Object-oriented programming, 867

Object recognition, 624

Objects
described by axiom systems, 797
and image compression, 1074

Observables
in quantum mechanics, 1056

Observational errors
and Gaussian distribution, 977
and origin of randomness, 967

Observers
and Anthropic Principle, 1026
computational abilities of, 737
definition of particles for, 1062
and origin of causal networks, 487

Occam's razor, 1025

Ocean currents
and weather, 1177

Ocean waves, 1001
as example of randomness, 301
nested breaking of, 997
pattern formation in, 947
repetitive behavior in, 988

Oceans
exploration of, 1191
finding artifacts under, 1183
whale songs in, 1180

Ocelot
pigmentation pattern of, 426

OCR (optical character recognition)
and cellular automata, 877

Octahedra
extraterrestrial in 2001, 1190
truncated, 930, 987

Octanes, 1194

Octaves (in music), 917, 1079

Octonions
as generalizing numbers, 1168

Octopus
communication by, 1181
form of tentacles of, 385
visual system of, 1075

Ocular dominance stripes
CA model for, 1012
maze-like patterns in, 1013
in visual system, 1075

OddQ (test for odd numbers)
and nested patterns, 931
and rule 132, 638

Odds and Evens game, 1105

ODEs, 922
see also Ordinary differential equations

Odlyzko, Andrew M. (USA, 1949– )
and additive CAs, 951
as coauthor of paper, 881
in Preface, xiii
and zeros of zeta function, 918

Odometer
and history of computing, 1107

Odor perception, 1105
and experience of dogs, 827

Offset lists
in general cellular automata, 927

OH (hydroxyl) frequency, 1189

Oil and water
separation of, 980

Oil fires
patterns in, 377
visible from space, 1187

Olaus Magnus (Sweden/Italy, 1490–1568)
and snowflakes, 992

Olbers' paradox, 1021

Old English
logic operations in, 1173

Olfactory perception, 1105
and experience of dogs, 827

Olive shell
growth of, 415

Ω (omega) (halting probability), 1067

ω (first transfinite ordinal), 1162

One-sided additivity
and cryptography, 1088
repetition from, 954

One-time pad (in cryptography), 1086

One-to-one mapping (injective), 960

Ones
expressing numbers with, 916
number of (DigitCount), 902

Onsager, Lars (USA, 1903–1976)
and 2D Ising model, 982

Onto cellular automata, 280, 959

Ontology
implications for, 1197
and notions of space, 1028

Opcodes
in practical computers, 1108
in register machines, 896

Open sets
and axioms of topology, 1155

Open systems
and thermodynamics, 455

Operating systems
and concept of halting, 1137
and history of computing, 1108
in practical computers, 1108

Operation counts in computations, 1143

Operator representations
for integers, 916

Operator systems, 801
automated proofs in, 1158
evolution rules for, 898
frameworks for proofs in, 1177
vs. multiway systems, 1169

Operators
as generalizing numbers, 1168
notation for, 1182
possible Boolean, 806
precedence of, 1104
precedence of in math, 1150
in quantum theory, 1056

Optic nerve, 1075

Optical amplifier, 310

Optical character recognition (OCR)
and history of CAs, 877

Optical computing
CA implemented with, 869
and quantum computers, 1148

Optical fibers
as technology component, 1195

Optical illusions, 1076

Optical SETI, 1189

Optics
and history of math in science, 859
path integral in, 1061

Optimal algorithms, 1141

Optimal evaluation chains, 1095

Optimality
and defining purpose, 829
and models of markets, 1015

Optimization
combinatorial, 985
and complexity in biology, 398
of designs in engineering, 1193
intuition about in programs, 872
and Occam's razor, 1025
in plant leaf arrangements, 408

Optimized circuit blocks, 1193

Optimizers
and proofs about programs, 1168
search strategies in, 1193

Options
market prices of, 1015

Or (∨)
axioms in terms of, 1151
and Boolean formulas, 1095
calculus of expressions, 1170
emulated by CAs, 662, 1112
encoded as integer equation, 1160
in multivalued logic, 1175
and P completeness, 1149
and satisfiability, 1146
table for, 806
theorems involving, 817
words in languages for, 1173

Oracles (in computation theory), 1126
and arithmetic hierarchy, 1139
based on infinite tables, 720
and intermediate degrees, 1131
and non-standard arithmetic, 1169
and second-order logic, 1167

Orbits
in Bohr atom, 1056
of comets and Gaussians, 977
and history of computing, 1107
randomness in, 313
as simple shapes, 1187
in three-body problem, 972

Order
emergence from randomness, 223

Order diagrams of posets
and causal networks, 1040

Order of updates
in sequential CAs, 1034
in sequential substitution systems, 894
in string rewriting, 497
in symbolic systems, 898
see also Causal invariance

Order parameters
in phase transitions, 981

Ordered fields
axioms for, 1153

Orderless (commutative)
and associative CAs, 956
Boolean functions, 1173
and generalized additivity, 952
and speedups in evolution, 1095

Orders of infinity, 1127, 1162

Ordinal numbers
and encoding of proofs, 1163
as generalizing numbers, 1168
and notions of infinity, 1162

Ordinary differential equations (ODEs), 922
for chemical processes, 1013

Oresme, Nicole (France, ~1320–1382)
and time as a dimension, 1031

Organic molecules
production of, 1179

Organization
see Self-organization

Organizational structures
and history of complexity, 862
and Occam's razor, 1025

Organogenesis in plants, 409

Organs
and biological evolution, 389, 1002
separate rules for different, 383

Origami (paperfolding)
NP completeness in, 1145
rules in, 875

Origin of Species, 1001

Ornament
examples of, 43
grammatical rules in, 1181
and notion of purpose, 830
rules in, 872
and theory of form, 967

Orreries, 1184

Orthogonal bases, 1072
and image compression, 573
and radio signals, 1188

Oscillations
in chemical reactions, 1013
of density in rule 73, 954
in fluid flow, 998
in math functions, 917, 1134
origins of, 354–356
in PDEs, 988

Oscillators
in Game of Life, 964
nonlinear, 971
in quantum history, 1056

Oseen, C. William (Sweden, 1879–1944)
and fluid flow past spheres, 998

Oster, George F. (USA, 1940–[2018])
and patterns on shells, 1012
in Preface, xiii

Other minds
problem of recognizing, 1197

Otter (theorem-proving system), 1158

Outer
and branching patterns, 1005
and enumerating expressions, 897
and metrics on surfaces, 1048
and shape tensors, 980

Outer totalistic rules
in cellular automata, 928

Outlying data, 1083

Overlaps
in networks, 510
in strings, 503, 1033

Overshooting
in gradient descent, 985

Ozsváth–Schücking rotating vacuum, 1053