Stephen Wolfram - A New Kind Of Science Episode 12 doc

sound of random sequence, 585
Historians
and free will, 1135
Historical perspectives, 844-846
see also history items in specific entries
History
accidents of in universe, 505
as central to human condition, 846, 1196
of complexity research, 862
computations done in human, 837
of definition of sciences, 863
effect on math of, 792
as explaining axiom systems, 816
general comments about, 850
of life on Earth, 387, 825
of math and ease of concepts, 1177
recorded by physical objects, 1195
role of in biological evolution, 388
theory of and chaos, 971
uniqueness of for universe, 504
History register
see Shift registers
Hobbes, Thomas (England, 1588-1679)
and free will, 1135
and nature of space, 1028
Hodgkin-Huxley equation
as example of PDE, 161
Hofmeister, Wilhelm F. B. (Germany, 1824-1877)
and phyllotaxis, 1007
Hofstadter, Douglas R. (USA, 1945- )
and recursive sequences, 880, 907

Hokusai, Katsushika (Japan, 1760-1849)
and fluid turbulence, 997
Hölder exponents, 959
Holland, John H. (USA, 1929- )
and genetic algorithms, 985
in Preface, xiii
Homeobox genes, 419, 1010
Homoclinic tangles, 972

Homogeneity
origins of, 353
Homogeneous structures
see Cellular automata
Homomorphisms
and generalized additivity, 952
Homoplectic processes (extrinsic randomness generation), 973
Homotopy
and structures in networks, 1046
and topological defects, 1045
Homunculus hypothesis, 1010
Honda, Hisao (Japan, 1943- )
and branching in plants, 1005
Honeycomb
form of, 1011
length minimization in, 988
Hooke, Robert (England, 1635-1703)
and snowflakes, 992
Hopfield, John J. (USA, 1933- )
and neural networks, 1100, 1102
in Preface, xiii

Hopi language
and Sapir-Whorf hypothesis, 1181
Hopper crystals, 993
Horizon problem (in cosmology), 1055
Hormones
in brains, 1099
in plants, 404
Horn shell
growth of, 415
Horner form
and evaluation of powers, 1093
Horns (animal)
as artificial-looking, 828
compared to antlers, 1008
growth of, 413
Horse races
as source of randomness, 968
Horton stream order (in tributaries), 1001
Hours-minutes-seconds
origin of, 902
Hoyle, Fred (England, 1915-2001)
and extraterrestrials, 1190
HP 700
and creation of this book, 854
HP calculators
reverse Polish notation in, 896
Huffman, David A. (USA, 1925-1999)
and data compression, 1069
Huffman coding, 564

adaptive, 1069
implementation of, 1071
and JPEG compression, 1074
Human behavior
and computational irreducibility, 1132
free will in, 750-753, 1135
and game theory, 1104
laws of, 1014
Occam's razor for, 1025
Human condition
defining features of, 1196
scientific approaches to, 845
Human form
parametrizations of, 1010
Human history
vs. biological evolution, 399
Human languages
see Languages (human)
Human relics in archeology, 1183
Human thinking, 620-631
and constructs in math, 1177
emulation by computers of, 733
and Gödel's Theorem, 1159
and Principle of Computational Equivalence, 733, 1130
Human will
see Free will
Humanities
relativism in, 1131
summary of relations to, 9, 863
Humans

artifacts of early, 1184
defining properties of intelligence of, 822
and future technology, 1195, 1196
genetic programs of, 383, 1002
height distributions of, 977, 1003

making randomness, 306, 970
mathematical power of, 1126
as only intelligence, 822
uniqueness of, 844, 1195
visual system of, 577
weight distributions of, 977
Humboldt, F. W. H. Alexander v. (Germany, 1769-1859)
and branching in plants, 1004
Hume, David (Scotland, 1711-1776)
and purposes in nature, 1185
Humidity
and weather, 1177
Huntington, Edward V. (USA, 1874-1952)
and axioms for logic, 1151
Huntington axioms for logic, 773, 1151
Hurwitz numbers, 914
Hydraulic jump, 377
Hydrocarbons
considered interesting, 1194
Hydrodynamics
see Fluid flow
Hydrogen atom
as exactly soluble, 1133
and history of quantum theory, 1056

Hydrology
and forms of landscapes, 1001
Hyena
pigmentation pattern of, 426
Hyperbolic equations, 923, 940
Hyperbolic space, 1049
Hypercube
Boolean, 1095
and trivalent networks, 1029
Hypercycles
and origin of life, 1179
Hyperelliptic Diophantine equations, 1164
Hyperelliptic functions
in solution of spinning top, 1133
Hypergeometric functions
asymptotic behavior of, 917
and continued fractions, 914
and curves from curvature, 1009
and nested patterns,
612
Hypergeometric1F1
in Calogero-Moser model, 1133
Hypergeometric2F1
and caps on hyperspheres, 1050
and quintic equations, 945
in Seiberg-Witten model, 1133
and special functions, 1092
and Thue-Morse sequence, 1092
and trinomial coefficients, 1091
HypergeometricPFQ

and 6j symbols, 1055
difficulty of evaluating, 1134
from rational integral, 916
Hypergraphs, 1029
Hyperreals
as generalizing numbers, 1168
Hypersonic flow
see Supersonic flow
Hypersphere
volume of, 479, 1050
Hyphenation
lack of in this book, 852
I-1 base, 932
I Ching
and substitution systems, 893
IBM 370 computer
my use of, 854
IBM 7090 computer
and pi, 911
ICBMs
radar used by, 970
radars for detecting, 1188
Ice
hopper crystals in, 993
snowflake crystals of, 370
Ice-ray style of Chinese lattice, 874
Icosahedral group, 938
Icosahedral symmetry
in quasicrystals, 994
Icosahedron

clusters in sphere packing, 986
and GoldenRatio, 891
IDEA cryptographic system, 1086
Ideal forms
and Platonism, 1176
Ideal gas
CA idealization of, 446
Idealism
and free will, 1135
and history of universality, 1109
and logic as foundation, 860
and models of math, 1149
and nature of space, 1028
and theories of communication, 1181
and universe as intelligence, 1196

Idealizations
of mathematics, 776
in modelling, 364
and Principle of Computational Equivalence, 728
in traditional science, 736
and ultimate theory of physics, 465
Ideals (algebraic)
as generalizing numbers, 1168
Idempotence
of And, 817
in operator systems, 801
of Or, 817
Identity cellular automaton, 883
Identity function

in combinators, 1122
IEEE floating-point standard
and chaos experiments, 919
If (conditional)
emulated by Gamma, 1129
Illative logic
and symbolic systems, 102
ILLIAC IV
and history of CAs, 877
Illinois, University of, xiii
Illusions
visual, 1076
Image compression
lossless, 560
lossy, 572
Image processing
by averaging, 1073
history of, 1077
and history of 2D CAs, 877, 928
and visual perception, 1076
and Walsh transforms, 1073
Imaginary time formalism, 1043
as example of model feature, 992
Imitation
and bird songs, 1180
see also Emulation
Immune system
and Boolean networks, 936
and combinatorial chemistry, 1194
and growth of animals, 418

randomness in, 970, 1002
shape space in, 1184
Impact craters
as landscape elements, 1001
Implementation
see also Emulation
see also entries for specific systems
Implicants
in Boolean formulas, 1095
Implicational calculus, 803, 1170
models in, 1171
possible models of, 805
theorems in, 813
Implicit rules
constraints as, 210, 940
Implies
axioms for, 803
axioms in terms of, 1151
in multivalued logic, 1175
non-confluence of, 1037
non-universality of, 1174
and structure of proofs, 1155
truth tables for, 802
Impossibilities
mathematical, 1137
Impulse response
for additive systems, 952
spectrum of, 1081
Incompleteness, 782
essential, 1159

in multiway systems, 783, 797
of Robinson arithmetic, 1169
see also Gödel's Theorem
Inconsistency
in axiom systems, 781
in multiway systems, 797
Increment function
in combinators, 1122
in register machines, 97
in Turing machines, 1144
Incremental improvement
and natural selection, 392

Incremental programming, 894
Independence
from axioms, 1167
of axioms, 803
from more powerful axioms, 1166
of theorems in logic, 818
Independence results
since Gödel's Theorem, 1163
Indeterminism
and free will, 750, 1135
Index
features of this, 852
names in this, 852
Indian studies of prosody, 875
Indigenous peoples
and animism, 1195
Indo-European languages, 1103

Induction (mathematical)
and automated proofs, 1158
in axioms of arithmetic, 1152
and recursive sequences, 907
in reduced arithmetic, 800
as scheme for proofs, 1177
statements unprovable without, 1169
Induction (scientific)
and experimental math, 899
and ultimate theory of physics, 466
Inequivalence
as name for Xor, 1173
Inertia
and uniformity of space, 1028
Inertial motion, 521
Infeasible computations, 1143
Inference
of cellular automaton rules, 1089
statistical, 589
Inference rules
in proofs, 1151
Infinite acceleration
in 5-body problem, 1130
Infinite evolution
and undecidability, 755
Infinite impulse response
cellular automata as, 1035
Infinite loops
in Mathematica, 1137
and proving undecidability, 1137

Infinite objects
and non-standard arithmetic, 1169
Infinite patterns
generated in finite time, 732
Infinite trees
and symbolic systems, 898
Infinitesimals
and computational irreducibility, 1132
as generalizing numbers, 1168
and non-standard analysis, 1172
and PDEs, 161
Infinities
in QED, 1057
Infinity
and the continuum, 1127
forms of, 1162
and symbolic representation, 788
and transfinite numbers, 1162
and undecidability, 788
Inflationary universe, 1026, 1055
Information
in axiom systems, 819
in block encoding, 1071
in cellular automata, 959
and definition of life, 1178
in multicolor encodings, 1111
and radiation in rule 37R, 455
in theorems of logic, 818
and thermodynamics, 1020
Information content

algorithmic, 1067
vs. amount of computation, 1133
of biological organisms, 1002
entropy as, 960
and history of chaos theory, 971
in initial conditions, 920
and lower bounds, 1143
measures of,
1181
Information dimension, 959
Information-preserving data compression, 560
Information theory
algorithmic, 1067
and cryptography, 1086
and data compression, 1069
and defining complexity, 1068
and defining randomness, 1068
and human languages, 1181
Information transmission
and causal networks, 520
in cellular automata, 252
in class 4 systems, 281
and data compression, 1069
and error-correcting codes, 1101
in financial markets, 1015
limited by speed of light, 518
in PDEs, 923, 925
and quantum entanglement, 1065
and universality, 694
Initial conditions

for cellular automaton fluids, 381
and complexity, 41
and computability, 1129
finding as NP problem, 1142
information content in, 920
as input for computations, 637
nested in rule 184, 272
NP completeness of finding, 769
and oracles, 1126, 1129
periodic, 266
persistent structures from, 283
and Principle of Computational Equivalence, 724
random, 223-
296
randomness from, 304-314
for rule 110 universality, 689
sensitive dependence on, 153
sensitivity to, 250-252
sets of for Turing machines, 1139
shape dependence on, 179
and thermodynamics, 443
and time travel, 1043
for Turing machines, 710
and undecidability, 756
for the universe, 1026
Initial value problems
in general relativity, 1053
in PDEs, 940
Injectivity
in cellular automata, 959

Ink
and printing of this book, 852
spreading of as diffusion, 978
Inner
and associative CAs, 956
InputForm
for programs in notes, 854
Insanity
and free will, 1136
Inscribed circles, 986
Insert (insert element)
and random networks, 1038
Insertion sort, 1142
Instabilities
in CA evolution, 950
and chaos theory, 153, 971, 972
in crowds of people, 1014
in financial systems, 430
of fluid vortex sheets, 997
in numerical PDEs, 924
in numerical turbulence, 997
in PDEs, 988
and quantum measurement, 542
in reaction-diffusion, 1013
and the Second Law, 1020
in self-gravitating systems, 1021
in splashes, 1000
Instantons
in path integrals, 1057
Instincts

and defining intelligence, 1178
Institute for Advanced Study, xiii
Institutions
and complexity research, 862
role of in science, 857
Instruments (musical)
as nonlinear oscillators, 971
sounds produced by, 1079
Insurance
and history of statistics, 1082
Integer equations, 790, 944
and undecidability, 787
see also Diophantine equations
Integer factoring, 1090
see also FactorInteger
Integer functions
patterns generated by, 870
Integer linear programming
NP completeness of, 1145
Integer sequences, 123, 128-131
IntegerDigits
basic examples of, 854
and computational reducibility, 747
concatenation of, 913
implementation of, 901
and rule 90 pattern, 870
and substitution systems, 889, 891
and sums of three squares, 910
IntegerExponent
and 3n+1 problem, 904

and additive CA attractors, 963
behavior of, 911
and binomial coefficients,
870
and cyclic multiplication, 950
and decimation systems, 909
and period doubling, 892
and a recursive sequence, 906
relation to DigitCount of, 902
and rule 150 pattern, 885
and spectra, 1081
and symbolic systems, 897
TM computation of, 1144
and TM enumeration, 1139
and Turing machine 600720, 1145
and Turing machine for increment, 759
IntegerQ (integer test)
and fraction systems, 1115
Integers
algorithmically simple, 916
as encodings of lists, 1120
and human experience, 1177
number of, 1127
as ordered set, 1152
represented by symbolic expressions, 1121
transfinite, 1162
Integrability
of PDEs, 1023
in three-body problem, 972
Integrals

as exact solutions, 1133
Integrate
and distance in curved space, 1048
impossibilities in, 1137
numbers generated by, 916
systematic methods for, 1177
Integrated circuits
see Chips
Intelligence
analysis using human, 620
and Anthropic Principle, 1026
attributed to universe, 1195
definition of, 822, 1178
in Drake equation, 1191
extraterrestrial, 635, 822
machine, 628, 1099
and Maxwell's demon, 1021
and philosophical implications, 1197
physics as showing, 1191
as special feature of humans, 844
testing of, 1104, 1178
in the universe, 822
see also Thinking
Intelligent design
and complexity, 861
and teleology, 1185
Intentionality
see Purpose
Interdependence
and definition of life, 1178

Interestingness
of chemicals, 1194
in mathematics, 793, 816
of theorems in general, 821,
1176
of theorems in logic, 817
Interfaces (boundaries)
effective rules for in CA, 980
Interfaces (computer)
and concept of halting, 1137
graphical vs. language, 631
history of, 1102
Interference
and quantum computers, 1148
in quantum mechanics, 1059, 1062
of string updates, 503
Interferometer
device like, 311
Intermediate degrees (in computation theory), 734, 1130
Intermediate growth groups, 938, 945
Intermediate steps
in multilevel logic, 1096
in proofs, 811, 1155, 1157
in solving PCP, 1140
Intermittency
and 1/f noise, 969
Internationalism
in mathematical notation, 1182
Internet businesses
and speculative stocks, 1015

Interpolation
of CA patterns, 1092
and fitting models, 1084
Interpretability
axiom system encoding, 816
Interpretation of quantum mechanics, 1058
Interpreted languages, 1109
Interrupts (computer)
as source of randomness, 970
Intersection
and finite set theory, 1171
Interstellar radio communication, 1189
Interstellar space
molecules in, 1179
turbulence in, 1188
Interstices
packings that use, 986
Interval, maps on the
see Iterated maps

Interval arithmetic
and generalizing numbers, 1168
Intractability (computational), 758-771
and cryptography, 1089
Intrinsic curvature, 1049
Intrinsic randomness generation, 315-326
and continuum behavior, 333
early reactions to, 971
in early universe, 1056
experiments showing, 976

in financial systems, 432
in fluid flow, 382
in fluttering, 971
and free will, 752
in Ising models, 982
my first paper on, 882
in quantum systems, 543, 1063
in reversible conserving systems, 462
and Second Law, 450
see also Randomness generators
Intrinsically defined curves, 1009
Introspection
and free will, 1135
Intuition
about objects in nature, 828
and computer experiments, 856
and continuous math, 925
development of my, 21
in human thinking, 627
mathematical and proofs, 1156
mechanism for in brain, 1136
need for new, 39-41
from practical computing, 716,
872
and Principle of Computational Equivalence, 726
and ultimate theory of physics, 468
Intuitionism
as foundation of math, 1176
Intuitionistic logic
and double negation, 1158

and Peano arithmetic, 1152
Invagination
in animal development, 418, 1009
Invariances
and cellular automata state transition graphs, 963
Invariant configurations
in 1D cellular automata, 941, 954
in 2D cellular automata, 942, 954
as attractors, 276
in Game of Life, 964
and satisfying constraints, 348
undecidability of, 1138
Invariant entropy (spacetime entropy), 960, 961
Invariant interval (in relativity theory), 1042
Invariants
and axiom system proofs, 1170
in dynamical systems, 961
for knots, 1046
see also Conserved quantities
Inventions
vs. discoveries in math, 1176
for randomness generation, 969
Inverse problems
and NP completeness, 771
and perception, 551
Inverse square law for gravity, 536, 1047
InverseFunction
and CA encodings, 1118
Invertible cellular automata
see Reversible cellular automata

Involute of circle, 418
Involutions
cellular automata as, 1017
Ion traps
for quantum gates, 1148

Ionosphere
radio emissions from, 1187
IQ (intelligence quotient), 1178
tests, 1104
Iran
ornamental art in, 874
Iraq
ornamental art in, 873
Iris (eye) patterns
randomness of, 1014
Iron Age
ornamental art from, 873
Irrational numbers
and 3D non-periodic tiling, 943
generating additive systems, 953
generating quantum gates, 1148
and idealized billiards, 1022
iterated multiplication by, 903
and musical chords, 1079
in Peano arithmetic, 1162
Irreducibility
see Computational Irreducibility
Irreducible representations (of groups)
and isotropy, 980

and spin, 1046
Irregularities
see Regularities
Irreversibility
in data compression, 572-576
and memory, 625
in physical systems, 441-457
and quantum measurement, 1063
Ising, Ernst (Germany/USA, 1900-1998)
and Ising model, 981
Ising model, 981
exact solution of 2D, 1133
ground state of, 944
and history of CAs, 876
lack of exact 3D solution, 1133
and lattice gas models, 999
and my work on CAs, 880
and P completeness, 1149
phase transition in, 982
as precursor to my work, 879
as probabilistic model, 1083
and randomness tests, 1085
undecidability in, 1138
see also Spin systems
Islam
and ornamental art, 872
as rejecting animism, 1195
Islamic art, 43
Isocorrelation textures, 1078
Isomers (variant molecular structures), 1194

Isotropic tensor, 980
Isotropy
in aggregation systems, 978
in biological growth, 1007, 1010
in code 746 CA, 334
of differences in 2D CAs, 950
in discrete systems, 980
in physics and CAs, 473
of random walks, 977
of sums of squares, 910
Iterated aliquot sums, 911
Iterated bitwise operations, 906
Iterated division
and continued fractions, 143
Iterated function systems, 191
and pictures of ferns, 1005
Iterated maps, 149-155
algebraic iterates in, 1098
attractors in, 961
of bit operations, 921
from CA densities, 953
on the Cantor set, 869
complex, 933
and computation universality, 1129
and computer experiments, 899
emulating discrete systems, 1129
as evidence of determinism, 972
and financial markets, 1015
and fluid flow, 998
history of, 918

and history of chaos, 971
and history of complexity, 49
of integers, 122
in Newton's method, 1101
periodic points in, 955
as precursors to my work, 879
and QCD, 1061
and recurrence relations, 906
spectra in, 1080
and substitution systems, 921
two-dimensional, 921
Iterated morphisms (substitution systems), 893
Iterated radicals, 915
and GoldenRatio, 891
Iterated rules
sizes of formulas for, 1096
Iterated run-length encoding, 905
Iteration
as basis for algorithms, 1141
compared to repetition, 990
examples of in Mathematica, 853
Iteration theory
and iterated maps, 918
$IterationLimit
and avoiding undecidability, 1138
Iterative automata
see Cellular automata
Iterative improvement
for engineering designs, 1193
in satisfying constraints, 344

IUPAC chemical nomenclature, 1194
J combinator, 1123
Jacobi, Carl G. J. (Germany, 1804-1851)
and sums of four squares, 910
JacobiP (Jacobi polynomial)
and trinomial coefficients, 1091
JacobiSN (Jacobi elliptic function)
asymptotic oscillation of, 917
and background for PDE, 924
and logistic map formulas, 1098
in solution of spinning top, 1133
JacobiSymbol
and Hadamard matrices, 1073
and quadratic residues, 1090
and spectra, 1081
Jacquard, Joseph-Marie (France, 1752-1834)
and programmable looms, 1107
Jamming-resistant radio
and shift registers, 878
Jansky, Karl G. (USA, 1905-1950)
and radio astronomy, 1188
Japanese work on CAs, 877
Java (computer language)
and chaos experiments, 919
and computer communication, 1182
as example of language, 1109
and register machines, 101
Jellies
large alkanes as, 1194
Jen, Erica (USA, 1952- )

in Preface, xiii
and rule 30, 871
Jet stream, 1001
Jets
turbulence in, 377
Jevons, W. Stanley (England, 1835-1882)
and logic machines, 1107
Job, Book of
and understanding nature, 876
Johnson noise, 968
Join
in lattice theory, 1153
Join (join lists)
as associative function, 1094
basic example of, 853
evaluation chains based on, 1095
theorems about, 1168
Joint denial
as name for Nor, 1173
Joints
characteristic shapes of, 1183
Jones, James P. (USA/Canada, 1941- )
and Diophantine equations, 1161
Josephus problem, 909
Joule, James P. (England, 1818-1889)
and nature of heat, 1019
JPEG compression, 573, 1074
Judaism
and free will, 1135
and human uniqueness, 1195

as rejecting animism, 1195
Judgement
as crucial in modelling, 365
in social sciences, 9
Julesz, Bela (Hungary/USA, 1928- )
and texture perception, 1076
Julia, Gaston M. (France, 1893-1978)
and iterated maps, 879
Julia sets, 933, 934
analog of in tree space set,
1006
Junctional calculus, 803, 1170
possible models of, 805
theorems in, 813
Jupiter
and extraterrestrial life, 1179
radio emissions from, 1187
Red Spot of, 377, 999
K combinator, 711, 1121
K^0 particle
and time reversal violation, 1019
K_33 non-planar graph, 527
Kabala (universal object), 1127
Kaluza, Theodor F. E. (Germany, 1885-1954)
and notions of space, 1028
Kampé de Fériet hypergeometric functions, 1009
Kaneko, Kunihiko (Japan, 1956- )
and continuous CAs, 922
in Preface, xiii
Kansas

grid of towns in, 1187
Kant, Immanuel (Germany, 1724-1804)
and complexity in biology, 861
and free will, 1135
and purposes in nature, 1185
Karatsuba, Anatolii A. (Russia, 1937- )
and multiplication, 1142
Kardar-Parisi-Zhang (KPZ) equation, 925
Kari, Jarkko J. (Finland/USA, 1964- )
and undecidability in 2D cellular automata, 1138
Kármán, Theodore von (Hungary/Germany/USA, 1881-1963)
and vortex streets, 998
Karnaugh maps, 1097
Kasiski, Friedrich W. (Poland, 1805-1881)
and cryptanalysis, 1086
Kauffman, Stuart A. (USA, 1939- )
and Boolean networks, 936
in Preface, xiii
Kazakhstan
straight railroad in, 1187
KdV equation
as exactly soluble, 1133
Kells
Book of, 873
Kelvin (William Thomson) (Scotland, 1824-1907)
and harmonic analyzer, 1107
and minimum area packings, 988
and models of space, 1027
and Monte Carlo methods, 1192
and thermodynamics, 1019

and vortices in ether, 1044
Kelvin-Helmholtz instabilities, 988
Kepler, Johannes (Germany, 1571-1630)
and 2-body problem, 972
and computers, 1107
and Fibonacci numbers, 891
and inhabitants of Moon, 1190
and rules of nature, 860
and snowflakes, 992

and sphere packings, 986
Keratin
repetitive structure of, 1003
Kernels
in visual perception, 579
Kerr solution
as exact solution, 1133
and time and computation, 1130
Key sequences
in cryptography, 598
Keyboard characters
for programs in notes, 854
Keystrokes
number to create this book, xiv
as source of randomness, 970
Khinchin (Khinchin's constant), 914
Kidney ducts
branching in, 1008
Kinetic theory
and diffusion equation, 1024

and thermodynamics, 1019
"Kingdom was lost"
and chaos theory, 971
Kingdoms of organisms, 1003
Kingsnake
pigmentation pattern of, 426
Kirby, Laurence A. S. (England/USA, 1952- )
and unprovable statements, 1163
Kirkpatrick, E. Scott (USA, 1941- )
in Preface, xiii
and simulated annealing, 985
Kites
characteristic shapes of, 1183
Kleene star (RepeatedNull), 957
Klein, Oskar (Sweden/Denmark/USA, 1894-1977)
and notions of space, 1028
Klein-Gordon equation
discretization of, 1060
vs. sine-Gordon, 922
Kleinian groups
and circle packings, 986
KleinInvariantJ (modular elliptic function)
and almost integers, 915
Knapping (of stone tools), 1183
Kneading
randomness in, 306
Kneading sequences, 955
Knossos
maze at, 873
Knots

Alexander moves in, 1038
as complex artifacts, 1183
as constructed by rules,
875
in ether as atoms, 1028, 1044
model of atoms as, 1024
as not in 1D systems, 927
theory of, 1046
Knuth, Donald E. (USA, 1938- )
and dragon curves, 893
and Knuth-Bendix procedure, 1037
in Preface, xiii
and tests of randomness, 1084
Knuth-Bendix procedure, 1037
Koch, N. F. Helge von (Sweden, 1870-1924)
and nested curves, 934
Koch curve, 191, 933
history of, 934
Kolakoski, William G. (USA, 1944-1997)
and cyclic tag systems, 895
Kollectiv (collective)
and defining randomness, 1068
Kolmogorov, Andrei N. (Russia, 1903-1987)
and algorithmic randomness, 1068
and fluid turbulence, 997, 997'
Kolmogorov complexity
see Algorithmic information
Kondo model, 1057
Koran
calligraphy from the, 874

Korec, Ivan (Slovakia, 1943-1998)
and encodings of arithmetic, 1163
and register machines, 1121
KPZ (Kardar-Parisi-Zhang) equation, 925
Krohn-Rhodes decompositions, 1069
Kronecker product, 1073
KroneckerDelta
and orthogonal bases, 1072
Kufi calligraphy, 874
Kummer, Ernst E. (Germany, 1810-1893)
and Fermat's Last Theorem, 1166
Kuramoto-Sivashinsky equation, 925
Kuratowski's theorem, 1045
Kurdyumov, Georgii L. (Russia, 1953- )
and transitions in CAs, 981
L-shaped tiles, 932
L systems, 82-87
2D, 187-189
and biological models, 1004
and plant branching, 1005
see also Substitution systems
Labyrinth
pattern of, 873
Labyrinthine patterns
from 2D cellular automata, 427
in code 746, 334
Lag systems, 894
Lagrange, J. Louis (La Grange) (Italy/Germany/France, 1736-1813)
and continued fractions, 915
Lagrange points

simple geometry of, 1187
in three-body problems, 972
Lagrangian density
in general relativity, 1053
for PDEs, 923
in quantum field theory, 1061
LaguerreL (Laguerre polynomials)
in hydrogen atom, 1133
Lambda calculus, 1121
and Church's Thesis, 1125
and combinators, 1122
confluence in, 1036
and history of universality, 1110
as idealization of math, 1150
undecidability in, 1136
Lamé equations, 995
Laminar fluid flow, 376
in the atmosphere, 1001
Laminated sphere packings, 986
Land grants
and regular grid patterns, 1187
Land use
effects on weather of, 1177
Landau, Lev D. (Russia, 1908-1968)
and fluid turbulence, 997
Landau-Ginzburg equation, 925
Landscapes (abstract)
constraint satisfaction on, 346
for memory, 1101
Landscapes (geological), 1001

Langton, Christopher G. (USA, 1948- )
and 2D Turing machines, 930
and CA classes, 948
in Preface, xiii
Langton's ant (2D Turing machine), 931
Language games, 1181
Language of mathematics
and science, 859
Languages (abstract)
as attractors in TMs, 961
context-free, 939, 1091, 1103
context-sensitive, 939
regular, 939
universal and Leibniz, 1149
unrestricted, 939
see also Formal languages
Languages (computer), 1103
as examples of software, 1108
functional and combinators, 898
and history of computing, 1108
intuition from existence of, 872
logic programming, 1158
Mathematica, 853
meaning in, 1182
notion of types in, 898
for parallel computing, 1035
thinking in, 1104
universals in, 1182

see also Computer languages

Languages (human), 1103
as analogy in atomism, 876
animals trained to use, 826
center in brain, 1099
conventional aspect of, 1181
and cryptanalysis, 599
and defining intelligence, 822, 1178
frequency of words in, 1014
and history of universality, 1109
influence of on thinking, 1181
information content in, 1181
logic in everyday, 806, 1173
and math thinking, 1177
reasons for features of, 1181
recognizing meaning in, 826
redundancy of, 1086, 1181
relativity hypothesis in, 1181
rules for and programs, 860
rules in grammar for, 875
and theories of communication, 1181
and thinking, 630
translation of and AI, 1100
translation of and universality, 643
understanding dead, 1184
and Zipf's law, 1014
Laplace, Pierre-Simon (France, 1749-1827)
and Central Limit Theorem, 977
and determinism, 1135
Laplace equation
as constraint, 940

and diffusion-limited aggregation, 994
and flatness of growing tissue, 1007
and network dimensions, 1031
Laplacian
in curved space, 1050
LARC computer
and gas simulation, 864
Large cardinal axioms, 1167
Lasers
as technology components, 1195
Latencies
in circuits, 1095
Latent heat
and crystal growth, 370, 372
Lateral inhibition
and pigmentation patterns, 427
in plant phyllotaxis, 409
in snowflake growth, 370
Latin
logic operations in, 1173
Latin squares
and quasigroups, 1171
as satisfying constraints, 945
Lattice dynamical systems, 155-160, 922
see also Continuous cellular automata
Lattice gases, 999
enumeration in, 959
see also Cellular automaton fluids
see also Spin systems
Lattice gauge theories, 1057, 1061

and computational irreducibility, 1133
and discrete space, 1027
and quantum gravity, 1054
Lattice models
see also Cellular automata
see also Spin systems
Lattice packings, 987
Lattice points
numbers inside spheres, 910
Lattice theory
axioms for, 1153
and posets, 1041
Lattice vectors
and linear congruential generators, 974

LatticeReduce
and digit sequence cryptanalysis, 1089
Lattices
isotropic behavior on, 980
lines at slopes on, 916
possible, 929
random walks on, 329, 977, 978
Law of large numbers, 976
Laws (human)
and defining randomness, 1068
and free will, 1135
and regularities in radio spectrum, 1188
and responsibility, 1136
Laws of Form
and Nand, 1173

Laws of physics
vs. mathematics, 772
vs. programs in biology, 397
and ultimate theory, 470
Layout of networks, 193
NP completeness of, 1145
LCDs
see Liquid crystals
LCGs
see Linear congruential generators
LCM (lowest common multiple)
and Lissajous figures, 917
and Riemann Hypothesis, 918
Leading digits, 914
and number compression, 1070
of powers, 903
Leaf shapes
see Leaves
LeafCount (expression size)
and algebraic expressions, 945
and combinator expressions, 1122
and number of recursive functions, 908
and symbolic systems, 897
for universal Diophantine equation, 1161
Learning
of bird songs, 1180
of computer languages,
1104
and defining intelligence, 823, 1178
human, 626

models of, 1099, 1102
the science in this book, 855
as trick in evolution, 1002
Learning curves, 1014, 1102
Least squares fitting, 1082, 1083
Leaves
arrangements of (phyllotaxis), 408-412, 1006
evolutionary theories of, 1005
flatness in growth of, 1008
growth of, 401, 412, 1005
shapes of, 401-407, 1005
LED segments
and textons, 1076
Leech lattice
isotropy of, 980
and sphere packings, 987
Leftmost innermost evaluation, 906
Leg (animal)
development of, 419
Legendre, Adrien-Marie (France, 1752-1833)
and sums of three squares, 910
LegendreP (Legendre polynomials)
and isotropy measures, 980
in solution of Mie scattering, 1133
Lehmer, Derrick H. (USA, 1905-1991)
and linear congruential generators, 974
Lehr, Eduard (Germany, ~1932)
and parametric curves, 1009
Leibniz, Gottfried W. v. (Germany, 1646-1716)
and binary numbers, 902

and calculators, 1107
and free will, 1135
and I Ching, 893
and idealizations of math, 1149
and logic, 1099
and logic as basis for science, 860
and math notation, 1182
and the nature of space, 1028
and origins of universality, 1109
and symbolic logic, 1151
and universal language, 1181
Lemmas
defining characteristics of, 1176
and lengths of proofs, 1175
in proof of axiom for logic, 811
in proof searching, 1157
in structure of proofs, 1155
Lempel, Abraham (Israel, 1936- )
and data compression, 1069
Lempel-Ziv encoding, 565
Lemur
pigmentation pattern on, 426
Length
basic example of, 853
Lenses
characteristic shapes of, 1183
Lenticular clouds, 1183
Lenz, Wilhelm (Germany, 1888-1957)
and Ising model, 981
Leonardo da Vinci

see da Vinci, Leonardo
Leopard
pigmentation pattern of, 426
Leptons
as types of particle, 1043
Less (<)
encoded as integer equation, 1160
Letters
frequencies of in English, 1069
visual perception of, 1076
Lettuce
ammonite septa like, 1008
growth of, 413
Leucippus (Greece, ~480 - ~420 BC)
and atomism, 876
Levin, Leonid A. (Russia/USA, 1948- )
and NP completeness, 1143
in Preface, xiii
and transitions in CAs, 981
Lévy, Paul P. (France, 1886-1971)
and nested C curve, 893, 934
Lévy flights, 969
lex
and regular languages, 958
Lexical analysis
and regular languages, 958
LFSRs (linear feedback shift registers)
see Shift registers
LHASA (chemical synthesis software), 1194
Li, Wentian (USA, 1959- )

and CA classes, 948
in Preface, xiii
Lichens
on Mars, 1180
Lichtenberg figures, 995
Lie groups
and additive systems, 953
and sphere packings, 987
Life
definition of, 557, 823, 1178
and definition of complexity, 1069
on Earth, 825
and history of complexity, 50
on Mars, 1179
origin of, 1179
Life, Game of
see Game of Life
Life force, 1178
Life insurance
and history of statistics, 1082

Lifeforms
alien, 825
LIFO (last-in first-out) stacks
and context-free languages, 939
Light
caustics in, 984
perception of, 577
and quantum history, 1056
Light bulb filaments, 842

Light cones, 519
for cellular automata, 949
differential geometry of, 1051
and random causal networks, 1052
Light levels
and object recognition, 1076
Lightning
and origin of life, 1179
pattern of from space, 1187
randomness in, 995
as source of radio emissions, 1187
Lights of cities
patterns of from space, 1187
Likelihood
maximum, 589, 1083
Lily pads, 401
Limit cycles
for differential equations, 961
in iterated maps, 961
see also Repetitive behavior
Limit ordinals, 1162
Limitations of science, 6, 748, 846
and ultimate theory of physics, 471
Limited size systems, 255-260, 267
Limits
of cellular automaton rules, 948
and fractal dimensions, 933
of networks as space, 1030
undecidability of, 1138
see also Continuum limits

Lincoln cathedral, 873
Lincos language, 1189
Lindenmayer, Aristid (USA/Netherlands, 1925-1989)
and L systems, 893
and plant branching, 1005
in Preface, xiii
Lindgren, Kristian (Sweden, 1960- )
in Preface, xiii
and universal CAs, 1115
Line defects, 1045
Line-printer output
of my early cellular automata, 19
Linear algebra
and CA invariances, 1022, 1023
and dimensions of networks, 1031
and models of memory, 1101
Linear cellular automata
see also Additive cellular automata
see also Elementary cellular automata
Linear codes, 1101
Linear congruential generators, 318, 974
based on powers of 3, 903
and chaos on calculators, 920
cryptanalysis of, 1089
historical use of, 968
and randomized algorithms, 1085
as shift register analogs, 975
Linear differential equations
and time series, 1083
Linear differential operators, 953

Linear Diophantine equations, 944, 1164
and polynomial time, 1146
Linear equations, 940
as defining rationals, 916
and free field theories, 1061
Linear feedback shift registers
see Shift registers

Linear filters
in sound compression, 1080
and spectra, 1082
in visual perception, 1076
Linear models
and computational irreducibility, 1132
in statistics, 589
Linear predictive coding
in sound compression, 1080
Linear programming
NP completeness of integer, 1145
for optimizing designs, 1193
Linear recurrences, 891, 906
and additive CAs, 1018
and history of CAs, 878
and time series, 1083
see also Linear congruential generators
Linear stability analysis, 988
Linear systems
vs. nonlinear, 940
Linearity
compared to additivity, 952

in quantum mechanics, 541, 1060
LinearSolve
iterative algorithm for, 1141
and LFSR cryptanalysis, 1087
and linear systems, 940
and NC computations, 1149
Linguistics
see Languages (human)
Linial (Gulden), Samuel (USA, 1927- )
and underivability of logic axioms, 1170
Linkages
in chaos toys, 1183
characteristic shapes of, 1183
and constructible reals, 1129
for randomness generation, 969
Linked lists
for Game of Life, 949
Linker
for register machines, 1114
Linnik's problem, 910
Linux
and creation of this book, 854
Liquid crystals
as components in technology, 1195
phase transitions in, 981
topological defects in, 1045
Liquid-gas phase transitions, 981
Liquids
flow of, 376
see also Fluid flow

Lisp (computer language)
grammar of,
1104
and network systems, 936
representing expressions in, 897
theorems about, 1158, 1168
Lissajous figures, 917
List manipulation
examples of in Mathematica, 853
Listable
and implementing CAs, 866
ListConvolve
and aggregation systems, 978
and autocorrelation, 1080
basic example of, 853
and CellularAutomaton, 886
and elementary CAs, 865
and fast evolution rules, 1094
and general CAs, 886
and hexagonal CAs, 992
and image processing, 1077
and integer multiplication, 1093
and Ising model, 981
and multidimensional CAs, 927
recursive algorithm for, 1142
in sandpile models, 989
and Sierpinski pattern, 931
and visual perception, 1076
ListCorrelate
and additive CAs, 870, 952

and visual feature extraction, 1077
ListPlay
and sound output, 1080
Lists
encoded as integers, 1120, 1127
see also Sequences
Lithic analysis, 1183
Little group (spacetime symmetries), 1046
Littlewood, John E. (England, 1885-1977)
and chaos theory, 971

and Goldbach's Conjecture, 911
and iterated maps, 919
and numbers of primes, 910
Livermore (Lawrence Radiation Laboratory), 864
Lizards
pigmentation patterns of, 426
Local conservation laws, 1023
see also Conservation laws
Local minima, 346
Localization
in class 1 and 2, 252
as origin of repetition, 355
Localization (of eigenstates)
and randomness tests, 1085
Localized structures
and 3n+1 problem, 904
in class 4 systems, 281-296
in code 20 CA, 285
in code 357 CA, 286

in code 1329 CA, 287
in continuous CAs, 160, 244
as engineering components, 678
finding in 2D CAs, 965
and long-range communication, 252
in network evolution, 525
origins of, 990
and particles in physics, 525
in PDEs, 164, 1129
in presence of randomness, 526
in QCD, 1061
from random initial conditions, 228
in reversible CAs, 440
in rule 30 CA, 700
in rule 37R CA, 455
in rule 45 CA, 701
and rule 110 axiom system, 1168
in rule 110 CA, 31, 292, 964
same in different 2D rules, 949
in three dimensions, 949
in turbulent fluids, 997
in Turing machines, 888
and universality, 692
see also Gliders
Locomotion in animals, 1011
Log (logarithm)
algebraic independence of, 1164
computation of nth digits in, 912
and computational reducibility, 744, 747
continued fractions for, 144

digits of, 141
leading digits in, 914
randomness in digits of, 967
as source of randomness, 968
uniform distribution mod 1, 904
and Weber's law, 1014
Log-periodic antennas, 1183
Logarithmic growth
of number representations, 1070
Logarithmic law
for leading digits, 914
Logarithmic series, 1134
Logarithmic spiral, 418
Logging
and messages to Moon, 1189
regular patterns of, 1187
Logic
as applied to math, 780
axioms for, 773, 808
and combinators, 711, 898
and communication,
1181
and computational irreducibility, 1196
and defining intelligence, 1178
difficult notation in, 1104
and discreteness in programs, 976
as example of lattice theory, 1153
as example of rules, 875
and finite set theory, 1171
as foundation of math, 1176

history of, 1097, 1099, 1151
history of notation for, 1182
and history of universality, 1109
in human languages, 1173
in human thinking, 627
implications for, 1197
interesting theorems in, 816
lengths of proofs in, 1175
machines for doing, 1107
multilayer, 1096
multivalued, 814, 1175
and natural science, 860
non-universality of basic, 784
as not mainstream math, 859
possible models of, 805
in practical computers, 1108
and quantum mechanics, 1058
reversible, 1097
shortest And, Or axioms for, 1174
shortest Nand axioms for, 808
ternary, 814
and Thue on substitution systems, 893
see also Predicate logic
Logic circuits
emulated by CAs, 662, 1112
emulated by Game of Life, 1117
and universality, 1119
Logic expressions
see Boolean expressions
Logic minimization, 1095

compared to searching, 1193
Logic programming languages, 1158
Logic Theory Machine, 1157
Logical depth
and defining complexity, 1069
for Turing machines, 1144
Logical functions
see Boolean functions
Logical positivism
and theories of communication, 1181
Logical rules
see Rules
Logicism
as foundation of math, 1176
LogIntegral (logarithmic integral li)
and distribution of primes, 133, 909
and Zeta, 918
Logistic differential equation
and iterated maps, 918
Logistic map, 920
closed forms for iterates of, 1098
and continuous CAs, 922
and history of chaos theory, 971
see also Iterated maps
Logistics
and causal networks, 1033
and complex systems, 862
Lognormal distribution
for human weights, 977,
1003

and multiplicative processes, 977
Logo (computer language)
and 2D TMs, 930, 931
Long division, 139
Long-term memory, 1102
Long-time tails
in molecular dynamics, 999
and shaking, 969
Longest paths
and NP completeness, 1146
Looms
and history of computer, 1107
Loop general relativity, 1048
Loop quantum gravity, 1054
and spin networks, 1055
Loop switching method
and network layout, 1031
Lorentz, Hendrik A. (Netherlands, 1853-1928)
and relativity theory, 1041
Lorentz contraction, 1041
Lorentz gas, 1022
Lorentz transformations, 1041, 1042
Lorentzian spaces, 1051
Lorenz, Edward N. (USA, 1917- )
and chaos theory, 971
and complex ODE, 879
and experimental math, 899
and fluid turbulence, 998
in Preface, xiii
Lorenz equations

as giving strange attractor, 922
and history of chaos theory, 971
and Lissajous figures, 917
and weather prediction, 1178
Los Alamos
and history of CAs, 877
and my work on CAs, xiii, 880
Lossless data compression, 560
Lossy data compression, 572
Lotka, Alfred J. (USA, 1880-1949)
and oscillatory chemical reactions, 1013
Lotteries, 969
Lovelace, A. Ada (Byron) K. (England, 1815-1852)
and universality, 1110
Low-level languages
and practical computing, 1109
and register machines, 101
Löwenheim-Skolem theorem, 1172
Lower bounds
on computations, 761, 1143
on parity, 1096
on sorting algorithms, 1142
LPC
in sound compression, 1080
Lucas, F. Edouard A. (France, 1842-1891)
and binomials mod k, 870
Lucas-Lehmer test
for primes, 911
Lucas numbers, 891
Lucian (Greece, 120 - ~180 AD)

and inhabitants of Moon, 1190
Lucretius, Titus Carus (Italy, ~95 - ~55 BC)
and atomism, 876
and models based on rules, 860
Lukasiewicz, Jan (Poland, 1878-1956)
and axioms for logic, 1151
and multivalued logic, 1175
Lukasiewicz representation, 896
Lunes
and ornamental art, 872
theory of, 873
Lungs
branching in, 1008
Lyapunov exponents
in cellular automata, 950
and entropies, 960
in iterated maps, 921
probabilistic estimates of, 953
LZW compression, 565, 1069, 1072
MacArthur Foundation, xiii
Mach, Ernst (Austria/Czech., 1838-1916)
and Mach's Principle, 1047, 1053
and origins of relativity, 1042
Mach numbers, 997
Machiavelli, Niccolò (Italy, 1469-1527)
and rules for military drill, 875
Machine 1447
function computed by, 761
and longest halting times, 1144
Machine 1507

as complicated TM, 1144
function computed by, 761
Machine 596440
as candidate for universality, 709
properties of, 1120
Machine 600720
behavior of, 763
and longest halting times, 1144
properties of, 1145
Machine code
and history of computing, 1109
and implementation of CAs, 866
of programs by Alan Turing, 1013
and register machines, 97
Machine integers
and linear congruential generators, 974
Machine intelligence, 628, 1099
Machine numbers
for Turing machines, 888
Machines
see also Register machines
see also Turing machines
Machining plans
and Voronoi diagrams, 987
Macintosh
and computer interfaces, 1102
Magic Eye random dot stereograms, 1076
Magic squares
as defined by constraints, 945
for SETI, 1190

Magnetic chaos toys, 1183
Magnetic field
compared to gravitational, 530
and gauge invariance, 1045
Magnetic moments
and size of particles, 1044
Magnetic monopoles, 1046
Magnetization
in Ising model, 981
phase transition in, 981

Magnetosphere
radio signals from, 827, 835, 1187
Magnus, Wilhelm (Germany/USA, 1907-1990)
and word problem for groups, 1141
Mainframes (computers)
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
Mallow leaves, 1006
Mammoth ivory, 873
Man
see Humans

Management
of machines that think, 629
Management science
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
in Preface, xiii
and random walks, 990
Mandelbrot set, 934
as precursor to my work, 880
and tree space set, 407, 1006
Manicouagan, Canada
circular crater in, 1187
Manifolds
in differential geometry, 1048
discrete approximations to, 1050
undecidability in equivalence of, 1051, 1138

Manneville, Paul (France, 1946- )
and CA classes, 948
and continuous CAs, 922
Manufacturing

causal network systems for, 1032
Many-fingered time, 1035
Many-to-one mappings, 960
Many-worlds models, 1035
for quantum mechanics, 1063
Map
basic example of, 853
MapAt
and implementing proofs, 1155
and representation of TMs, 888
MapIndexed
basic example of, 853
and causal networks, 1033
Mappings
on Cantor set, 869
cellular automata as, 959
complex, 933
functions as, 898
history of iterated, 918
iterated, 149-155
from linear congruences, 974
local and linear, 953
many-to-one, 960
piecewise linear, 921
Marconi, Guglielmo (Italy, 1874-1937)
and SETI, 1188
Margolus, Norman H. (USA, 1955- )
and 2D CA simulators, 928
in Preface, xiii
Mark I computer, 1107

Market research
and random sampling, 1192
Markets
price fluctuations in, 429
Markov, Andrei A. (Russia, 1903-1979)
and sequential substitution systems, 894
and statistics of language, 1181
and undecidability, 1136
and word problem for semigroups,
1141
Markov partitions
and digit sequences, 901
Markov processes, 1084
and 3n+1 problem, 904
and finite automata, 958
and game strategies, 1105
history of, 1181
spectra of, 1081
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
function names in index, 852
general notes about, 853
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
theorems about, 1168
thinking in, 1104
ultimate theory of physics in, 468
undecidability in, 1138
as universal system, 642
universality in, 1110
and use of digit sequences, 902
use of hashing in, 1100
Mathematica Book, 853
writing style in, 849
Mathematical constants, 136-144

Mathematical equations
as foundation for science, 1
in history of science, 44
see also Equations
Mathematical formulas
and computational irreducibility, 737
and engineering, 829
as results of models, 368
and science education, 1133
Mathematical functions, 145-148
as computable, 1128
difficulty of evaluating, 1134
evaluated by CAs, 639
evaluation of as P computations, 1142
interpretation of TMs as, 1144
see also Standard mathematical functions
Mathematical impossibilities, 1137
Mathematical induction, 1152
Mathematical logic
combinators in, 898
and defining complexity, 1068

and DNF, 1097
history of, 1150
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
Mathematical models
history of, 992
and Principle of Computational Equivalence, 728
Mathematical morphology, 1077
Mathematical notation, 1182
grammar of, 1104
influence on thinking of, 1181
for logic, 1173
vs. Mathematica, 853
relations vs. axioms in, 1169
Mathematical physics
functions of, 145
Mathematical theories
distributions of theorems in, 1175
Mathematicizing power
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, 5, 859
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, 7, 863
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

Maxwell, James Clerk (Scotland/England, 1831-1879)
and chaos theory, 971
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
Maze designs, 873, 875
Maze-like patterns, 1013
from 2D cellular automata, 427

in code 746, 334
in visual cortex, 1075
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
Meaning
of axiom systems, 797
and intelligence, 826
in languages, 1181
in programs, 1182
and regularity, 1183
of the universe, 1027
Measure entropy
for cellular automata, 959

computing, 1084
spacetime, 960
Measure theory
and defining randomness, 1067
Measurement
in quantum theory, 542, 1062
in thermodynamics, 448, 1020
Measures
and fractal dimensions, 934
and spectra, 1081
Mechanical automata, 1183
Mechanical calculators, 1110
Mechanical computers, 1107
Mechanical devices
as not universal, 643
Mechanical perception, 1105
Mechanical processes
for real numbers, 1129
Mechanical systems
for cyclic tag systems, 895
proofs in, 1157
randomness in, 969
and repetitive motion, 829
self-reproducing, 1179
Mechanical toys
randomness in, 969
Mechanics
and history of mathematics in science, 859
Mechanisms
captured in models, 367

compared to purposes, 830
in programs and nature, 297
for randomness, 299
Mechanistic explanations
for complexity in nature, 861
in physics,
860, 1026
Medians
and robust statistics, 1083
Medical applications, 9, 842
Medicinal plants
leaf shapes for, 1005
Medieval logic, 1151
Mediterranean
patterns of lights around, 1187
Meet (in lattice theory), 1153
Meier, Willi (Switzerland, 1948- )
and cryptanalysis of rule 30, 1087
MeijerG (Meijer G function)
and Integrate, 1177
Meinhardt, Hans (Germany, 1938- )
and patterns on shells, 1012
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
Methane, 1194
Method of lines
for PDEs, 924
Methodology
of computer experiments, 108
and definition of math, 860
development of my, 21
of math in science, 859
for studying sequences, 1091
in this book, 1193
Metric
in differential geometry, 1048
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
Michelson-Morley experiment, 1041
Microcanonical ensemble
for 2D Ising model, 982
Microcosm, 1196
Microorganisms
random motion of, 970
Microprocessors
randomness instructions in, 970
Microsoft Windows
and creation of this book, 854
Microspine
pattern from, 385
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

Middle-square method, 975
for molecular dynamics, 864

Midwest
field patterns in, 1187
Mie scattering
as exactly soluble, 1133
Military camouflage, 1077
Military cryptography, 1085
and shift registers, 878, 1088
Military drill
use of rules in, 875
Military GPS (P-code), 1086
Military secrets
and history of complexity, 49
Military SETI, 1188
Military vehicles
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
Mind-body problem, 1197
Minds
of extraterrestrials, 1191
and free will, 1135

in inanimate objects, 822
and theories of communication, 1181
Mineralogy
and forms of crystals, 993
Minimal surfaces
computing forms of, 1009
and deformable packings, 988
vs. Einstein equations, 1052
and general study of form, 967
and radiolarians, 1011
as shapes of exoskeletons, 1011

Minimal systems
for particular purposes, 1186
for particular sequences, 1186
see also Shortest programs
Minimal theories, 1186
Minimality
and recognition of purpose, 831
Minimization
of Boolean formulas, 617, 1095
of energy, 940
of finite automata, 957
as model of drainage, 1001
of multilevel logic, 1096
and variational principles, 1185
Minkowski, Hermann (Germany, 1864-1909)
and relativity theory, 1042
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 Rule 90
Mod k rules, 870, 952
see also Additive cellular automata
Modal logic, 1158
Model theory (in mathematical logic), 1172
of operator systems, 805
and Peano arithmetic, 1169
Models
based on grammars, 1084
based on programs, 860
compared to reality, 991
complexity of, 1068, 1083
of data, 550
and epistemology, 1196
general issues about, 363-369, 857
history of making, 992
of mathematics, 776
mechanistic for physics, 1026
origins of physical, 1025
personal experiences of, 991, 992
and Principle of Computational Equivalence, 728

probabilistic, 588
programs as more than, 434
rarity of new, 368
undecidability in, 1138
of universe, 466
see also Ultimate theory of physics
see also Simulations
Modems
data compression in, 1070
Modes
in PDEs, 988
Modesty
and clarity of explanation, 849
Modular addition, 255,
950
state transition graphs for, 961
Modular functions
difficulty of evaluating, 1134
and nested patterns, 1093
Modular groups, 933
Modular multiplication, 257, 950
state transition graphs for, 961
ModularLambda
and nested patterns, 1093
Modulo arithmetic
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
Moire patterns, 1078
Molding profiles
as examples of patterns, 874
Molds
and self-reproduction, 1179
Molecular biology
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
as precursor to my work, 879
Molecules
and Brownian motion, 302
defining life, 1178

in fluids, 376
idealized in CA fluids, 379
and quantum theory, 1056
in smell, 1105

unpredictable properties of, 1194
vibrational modes of, 1081
Mollusc shells
see Shells
Molluscs
classified by shell patterns, 1012
pigmentation patterns of, 389
Moments of data, 1083
Moments of inertia
and isotropy, 980
Momentum
basic mechanism for, 537
Momentum conservation
in 2D cellular automata, 1024
in cellular automaton fluids, 379
in Einstein equations, 1052
in network evolution, 529
in physics, 1022
Monadic pure predicate logic, 1152
Monads
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)
and shell shapes, 1008
Mosses
growth of, 1004
Motion
absolute, 1042
of class 4 structures, 281
concept of in physics, 521
Motivation
Occam's razor for, 1025
and thinking, 629
Motor skills
memory for, 1102
Mountains
patterns of, 1001
Mouse motion
miles of in creating this book, xiv
as source of randomness, 970
Movie effects
and substitution systems, 893
MP3 sound compression, 1080
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
Multiplication
CAs based on generalized, 886
computation of, 1093
in digit sequences, 118
lower bound on, 1143
NC algorithms for, 1142
number of steps required for, 1134
performed by CAs, 661,
1112
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
Multivalued logic, 814, 1175
Multivariate hypergeometric functions, 916, 1092
Multiverse (many-worlds), 1035, 1063
Multiway systems, 204-209
based on numbers, 939

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
completion algorithms for, 1037
computational reducibility in, 1134
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
in Fourier transforms, 1074
for multiplication, 1142
for pi, 912
for PrimeQ, 1090
in sorting, 1142
in Walsh transforms, 1073
see also Computational complexity theory
Nakaya, Ukichiro (Japan, 1900-1962)
and snowflakes, 992
Name-value pairs, 1182
Named theorems in logic, 817
Names
of functions as symbolic expressions, 896
in the index, 852
lookup by sound, 623
of variables in math logic, 1150
Nand
axioms for, 803, 808, 1151
emulated by Life, 1117
expression trees, 619, 1096
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
theorems for, 1175
truth table for, 802
as universal function, 807, 1096, 1173
and universality, 1119
words in languages for,
1173
Nanotechnology, 1193
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
Natural order
for Walsh functions, 1073
Natural selection
Biological evolution, 415
difficulties in simulating, 1002
as engineering, 842
vs. engineering, 393

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
as origin of complexity, 383, 861
and pigmentation patterns, 423, 1012
as predictive theory, 397
and Principle of Computational Equivalence, 1002
and purpose, 831, 1185
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
Nature
compared to artifacts, 40, 828, 967
computations in, 716
mimicked by technology, 1193
undecidability in, 1138
universality in, 718

Nature
article of mine in, 882
Nature's God, 1196
Nautilus shell
chamber structure of, 1008
growth of, 414
shape of, 385
Navier-Stokes equations, 996
and computational fluid dynamics, 1000
vs. Einstein equations, 535
next-order corrections to, 997
simplified versions of, 997
singularities in, 923
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 2D cellular automata, 171
in 2D substitution systems, 187
in additive cellular automata, 955
in animal skeletons, 420
at phase transitions, 983
and auditory perception, 586
as basis for algorithms, 1141
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
in digit sequences, 117, 913
display hacks for, 871, 932
in eddies in fluids, 997
in elliptic functions, 1093
in erosion patterns, 1001
in firing squad problem, 1035
in flocks of birds, 1011
in forms of artifacts, 1183
formulas for, 608
in Game of Life, 965

with general associative rules, 956
history of, 934
and history of math, 735
in Ising model, 983
in iterated maps, 921, 961
and lack of universality, 694, 734
in lists, 931, 989
in mobile automata, 73

in multiway systems, 207, 937
and musical scores, 1080
in networks, 197, 509
in Newton iteration, 1101
and Nim, 939
as only recently familiar, 1106
origins of, 357-360
in ornamental art, 872
in pattern-avoiding sequences, 944
in patterns of cracks, 995
and Principle of Computational Equivalence, 722
from projections of lattices, 932
in quasicrystals, 994
from random initial conditions, 273
recognizing visual, 582
in recursive sequences, 130
in rule 45, 701
in rule 90, 25, 270
in sequential substitution systems, 91
in snowflakes, 371
in sorting networks, 1142

in structure of attractors, 959
in substitution systems, 83
in symbolic systems, 104
in systems based on numbers, 988
in Turing machines, 79, 1119
and visual uniformity, 1078
in Walsh functions, 1073
in Weierstrass functions, 918
see also Fractals
NestList
basic example of, 853
in CA evolution, 865
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
Nets of polyhedra, 476
Network constraint systems, 483, 1032
and self-assembly, 1011
Network systems, 193-203
dimensions in, 936
implementation of, 935
random initial conditions in, 949

sequential, 936
Networks
algebraic systems based on, 1172
approximating flat space, 477
approximating spheres, 1049
for attractors, 276
and Boolean functions, 619, 1096, 1097
of CA emulations, 1118
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
as rule-based systems, 860
and undecidability, 1138
and visual perception, 1076
Neurons
in brain, 1098
in cybernetics, 862
properties of, 1075
repeatable randomness in, 976
see also Nerve cells
Neurotoxins
from cone shells, 1011
Neurotransmitters, 1099
Neutral traits, 399
Neutrinos, 1043
identified with gravitons, 1054
limit on number of, 1047
and parity violation, 1019
for SETI, 1189
and speed of light, 1041
Neutron diffusion
Monte Carlo of, 968, 1192
Neutron stars
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
in quote by Kant, 861
and theology, 861
Newtonian physics
and determinism, 967
Newton's Laws
character of as principle, 1126
Newton's method
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
Nilpotent groups
Cayley graphs of, 1032
Nim (game), 939
Nine-neighbor CA rules, 927
NIntegrate
difficulty of evaluating, 1134
NK models (random Boolean networks), 936
NLS equation, 925
NMR
and quantum computers, 1148
Noise
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)
Non-Abelian plane waves, 1061
Non-Archimedean fields
as generalizing numbers, 1168
Non-commutative geometry
and general relativity, 1048
Non-compact groups
and tachyon spin, 1046
Non-computability
of entropies, 958
of fractal dimension, 933
see also Undecidability
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
see Gaussian distributions
Normal forms
in multiway systems, 1036
see also Canonical forms
Normal modes
of nested systems, 1081
Normal numbers, 912
and concatenation sequences, 913
and defining randomness, 1068
Normal ordering (in quantum field theory), 1062
Norms of tensors, 1050
Nose (human), 1105
Not
in multivalued logic, 1175
and satisfiability, 1146
table for, 806
theorems involving, 817
words in languages for, 1173
Notation
for chemical compounds, 1194
in logic, 1173
mathematical, 1182
for numbers, 1182
for operators, 1182
for symbolic expressions, 896

used in this book, 852
Notes (musical), 585, 1080
Noughts-and-crosses (tic-tac-toe), 939
Nouns
in human languages, 1103
in mathematical notation, 1182
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
in rule 30, 770
of satisfiability, 768

and systems in nature, 1146
of tensor simplification, 1048
of tiling problems, 942, 984
Nqthm (theorem prover), 1158
NSolve
difficulty of evaluating, 1134
NTSC television signals
bandwidth of, 1079
Nuclear explosions
and computer simulations, 1133
and extraterrestrial contact, 1190
patterns of, 377
Nuclear physics
and quantum theory, 1056
Nucleation of crystals, 992
Nucleus (atomic)
discovery of, 1044
forces in, 1057
spins of,
1046
NullSpace
and CA conservation laws, 1022
Number classes
and transfinite numbers, 1162
Number conservation
in cellular automata, 458
in cellular automaton fluids, 379
Number representations, 142
and data compression, 561
length prefixed, 1070

in symbolic systems, 897
Number theory, 135
and acoustic diffusers, 1183
analytic, 909
and cryptography, 1090
and Diophantine encodings, 1160
and Diophantine equations, 1164
evaluating functions from, 1134
experimental math in, 899
functions in, 1092
history of, 909
and history of complexity, 862
and history of iterated maps, 919
and linear congruential generators, 974
network of theorems in, 820
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
OCR (optical character recognition)
and cellular automata, 877
Octahedra
extraterrestrial in 2001, 1190
truncated, 930, 987
Octanes, 1194
Octaves (in music), 917, 1079
Octonions
as generalizing numbers, 1168
Octopus
communication by, 1181
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
as coauthor of paper, 881
in Preface, xiii

and zeros of zeta function, 918
Odometer
and history of computing, 1107
Odor perception, 1105
and experience of dogs, 827
Offset lists
in general cellular automata, 927
OH (hydroxyl) frequency, 1189
Oil and water
separation of, 980
Oil fires
patterns in, 377
visible from space, 1187
Olaus Magnus (Sweden/Italy, 1490-1568)
and snowflakes, 992
Olbers' paradox, 1021
Old English
logic operations in,
1173
Olfactory perception, 1105
and experience of dogs, 827
Olive shell
growth of, 415
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
Ontology
implications for, 1197
and notions of space, 1028
Opcodes
in practical computers, 1108
in register machines, 896
Open sets
and axioms of topology, 1155
Open systems
and thermodynamics, 455
Operating systems
and concept of halting, 1137
and history of computing, 1108
in practical computers, 1108
Operation counts in computations, 1143
Operator representations
for integers, 916
Operator systems, 801
automated proofs in, 1158
evolution rules for, 898
frameworks for proofs in, 1177
vs. multiway systems, 1169
Operators
as generalizing numbers, 1168

notation for, 1182
possible Boolean, 806
precedence of, 1104
precedence of in math, 1150
in quantum theory, 1056
Optic nerve, 1075
Optical amplifier, 310
Optical character recognition (OCR)
and history of CAs, 877
Optical computing
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
Organic molecules
production of, 1179
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
Origami (paperfolding)
NP completeness in, 1145
rules in, 875
Origin of Species, 1001
Ornament
examples of, 43
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
Oscillations
in chemical reactions, 1013
of density in rule 73, 954

in fluid flow, 998
in math functions, 917, 1134
origins of, 354-356
in PDEs, 988
Oscillators
in Game of Life, 964
nonlinear, 971
in quantum history, 1056
Oseen, C. William (Sweden, 1879-1944)
and fluid flow past spheres, 998
Oster, George F. (USA, 1940- )
and patterns on shells, 1012
in Preface,
xiii
Other minds
problem of recognizing, 1197
Otter (theorem-proving system), 1158
Outer
and branching patterns, 1005
and enumerating expressions, 897
and metrics on surfaces, 1048
and shape tensors, 980
Outer totalistic rules
in cellular automata, 928
Outlying data, 1083
Overlaps
in networks, 510
in strings, 503, 1033
Overshooting
in gradient descent, 985

Ozsváth-Schücking rotating vacuum, 1053
P = NP problem, 765
and SETI, 1190
and systems in nature, 770
undecidability of, 1146
P (polynomial time), 1142
p-adic numbers
as generalizing numbers, 1168
P-code for GPS, 1086
P completeness, 1149
Packard, Norman H. (USA, 1954- )
and CA classes, 948
as coauthor of paper, 882
in Preface, xiii
Packings
of bits in CA implementation, 866
of deformable objects, 988
discrete, 987
history of circle, 985
and shapes of cells, 1007
of spheres, 349
PadLeft
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
Pair annihilation
and nested patterns, 358
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 iterated maps, 921
and Mandelbrot set, 934
for shell growth, 415
for shell shapes, 1008
for trees, 405-407, 1006
and universal objects, 1127
Parametric differentiation
as framework for integrals, 1177
Parametric surfaces
in shell models, 1008
Paramodulation
as step in proof, 1156
Parasitism
and biological evolution, 1002
Parastichies (phyllotaxis spirals), 1007
Parentheses
language of balanced, 939, 1103
pattern from balanced, 1091
pattern of balanced, 989
and single combinators, 1123
Paris, Jeffrey B. (England, 1944- )
and unprovable statements, 1163
Paris-Harrington theorem
unprovability of, 1163
Parity function (Xor)
DNF for, 1096
from rule 132, 638
Parity sequence, 83, 890
see also Thue-Morse sequence

Parity violation, 1019
difficult to get from gravity, 1054
Parrots
communication with, 1180
Parsley
shape of cow, 385
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
and universality, 1129
Partial fractions
as framework for integrals, 1177
Partial functions
general recursive as, 907
in Turing machines, 1143
Partial quotients
in continued fractions, 914
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
Particle counting
and history of CAs, 877
Particle detectors
Monte Carlo studies of, 1192
Particle physics
my work in, 17, 864
relativistic invariance in, 1043
time reversal violation in, 1019
Particle production
in vacuum field theory, 1062
Particle showers
as examples of nesting, 988
multiway system models for, 938

stochastic models for, 968
Particles
thermodynamic behavior with, 443
see also Defects
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
see Principle of Computational Equivalence
PCP (Post Correspondence Problem), 757, 1139

PCs (personal computers)
and history of computing, 1108
my use of, 854
PDEs, 161-164
see also Partial differential equations
PDF
and production of this book, 852
PDP-1 computer
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
and Nand, 1173
and theories of communication, 1181
Pell equation, 944
and equation for Power, 1161
properties of, 1165
and quadratic Diophantine equations, 1164
Pendulum
attractor for, 275
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
Pentagons
in deformable packings, 988
and Fibonacci numbers, 891
and GoldenRatio, 891
as inducing curvature, 532
lattices from, 930
see also Five-fold symmetry
Pentodes
and Nand operation, 1173
Perception, 547-635
atomic-scale,
1195
auditory, 585-588
in biological evolution, 1105
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
traditional idealization of, 736
Perceptrons, 1076, 1099, 1102

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
and causal invariance, 1035
and causal networks, 1033
Petroleum, 1194
PGP cryptographic system, 1086

Phase factors
and gauge invariance, 1045
Phase-locked loops, 1188
Phase space
of cellular automata, 278
for finite-size systems, 961
Phase transitions, 981, 983
in cellular automata, 339
in continuous CAs, 948
discreteness from, 337
and inflationary universe, 1055
in Ising model, 982
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
Pheromones
alkanes as, 1194

and animal communication, 1180
phi
see GoldenRatio
Philippines
as source of shells, 1011
Philosopher's stone
and alchemy, 861

as universal object, 1127
Philosophy
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
Phoenician art, 43, 873
Phonemes
in sound compression, 1080
Photoconductivity

as element in technology, 1195
Photoelectric effect
in quantum theory, 1056
Photographs
and concept of intelligence, 838
Photons
and Bell's inequalities, 1064
history of, 1056
randomness from, 970
as source of decoherence, 1063
as type of particles, 1043
and vacuum fluctuations, 1062
zero mass of, 1046
Photorealistic graphics
textures in, 1077
Photoshop (software)
and image processing, 1077
Phrases
in human languages, 1103
Phyla of organisms, 1003
Phyllotaxis, 408-412, 1006
purpose invented for, 387
Physical constants
numerology for, 1025
Physicists
and Church's Thesis, 1126
and defining life, 1178
and effectiveness of math, 860
and free will, 1135
and measuring complexity, 1069

note for, 1043
and quantum measurement, 1058
and SETI, 1180
and study of complexity, 862
and studying hearing, 1080
Physics, 369-382, 433-545
avoidance of complexity in, 862
and Church's Thesis, 1126
compared to math, 821
conserved quantities in, 1022
continuous systems in, 729
fundamental, 433-545
and laws of human behavior, 1135
learning compared to science in this book, 856
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, 8, 863
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
Pions (in particle physics), 1057