H theorem of Boltzmann, 1020

Haar wavelets, 1074

Hackers
and precursors to my work, 879
and universe as computer, 860

Hadamard, Jacques S. (France, 1865–1963)
and chaos theory, 971
and Hadamard matrices, 1073

Hadamard matrices, 1073

Hadamard order
for Walsh functions, 1073

Hadrons
as particles, 1043
in QCD, 1061

Hailstone numbers (3n+1 problem), 904

Hair cells
in inner ear, 585, 1079

Hair follicles
and animal coloration, 1012

Haken, Hermann P. J. (Germany, 1927– )
and reaction-diffusion, 1013

Hales, Stephen (England, 1677–1761)
and leaf growth, 1005

Hales, Thomas C. (USA, 1958– )
and sphere packings, 986

Halftoning, 1077

Halite (rock salt) hopper crystals, 993

Halting
in aggregation systems, 979
in function definitions, 1163
longest times for, 1144
in one-way TMs, 759, 1143
in register machines, 896
in small Turing machines, 1119
in symbolic systems, 897
in Turing machines, 889

Halting probabilities
and algorithmic randomness, 1067
for code 20, 964
for Turing machines, 1143
and universal objects, 1127

Halting problems, 1137
analog of, 754
axioms for proofs about, 1167
for cellular automata, 1137
and correspondence systems, 1140
and integer equations, 788
in Mathematica, 1138
for multiway systems, 1137
and negation, 1158
and oracles, 1126
proofs of undecidability of, 1137
and proving undecidability, 1130
and sets, 1139
in specific axiom systems, 1163
trivial for primitive recursion, 907
for Turing machines, 1137
undecidability of, 1128, 1136

Halton (digit reversal) sequences, 905

Hamiltonian paths
and NP completeness, 1146
trivalent networks with, 1029

Hamiltonians
in discrete quantum systems, 1060
for PDEs, 923

Hamming, Richard W. (USA, 1915–1998)
and error-correcting codes, 1101

Hamming codes, 1101

Hamming distance
and error correction, 1101
and layout of networks, 1031

Hamming weight (DigitCount), 902

Handles on networks, 527, 1045

Handprints
purpose of in cave paintings, 1184

Handwriting
randomness in, 1192

Hankel, Hermann (Germany, 1839–1873)
and generalization in math, 1168

Haploid cells
and randomness in biology, 970

Hard hexagon model
and 2D entropies, 959
as exactly soluble, 1133

Hard sphere gas, 1022
and CA fluids history, 999
CA idealization of, 446
and chaos theory, 971
on inspirational book cover, 864
with stationary scattering, 312
and thermodynamics, 1020

Hard square models (lattice gases), 999

Hardware
for 1D cellular automata, 868
for 2D cellular automata, 928
structure of computer, 1108

Hardy, G. Harold (England, 1877–1947)
and Goldbach's Conjecture, 911
and iterated maps, 919

Harmonic analysis
of substitution systems, 1081
and Walsh functions, 1073

Harmonic analyzer, 1107

Harmonic functions
and flatness in tissue growth, 1007, 1010

Harmonic oscillator
as exactly soluble, 1133

Harmony (musical)
and math thinking, 1177
perception of, 1079

Harmony in nature, 967

Harrington, Leo A. (USA, 1946– )
and unprovable statements, 1163

Hartley, Ralph V. L. (USA, 1888–1970)
and information theory, 1181

Hartman, Hyman (USA, 1936– )
and CA classes, 948
in Preface, xiii

Harvard Mark I computer, 1107

Hashing, 622, 1100
as application of randomness, 1192
to speed up evolution, 1134

Hasse diagrams of posets
and causal networks, 1040

Hasse's algorithm (3n+1 problem), 904

Hasse's Principle
for Diophantine equations, 1165

Hastings, Stuart P. (USA, 1937– )
and reaction-diffusion, 1013

Hauptvermutung (of combinatorial topology), 1051

Hausdorff dimension, 933, 958, 1030

Hawking, Stephen W. (England, 1942–[2018])
and black hole radiation, 1062

Hawking radiation, 1062
and uniform spectra, 988

Hayek, Friedrich A. (Austria/England/USA/Germany, 1899–1992)
and free will, 1135

hcp (hexagonal close packing), 986

Head size
in animal growth, 421

Heads (of expressions), 896
depth of in trees, 897
as type specifications, 898

Hearing, 585–588
of elephants, 1181
human, 1079

Heart (human)
1/f noise in beating of, 969
as excitable medium, 1013
modelling and history of CAs, 878

Hearts of palm, 1004

Heat
analogies with computation, 726
character of, 445
history of theories of, 1019

Heat bath
and 2D Ising model, 982

Heat capacities
and quantum history, 1056

Heat exchangers
geometrical forms of, 1183

Heated fluids
and convection, 1000

Heaven
and form of extraterrestrials, 1191

Heavenly bodies
perfection of, 875

Hebb, Donald O. (Canada, 1904–1985)
and neural networks, 1099

Hebrew religion
and free will, 1135

Hedlund, Gustav A. (USA, 1904–1993)
in Preface, xiii
and shift-commuting block maps, 878
and surjective CAs, 960

Heights of humans
distribution of, 977, 1003

Heighway, John E. (USA, 1930– )
and dragon curves, 893

Heisenberg, Werner K. (Germany, 1901–1976)
and matrix mechanics, 1056

Heliozoan
pattern formed in, 385

Helium
lack of quantum solution to, 1133
randomness in convection of, 976

Helix
in shell models, 1008

Hellman, Martin E. (USA, 1945– )
and math cryptography, 1089

Helmholtz, Hermann L. F. v. (Germany, 1821–1894)
and sound perception, 1080

Henkin semantics (for higher-order logics), 1167

Heptagons
hyperbolic tiling with, 1050
and negative curvature, 532

Herbs
leaf shapes for identifying, 1005

Herds
and randomness in markets, 1015

Heredity
use of simple rules in, 860

HermiteH (Hermite polynomials)
in solution of harmonic oscillator, 1133

Hero (Heron) (Egypt, ~62 AD)
and computing devices, 1107

Heterostructures, 1082

Hewlett-Packard calculators
reverse Polish in, 896

Hexagonal cellular automata
and fluid flow, 378
implementation of, 992
as models of crystal growth, 369
and momentum conservation, 1024
number of rules in, 928

Hexagonal close packing, 986

Hexagonal lattice
of animal pigment cells, 1012
circle packings in, 349, 985
of circles in ornament, 873
of deformable objects, 988
enumeration on, 959
implementation of CAs on, 992
isotropy on, 980
and r=1/2 1D CA, 885
random walks on, 329
and shapes of cells, 1007
Turing machines on, 930

Hexagonal prism, 929

Hexagrams
of I Ching, 893

Hidden Markov models, 590, 1084

Hidden units
in neural networks, 1102

Hidden-variables theories, 543, 1058, 1064

Hierarchies
as avoiding thermodynamics, 453
in computational complexity theory, 1142
and defining complexity, 1069
in embryo development, 419
of function growth rates, 1163
and history of complexity, 862
in probabilistic models, 1084
and randomness in markets, 1015

Higgs field
and expansion of universe, 1055
and notion of ether, 1042
particle associated with, 1047
and particle masses, 1047

High-energy collisions
Monte Carlo simulation of, 968

High-level programming
and programs in notes, 854

High-school algebra
axioms of, 1153

Higher-order logics, 1167

Higher organisms
phenomenon of existence of, 398

Highest common factor
see GCD (greatest common divisor)

HighLife (2D class 4 CA), 949

Higman, Graham (England, 1917–[2008])
and axioms for groups, 1153

Hilbert, David (Germany, 1862–1943)
and axioms for geometry, 1154
and character of math, 1176
and Diophantine equations, 1161
and foundations of math, 1149
and general relativity, 1052
and math in science, 859
and metamathematics, 1150
and space-filling curves, 893
and undecidability, 1136
and zeros of zeta function, 918

Hilbert space
in quantum theory, 1059

Hilbert's Tenth Problem, 1161

Hilbert's Thirteenth Problem, 1128

Hill climbing, 346

Hillenbrand, Thomas (Germany, 1970– )
and automated proofs, 1158

Hillman, David (USA, 1955– )
and axioms for logic, 1151
in Preface, xii

Hindu-Arabic number system, 902, 1182

Hindu temples
nesting in, 874

Hinton, Geoffrey E. (England/USA/Canada, 1947– )
and neural networks, 1100

Hippocampus (in brain), 1102

Hippocrates of Chios (Greece, ~460 BC)
and embryology, 1010

Hiss
sound of random sequence, 585

Historians
and free will, 1135

Historical perspectives, 844–846
see also history items in specific entries

History
accidents of in universe, 505
as central to human condition, 846, 1196
of complexity research, 862
computations done in human, 837
of definition of sciences, 863
effect on math of, 792
as explaining axiom systems, 816
general comments about, 850
of life on Earth, 387, 825
of math and ease of concepts, 1177
recorded by physical objects, 1195
role of in biological evolution, 388
theory of and chaos, 971
uniqueness of for universe, 504

History register
see Shift registers

Hobbes, Thomas (England, 1588–1679)
and free will, 1135
and nature of space, 1028

Hodgkin–Huxley equation
as example of PDE, 161

Hofmeister, Wilhelm F. B. (Germany, 1824–1877)
and phyllotaxis, 1007

Hofstadter, Douglas R. (USA, 1945– )
and recursive sequences, 880, 907

Hokusai, Katsushika (Japan, 1760–1849)
and fluid turbulence, 997

Hölder exponents, 959

Holland, John H. (USA, 1929–[2015])
and genetic algorithms, 985
in Preface, xiii

Homeobox genes, 419, 1010

Homoclinic tangles, 972

Homogeneity
origins of, 353

Homogeneous structures
see Cellular automata

Homomorphisms
and generalized additivity, 952

Homoplectic processes (extrinsic randomness generation), 973

Homotopy
and structures in networks, 1046
and topological defects, 1045

Homunculus hypothesis, 1010

Honda, Hisao (Japan, 1943– )
and branching in plants, 1005

Honeycomb
form of, 1011
length minimization in, 988

Hooke, Robert (England, 1635–1703)
and snowflakes, 992

Hopfield, John J. (USA, 1933– )
and neural networks, 1100, 1102
in Preface, xiii

Hopi language
and Sapir–Whorf hypothesis, 1181

Hopper crystals, 993

Horizon problem (in cosmology), 1055

Hormones
in brains, 1099
in plants, 404

Horn shell
growth of, 415

Horner form
and evaluation of powers, 1093

Horns (animal)
as artificial-looking, 828
compared to antlers, 1008
growth of, 413

Horse races
as source of randomness, 968

Horton stream order (in tributaries), 1001

Hours-minutes-seconds
origin of, 902

Hoyle, Fred (England, 1915–2001)
and extraterrestrials, 1190

HP 700
and creation of this book, 854

HP calculators
reverse Polish notation in, 896

Huffman, David A. (USA, 1925–1999)
and data compression, 1069

Huffman coding, 564
adaptive, 1069
implementation of, 1071
and JPEG compression, 1074

Human behavior
and computational irreducibility, 1132
free will in, 750–753, 1135
and game theory, 1104
laws of, 1014
Occam's razor for, 1025

Human condition
defining features of, 1196
scientific approaches to, 845

Human form
parametrizations of, 1010

Human history
vs. biological evolution, 399

Human languages
see Languages (human)

Human relics in archeology, 1183

Human thinking, 620–631
and constructs in math, 1177
emulation by computers of, 733
and Gödel's Theorem, 1159
and Principle of Computational Equivalence, 733, 1130

Human will
see Free will

Humanities
relativism in, 1131
summary of relations to, 9, 863

Humans
artifacts of early, 1184
defining properties of intelligence of, 822
and future technology, 1195, 1196
genetic programs of, 383, 1002
height distributions of, 977, 1003
making randomness, 306, 970
mathematical power of, 1126
as only intelligence, 822
uniqueness of, 844, 1195
visual system of, 577
weight distributions of, 977

Humboldt, F. W. H. Alexander v. (Germany, 1769–1859)
and branching in plants, 1004

Hume, David (Scotland, 1711–1776)
and purposes in nature, 1185

Humidity
and weather, 1177

Huntington, Edward V. (USA, 1874–1952)
and axioms for logic, 1151

Huntington axioms for logic, 773, 1151

Hurwitz numbers, 914

Hydraulic jump, 377

Hydrocarbons
considered interesting, 1194

Hydrodynamics
see Fluid flow

Hydrogen atom
as exactly soluble, 1133
and history of quantum theory, 1056

Hydrology
and forms of landscapes, 1001

Hyena
pigmentation pattern of, 426

Hyperbolic equations, 923, 940

Hyperbolic space, 1049

Hypercube
Boolean, 1095
and trivalent networks, 1029

Hypercycles
and origin of life, 1179

Hyperelliptic Diophantine equations, 1164

Hyperelliptic functions
in solution of spinning top, 1133

Hypergeometric functions
asymptotic behavior of, 917
and continued fractions, 914
and curves from curvature, 1009
and nested patterns, 612

Hypergeometric1F1
in Calogero–Moser model, 1133

Hypergeometric2F1
and caps on hyperspheres, 1050
and quintic equations, 945
in Seiberg–Witten model, 1133
and special functions, 1092
and Thue–Morse sequence, 1092
and trinomial coefficients, 1091

HypergeometricPFQ
and 6j symbols, 1055
difficulty of evaluating, 1134
from rational integral, 916

Hypergraphs, 1029

Hyperreals
as generalizing numbers, 1168

Hypersonic flow
see Supersonic flow

Hypersphere
volume of, 479, 1050

Hyphenation
lack of in this book, 852