# Index

*g* (general intelligence factor), 1178

Gabor functions

in visual system, 1075

Gacs, Peter (USA, 1947– )

in Preface, xiii

and transitions in CAs, 981

Gaia Hypothesis, 1195

Galaxies

distribution of, 1021

and expansion of universe, 1055

formation of, 455, 1021

formation of and my work, 880

formation of and Voronoi diagrams, 987

patterns in, 835

and Second Law, 1021

and simple initial conditions, 1026

Galaxy

colonization of, 1191

radio emission in, 1188

Galena

hopper crystals in, 993

Galilean invariance

in cellular automata fluids, 999

in fundamental physics, 522

and relativity, 1042

Galilei, Galileo (Italy, 1564–1642)

and Galilean invariance, 1041

and math in science, 859

and models vs. reality, 991

and nature of gravity, 1047

and the nature of space, 1028

and theoretical biology, 1003

Galois, Evariste (France, 1811–1832)

and group theory, 1153

and quadratic continued fractions, 915

Galois fields, 1153

see also Finite fields

Galton board

randomness in, 312

Gambling

and defining randomness, 1068

and Gaussian distribution, 977

and randomness sources, 968

Game of Go, 875

Game of Life

and history of 2D CAs, 928

and history of CAs, 877

and history of complexity, 50

implementation of, 949

persistent structures in, 964

as precursor to my work, 880

random initial conditions in, 249

unbounded growth in, 965

universality in, 693, 1117

Game theory, 1104

and computational irreducibility, 1132

and laws in economics, 1014

and models of markets, 1015

and relation to cybernetics, 862

Games

and AI, 1099

between programs, 1105

and history of CAs, 877

and multiway systems, 939

randomness in, 968

rules in, 875

Games of chance

and defining randomness, 1067

and history of statistics, 1082

randomness in, 305, 968, 1192

and tests of randomness, 1084

Gaming regulations, 969

`Gamma`

(gamma function)

argument structures for, 896

and curves from curvature, 1009

emulating `If`

, 1129

from rational integrals, 916

as special function, 1092

and transcendental numbers, 912

see also `Factorial`

Gamma ray bursts, 1187

Gamuts

color, 1074

Gap test, 1085

Garbage collection

in network systems, 199, 935

Garden of Eden

in cellular automata, 876, 961

Gardner, Martin (USA, 1914– )

and Game of Life, 877

Gas clouds

as source of radio emissions, 1187

Gases

CA idealization of, 446

and chaos theory, 971

discrete models for, 999

extraterrestrials based on, 1180, 1190

flow of, 376

see also Fluid flow

as main application of Second Law, 451

thermodynamic theories of, 1019

Gasket

see Sierpiński pattern

Gastrulation (biological folding), 417, 1009

Gates

in practical computers, 1108

quantum, 1147

reversible logic, 1098

see also Boolean functions

Gauge bosons, 1043

Gauge invariance, 1045

and causal invariance, 527

Gauge theories

origin of, 1028

and quantum history, 1057

Gauss, J. Carl Friedrich (Germany, 1777–1855)

and Central Limit Theorem, 977

and curvature of surfaces, 1049

and Diophantine equations, 1164

and experimental math, 899

and extraterrestrials, 1189

and iterated maps, 919

and primes, 909

and sphere packings, 985, 986

Gauss circle problem, 910

Gauss map

and iterated maps, 919

Gaussian curvature

and biological growth, 1009

as invariant quantity, 1049

Gaussian distributions

as assumed in statistics, 1083

and Central Limit Theorem, 976

of constraint violations, 984

in laws of human behavior, 1014

vs. non-standard diffusion, 1024

not from self-avoiding walks, 978

as PDE initial conditions, 163

of price fluctuations, 1014

in quantitative traits, 1003

from random walks, 329

in visual system responses, 1075

Gaussian elimination

as iterative algorithm, 1141

Gauss's law

in cellular automata, 1023

`GCD`

(greatest common divisor)

as associative function, 1094

and computational reducibility, 747

and correspondence systems, 1140

and cyclic addition periods, 950

encoded as integer equation, 1160

and Euclid's algorithm, 915

and `FactorInteger`

, 1090

iterative algorithm for, 1141

and limited size periods, 257

pattern from, 613, 1093

and primitive recursion, 907

and sine curves, 917

sounds from, 1080

Gears

and computing devices, 1107

Geese

patterns in flocks of, 1011

Gegenbauer, Leopold B. (Austria, 1849–1903)

and `GegenbauerC`

, 1091

`GegenbauerC`

(Gegenbauer polynomials)

history of, 1091

and isotropy measures, 980

and rule 150 pattern, 612

and trinomial coefficients, 1091

Geiger counters, 969

Gels

in living systems, 1180

Gemstones

as faceted crystals, 370

synthetic, 993

Genealogical trees

as nested, 988

from substitution systems, 893, 1002

General intelligence factor, 1178

General-purpose computers

and history of universality, 1110

and universality, 642

General recursive functions, 907, 1121

and Church's Thesis, 1125

General relativity

and animal growth models, 1010

difficulty of solving, 1132

and gauge invariance, 1045

history of, 1048

need for explicit matter in, 537

as simple physical model, 1025

and space vs. contents, 1028

as theory of gravity, 530

and time and computation, 1130

and time travel, 1043

and ultimate theory of physics, 1025

variants of, 1048

with varying spacetime dimension, 1056

General systems theory

and history of complexity, 862

summary of relations to, 15

Generalization

in human thinking, 627

in mathematics, 792, 821, 1168

Generalization rule

in axioms, 1150

Generalized mobile automata, 76

Generating function

for additive cellular automata, 951

for balanced parentheses, 989

for cycles, 958

for `DigitCount`

, 902

for `Fibonacci`

, 890

modulo 2, 1092

for nested patterns, 1091

for regular languages, 957

for Thue–Morse sequence, 890

for Thue–Morse spectrum, 1081

Generative grammars, 938

Generators

in groups, 1141

and relations, 1169

Generic behavior

in chaos theory, 971

Genes

as analogs of functions in software, 1003

as components of genetic programs, 1002

Genetic algorithms, 1002

and combinatorial optimization, 985

for design optimization, 1193

and sequential substitution systems, 894

Genetic analysis

and multiway systems, 938

Genetic code

and theoretical biology, 1004

Genetic networks

and Boolean networks, 936

Genetic programming

and methods of prediction, 1105

Genetic programs

and cell positions, 1009

compared to laws of physics, 397

as foundation of biology, 383

and growth of animals, 419

and parts of animals, 417

searching for in biology, 842

smooth variations in, 394

structure of, 1002

Genetic sequences

matching of, 1101

Genetics

and definition of life, 1178

discrete features of, 1003

and mollusc shell patterns, 1012

randomness assumed in, 970

use of simple rules in, 860

Genomes

as programs, 383

Gentzen, Gerhard (Germany, 1909–1945)

and consistency of arithmetic, 1168

and transfinite induction, 1163

Genus

of algebraic surfaces, 1164

of networks, 1045

Geodesic balls

volumes of, 1050

Geodesic deviation

and Riemann tensor, 1049

Geodesic domes

characteristic shapes of, 1183

and curved space, 532

and spherical networks, 1049

Geodesics

and definition of dimension, 1031

distance defined by, 1048

divergence of and chaos, 971

in flat space, 1048

and paths of particles, 531

on surfaces, 1049

Geographic cycle (in landscapes), 1001

Geological fractures, 374, 995

Geometric mean

of continued fraction terms, 914

Geometric series

and branching patterns, 1005

Geometrical constructions

and history of universality, 1109

of real numbers, 1129

rules in, 875

Geometrical forms

in aggregation systems, 979

of artifacts, 835

in astronomy, 1187

in biology, 384

and diversity in biology, 395

on Earth seen from space, 1187

molecules made for their, 1194

of shocks in fluids, 1000

Geometrical optics, 1061

Geometrical Play

of Leonardo da Vinci, 875

Geometrical substitution systems, 189–192

implementation of, 933

visualizing evolution of, 933

Geometry

of animal shapes, 421

applied to biology, 1004

of attractors for ODEs, 961

axioms for, 774, 1154

as basis for science, 44

of branching patterns, 401

of cellular automaton lattices, 929

curvature in, 532

curves in Greek, 875

differential, 1048

as historical basis of math, 792

labelling in diagrams, 1182

and math in science, 859

network of theorems in Euclid's, 1176

non-universality in, 784

and origin of math, 859

in parametrizing biological forms, 1010

of plant phyllotaxis, 409

and proof in math, 1149

puzzles, 875

reasoning in terms of, 1177

rules for planet motion using, 860

of shell shapes, 417

space as uniform in early, 1028

of space of CA rules, 948

theorems and computational irreducibility, 1132

Geomorphology, 1001

Geons

and matter from gravity, 1054

and unified field theory, 1028

Gestalt (in psychology), 1076

Gestures

in animal communication, 1180

grammatical rules for, 1181

Gibbs, J. Willard (USA, 1839–1903)

and statistical mechanics, 1020

Gibbs phenomenon, 917

and experimental math, 899

and image compression, 1074

Gibbs–Thomson effect, 993

GIF compression, 1070

Gila monster

pigmentation pattern of, 426

Gilbreth, Frank B. (USA, 1868–1924)

and causal networks, 1033

Gilbreth, Lillian E. M. (USA, 1878–1972)

and causal networks, 1033

Gilgamesh, Epic of, 873

Ginger leaves, 1006

Giraffe

pigmentation pattern of, 426

Girth

of networks, 1029