# Index

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

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