Concatenation
as semigroup operation, 938, 1172

Concatenation sequences, 913
block statistics in, 594
continued fractions for, 915
difficulty of CAs producing, 1186
and pointer-based encoding, 1072
sounds from, 1080

Conchoids (math curves), 875

Concurrent systems
and causal invariance, 1035

Condensates
topological defects in, 1045

Conditional entropies
and texture discrimination, 1077

Conditional probabilities
and Bayes's theorem, 1083

Cone cells
in color vision, 1074

Cone shells
growth of, 414
neurotoxins from, 1011
patterns on, 423

Cones
in Lorentzian spaces, 1051

Confinement in QCD, 1061

Confluence, 1036
for combinators, 1122
and generalized aggregation rules, 979
getting using completion, 1037
in multiway systems, 507
in symbolic systems, 1113
see also Canonical forms
see also Church–Rosser property

Conformal mappings
and flatness in growth, 1007
and hypergeometric functions, 1092
and transformation of biological forms, 1010

Conformal transformations
and gauge invariance, 1045

Conformations of proteins, 1184

Congruence
in axioms of geometry, 1154

Conifers
branching in, 1004

Conjugacy
in additive systems, 952
and CA conservation laws, 1023
in iterated maps, 921

Conjugate systems
and reversible cellular automata, 961

Conjunction
as name for And, 1173

Conjunctions
in human languages, 1173
starting sentences with, 849
as words for logic functions, 1173

Conjunctive Normal Form (CNF), 1095
and resolution method, 1157

Connected clusters
in percolation theory, 983

Connection Machine computer
my use of, 854, 881

Connections (on fiber bundles), 1045

Connectives
possible Boolean, 806
in well-formed formulas, 1150

Connectivity
of branching patterns, 1006
in network evolution, 1039
of networks, 1029

Conscious thinking
and free will, 1135
in visual perception, 582

Consciousness
and quantum theory, 1058, 1063
see also Intelligence
see also Thinking

Conservation laws, 458–464
in cellular automaton fluids, 379
finding in CAs, 1022
for particles in 2D CA, 445
in PDEs, 923
in physics, 1022
for quantum probabilities, 1059
in three-body problem, 972
topological, 1045
undecidability of, 1138
for vector quantities, 1024

Conserved quantities
from algebraic structures, 1023

Conserved traits, 395

Consistency
of arithmetic, 1160, 1163
of axiom systems, 1168
in multiway systems, 797

Consistency of action
and human will, 1136

Consonants
sound of, 1079
and soundex system, 1100

Constancy
color, 1074

Constancy (K) combinator, 1121

Constants
mathematical, 136–144
in operator systems, 1169

Constants of motion
in three-body problem, 972
see also Conserved quantities

Constellations
patterns of, 583, 1187
and SETI, 1189

Constraints
in 1D systems, 940
axioms as, 803
as basis for uniformity, 354
biological schemes for satisfying, 985
and concept of purpose, 829, 1185
in correspondence systems, 1139
Darwinian evolution and, 386
defining networks using, 483
as determining spacetime networks, 482
difficulties in modelling with, 368
enumerating cases satisfying, 959
and mathematical thinking, 1177
vs. mechanisms, 940
in multiway systems, 766
as NP problems, 1142
on operators, 1172
PDEs as, 923
for periodic behavior in CAs, 954
and Principle of Computational Equivalence, 729
probability of violating, 343
problem of satisfying, 342–351
producing complexity with, 351
proofs as systems based on, 1157
and quasicrystal growth, 994
in random systems
see Markov random fields
rule 30 repetition as, 266
satisfied by operators, 805
self-assembly and, 1011, 1193
systems based on, 210–221
undecidability of satisfying, 756
and undecidability of tiling, 1139

Constructibility
and intuitionism, 1158

Constructible reals, 1129

Constructions
of 48-gon, 873
vs. computer experiments, 108
and purposes, 830
rules in geometrical, 875
of universal CA, 648, 675

Constructive proofs
and computer experiments, 899

Consumer electronics
use of randomness in, 1192
use of statistics in, 1082

Contact (movie)
extraterrestrial contact in, 1190

Context
and recognition of purpose, 830
and theories of communication, 1181

Context-free languages, 939, 1103
math notation as, 1182
parentheses and, 989
patterns generated by, 1091
PCP and, 1139
pumping lemma for, 944
random, 1084
rule 184 as recognizing, 1109
undecidability in, 1137

Context-sensitive languages, 939

Continental drift
as non-math theory, 859

Continued fraction map (Gauss map), 914, 955

Continued fractions, 143, 914–915
and billiard systems, 971
computation of, 914
and digital slopes, 916
and encoding of reals, 1127
generalized, 914
history of, 915
and iterated maps, 919
and multiplication systems, 903
and Pell equation, 944
and phyllotaxis, 411
as precursors to my work, 878
and quadratic Diophantine equations, 1164
and sine curves, 147

ContinuedFraction, 914
see also Continued fractions

Continuity
and axioms of topology, 1155
of CA mappings, 869
and confusion about Gödel's Theorem, 1132
and the continuum, 1127
and difficulty of computer experiments, 899
discreteness from, 337
in genetic programs, 1002
in geometry axioms, 1154
and mathematical intuition, 925
in natural selection and engineering, 394
nesting as historical counterexamples in, 934
origins of, 327–336, 463–464
origins of in fluids, 379
and Principle of Computational Equivalence, 729
of space, 472

Continuous cellular automata, 155–160
additive, 1092
cardinality of, 1128
classification of, 948
history of, 921
implementation of, 921
as models of boiling, 994
as models of phyllotaxis, 1007
and probabilistic rules, 976
for quantum mechanics, 1059
with random initial conditions, 243

Continuous computation, 730, 1128

Continuous functions
additivity of, 953
cellular automaton maps as, 961
evaluation of, 1134
number of, 1127

Continuous models
of crystal growth, 993
of fracture, 995

Continuous Pascal's triangle, 922

Continuous systems
and Church's Thesis, 1126
vs. discrete systems, 167
emulating discrete, 731, 1129
isotropy in, 980
in physics, 729
and undecidability, 1138

Continuous truth values, 1175

Continuous variables
invention of, 901
models with, 368

Continuum, the, 1127

Continuum equations
see Partial differential equations

Continuum Hypothesis, 1127
in set theory, 1155
and truth from incompleteness, 1167
unprovability of, 1163

Continuum limits
different forms of, 1023
for diffusion equation, 1024
of networks, 1030

Contours
in landscapes, 1001
in visual perception, 578

Contractive mappings, 277, 959

Contradictions
and proofs of undecidability, 1137
as scheme for proofs, 1177

Control theory
and artificial intelligence, 1099
and complex behavior, 841
and cybernetics, 862
time series in, 1083

Convection, 1000
and history of pattern, 947
maze patterns in, 1013
patterns of, 377
repeatable randomness in, 976
repetitive cells in, 988
and snowflake development, 372

Convention in languages, 1181

Convergence
of fractal dimensions, 933
of isotropy in finite differences, 980
of iterated maps, 918
of metric expansion, 1050
in multiway systems, 507, 1036
in natural selection, 392
in negotiation processes, 430
of PDE solutions, 924
in plant phyllotaxis, 410

Convergent evolution, 397

Convergent series
as exact solutions, 1133

Conversation
and defining intelligence, 1178

Convolution
in visual perception, 1075, 1076
see also ListConvolve

Convolutional codes
and CA encodings, 1119

Conway, John H. (England/USA, 1937–[2020])
and 2D Turing machines, 930
and arithmetic recurrences, 1115
and Game of Life, 877, 880, 949
and iterated run-length encoding, 905
and non-periodic tilings, 943
in Preface, xiii
and recursive sequences, 907
and universality of Life, 1117

Cook, Matthew (USA, 1970– )
and cyclic tag systems, 895
and polyomino tiling, 943
in Preface, xii
and rule 110, 1115

Cook, Stephen A. (USA/Canada, 1939– )
and NP completeness, 1143

Cooking recipes, 1193

Cooley–Tukey transform, 1074

Cooperation
theories of evolution of, 1104

Coordinate singularities
in general relativity, 1048, 1053

Coordinates
and multiway strings, 1036
for network nodes, 478

Copernican astronomy
and extraterrestrial life, 1180
and history of gravity, 1047
vs. ideas of this book, 844

Copy operations
and Nand expressions, 1096

Coral
as animal artifacts, 1184

Coral reefs
symmetry of, 1187

Coralbells (plant), 1006

Córdoba, Great Mosque of, 873

Core collapse
in self-gravitating systems, 1021

Core memory
of early computer, 864

Cores
of elementary particles, 527

Corn
structure of, 385

Corner-overlapping patterns, 941

Corners
in code 976, 980

Cornu spiral, 418

Corollaries
defining features of, 1176

Coronas (in splashes), 988, 1000

Corpuscular theories, 876, 1024

Correlation entropy, 959

Correlations
in 2D Ising model, 982
and Bell's inequalities, 1064
between digits, 902
and CA difference patterns, 950
in CA evolution, 953
in class 3 CAs, 949
and defining communication, 827
and defining complexity, 1069
in DNA sequences, 1184
in H theorem, 1020
in human languages, 1086
in noise with given spectrum, 969
and non-standard diffusion processes, 1024
in phase transitions, 989
in quantum theory, 543
in random environments, 304
and random networks, 1039
in random patterns, 584
and randomness tests, 595, 1085
and spectra, 587, 1082
in stirring and shaking, 969
in visual textures, 1078
see also ListCorrelate

Correspondence systems (PCP)
NP completeness in, 1145
and undecidability, 757, 1139

Corrugation
in ammonite septa, 1008
in animal development, 418
characteristic shape of, 1183
in tissue sheets, 413

Cortex
auditory, 1079
sensorimotor, 1099
visual, 580

Cos (cosine)
and area of circle on sphere, 1050
and Bell's inequalities, 1064
and limit of 6j symbols, 1055
and logistic map formulas, 1098
and nested radicals, 915
and tent map, 919

Cosets
and state transition graphs, 963

Cosh
in solution of catenary, 1133
in solution of logistic map, 1098

Cosine transform
in data compression, 1070, 1074

Cosmas (Italy, ~1210 – ~1235)
and mosaic patterns, 873

Cosmati mosaic work, 873

Cosmic microwave background
and alignment of time, 1022
as analog of ether, 1042
and galaxy formation, 1021
and history of gravity, 1048
and initial conditions, 1026
origin of uniformity of, 1055
as source of radio noise, 1187

Cosmic rays
particles produced by, 1044, 1047
as possible artifacts, 1187

Cosmological principle (uniformity of universe), 1055

Cosmological term (in Einstein equations), 1052
and inflationary universe, 1055
and vacuum solutions, 1053

Cosmology
and Anthropic Principle, 1185
basic issues in, 1055
and Einstein equations, 1053
and ether in, 1042
formation of galaxies in, 1021
and history of gravity, 1048
my work in, 17, 864
network models of, 1056
and Second Law, 1020, 1021
and simple initial conditions, 1026
topological defects in, 1045
see also Universe

Cot (cotangent)
and shift map, 919

Coth
continued fraction for, 914
and solution of Ising model, 982

Countable models
of operator systems, 1172

Counter
Turing machine acting as, 888

Counter machines (register machines), 97–102, 896

Counterfactuals (quantum values), 1064

Counterpoint
rules of musical, 875

Countries
of people in the index, 852

Coupled map lattices, 155–160, 922
see also Continuous cellular automata

Coupling constants
in quantum field theory, 1057
and renormalization group, 955

Coupon collector's test, 1085

Courant condition (for PDEs), 924

Covalent forces
and virtual electrons, 1060

Cover
of inspirational book, 17, 864
of this book, 851

Covers
in Boolean minimization, 1095

Cow parsley
shape of, 385

Cowe, Russell J. (UK, 1946– )
and shell patterns, 1012

Cowfish
pigmentation pattern of, 426

Cowrie shells
growth of, 415, 1008
patterns on, 1012

CP violation (time reversal violation), 1019

CPT invariance, 1019