Chapter 6: Starting from Randomness

Section 1: The Emergence of Order

Properties of [initially random cellular automaton] patterns Continual injection of randomness [in cellular automata] History [of emergence of order]

Section 2: Four Classes of Behavior

Different runs [of initially random cellular automata] [Subset of] elementary rules States of matter [and cellular automaton classes] Class 4 [cellular automaton] rules Frequencies of [cellular automaton] classes History [of cellular automaton classes] Subclasses within class 4 Undecidability [of cellular automaton classes] [Classes in] continuous cellular automata Nearby cellular automaton rules 2D class 4 cellular automata Game of Life [cellular automaton] 3D class 4 [cellular automaton] rules Random initial conditions in other systems

Section 3: Sensitivity to Initial Conditions

Properties [of difference patterns] Difference patterns [in cellular automata] Lyapunov exponents

Section 4: Systems of Limited Size and Class 2 Behavior

Cyclic addition Cyclic multiplication Maximum periods [in cellular automata] Additive cellular automata [in finite regions] [Periods in] rules 30 and 45 Comparison of [periods for cellular automaton] rules Implementing boundary conditions [in cellular automata]

Section 5: Randomness in Class 3 Systems

Rule 22 [with simple initial conditions] Rule 225 [with simple initial conditions] Rule 94 [with simple initial conditions] Rule 218 [with simple initial conditions] Additive [cellular automaton] rules Generalized additivity [in cellular automata] Probabilistic estimates [of cellular automaton properties] Density in rule 90 Densities in other [cellular automaton] rules Density oscillations in rule 73

Section 6: Special Initial Conditions

Repeating blocks [in cellular automata] Localized structures [in cellular automata] [Repetition in] 2D cellular automata [Repetition in] systems based on numbers Sarkovskii's theorem Rule emulations Renormalization group Self-similarity of additive [cellular automaton] rules Fractal dimensions [of additive cellular automata] General associative [cellular automaton] rules Nesting in rule 45 Uniqueness of patterns [in cellular automata] Square root of rule 30 Nested initial conditions [in cellular automata]

Section 7: The Notion of Attractors

[Attractors in] discrete systems Implementation [of cellular automaton state networks] Finite automata Regular languages Regular expressions Generating functions [for regular languages] History [of finite automata] [Cellular automaton state] network properties Excluded blocks [in cellular automaton evolution] Entropies and dimensions [in cellular automata] Cycles and zeta functions 2D generalizations [of entropies] Probability-based entropies Entropy estimates Nested structure of attractors Surjectivity and injectivity [of cellular automaton maps] Temporal sequences [in cellular automata] Spacetime patches [in cellular automata] History [of dynamical systems approaches] Attractors in systems based on numbers Iterated maps Attractors in Turing machines [State networks for] systems of limited size Symmetries [and state networks] [State networks for] shift rules [State networks for] additive rules [Properties of] random networks

Section 8: Structures in Class 4 Systems

Survival data [for initial conditions in code 20] Background [in rule 110] Structures [in rule 110] Glider gun [in rule 110] Collisions [in rule 110] [Structures in] the Game of Life [Structures in] other 2D cellular automata Structures in Turing machines

From Stephen Wolfram: A New Kind of Science [citation]