Fabric
wrinkling of, 996
Face-centered cubic (fcc)
lattice, 930
sphere packing, 986
Voronoi diagram, 987
Faces (geometrical)
in deformable packings, 988
and discrete spaces, 1051
in planar networks, 1038
Faces (human)
development of bones in, 1010
parametrization of, 1010
random variations in, 1192
recognition of, 626
Faceted crystal growth, 370
Factor (polynomial factorization)
analog for Boolean formulas, 1095
and cycles in shift rules, 963
and quadratic congruential generators, 975
randomized algorithm for, 1192
and shift register periods, 975
Factorial (n!)
encoded as integer equation, 1160
and examples of universality, 1110
and hypersphere volume, 1050
leading digits in, 914
and number of reversible block cellular automata, 1023
number with digits at, 914
and QED perturbation series, 1060
as recursive sequence, 906
and transcendental numbers, 914
see also Gamma
Factorial2 (double factorial)
and isotropic tensors, 980
Factories
and self-reproducing systems, 876
FactorInteger (integer factorization)
and arithmetic system encoding, 1115
basic methods for, 1090
difficulty of evaluating, 1134
and Fermat's Last Theorem, 1166
and multiplicative digit sequences, 902
and MultiplicativeOrder, 1147
and NP completeness, 1146
and number of commutative groups, 1172
and quantum computers, 1148
running times for, 1090
unique domains for, 915
Factorization
of cellular automata, 956
False
definition from And and Not, 817
Falsifiability of models, 365
Faltings theorem (Mordell conjecture), 1164
Family trees
and substitution systems, 893
Fano, Robert M. (USA, 1917- )
and data compression, 1069
Farey sequences, 916
Farm animals
growth of, 1010
Fast algorithms, 1141
Fast Fourier transform (FFT)
recursive algorithm for, 1142
see also Fourier
Fast Walsh transform, 1073
Fastest programs, 760
Fate
vs. randomness, 967, 1135
Fax
data compression in, 1070
fcc (face-centered cubic)
lattice, 930
sphere packing, 986
Voronoi diagram, 987
Feasible computations, 1143
see also Polynomial time
Feathers
coloration of, 1012
Feature detection
as basis for perception, 549
in human visual system, 577
Fedorov, Evgraf S. (Russia, 1853-1919)
and shapes of 3D domains, 929
Feedback
control and cybernetics, 862
in growth of bones, 1010
in visual system, 1075
Feedback shift registers, 974
see also Shift registers
Feigenbaum, Mitchell J. (USA, 1944- )
and computer experiments, 899
and iterated maps, 921
in Preface, xiii
Feigenbaum's constant, 913
Fejes Tóth, László (Hungary, 1915- )
and circle packing, 985
Fermat, Pierre de (France, 1601-1665)
and Fermat's Last Theorem, 1166
and primes, 909
Fermat primes conjecture
as unsolved problem, 1166
Fermat's Last Theorem, 1166
and Diophantine equations, 1164
as having long proof, 779
and number theory history, 910
and sparse solutions, 945
Fermat's little theorem, 909
Fermat's principle (in optics), 1061
Fermi, Enrico (Italy/USA, 1901-1954)
and computer experiments, 879, 899
and extraterrestrial contact, 1191
and statistical mechanics, 1020
Fermi-Pasta-Ulam experiment, 1020
Fermilab
circular shape of, 1187
Fermions
and basic quantum effects, 1059
and classical limits, 1059
spin-statistics of, 1046
Ferns
branching in, 401, 1004
and iterated function systems, 1005
Ferrite core memory, 864
Ferrites
as components in technology, 1195
Ferrofluids
maze-like patterns in, 1013
phyllotaxis-like patterns in, 1007
Ferromagnetic materials
Ising model for, 981
phase transitions in, 981
Fertilization
randomness in, 970
Feynman, Carl R. (USA, 1962- )
and cryptanalysis of rule 30, 1087
in Preface, xiii
Feynman, Richard P. (USA, 1918-1988)
and cryptanalysis of rule 30, 1087
and discrete quantum models, 1060
and discreteness of space, 1027
and path integrals, 1057
and physics and computation, 1126
in Preface, xii, xiii, xiv
Feynman diagrams, 1060
exact calculations of, 1133
history of, 1057
Monte Carlo methods in, 1192
not as mechanistic model, 1026
in QCD, 1061
for quantum gravity, 1054
and random networks, 1039
vs. space networks, 1040
and spin networks, 1055
FFT (fast Fourier transform)
digit reversal sequences in, 905
implementation of, 1074
multiplication using, 1093
recursive algorithm for, 1142
see also Fourier
Fiber bundles
and continuum limits of networks, 1030
and gauge theories, 1045
Fiber optics
vs. broadcasting, 1188
Fibers (biological)
and folding of tissue, 417
Fibonacci, Leonardo (Pisano) (Italy, ~1170 - ~1250)
and digital numbers, 902
and Fibonacci numbers, 891
and rabbit populations, 1002
and tables of primes, 910
and trees, 893
Fibonacci (Fibonacci sequence)
difficulty of making with CAs, 1186
and entropy in rule 32, 958
generalized, 891
generalized for randomness generation, 975
generating function for, 1091
as initial condition for rule 60, 1091
leading digits in, 914
and multiway system states, 205
and multiway systems based on numbers, 939
number with digits at, 914, 1070
in ordering of math constructs, 1177
and plant phyllotaxis, 1006
and polyominoes, 943
as precursors to my work, 878
properties of, 890
and prosody, 875
and randomness generators, 975
and recursion history, 907
as recursive sequence, 128
and rule 150 pattern, 885
as solution to Diophantine equation, 1161
and spectral maxima, 1081
and substitution systems, 82, 890
as term in continued fraction, 913
uniform distribution mod 1, 904
as values of polynomial, 1161
Fibonacci multiway system, 205
Fibonacci number representation, 560, 1070
and multicolor encodings, 1111
as non-power base, 902
and substitution systems, 892
Fibonacci substitution system, 83
and cyclic tag system, 895
and cyclic tag systems, 96
in music, 875
and Penrose tilings, 932
properties of, 890
and sine curves, 147
spectrum of, 586, 1081
two-dimensional, 932
|  | Field Museum of Natural History, 1011
Field operators
measurement of, 1065
Field theory (in abstract algebra), 1153
axioms for, 773
decidability with ordering predicates, 1159
and generalizing numbers, 1168
universality of, 1159
see also Finite fields
Field theory (physics)
history of, 1024
nonlinear PDE as, 923
quantum, 1061
see also Quantum field theory
Fields (agricultural)
patterns of from space, 1187
Fifths (musical chords)
curves of, 146
perfect, 1079
Fig leaves, 1005
Figurate numbers, 911
Filters (for data)
cellular automata as, 225
in visual perception, 1076
Filters in posets, 1040
Final cause, 1185
Final theory
see Ultimate theory of physics
Financial systems
applications to, 429-432
data from as source of randomness, 969
history of models of, 1015
meaning of random data in, 1183
simulations of, 968
using randomness to verify contracts in, 968
FindMinimum
and network layouts, 1031
FindRoot
difficulty of evaluating, 1134
iterative algorithm for, 1141
Fine structure constant (alpha)
numerology for, 1025
and perturbation theory, 1057
Fine tuning
and self-organized criticality, 989
Fingerprints
origin of patterns in, 1013
randomness in, 1014
Fingers
formation of human, 419
Finite automata
and attractors for CAs, 277
and Boolean functions, 1097
and CA encodings, 1119
and defining complexity, 1069
and formulas for nesting, 608
and groups, 1141
history of, 958
lower bounds in, 1143
minimization of, 957
as networks, 957
probabilistic, 1084
and regular languages, 939
and semigroups, 1153
and soundex system, 1100
spectra of, 1081
and substitution systems, 891, 931, 1091
total number of, 957
Finite axiomatizability, 1176
Finite complement languages, 941
2D generalizations, 959
and excluded blocks, 958
and repetition in 1D CAs, 954
Finite differences
and CA diffusion, 1024
and continuous CAs, 921
explicit schemes and sequential cellular automata, 1035
and history of CAs, 876
isotropy in, 980
methods based on, 940
neighborhood compared to cellular automata, 928
for PDEs, 924
and reaction-diffusion, 1013
Finite element methods, 940
Finite fields
as not universal, 1160
see also Additive cellular automata
see also Field theory (in abstract algebra)
Finite groups
axioms for, 1176
Cayley graphs of, 1032
as extraterrestrial messages, 1190
rules for, 938
Finite impulse response
and sequential CAs, 1035
Finite-size scaling
in Ising models, 983
Finite-size systems, 255-260, 961
Finite state machines
see Finite automata
Finitely presented groups
see Groups
Finitistic mathematics, 1158
Finkelstein, David R. (USA, 1929- )
and discreteness of space, 1027
in Preface, xiii
Finnish
logic operations in, 1173
Fins (heat exchanger)
characteristic shapes of, 1183
Fire
as artifact, 1183
as basis for universe, 1125
as visible from space, 1187
Firing of neurons
in neural networks, 1102
repeatable randomness in, 976
Firing squad problem (in cellular automata), 876, 1035
First digits, 914
of powers, 903
First Law of Thermodynamics, 1019
First-order phase transitions, 981
Fisher, Ronald A. (England, 1890-1962)
and random number tables, 968
Fisher-Tippett distribution, 977
Fishes
color vision in, 1075
iterated maps for populations of, 918
pigmentation patterns of, 389, 426
shapes of related by transformations, 1010
Fishhooks
characteristic shapes of, 1183
Fit (least-squares fit), 1083
Fitness
optimization of in biology, 386
and purpose, 1185
Fitting
of models, 589, 1083
Fitting of clothes
and allometry, 1010
FitzGerald, George F. (Ireland, 1851-1901)
and relativity theory, 1041
FitzGerald-Lorentz contraction, 1041
Five-body problem, 1130
Five-fold symmetry
in biology, 1007
cellular automata with, 930
and Penrose tilings, 932, 943
in quasicrystals, 994
Five-neighbor rules (in CAs), 927
Fixed points
in 1D cellular automata, 224, 954
in 2D cellular automata, 954
as attractors, 276
of blocking transformations, 270, 955
in combinator evolution, 712, 1122
for differential equations, 961
in Game of Life, 964
in iterated maps, 955, 961
in Mathematica, 1137
of pigmentation patterns, 427
in recursive evaluation, 906
of sandpile model, 989
and satisfying constraints, 348
in symbolic systems, 897
undecidability of, 1138
FixedPoint
and gradient descent, 985
undecidability in, 1138
FixedPointList
and length prefixed numbers, 1070
and prime implicants, 1095
Flagellated microorganisms
random motion of, 970
Flames
as self-organizing systems, 824
as self-reproducing, 1178
Flash photography
and forms of splashes, 1000
Flat (associative)
and associative CAs, 886, 956
and Boolean functions, 1173
and confluence property, 1037
and fast evolution rules, 1094
and generalized additivity, 952
and sequential substitution systems, 894
Flat spectra, 1081
Flatness (geometry)
of growing sheets of tissue, 1007
Flatten
basic examples of, 853
and correspondence systems, 1139
generalized to 2D, 931, 1073
generalized to any dimension, 932
and sequence equations, 944, 1141
Flatworm
pigmentation pattern on, 426
Flicker noise, 969
see also 1/f noise
Flint arrowheads, 1183
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