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

Learning curves, 1014, 1102

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