

gravitational, 1021

n Log[n] algorithms and associative evolution, 1095

in automaton minimization, 957

in evaluating powers, 1093

for pi, 912

for PrimeQ, 1090

in sorting, 1142

in Walsh transforms, 1073

see also Computational complexity theory

Nakaya, Ukichiro (Japan, 1900-1962)

forms for cellular automata, 619

interesting theorems for, 819

lengths of proofs involving, 1175

as motivating combinators, 1121

in multivalued logic, 1175

as operation on sets, 1171

and reversible logic, 1098

single axiom for, 808

tautologies based on, 781, 1157

cellular automata and, 841, 869

and human future, 1196

and new forms of perception, 1106

summary of relations to, 15

Napier, John (Scotland, 1550-1617)

and binary numbers, 902

Natural equations (curves defined by curvature), 1009

Natural language

see Languages (human)

Natural language query systems, 1100

and extraterrestrial life, 1180

as force of limited power, 392

and games, 1104

history of, 1001

and leaf arrangements, 408

as non-math theory, 859

as not producing complexity, 392

of orbits in solar system, 973

and shapes of leaves, 404

and shapes of shells, 417, 1008

and shell patterns, 425

summary of relations to, 14

and teleology, 1185

vs thermodynamics, 1003

value of intelligence in, 1191

Natural theology

and history of complexity, 861

and intelligent universe, 1195

as subsonic approximations, 997

in two dimensions, 999 Navigation beacons radio signals from, 1188

NC (computational complexity class), 1142, 1149 and Boolean minimization, 1096

NDFA (non-deterministic finite automaton), 957 see also Finite automata

NDSolve and curves from curvature, 1009 difficulty of evaluating, 1134 and PDEs, 924

and Sitnikov problem, 973 and three-body problem, 972 NDTM

see Non-deterministic Turing machines Nearest-neighbor algorithms, 1101 Nebulas

patterns in, 835, 1187 Necklaces

and maximal periods, 950 and shift rules, 963 Needle

Buffon's for pi, 1192 Needle-like crystals, 372 Negation

in multivalued logic, 1175

in multiway systems, 796 notion of, 780, 1158 Negative bases, 902 power CAs and, 1093 Negative numbers representation of, 902, 942 Negatively curved space, 531 and automatic groups, 1141 and chaos theory, 971 divergence of geodesics on, 1049 and hyperbolic networks, 1049 Negotiations

and price determination, 430 Neighbor-dependent substitution systems, 85-87 Neighbor-independent network rules, 509, 1037 Neighbor-independent substitution systems, 82-85 Neighborhoods

possible in cellular automata, 929 symmetry classes of, 928 Neon discharge tubes randomness from, 969 Nerve cells, 1075 determinism in, 1135

as excitable media, 1013 and feature extraction, 623

in inner ear, 1079 and mollusc shell patterns, 1012 monitoring with sound, 1080 repeatable randomness in, 976 and song generation in birds, 1180 structure of and free will, 751

in visual system, 578 and Weber's law, 1014 Nest

basic example of, 853 and branching patterns, 1006 constructing expressions with, 897 and lambda calculus integers, 1121 and recursive functions, 1121 Nested expressions, 102 Nested radicals, 915 Nested sequences block frequencies in, 594, 1084 compression of, 566

as initial conditions, 272, 956 pointer-based encoding of, 1071 shown in scan lines, 892 Nested tilings, 219 Nesting and 1/f noise, 969

in biological systems, 384

in bitwise functions, 871

in cellular automata, 25, 57

and circle packings, 986

and computational reducibility, 741

in continuous systems, 1130

in Cosmati mosaics, 873

in curves from curvature, 1009

in cyclic tag systems, 96

and defining arithmetic, 1164

in digit count sequences, 905

and Principle of Computational Equivalence, 722

from projections of lattices, 932

and random walks, 977

understanding operation of, 1177

NestWhile

and computation of pi, 912

and concatenation sequences, 913

and context-free parsing, 1103

and encoding of lists, 1121

and general recursion, 907

NestWhileList

and length prefixed numbers, 1070

and network distances, 1031

algebraic systems based on, 1172

approximating flat space, 477

approximating spheres, 1049

for attractors, 276

causal invariance for, 515 and Cayley graphs, 1032 cellular automata on, 930, 936 chemical analogies for, 1040 and chemical properties, 1195 and chemical synthesis, 1194 chromatic number of, 1029 circumference of, 1029 colored, 1029, 1039 coloring of and dimension, 1031 coloring of and planarity, 1040 conditions for planarity of, 1045 connectedness of, 1039

of contacts between circles, 986 from continuous space, 533 from continuous systems, 1031 continuum limits of, 1030 and cosmological horizon problem, 1055 curvature in, 532

and cybernetics, 862

de Bruijn, 940 defined by constraints, 482 diameter of, 1029

as difficult to understand, 1177 dimensions of and growth rates, 478 eigenvalues of distance matrix for, 1031 evolution of, 508-515

face distribution in random, 1038 Feynman diagrams as, 1060 girth of, 1029

growth rates on, 1031 homogenous, 1032 implementation of, 1031, 1037 layouts of, 193, 476, 1031 localized structures in, 525, 1045 and Markov processes, 590, 1084 mobile automata on, 1040

as models of space, 475-480 molecules as, 1194 from multiway systems, 209 Nand, 1096

nested, 197 non-overlapping, 515 non-planarity in planar, 527

NP completeness of equivalence of, 1146

NP completeness of matching in, 1038, 1145

NP completeness of path finding in, 1146 number of replacements for, 1038 numbers of trivalent, 1029 overlaps in, 515 planar, 1038 planarity in evolution of, 515 and quantum information, 544 random, 963, 1038

random causal, 1052 random replacements in, 1038 random walks on, 1030 relations between types of, 1037 replacements in directed, 1040 reversible evolution of, 1040 rules for getting to any, 1038

of shift register states, 1089

of signs in languages, 1181 sorting, 1142

state transition, 961 substitution systems, 508-515 symbolic representation of, 1040 symbolic systems based on, 898 and systems theory, 862

of theorems in math, 820, 1176 trivalent as covering all, 476 trivalent examples of, 476 trivalent in 3D, 1030 see also Boolean networks Neumann, Bernhard H (England/Australia, 1909- ) and axioms for groups, 1153

Neural networks, 1102 for bird songs, 826 for design optimization, 1193 and finite automata, 958 history of, 1099 and history of CAs, 878 and history of statistics, 1083 and history of universality, 1110 and memory, 624

and my work on CAs, 880

and visual perception, 1076

Neurons

in brain, 1098

in cybernetics, 862

properties of, 1075

repeatable randomness in, 976

see also Nerve cells

identified with gravitons, 1054

limit on number of, 1047

and parity violation, 1019

radio signals from, 835, 1188

New Age movement

and animism, 1195

New moons, 973

New Zealand

circular shape in, 1187

Newcomb, Simon (USA, 1835-1909)

and leading digits, 914

Newell, Allen (USA, 1927-1992)

and automated proofs, 1157

Newspapers

reporting canals on Mars, 1180

Newton, Isaac (England, 1642-1727)

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

attractor basins in, 1101

for computing square roots, 913

and gradient descent, 985

and iterated maps, 918

as iterative algorithm, 1141

NeXT (computer)

and creation of this book, 854

Nicod, Jean (France, 1893-1924)

and axioms for logic, 1151

Night sky

observed darkness of, 455

and Olbers' paradox, 1021

effect of on transition to turbulence, 996

effect on CAs of, 325, 976

in electronic devices, 302, 968

in natural radio emissions, 1187

produced by crack propagation, 374

radio in galaxy, 836

and time series, 1083

in vacuum tubes, 971

see also Randomness

Nomography (Hilbert's thirteenth problem), 1128

Non-Abelian groups

CAs based on, 887, 956

and charge quantization, 1046

in gauge theories, 1045, 1057

see also Groups (mathematical)

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 Non-computable patterns from constraints, 943 Non-computable reals, 1128 Non-constructibility and additive functions, 953

in set theory, 1154 Non-determinism and multiway systems, 939 Non-deterministic finite automata, 957 Non-deterministic polynomial time (NP), 1142 Non-deterministic Turing machines and aggregation systems, 979 emulated by CAs, 766, 1146 Non-equilibrium thermodynamics and history of complexity, 862 Non-Euclidean geometry, 1149 and abstraction in math, 792 and history of space, 1028 Non-locality

and Bell's inequalities, 1064

in digit sequences, 730 Non-measurable sets, 1154 Non-overlapping networks, 515 Non-overlapping strings, 503, 1033 and completion algorithms, 1037 total number of, 1036

Non-periodic crystals (quasicrystals), 994 Non-periodic tilings, 219

Non-standard analysis, 1172 and axioms of analysis, 1155 Non-standard arithmetic, 800, 1169 Non-standard integers

as generalizing numbers, 1168 Non-verbal communication, 1181 Non-Western thinking, 1196 Noncontradiction law

in logic, 817 Nonlinear differential equations and history of complexity, 862 for ModularLambda, 1093 and special functions, 1092 Nonlinear dynamics summary of relations to, 15 Nonlinear feedback shift registers, 1088 and cryptography, 1085

and history of CAs, 878

as precursors to my work, 879 see also Shift registers Nonlinear field theories topological defects in, 1045 Nonlinear maps

in complex plane, 933

on the interval, 918 Nonlinear models and time series, 1083 Nonlinear oscillators arrays of, 921 and history of chaos, 971 Nonlinear Schrödinger equation, 925 Nonlinear wave equations, 165, 923 Nonlinearity

and computational irreducibility, 1133 and equation solving, 940

and interactions in field theory, 1061

in PDEs, 923 Noodles and chaos theory, 306 Nor

as universal function, 807, 1173 words for in languages, 1173 Nordahl, Mats G (Sweden, 1958- )

in Preface, xiii and universal CAs, 1115 Normal algorithms (sequential substitution systems), 894 Normal coordinates

and Riemann tensor, 1049 Normal distribution

and concatenation sequences, 913

and defining randomness, 1068

Normal ordering (in quantum field theory), 1062

for symbolic expressions, 896

used in this book, 852

Novikov, Petr S (Russia, 1901-1975)

and word problem for groups, 1141

NP (non-deterministic polynomial time), 1142

NP completeness, 766, 1145

and approximate solutions, 984

and automated proofs, 1157

in Boolean minimization, 1096

in cellular automata, 767

and combinatorial optimization, 985

and cryptography, 1086, 1089

density of difficulty in, 1147

of finding addition chains, 1095

of finding discrete packings, 987

of finding network genus, 1045

of finite PCP, 1139

of knot equivalence, 1046

in Mathematica, 1143

and network evolution rules, 1038

and proof lengths in logic, 1175

and proof searching, 1157

and quantum computers, 771, 1148

relation to undecidability of, 769

and computer simulations, 1133

and extraterrestrial contact, 1190

and origin of substitution systems, 893 and Peano axioms, 1152

and quadratic congruential generators, 975 spectra in, 1081

and Turing machine 600720, 1145 undecidability in, 791, 1138 unsolved problems in, 1166 Numbering

of CA rules, 866, 883

of TM rules, 888, 1143 Numbers

bases of, 116 chaos theory and, 309 and computational reducibility, 743 equivalence problem for, 916, 1138 generalizations of, 1168

historical notation for, 1182 history of, 901

models based on, 365 natural distributions of, 1070 normal, 912

operator representations for, 916 prime, 132

and Pythagoreans, 859 recognition of, 1089 systems based on, 115-168 tree representations for, 916 Numbers games, 969 Numerical analysis and chaos experiments, 919 and continuous CAs, 921 for differential equations, 924 and repeatability, 898 Numerical computation artifacts in, 1184 characterizing difficulty of, 1143

in fluid flow, 996

of gravitational fields, 1053 and history of computing, 1108 and weather prediction, 1178 Numerical data

lookup of, 1101 Numerical integration Monte Carlo methods in, 1192 searches for optimal algorithms in, 1193 Numerology, 1025

Nuts segregation of mixed, 986 Object-oriented programming, 867 Object recognition, 624

Objects described by axiom systems, 797 and image compression, 1074 Observables

in quantum mechanics, 1056 Observational errors

and Gaussian distribution, 977 and origin of randomness, 967 Observers

and Anthropic Principle, 1026 computational abilities of, 737 definition of particles for, 1062 and origin of causal networks, 487 Occam's razor, 1025

Ocean currents and weather, 1177 Ocean waves, 1001

as example of randomness, 301 nested breaking of, 997 pattern formation in, 947 repetitive behavior in, 988 Oceans

exploration of, 1191 finding artifacts under, 1183 whale songs in, 1180 Ocelot

pigmentation pattern of, 426

form of tentacles of, 385

visual system of, 1075

Ocular dominance stripes

CA model for, 1012

maze-like patterns in, 1013

in visual system, 1075

OddQ (test for odd numbers)

and nested patterns, 931

and rule 132, 638

Odds and Evens game, 1105

ODEs, 922

see also Ordinary differential equations

Odlyzko, Andrew M (USA, 1949- )

and additive CAs, 951

visible from space, 1187

Olaus Magnus (Sweden/Italy, 1490-1568)

Omega (halting probability), 1067

omega (first transfinite ordinal), 1162

One-sided additivity

and cryptography, 1088

repetition from, 954

One-time pad (in cryptography), 1086

One-to-one mapping (injective), 960

Ones

expressing numbers with, 916

number of (DigitCount), 902

Onsager, Lars (USA, 1903-1976)

and 2D Ising model, 982

Onto cellular automata, 280, 959

and concept of halting, 1137

and history of computing, 1108

automated proofs in, 1158

evolution rules for, 898

frameworks for proofs in, 1177

CA implemented with, 869 and quantum computers, 1148 Optical fibers

as technology component, 1195 Optical illusions, 1076

Optical SETI, 1189 Optics

and history of math in science, 859 path integral in, 1061

Optimal algorithms, 1141 Optimal evaluation chains, 1095 Optimality

and defining purpose, 829 and models of markets, 1015 Optimization

combinatorial, 985 and complexity in biology, 398

of designs in engineering, 1193 intuition about in programs, 872 and Occam's razor, 1025

in plant leaf arrangements, 408 Optimized circuit blocks, 1193 Optimizers

and proofs about programs, 1168 search strategies in, 1193 Options

market prices of, 1015

Or axioms in terms of, 1151 and Boolean formulas, 1095 calculus of expressions, 1170 emulated by CAs, 662, 1112 encoded as integer equation, 1160

in multivalued logic, 1175 and P completeness, 1149 and satisfiability, 1146 table for, 806 theorems involving, 817 words in languages for, 1173 Oracles (in computation theory), 1126 and arithmetic hierarchy, 1139 based on infinite tables, 720 and intermediate degrees, 1131 and non-standard arithmetic, 1169 and second-order logic, 1167 Orbits

in Bohr atom, 1056

of comets and Gaussians, 977 and history of computing, 1107 randomness in, 313

as simple shapes, 1187

in three-body problem, 972 Order

emergence from randomness, 223 Order diagrams of posets

and causal networks, 1040 Order of updates

in sequential CAs, 1034

in sequential substitution systems, 894

in string rewriting, 497

in symbolic systems, 898 see also Causal invariance Order parameters

in phase transitions, 981 Ordered fields

axioms for, 1153 Orderless (commutative) and associative CAs, 956 Boolean functions, 1173 and generalized additivity, 952 and speedups in evolution, 1095 Orders of infinity, 1127, 1162 Ordinal numbers

and encoding of proofs, 1163

as generalizing numbers, 1168 and notions of infinity, 1162 Ordinary differential equations (ODEs), 922 for chemical processes, 1013 Oresme, Nicole (France, ~1320-1382) and time as a dimension, 1031

Organization

see Self-organization

Organizational structures

and history of complexity, 862

and Occam's razor, 1025

Organogenesis in plants, 409

Organs

and biological evolution, 389, 1002

separate rules for different, 383

grammatical rules in, 1181

and notion of purpose, 830

rules in, 872

and theory of form, 967

Orreries, 1184

Orthogonal bases, 1072

and image compression, 573

and radio signals, 1188

Oseen, C William (Sweden, 1879-1944)

and fluid flow past spheres, 998

Oster, George F (USA, 1940- )

and patterns on shells, 1012

and branching patterns, 1005

and enumerating expressions, 897

and metrics on surfaces, 1048

and shape tensors, 980

Outer totalistic rules

and initial conditions, 866

Painlevé, Paul (France, 1863-1933)

and Painlevé functions, 1092

Painlevé functions

and 2D Ising model, 982

and nonlinear ODEs, 1092

and random matrices, 977

Paintings

and concept of intelligence, 838

nesting from, 989 Pair correlations and isotropy measures, 980 Pair production

in 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, 1907-1933) and Walsh transforms, 1073

Paley, William (England, 1743-1805) and natural theology, 861 Paley family

of Hadamard matrices, 1073 Paley order

for Walsh functions, 1073 Palindrome systems, 125-127 Palindromes

in period-doubling sequences, 892 Palm trees

growth of, 1004, 1004' phyllotaxis in, 409 Pandas

patterns on, 426, 1013 Pangolins

patterns in scales of, 1007 Panini (India, ~500 BC) and grammars, 875 Pantheism, 1196 and physics as intelligence, 1192 Paper

crumpling of, 996 fluttering of, 971 Paperfolding and dragon curve, 932 rules in, 875 sequences from, 892 Parabolic equations, 925, 940 Parabolic reflectors characteristic shapes of, 1183 Paraboloids

geodesics on, 531 plant stem tips as, 1007 Paradoxes

and Gödel's Theorem, 1159 and history of math, 1149

in set theory, 1154 and undecidability proofs, 1137 Paraffins, 1194

Parallel bit operations, 866 Parallel computation

in cellular automata, 1109 and creation of this book, 854 difficulty of understanding, 1177 and history of CAs, 876 and implementing CAs, 868 and machine intelligence, 1100

of mathematical functions, 1134 and memory lookup, 622 and my work on CAs, 46 and NC, 1142 and network evolution, 1035 and P completeness, 1149 randomized methods in, 1192

of repetition periods, 1147

vs sequential, 765

in sorting networks, 1142 and work on CA fluids, 999 Parallel postulate

in Euclidean geometry, 1154 Parallel transport (in curved spaces), 1049

on networks, 1051 Parallel universes and branching in time, 1035 Parallel updating

in CA programs, 866 perceived by observer, 487 Parameter estimation, 1083, 1083' Parameter space sets

for activator-inhibitor systems, 428 for animal shapes, 421

for CA patterns, 956 for CA rules, 948 for continuous CAs, 158, 243, 922

for shell growth, 415

for shell shapes, 1008

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, Jeffrey B (England, 1944- )

and unprovable statements, 1163

Partial differential equations (PDEs), 161-164

for 3D tearing-free growth, 1010

compression of time in, 732

computation with, 732

conservation laws in, 1023

derived from CAs, 1024

Einstein equations as, 1053

emulating cellular automata, 1129

existence of solutions to, 940

for fluid flow, 996

and fracture processes, 995

higher-dimensional, 923

higher-order, 940

and history of CAs, 877

isotropy in, 980

and noisy cellular automata, 976

nonlinear studied by Turing, 1012

origin of, 923

and quantum field theories, 1061

in quantum theory, 1056

for reaction-diffusion, 1012

repetitive behavior in, 988

and self-reproduction models, 876

for solidification fronts, 993

for surfaces from curvature, 1009

Partially additive rules

repetitive behavior in, 954

spacetime entropies for, 960

Partially ordered sets (posets), 1040

as example in lattice theory, 1153

Particle accelerators

and discovery of particles, 1044

time dilation in, 524

relativistic invariance in, 1043

time reversal violation in, 1019

see also Elementary particles see also Localized structures Partition

and 1D constraints, 940 basic example of, 853 and counting blocks, 1022 and implementation of CAs, 867 and recursive multiplication algorithm, 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, 1623-1662) 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 nested patterns, 610, 870

as precursor to my work, 878 and prosody, 875

Passerines (songbirds) songs of, 1180 Passwords

as random seeds, 970 Patch entropies, 960 Patches of texture, 578 Patching

of programs, 872 Patent

related to rule 30, 973 Paterson, Michael S (England, 1942- ) and 2D Turing machines, 880, 930 Paterson worms

as precursors to my work, 880 Path independence

see Confluence Path integrals history of, 1057

in quantum field theory, 1061 for quantum gravity, 1054 and random networks, 1039 and statistical mechanics, 1061 Path metric spaces

networks as, 1030 Paths

representing substitution systems, 892 Paths in multiway systems

convergence of, 1036 independence on, 507 and non-determinism, 765 Paths in symbolic systems dependence on, 898 Pattern-avoiding sequences, 944

as extraterrestrial signals, 1190

vs Ramsey theory, 1068 Pattern formation

in biology, 422-428 history of, 947 and history of complexity, 862

in physics, 369-382 from randomness, 223 see also Cellular automata, etc

Pattern matching and game strategies, 1105 see also Matching Pattern recognition, 549

Patterns (in Mathematica)

with as NP-complete, 1143 basic examples of, 854 and human thinking, 1103 and implementing CAs, 867 and network evolution, 1037 and proof strategies, 1156 Patterns (visual)

perception of, 577 Patterns of pigmentation, 422-429 PCE

and history of CAs, 877

and munching squares, 871

PDP-7 computer

and history of CAs, 877

Peacock

pigmentation pattern of, 426

Peacock, George (England, 1791-1858)

and generalization in math, 1168

Peaks

in spectra, 586

Peano, Giuseppe (Italy, 1858-1932)

and axioms for arithmetic, 1152

and foundations of math, 1149

and math notation, 1182

and space-filling curves, 893

Peano arithmetic

axioms for, 773, 1152

emulating with groups, 1159

encoding proofs by ordinals, 1163

and Fermat's Last Theorem, 1166

and finite axiomatizability, 1176

and Gödel's Theorem, 1159

and Goodstein sequences, 1163

non-standard models for, 1169

and proofs of universality, 1127

undecidability of consistency of, 1168

see also Arithmetic

Pebbles

coming to surface of sand, 986

and origin of word "calculus", 925

prehistoric computing with, 1107

Peephole set (parameter space set for trees), 407, 935, 1006

Pegboards

randomness in, 312, 968

Peirce, Charles S (USA, 1839-1914)

and axioms for logic, 1151

Penny Matching game, 1105

Penrose, Lionel S (England, 1898-1972)

and mechanical self-reproduction, 1179

Penrose, Roger (England, 1931- )

and discreteness of space, 1027

and Penrose tiles, 932, 943

and polyomino tilings, 943

in Preface, xiii

and spin networks, 1055

Penrose tilings, 932

cellular automata on, 930, 1028

diffraction patterns of, 1082

see also Quasicrystals

and branching in time, 505

and extraterrestrial intelligence, 836

and familiarity of features, 1105

higher forms of, 632-635

and Principle of Computational Equivalence, 722

and recognizing meaning, 826

relations to NP completeness, 771

of time in universe, 487

Percolation and Boolean networks, 937

as model of drainage, 1001

as phase transition, 983 and structure of class 4, 948 Percussion instruments, 1079 Perfect cuboid

as unsolved problem, 1166 Perfect fifth (musical chord) curve of, 146

perception of, 1079 Perfect numbers, 911 and iterated aliquot sums, 911

in ordering of math constructs, 1177 unsolved problem about, 1166 and zeros of sequence, 135 Perfect shuffles, 974 Perfumes

odors in, 1105 Period 3 in iterated maps, 955 Period-doubling sequence

in iterated maps, 921 spectrum of, 1081 from substitution sequence, 892 Periodic behavior

origins of, 354-356 see also Repetitive behavior Periodic points

enumeration of, 958 enumeration of in CAs, 954

in shift rules, 963 Periodic structures see Localized structures Periods

see Repetition periods Periods (number type), 916 Periostracum, 1011 Peristalsis

as repetitive process, 1011 Periwinkle shell

growth of, 415 perl (computer language), 894 Permanence

Principle of, 1168 Permanents and NP completeness, 1145 Permutation test, 1085 Permutations and group theory, 1153 and reversible block CAs, 1023 and reversible mobile automata, 1018

in sestina rhymes, 875 Perpetual motion machines

in quantum field theory, 1062

in thermodynamics, 447 Perrin sequence (generalized Fibonacci sequence), 891 Perron numbers, 958

Persian (Islamic) art, 874 Persian gardens and nested patterns, 874 Persian religions

and free will, 1135 Persistent structures

in class 4 systems, 281-296 see also Localized structures Personal computers (PCs) and history of computing, 1108

my use of, 854 Perturbation series computational difficulty in, 1146 and computational irreducibility, 1133

in quantum electrodynamics, 1056, 1060 Perturbations

continual in CAs, 947 and effect on turbulence, 996

in initial conditions, 250 and randomness generation, 323 sensitivity of complexity to, 1002 Peru

mazes in, 873 Petals (in flowers) arrangement of, 409 growth of, 412 Petersen network examples of drawing, 476 Petri nets

melting in alkanes as, 1194

from molecular dynamics, 999

and nesting, 989

in one dimension, 983

in percolation, 983

in probabilistic CAs, 591, 976

and randomness tests, 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, 1933- )

and minimum area packings, 988

discussion of math in, 860

and human thinking, 1099

and idea of math in science, 859

implications for, 1196

of language, 1181

and pragmatic theory of law, 1136

and quantum mechanics, 1058, 1135

structure of proofs in, 1151

summary of relations to, 10, 863

and theories of communication, 1181

and universe as intelligent, 1195

and visual perception, 1076

writing style in, 849

Philosophy of science

issues of modelling in, 991

and methodology of this book, 1197

and vacuum fluctuations, 1062

zero mass of, 1046

vs mathematics, 1026 mechanistic explanations in, 860 methods applied to biology, 1003

my early work in, 18 notions of purpose in, 1185

as origin of PDEs, 923 and Principle of Computational Equivalence, 720 quantum, 537, 1056-1065

reversibility of laws of, 435, 1019 satisfying constraints in, 349 success of math in, 859 summary of relations to, 8863 textbooks and Second Law, 1020 ultimate theory of

see Ultimate theory of physics Physiology

and free will, 1135 see also Biology

Pi approximations to, 912 block frequencies in digits, 594 Buffon's needle and, 1192 computation of nth digits in, 912 continued fraction for, 143, 914 difficulty of evaluating, 1134 digit sequence of, 136

as early example of complexity, 48 frequencies of digits in, 912 generating as purpose, 1185 and history of randomness, 967 and random networks, 963 rational approximations to, 1134 from rational integral, 916

as rule 60 initial condition, 1091 and SETI, 836, 1190

Pi_2 questions, 1126 Pi_n sets, 1139 Piaget, Jean (Switzerland, 1896-1980) and animism in children, 1195 Piano

CA cells as keys on, 869 frequencies on, 1079 Pictures

as alternative to notation, 853 deductions from, 577 printing of in this book, 852 Pie charts

of classes of behavior, 948 Piecewise linear maps, 921 Piecewise linear spaces (cell complexes), 1050 Piezoelectricity

as element in technology, 1195 Pigmentation patterns (of organisms), 422-429 communication through, 1181

and complexity in biology, 384 randomness in, 970

in similar organisms, 389 Pilot waves (hidden variables), 1058 Pinball

and randomness from initial conditions, 312 Pinecones

phyllotaxis in, 409 Pingala (India, ~200 BC) and Fibonacci numbers, 890 Pingala's method

for computing powers, 1093 Pink (1/f) noise, 969

see also 1/f noise Pioneer 10 spacecraft, 1189

Pitting (hoppering) in crystals, 993

Pitts, Walter H (USA, 1923-1969)

and neural networks, 880, 1099

and universality, 1110

Pixels

in bitmaps, 1108

displays made from, 46

and printing of this book, 852

PKZIP (compression program), 1069

Planar Feynman diagrams

and QCD, 1040

and quantum gravity, 1054

and random networks, 1039

Planar networks, 195, 1038

Planarity

vs dimension for networks, 1038

generalizations of, 1045

and network evolution, 515

particles and in networks, 526

Planck, Max K E L (Germany, 1858-1947)

and quantum theory, 1056

new and gravity theory, 1047

orbits and role of God, 861

rules for motion of, 860

diversity of pollen in, 1011

growth compared to animals, 420

and electric breakdown, 995

and natural radio emissions, 1187

and pulsar signals, 1188

metric for surface in, 1048 Plouffe, Simon (Canada, 1956- ) and computation of pi, 912

in Preface, xiii Pluperfect numbers, 911 Plus (+)

combinator for, 1122 and NC computations, 1149 primitive recursive definition of, 907 see also Addition

PN (pseudonoise) sequences, 1084 see also Shift registers Podolsky, Boris (USA, 1896-1966) and EPR experiment, 1058 Poetry

and Fibonacci numbers, 891 rules in, 875

Poincaré, J Henri (France, 1854-1912) and 3-body problem, 972

and cell complexes, 1050 and chaos theory, 971 and iterated maps, 918 and notion of complexity, 1068 and Poincaré recurrence, 1022 Poincaré disk (negatively curved space), 1049 Poincaré recurrence

in cellular automata, 267 and thermodynamics, 1022 Point defects, 1045

Point location and Voronoi diagrams, 987 Pointer-based encoding, 565 implementation of, 1071 and memory, 1100 Pointers

and hashing, 622, 1100

in implementation of CAs, 866 Poisson's equation

and Einstein equations, 1052 Poker test, 1085

Polar plots and shapes of leaves, 1006 Polarization

and Bell's inequalities, 1064

as spin direction, 1046 Poles (in complex plane) undecidability of, 1177 Polish notation for expressions, 896

in logic, 1173 Political boundaries

as visible from space, 1187 Political history

and human languages, 1103 Politzer, H David (USA, 1949- ) and particle masses, 1047

in Preface, xiv Pollard, John M (England, 1941- ) and integer factoring, 1090 Pollard rho method, 1090 Pollen

and Brownian motion, 302 and circle packings, 985 forms of, 385, 1011 Polling

as application of randomness, 1192 Pólya, George (Hungary/Switzerland/USA, 1887-1985) and zeta function zeros, 918

Polyadic groups, 1171 Polycrystalline materials, 993 PolyGamma (polygamma functions) from sums, 917

Polygenes, 1003 Polygons and constructible reals, 1129 produced by 2D CAs, 929 regular allowing tilings, 943

in Voronoi diagrams, 987 Polyhedra

as Plato's model for space, 1028

as shapes of dice, 971

as shapes of pollen grains, 1011

in Voronoi diagrams, 987

and Feynman diagrams, 1060, 1133

see also Quartic equations

see also Quintic equations

Polynomial factoring

see Factor

Polynomial growth

in substitution systems, 890

Polynomial space (PSPACE), 1142

Polynomial time (P) computations, 762, 1142

and additive CAs, 610, 951

canonical forms of, 1037

and continued fractions, 914

and difference tables, 1091

digit sequences from, 914

as generalizing numbers, 1168

generating primes with, 909

iterated, 1098

iteration to find roots of, 1101

machines for computing, 1107

as models, 1083

and nested patterns, 610

sets of values taken on by, 1161

and shift registers, 975

universal, 1161

Polyominoes, 943

Polytopes

in color space, 1074

and trivalent networks, 1029

Pomeau, Yves (France, 1942- )

and cellular automaton fluids, 999

in Preface, xiii

Pompeii

maze patterns at, 873

Pond snail shell

growth of, 415

Ponzano, Giorgio E (Italy, 1939- )

and spin networks, 1055

Popcorn

segregation of sizes in, 986

Popper, Karl R (Austria/New Zealand/England, 1902-1994)

and free will, 1135

Popping

in natural radio emissions, 1187

Population count (DigitCount), 902

Population studies

and history of complexity, 861

and history of statistics, 1082

Pores in sphere packings, 986

Posets (partially ordered sets), 1040

as example in lattice theory, 1153

Position

and aggregation models, 978

basic example of, 853

and nested patterns, 931

and truth tables, 1170 and undecidability, 1136 and undecidability of word problem, 1141 and underivability of logic axioms, 1170 and universality, 1125

Post Correspondence Problem (PCP), 757, 1139 density of difficult instances of, 1147 Post tag systems

see Tag systems Postmodernism, 1196 Post's problem and intermediate degrees, 1130 PostScript (page description language) and computer communication, 1182 and production of this book, 852 Power

see Powers Power cellular automata, 1093 Power-free sequences (repetition-free sequences), 944 Power laws

for crushed rock, 995 and fractal dimensions, 933

in spectra, 969 for turbulent fluid flow, 997 Zipf's law as example of, 1014 Power lines

radio emissions from, 1188 Power of axiom systems and ordinal numbers, 1163 Power spectra

of natural processes, 969 properties of, 1080 see also Spectra Power towers and Ackermann function, 906 and symbolic systems, 897, 897' Power trees, 615

PowerMod and randomized primality testing, 1192 and RSA cryptography, 1090 Powers

combinator for, 1122

of complex numbers, 1094 computation of, 615, 1093 computational reducibility in, 749 digit sequences of, 119, 614, 749 encoded as integer equation, 1161 fractional parts of, 121, 903 and linear congruential generators, 318 with nearby values, 1166

in ordering of math constructs, 1177 and primitive recursion, 907 see also Arrow function (nested power) Powers of 2

cellular automaton for, 614, 749 Powers of 3

cellular automaton for, 661 digits in, 119, 903 and linear congruential generators, 318 Powers of 3/2

in continuous CAs, 157 digits in, 903 Powersets and cardinality of reals, 1127

in finite set theory, 1171 Practical computers and concept of halting, 1137 and discrete models, 369 emulated by CAs, 661 history of, 1107 intuition from, 716 and universality, 644 workings of, 1108 Pragmatic theory of law, 1136 Pragmatics

and theories of communication, 1181

automated proofs in, 1158

axioms based on, 1159

and Principle of Computational Equivalence, 6

as requirement for science, 1196

of robots in science fiction, 629

and thermodynamics, 447

of weather, 1177

Prediction

of data, 549

evolving methods for, 1105

and game strategies, 1105

identifying purposes of, 1184

Premium bonds (lottery)

randomness in, 969

Prepend (prepend element)

theorems about, 1168

Presburger, Mojzesz (Poland, 1904 - ~1943)

and axioms for arithmetic, 1152

Presburger arithmetic, 1152

algorithms for, 1143

Price fluctuations in markets, 429

Prigogine, Ilya (Belgium/USA, 1917-[2003])

auditory perception in, 1079

Prime (prime numbers)

see Primes

Prime implicants

in Boolean formulas, 1095

PrimePi (prime count), 133, 909

and encoding of lists, 1120

and Zeta, 918

PrimeQ (primality test), 909

encoded as integer equation, 1160

and integer factoring, 1090

randomized methods for, 1192

and shift register periods, 975

in ordering of math constructs, 1177 and periods of limited size systems, 256

as precursors to my work, 878 and primitive recursion, 907, 908 and quadratic congruential generators, 975 and register machines, 1114

as rule 60 initial condition, 1091 sequence of, 132-135, 909 simplest CA for producing, 1186 tables of, 910

trapezoidal, 911

as values of polynomials, 1162 and Zeta, 148, 918

Primitive cells and CA lattices, 929 Primitive operations

in logic, 807, 1096 and natural phenomena, 18 and universality, 642 Primitive polynomials and shift register periods, 975 Primitive recursive functions, 907 and Church's Thesis, 1125 undecidability in, 1136 and universality, 1121 Princeton (Institute for Advanced Study), xiii Principal curvatures, 1049

Principia (of Newton), 859 Principia Mathematica (of Whitehead and Russell) and foundations of math, 1149

and Gödel's Theorem, 1158 logic axioms in, 1151 and orders of logic, 1167

as origin of tag systems, 894 Principle of Computational Equivalence, 715-846 antecedents to, 1125

and computational complexity theory, 766 and computational irreducibility, 738 and concept of microcosm, 1196 and continuity, 729

and epistemology, 1196 and future of computers, 841 and future of technology, 1196 and Gödel's Theorem, 782 historical perspectives on, 844-846 and human uniqueness, 844 implications for ontology, 1197 implications for philosophy, 1196 implications for technology, 840 initial conditions violating, 1129 introduction to, 5

and mathematicians, 1125 and mathematics, 772 and modelling, 728 and natural selection, 1002 and operator systems, 815 and origin of complexity, 736 outline of, 716-719 and philosophy of science, 1197 and postmodernism, 1196 process of understanding, 856 and quantum measurement, 1064 and thermodynamics, 444 and ubiquity of intelligence, 822 and universe as intelligent, 1191 validity of, 726-735

Principle of Entropy Increase character of as principle, 1126 see also Second Law of Thermodynamics Principle of Equivalence (for gravity), 530 Principle of Least Action, 1185 Principle of Least Effort and Zipf's law, 1014 Principle of Natural Selection character of as principle, 1126 see also Natural Selection

aggregation systems as, 332

and external randomness, 299

and fractal dimensions, 934

for random walk, 977

Probability theory

and defining randomness, 1067

foundations of, 967

Problem-solving

and defining intelligence, 1178

Procedural computer languages, 627

Procedures

see Programs

Process charts (flowcharts)

and causal networks, 1032

and computer languages, 1104

and formal languages, 939

see also Multiway systems

see also Sequential substitution systems

Programmable logic arrays (PLAs), 1095, 1097

Programmed cell death, 419

Programmed trading systems, 431

Programming

and proofs of universality, 698

and study of programs in the notes, 854

as foundation for science, 3

history of concepts related to, 860

and human thinking, 628

making models based on, 368, 992

in notes to this book, 853, 854

in practical computers, 1108

proving theorems about, 1168

random, 23, 101

as reproducing the universe, 465

see also Computation Projection method for Penrose tilings, 932 Prolog (computer language), 1158 Pronunciation

lookup based on, 623 Proof theory

and number theory, 1166 Proofs, 775

as application of rules, 875

of axiom system correctness, 1170 and computer experiments, 899 and definition of math, 860 encoded by ordinals, 1163 general frameworks for, 1177

of halting for TMs, 1145 importance of relative to theorems, 1177 inference rules in, 1151

lengths of, 1143 lengths of in logic, 1175

of NP completeness, 1145

of P!=NP, 1146 role of in current math, 1156 searching for, 1157

of shortest axiom for logic, 811 structures of, 1155

of undecidability, 1137

of undecidability of PCP, 1140 and universal language of Leibniz, 1109

of universality, 664-673, 722, 1127

of universality by computer, 1115

of universality of Life, 1117

of universality of rule 110, 678-689, 1116 Propagation

of class 4 structures, 281

of effects in CAs, 252, 950

of effects in PDEs, 923 Propellers

characteristic shapes of, 1183 Proper time (in relativity theory), 1042, 1051 Proportion

theory of in art, 872 Propositional logic, 1151 axioms for, 773, 803 see also Logic Prosody (study of verse) and Fibonacci numbers, 891 Protein folding

and biological evolution, 1003 and NP completeness, 1146 and optimization, 988 randomness in, 1184 Proteins

as biological artifacts, 1184 made from genes, 1002 and nanotechnology, 1193 shapes of, 1003 statistics of sequences in, 1184 symmetries in shapes of, 1007 Protestantism

and free will, 1135 Protists

pattern in, 385 see also Radiolarians Protocols

for computers, 1182 Protons

decay of, 1025 size of, 472, 1044 Protoplasm, 1178 Protozoa

as fairly optimal organisms, 398 Prouhet, Eugène (France, 1817-1867) and Thue-Morse sequence, 893 Prouhet-Thue-Morse sequence see Thue-Morse sequence Prusinkiewicz, Przemyslaw (Poland/Canada, 1952- ) and branching in plants, 1005

Pseudocode

avoidance by use of Mathematica, 853

PseudoInverse and matrix memories, 1101 Pseudonoise (PN) sequences, 1084 and CDMA signals, 1188 see also Shift registers Pseudorandom generators, 317

as precursors to my work, 879

randomness tests on, 1085

see also Intrinsic randomness generation

see also Random number generators

Pseudospectral methods for PDEs, 924

PSPACE (polynomial space) computations, 1142

experimental difficulties in, 1077

Ptolemy (Egypt, ~100 - ~170 AD)

and math in science, 859

and models based on rules, 860

Public-key cryptography, 1086

and quadratic congruential generators, 975

Publication counts

for axiom systems, 1153

for cellular automata, 878

Publication dates

conventions for, 851

Publicon

and authoring this book, 852

Puffer train (in Game of Life), 965

Pulsar (in Game of Life), 964

Pulsar puffer (in Game of Life), 965

Pulsars

and general relativity, 1048

and history of SETI, 1189

on Pioneer 10 plaque, 1189

radio emissions from, 1188

thought to be intelligent, 835

Pulse code modulation (PCM), 1188

Pumping lemmas (for formal languages), 944

Punched cards

and history of computing, 1107

and history of universality, 1110

Pure functions

and lambda calculus, 1121

see also Function

for extraterrestrials in science fiction, 1191

and free will, 1136

Putnam, Hilary W (USA, 1926- )

and Diophantine equations, 1161

training animals to solve, 826

use of base 2 in, 902

and math in science, 859, 860

and perfect numbers, 911

sending signals based on, 1189 Pythagorean triples

and Moire patterns, 1078

as satisfying constraints, 945 Pythagoreans

and aliquot sums, 910 and irrationality of Sqrt[2], 912 and music, 1080

and numerology, 1025 and primes, 908 Python

pigmentation pattern of, 426

Q see Robinson arithmetic q-hypergeometric functions

in hard hexagon problem, 1133 q-numbers

as generalizing numbers, 1168 QCD (quantum chromodynamics), 1057, 1061 and computational irreducibility, 1133 and experience of models in, 991

as simple physical model, 1025 QED

see Quantum electrodynamics QSPR (quantitative structure-property relations), 1195 Quadratic congruential generators, 975

Quadratic Diophantine equations, 944

NP completeness of, 1146 Quadratic equations

as exactly soluble, 1133 Quadratic forms

and Diophantine equations, 1164 Quadratic functions

in Turing machines, 1144 Quadratic irrationals and constructible reals, 1129 and continued fraction map, 955 and Penrose tilings, 932 and spectra, 1081 see also Sqrt (square root) Quadratic maps, 921 and history of chaos, 971 Quadratic primes conjecture

as unsolved problem, 1166 Quadratic programming and circle packings, 986 Quadratic residue sequences, 1090

in acoustic diffusers, 1183 and flat spectra, 1081

NP completeness in, 1145 Quadratures

as exact solutions, 1133 Quadrics

fit to curved surfaces, 1049 Quadtrees, 568, 1101 Quality factor

in image compression, 573 Quantifier elimination and commutative groups, 1160 and real algebra, 1154 Quantifiers

higher-order, 1167

in predicate logic, 1151

in well-formed formulas, 1150 Quantitative traits, 1003 Quantization

and discreteness of networks, 528

of volume in spin networks, 1055 Quantized vortices

in neutron stars, 1188 Quantum chaos, 1056 and decoherence, 1064 Quantum chromodynamics see QCD

Quantum computers, 1147 and NP completeness of factoring, 1090 and quantum history, 1058

vs Turing machines, 771 Quantum cosmology, 1026, 1063 Quantum cryptography, 1058 Quantum-dot cellular automata, 1193 Quantum electrodynamics (QED)

as analogy for quantum gravity, 1054 effective coupling in, 1062

Feynman diagrams in, 1060 history of, 1056

Quantum field theory, 1061

and computational irreducibility, 1133

difficulty of computations in, 1146

and discrete space, 1027

and energy conditions, 1053

planar diagrams in, 1040

and quantum computers, 1148

and quantum measurement, 1064

and random networks, 1038

simple effective theories in, 1026

and space as background, 1028

and ultimate theory of physics, 1025

and uniform spectra, 988

vacuum fluctuations in, 1062

Quantum fluctuations

and noise, 969

Quantum gravity, 1054

and discrete space, 1027

and Hawking radiation, 1062

history of, 1025

path integrals in, 1057

as source of decoherence, 1063

and space vs contents, 1028

and spin networks, 1055

undecidability in models of, 1138

Quantum groups

and spin networks, 1055

Quantum information theory, 1058

Brillouin zones in, 988

and character of programs, 538

and Church's Thesis, 1126

discretization of, 1059

and existence of formulas, 1132

and free will, 752, 1135

and gauge invariance, 1045

history of, 1056

and history of randomness, 967

and history of statistics, 1082

and mechanistic explanations, 860

and new types of perception, 1106

search for defining features of, 1059

in ultimate theory of physics, 469

and use of abstract models, 1026

and wave-particle duality, 1044

Quasi-Monte Carlo methods

and digit reversal, 905

in combinations of sines, 146 and turbulence, 998 Quaternions

and abstraction in math, 792 cellular automata based on, 886

as generalizing numbers, 1168

as non-commutative rings, 1153 Qubits (in quantum systems), 1147 Queen Anne's Lace, 1006 Quicksort algorithm, 1142, 1142' Quincunx board

randomness in, 312 Quine-McCluskey procedure, 1095 Quintic equations, 945

as exactly soluble, 1133 numbers defined by, 916 solving as impossibility, 1137 unsolvability of, 1132 Quotient

basic example of, 854 and computational reducibility, 747 encoded as integer equation, 1160 Qur'an

calligraphy from, 874 Rabbit (Fibonacci) sequences, 891 Racah coefficients

see 6j symbols Radar

random pulses for, 970, 1192

as source of radio signals, 1188 Radiation

of structures from rule 37R, 456 and thermodynamics, 1021

in vacuum field theory, 1062 Radicals

continued fractions for, 144 digit sequences for, 139 and linkages, 1129 nested, 915 and periodic points of iterated maps, 955 solving equations in, 945

see also Quadratic irrationals see also Root

see also Sqrt Radio astronomy noise in, 1187 and SETI, 835, 1180, 1188 Radio noise, 302

and SETI, 836 Radio signals extraterrestrial, 835, 1180

of human origin, 1188

of natural origin, 1187 and shift registers, 1086 Radioactive decay and quantum field theory, 1057 randomness in, 970

relativity theory applied to, 1043 Radiolarians

and circle packings, 985 forms of, 385, 1011 skeletons as animal artifacts, 1184 Radix of numbers

see Bases (number) Radó, Tibor (Hungary/USA, 1895-1965) and Busy Beaver Problem, 889 Railroads

as speculative stocks, 1015

as visible from space, 1187 RAM (random-access memory) emulated by CA, 663, 1112 Ramanujan, Srinivasa A (India/England, 1887-1920) and almost integers, 915

and divisor sequences, 910, 911 and experimental math, 899 Ramsey theory, 1068 unprovability related to, 1163 RAND Corporation

and random number table, 968 Random

as intrinsic randomness generator, 317 Random-access memory (RAM) emulated by CA, 663, 1112 Random-dot stereograms, 1076