Anagni, Italy
and Cosmati mosaics, 873
Analog computers
and chaos theory, 971
and Gibbs phenomenon, 899
and history of computing, 1107
and history of numbers, 902
linkages in, 1129
Shannon's general purpose, 1129
sound output from, 1080
theoretical power of, 730, 1128
Analog electronics
and form of radio signals, 1188
Analogy
in human thinking, 627
Analysis
by extraterrestrials, 836
definition of, 549
and NP completeness, 771
and Principle of Computational Equivalence, 722
processes of, 547-635
and recognizing meaning, 826
statistical, 588-596
traditional idealization of, 736
using mathematics, 606-620
Analysis (mathematical)
axioms for, 774, 1155
non-standard, 1172
Analysis of variance (ANOVA), 977
Analytic continuation
of cellular automata, 1092
of mathematical functions, 1092
in quantum field theory, 1061
Analytic number theory
axioms needed for, 1166
Analytical engine, 1107
Analytical results
and computational reducibility, 1133
Analyticity
of S matrices, 1057
in three-body problem, 972
see also Continuity
Anatomy
and studies of form, 967
Anaxagoras (Greece, ~500 - ~428 BC)
and purpose in nature, 1185
And
as associative function, 1094
axioms for, 803
axioms in terms of, 1151
and Boolean formulas, 1095
confluence of, 1037
emulated by CAs, 662, 1112
encoded as integer equation, 1160
and junctional calculus, 1170
in multivalued logic, 1175
and P completeness, 1149
and satisfiability, 1146
theorems involving, 817
truth tables for, 802, 806
words in languages for, 1173
Angelfish
pigmentation pattern of, 426
Angle bisection
in decorative constructions, 873
Angles
of CAs on lattices, 473
in differential geometry, 1048
of images on retina, 1076
Moire patterns from, 1078
of stems in plants, 408
Angular momentum
intrinsic of particles, 1046
quantization of, 1056
and spin networks, 1055
Anharmonic oscillator
lack of solution to, 1133
Animal behavior, 1011
and game theory, 1104
randomness in, 970
Animal breeding
as randomized algorithm, 1193
Animals
branching in, 1008
camouflage of, 1077
color vision in, 1075
communication among, 1180
evolution of plans for, 395
evolution of prediction in, 1105
free will in, 752, 1135, 1136
growth of, 413-422
growth of farm, 1010
intelligence in, 825, 1178
as models for extraterrestrials, 1190
objects created by, 1184
overall forms of, 417
perception in, 827
pigmentation patterns of, 422-429
and security systems, 1183
shapes of and mathematics, 859
similarity of forms in, 421
territories of, 987
use of randomness by, 1192
Animism, 845, 1195
Anisotropy
in aggregation systems, 978
in code 746, 979
of patterns on lattices, 980
of random walks, 977
Annualized models
and iterated maps, 918
Anomalies (quantum field theory)
as basic quantum effects, 1059
and high-mass particles, 1047
Anomalous magnetic moments, 1057, 1060
Anonymous functions
and lambda calculus, 1121
Anosov maps, 921
periodic points of, 955
ANOVA (analysis of variance), 977
Ant pheromones, 1194
Antarctica
Mars meteorite from, 1179
Anteaters
patterns on, 1013
Antennas
nesting in shapes of, 1183
Anthropic Principle, 1026
and teleology, 1185
Anthropology
and history of complexity, 862
and theories of communication, 1181
Anti-reflection in acoustic diffusers, 1183
Antibodies
randomness in, 970, 1184
Antichains (in posets), 1041
Antidiffusion equation, 923
Antikythera device
and history of computing, 1107
identifying purpose of, 1184
Antimatter
rarity of and T violation, 1019
Antiparticles
and T violation, 1019
and vacuum fluctuations, 1062
Antlers
growth of, 1008
Ants
artificial, 931
Aphorisms, 1196
Apical meristems (in plants), 409
APL (computer language), 1104
Apollo astronauts, 1180
Apollonian packing, 986
vs. Cosmati pattern, 873
and network evolution, 1037
Apollonius (Turkey/Egypt, ~262 BC - ~190 BC)
and circle packing, 986
Apoptosis (cell death), 419
Applications
to biology, 383-428
to financial systems, 429-432
general issues about, 857
to mathematics, 772-821
to physics, 369-382, 433-545
summaries of, 7-11
to technology, 840-843
Apply
basic example of, 853
for computing factorial, 1110
Aquinas, Thomas (Italy, 1225-1274)
and argument by design, 861
AR (automated reasoning)
see Automated theorem proving
Arab-Norman style, 873
Arabic script, 43, 874
Arc length, 418, 1048
ArcCos
and tent map, 919
Archeology
and extraterrestrials, 1190
identifying purposes in, 1184
recognizing artifacts in, 1183
recognizing purposes in, 829
Arches
characteristic shapes of, 1183
Archimedean solids
and trivalent networks, 1029
Archimedes (Sicily, 287-212 BC)
and pi, 911
and use of mathematics, 859
Architecture (building)
as affecting perception, 1106
ornament in, 872, 873
rules in, 874
study of form in, 967, 1010
Architecture (computer), 869
ArcSin
and logistic map formulas, 1098
ArcTan
and shift map, 919
Area
and axioms for geometry, 1154
Area law
for Nand expressions, 1097
Arecibo radio telescope, 1190
Argonne National Laboratory
and theorem proving, 1158
Argument by design
and complexity, 861
and teleology, 1185
Arguments of functions
currying of, 896, 1122
and Hilbert's 13th problem, 1128
and primitive recursion, 908
Aristotle (Greece, 384-322 BC)
and complexity, 861
and definition of life, 1178
and embryology, 1010
and gnomons, 1008
and history of logic, 1099
and logic in science, 860
and nature of space, 1028
and origins of randomness, 967
and purpose in nature, 1185
and theoretical biology, 1003
Arithmetic
algorithmic randomness in, 1067
axioms for, 773, 1152
of Babylonians, 859
as basis for science, 44
cellular automaton based on, 614
emulated by set theory, 1160
and history of proof, 1149
and intelligence, 1178
non-locality of, 124, 731
non-standard models of, 800, 1169
and origin of math, 792, 859
Presburger, 1143, 1152
processes of, 117
recursive functions and, 907
rules in, 875
universal equation in, 786
universality in, 784
see also Peano arithmetic
Arithmetic coding, 1071
Arithmetic hierarchy
and oracles, 1139
Arithmetic operations
and computational power, 730
performed by CAs, 639
Arithmetic progressions
and Ramsey theory, 1068
|  | Arithmetic puzzles, 875
Arithmetic recurrences, 123
and register machines, 100
Arithmetic systems, 122-124
emulating register machines, 673, 1114
history of, 1115
and Turing machine 600720, 1145
see also 3n+1 problem
Arithmetization of metamathematics, 1158
ARMA processes
spectra of, 1082
and texture discrimination, 1077
Armadillo skin
texture of, 385
Array computers
and history of CAs, 877
Arrow function (nested power)
and Ackermann function, 906
and symbolic systems, 103
Arrowheads
characteristic shapes of, 1183
Art
as affecting perception, 1106
applications to, 841, 872
astronomical systems like, 835
and cellular automata, 872, 929
examples of ornamental, 43
and form of extraterrestrials, 1191
recognition of, 839, 874
shell patterns in, 1012
and simple rules, 42, 872
study of form in, 967, 1003, 1010
summary of relations to, 11
and visual perception, 1076
Arthropods
growth of, 1011
Artichoke
phyllotaxis in, 409
Artifacts
on astronomical scales, 834
compared to nature, 3
created by animals, 1184
in data, 1184
on Earth from space, 1187
historical forms of, 1183
problem of recognizing, 828, 1183
in turbulence computations, 997
universe as, 1191
Artificial intelligence (AI), 628, 1099
and complexity history, 862
and defining intelligence, 1178
and extraterrestrial intelligence, 839
and free will, 1135
predicate logic and, 1152
summary of relations to, 12
and theorem proving, 1157
and visual perception, 1076
Artificial life
and biological form, 1004
and natural selection, 1001
summary of relations to, 12
Artificial organs, 842
Artificial snowflakes, 992, 992'
Artificial tissues, 842
ASCII versions (InputForm)
of programs in notes, 854
ASICs (Application-Specific Integrated Circuits), 1097
Asparagus
phyllotaxis in, 409
AspectRatio
and GoldenRatio, 891
Assembler
and my early programs, 854, 864
and register machines, 101, 1114
Associative algebras, 801
as generalizing numbers, 1168
Associative calculi
see Multiway systems
Associativity
3-argument analog of, 1171
of And and Or, 817
in Boolean functions, 1173
cellular automata based on, 886
and fast evolution, 1094
and generalized additivity, 952
in operator systems, 801
as source of self-similarity, 956
see also Flat
Assyrian military drill, 875
Asteroids
and chaos theory, 973
Astrology, 1135
Astronauts
quarantine of, 1180
Astronomical clocks, 1184
Astronomy
Babylonian, 859
extraterrestrial artifacts in, 834
geometrical patterns in, 1187
Greek, 861
and history of computing, 1107
and history of statistics, 1082
measurement errors in, 967
and prehistoric structures, 1184
radio phenomena in, 1187
randomness in, 313
and time in universe, 1021
Asymptotic freedom in QCD, 1057, 1062
Asymptotic growth rates, 1162
difficulty of finding, 1146
Asymptotic series
in perturbation theory, 1060
Asymptotic states
and quantum field theory, 1130
Asynchronous cellular automata
see Sequential cellular automata
Atanasoff, John V. (USA, 1903-1995)
and early electronic computers, 1107
Atmosphere
2D fluid flow in, 999
of extrasolar planets, 1179
and origin of life, 1179
and randomness in lightning, 995
turbulence in, 1001
and weather patterns, 1177
Atomic models
of crystal growth, 993
of fracture, 995
Atomic physics
6j symbols in, 1055
and quantum theory, 1056
Atomic-scale devices, 841, 1193
Atomic-scale features
and future forms of perception, 1106
Atomic-scale mapping of wear, 1184
Atomism
and elementary particles, 1044
as general theory, 1125
history of, 876
Atoms
arrangement of in crystals, 369
and chaos theory, 310
and concept of programs, 860
early models of, 1026
and history of complexity, 861
and history of quantum theory, 1056
as knots in the ether, 1028
in models of space, 1028
as source of randomness, 967
and thermodynamics, 445, 1019
ATP
and definition of life, 825
Attractors, 275
effective reversibility on, 1018
in finite systems, 961
in iterated maps, 961
and memory, 625
nested structure of, 959
and phase transitions, 341
in root finding, 1101
and structures in PDEs, 1129
in systems based on numbers, 961
in Turing machines, 961
in visual perception, 1076
Auctions
game theory and, 1105
Audio representation
for cellular automata, 869
for data, 1080
Auditory cortex, 1079
Auditory perception, 585-588
Augustine (Algeria, 354-430 AD)
and free will, 1135
Aurora
and natural radio emissions, 1187
Australia
straight road in, 1187
Authentication
and hashing, 1100
Autoactive decay processes, 905
Autocorrelations
in shift registers, 1084
and spectra, 1080
in textures, 1076
Automata
see also Cellular automata
see also Finite automata
see also Mobile automata
Automated theorem proving, 1157
for axioms of logic, 811
and chemical synthesis, 1194
and group axioms, 1153
and paramodulation, 1156
and proof frameworks, 1177
proofs not useful from, 811, 1156
for Robbins axioms, 1151
of universality, 1115, 1127
Automatic groups, 1141
Autoplectic processes (intrinsic randomness generation), 973
Auxins (plant hormones), 1004
Auxology
and recursive growth, 877, 929
Avalanches
in sandpile models, 989
in snow, 993
Average-case difficulty, 1143
Averaging
in diffusion equation, 1024
and Gaussian distributions, 976
as source of continuity, 327
as source of uniformity, 353
Avoidable patterns (repetition-free sequences), 944
Axelrod, Robert M. (USA, 1943- )
and prisoner's dilemma, 1104
Axiom of choice, 774, 1154
and additive functions, 953
Axiom of infinity, 1154
Axiom of replacement, 1154
Axiom schemas, 1156, 1176
Axiom systems
based on multiway systems, 1173
for basic logic, 808
for cellular automata, 794, 1168
and defining complexity, 1068
effect of expanding, 1159
enumeration of, 809
for extended set theory, 1167
forcing an operator in, 1172
general structure of, 1150
vs. generators and relations, 1169
interpreting algebraic, 1153, 1159
model theory of, 1172
for multivalued logic, 1175
and operator evolution systems, 898
for operator systems, 801
and P=NP, 1146
proving correctness of, 802, 1170
sampled by mathematics, 795
table of in mathematics, 773
theorems holding in simple, 812
and Turing machines, 1145
and universality proofs, 1127
as way to handle infinity, 788
Axiomatic quantum field theory, 1057, 1065
Axiomless formulations
of logic, 1156
Axons
of nerve cells, 1075
B particles
time reversal violation for, 1019
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