# Index

ⅇ (exponential constant)

continued fraction for, 144

digits of, 142

as not from rational integral, 916

and random networks, 963

e-commerce

use of randomness in, 1192

E_{8} (group) lattice

isotropy of, 980

and sphere packings, 987

Eanna temple, 873

Ear lobes

folding in formation of, 417

Early computers

and cellular automata, 45

Early universe

particles surviving from, 1047

and thermodynamics, 1021

Ears

operation of, 585, 1079

see also Auditory perception

Earth

conditions for life on, 1179

and extrasolar planets, 1179

formation of the, 1179

origin of life on, 1179

viewed from space, 835, 1187

Earth-Moon-Sun system, 313, 972

Earthquakes

*1/f* noise in, 969

and fracture, 374

Earthworks

as artifacts, 1184

Eastern philosophy, 1195, 1196

ECAD

multilevel logic in, 1096

Ecology

and complexity research, 863

of extraterrestrial colonization, 1191

iterated maps and models in, 918

Economics

applications to, 429–432

and complexity in science, 861

and computational irreducibility, 1132

and extraterrestrial trade, 1191

and free will, 1135

history of models from, 1015

iterated maps and models in, 918

and notion of purpose, 1185

and pragmatic theory of law, 1136

quantitative laws in, 1014

successes of math in, 859

ed (text editor)

and regular expressions, 958

and substitution systems, 894

Eddies

in fluid flow, 376

Eddington, Arthur S. (England, 1882–1944)

and models of the universe, 1025

Eddy viscosities, 996

Eden, Murray (USA, 1920– )

and aggregation systems, 978

Eden model, 331, 978

see also Aggregation systems

Edge detection in images, 1077

and scale invariance, 1076

Edgerton, Harold E. (USA, 1903–1990)

and splashes, 1000

Edison, Thomas A. (USA, 1847–1931)

and searching for materials, 1193

Education

and defining intelligence, 1178

and exact solutions, 1133

and recognizing intelligence, 825

and the science in this book, 855

EDVAC (computer)

and early CA, 1117

EEG, 1099

Eel

pigmentation pattern of, 426

Effective computability

see Decidability

Effectively computable sets, 1138

Efficiency of computations, 758

Efficient causes, 1185

Efficient Market Hypothesis, 432, 1015

Egg

randomness in fertilization of, 970

Egmont volcano

circular shape around, 1187

Egyptian art, 873

Egyptian fractions, 915

Egyptian mathematics, 1149

and evaluation chains, 1095

and pi, 911

Egyptian pyramids, 874

Ehrlich, Paul (Germany, 1854–1915)

and chemotherapy, 1193

`Eigenvalues`

and dimension of networks, 1031

of distance matrices, 1031

and entropies of CAs, 958

and fractal dimensions for additive rules, 955

of nested systems, 1081

nesting in pattern of, 1130

of random matrices, 918, 977

Eikonal approximation, 1061

Einstein, Albert (Germany/Switzerland/USA, 1879–1955)

and EPR experiment, 1058

and general relativity, 1047

and math in science, 860

and particles as gravity, 1054

and quantum theory, 1056

and special relativity, 1041

and unified field theory, 1028

Einstein equations, 1052

exact solutions to, 1133

as examples of PDEs, 161, 923

history of, 1047

matter vs. space in, 1028

quantization of, 1054

specifying curvature of space, 531

vacuum solutions to, 1053

Einstein–Hilbert action

and spin networks, 1055

Einstein–Podolsky–Rosen (EPR) experiment, 1064

Elasticity theory

and fracture, 995

Electric breakdown, 302, 995

see also Lightning

Electric fields

particle production in, 1062

Electric light bulb filaments, 1193

Electricity

as defining feature of life, 1178

Electromagnetic fields

and gauge invariance, 1045

quantization of, 1056

and radioactive randomness, 970

and relativistic invariance, 1041

theories of based on space, 1028

as ultimate constituents, 1024

and virtual photons, 1060

Electromagnetic radiation

and information transmission, 1069

Electromagnetic signals

see Radio signals

Electromechanical machines

as computers, 1107

and finite automata, 958

Electron diffraction

and visualizing fracture, 995

Electron levels

in nested potentials, 1081

Electron-photon scattering, 1060

Electronic computers

and Church's Thesis, 1125

history of, 1107

and history of universality, 1110

numbers in history of, 902

and proofs of universality, 1117

Electronic design

and Boolean formulas, 1097

Electronic noise, 302, 968

randomness from, 969

Electronics

making reliable, 981

and studying complexity, 862

Electrons

discovery of, 1024, 1044

as initiators of randomness, 303

lack of intuition about, 537

limits on sizes of, 1044

magnetic moment of in QED, 1060

mass of, 1046

in practical computers, 1108

theories of structure of, 1044

wave-particle duality of, 1056

Elegant theorem

definition of, 1176

`Element`

in set theory, 1155

Elementary cellular automata, 53, 883

emulation of rules by, 705

legal rules for, 948

my first pictures of, 19, 864

with random initial conditions, 232, 424

rule equivalences, 883

table of Boolean forms for, 884

uniqueness of patterns from, 956

as universal logic functions, 1174

see also Cellular automata

see also Rule 30, Rule 90, etc.

Elementary functions

difficulty of evaluating, 1134

and math impossibilities, 1137

numbers generated from, 916

vs. special functions, 1092

Elementary particles, 525–530

decays and time quantization, 1032

detection of, 969

discovery dates of, 1043

as dual to black holes, 1045

enumerating in networks, 1045

interactions between, 539

known types of, 1043

and limit of networks, 1030

masses of, 528, 1046

motion of, 528

motion of along geodesics, 531

motion of through network, 536

quantum threads between, 544

searching for additional, 1047

spin of, 1046

in unified field theory, 1028

upper limits on mass of, 1047

*Elements*

of Euclid, 909

Elephants

communication by, 1181

Elkies, Noam D. (USA, 1966– )

and nearby powers, 1166

in Preface, xiii

Elliott 903 computer

my use of, 854, 864

Elliptic curves

and Diophantine equations, 1164, 1165

and Fermat's Last Theorem, 1166

Elliptic equations (PDEs), 923, 940

Elliptic functions

and background for PDE, 924

difficulty of evaluating, 1134

and logistic map formulas, 1098

`EllipticK`

(elliptic integral)

in 2D Ising model, 982, 1133

`EllipticTheta`

(elliptic theta functions)

and algebraic equations, 1092

as generating function, 1092

nested patterns from, 1093

and quintic equations, 945, 1133

and runs of digits, 914

Embeddings of networks, 193, 476, 1031

Embryos

development of, 417, 419, 1009

history of study of, 1004, 1010

pigmentation precursors in, 425

randomness in, 970

Emergence

of order from randomness, 223

Emergent behavior

general theory of, 3

Emotions

and learning, 1102

Empirical mathematics

see Experimental mathematics

Empiricism

and theories of communication, 1181

and visual perception, 1076

Emulation

between cellular automata, 1118

by universal CA, 652

and computational irreducibility, 742

rates of, 765

and universality, 642

see also entries for specific systems

Encoding

arithmetic, 1071

of arithmetic by set theory, 1160

block, 563

dictionary-based, 565

Huffman, 564

in mathematics, 815

pointer-based, 565

run-length, 560

theory of cellular automaton, 1118

of three colors by two, 655, 1111

in universal CA, 652

and universality proofs, 1126

Encryption

see Cryptography

*Encyclopedia Britannica*

leaf shapes in first edition of, 1005

Endogenous randomness

see Intrinsic randomness generation

Endomorphisms

cellular automata as, 959

and generalized additivity, 952

Endpapers in this book, 851

Energy

and advanced technology, 1191

in gravity theory, 537

heat as form of, 1019

Energy conditions

and general relativity, 1053

Energy conservation

in cellular automata, 458

in Einstein equations, 1052

in Ising model CAs, 982

in network evolution, 529

in PDE, 923

in physics, 1022

and thermodynamics, 445

in three-body problem, 972

Energy levels

in nested potentials, 1081

Energy minimization

and adaptation, 823

in chemistry, 1194

and NP completeness, 1146

in proteins, 988, 1003

and purpose, 831

and quasicrystals, 994

Energy-momentum tensor

in Einstein equations, 1052

Engineering

avoiding complexity in, 723

vs. biological evolution, 393

vs. biological forms, 1004, 1180

and computational reducibility, 829

correctness proofs in, 1157

fracture as issue in, 995

future of, 840

historical forms in, 828, 1183

and history of complexity, 862

intuition from, 40

optimization of designs in, 1193

of universal systems, 678

and universality proofs, 698

used in building computers, 822

English

and cryptanalysis, 599

frequency of letters in, 1069

hashing of words in, 1100

logic operations in, 1173

redundancy of, 1086

ENIAC (computer)

and digits of pi, 911

and history of computing, 1107

Enigma rotor machine, 1085

Ensembles

evolution of in CAs, 278

in Ising model, 982

in rule 37R, 457

in statistical mechanics, 1020

in statistics, 593

Entanglement (in quantum theory), 544

as basic quantum effect, 1059

and Bell's inequalities, 1064

Entrails of goats

and fortune-telling, 968

Entropy

as algebraic number in CA, 958

analogy of to amount of computation, 1133

vs. biological complexity, 1003

in block encoding, 1071

in cellular automata, 958

as defining CA classes, 960

and definition of life, 1178

estimation of, 959, 1084

history of, 1019

and Lyapunov exponents, 960

measure, 959

origins of increasing, 441–457

of particles in box, 1022

and redundancy of text, 1086

and repetitive CA behavior, 954

of rule 30 mapping, 1087

and self-organization, 948

spacetime, 960

temporal, 960

in thermodynamic systems, 448

undecidability of in CAs, 958, 1138

see also Second Law of Thermodynamics

Entscheidungsproblem (decision problem), 1110

Enumeration of

alkanes, 1194

balanced parenthesis strings, 989

CA periodic points, 954

CAs to find doubling rules, 832

cases in proofs, 1177

commutative monoids, 952

cycles in shift rules, 963

finite automata, 957

programs, 758

reachable states in CAs, 958

string constraints, 1140

surjective cellular automata, 959

tiling configurations, 959

Turing machines, 888, 1120

Envelopes

masking patterns on, 1077

Environmental sciences

math morphology in, 1077

and systems theory, 862