Notes
Chapter 6: Starting from Randomness
Section 1: The Emergence of Order
https://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-1--the-emergence-of-order
![Properties of [initially random cellular automaton] patterns](/nks/img/thumbnails/notes-6-1--properties-of-initially-random-cellular-automaton-patterns--textonly.png "Properties of [initially random cellular automaton] patterns")
![Continual injection of randomness [in cellular automata]](/nks/img/thumbnails/notes-6-1--continual-injection-of-randomness-in-cellular-automata--textonly.png "Continual injection of randomness [in cellular automata]")
![History [of emergence of order]](/nks/img/thumbnails/notes-6-1--history-of-emergence-of-order--textonly.png "History [of emergence of order]")
Properties of [initially random cellular automaton] patterns
Section 2: Four Classes of Behaviorhttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-2--four-classes-of-behavior
![Different runs [of initially random cellular automata]](/nks/img/thumbnails/notes-6-2--different-runs-of-initially-random-cellular-automata--textonly.png "Different runs [of initially random cellular automata]")
![[Subset of] elementary rules](/nks/img/thumbnails/notes-6-2--subset-of-elementary-rules--textonly.png "[Subset of] elementary rules")
![States of matter [and cellular automaton classes]](/nks/img/thumbnails/notes-6-2--states-of-matter-and-cellular-automaton-classes--textonly.png "States of matter [and cellular automaton classes]")
![Class 4 [cellular automaton] rules](/nks/img/thumbnails/notes-6-2--class-4-cellular-automaton-rules--textonly.png "Class 4 [cellular automaton] rules")
![Frequencies of [cellular automaton] classes](/nks/img/thumbnails/notes-6-2--frequencies-of-cellular-automaton-classes--textonly.png "Frequencies of [cellular automaton] classes")
![History [of cellular automaton classes]](/nks/img/thumbnails/notes-6-2--history-of-cellular-automaton-classes--textonly.png "History [of cellular automaton classes]")
![Undecidability [of cellular automaton classes]](/nks/img/thumbnails/notes-6-2--undecidability-of-cellular-automaton-classes--textonly.png "Undecidability [of cellular automaton classes]")
![[Classes in] continuous cellular automata](/nks/img/thumbnails/notes-6-2--classes-in-continuous-cellular-automata--textonly.png "[Classes in] continuous cellular automata")
![Game of Life [cellular automaton]](/nks/img/thumbnails/notes-6-2--game-of-life-cellular-automaton--textonly.png "Game of Life [cellular automaton]")
![3D class 4 [cellular automaton] rules](/nks/img/thumbnails/notes-6-2--3d-class-4-cellular-automaton-rules--textonly.png "3D class 4 [cellular automaton] rules")
Different runs [of initially random cellular automata]
States of matter [and cellular automaton classes]
Class 4 [cellular automaton] rules
Frequencies of [cellular automaton] classes
History [of cellular automaton classes]
Undecidability [of cellular automaton classes]
[Classes in] continuous cellular automata
Nearby cellular automaton rules
Game of Life [cellular automaton]
Section 3: Sensitivity to Initial Conditionshttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-3--sensitivity-to-initial-conditions
![Properties [of difference patterns]](/nks/img/thumbnails/notes-6-3--properties-of-difference-patterns--textonly.png "Properties [of difference patterns]")
![Difference patterns [in cellular automata]](/nks/img/thumbnails/notes-6-3--difference-patterns-in-cellular-automata--textonly.png "Difference patterns [in cellular automata]")
Section 4: Systems of Limited Size and Class 2 Behaviorhttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-4--systems-of-limited-size-and-class-2-behavior
![Maximum periods [in cellular automata]](/nks/img/thumbnails/notes-6-4--maximum-periods-in-cellular-automata--textonly.png "Maximum periods [in cellular automata]")
![Additive cellular automata [in finite regions]](/nks/img/thumbnails/notes-6-4--additive-cellular-automata-in-finite-regions--textonly.png "Additive cellular automata [in finite regions]")
![[Periods in] rules 30 and 45](/nks/img/thumbnails/notes-6-4--periods-in-rules-30-and-45--textonly.png "[Periods in] rules 30 and 45")
![Comparison of [periods for cellular automaton] rules](/nks/img/thumbnails/notes-6-4--comparison-of-periods-for-cellular-automaton-rules--textonly.png "Comparison of [periods for cellular automaton] rules")
![Implementing boundary conditions [in cellular automata]](/nks/img/thumbnails/notes-6-4--implementing-boundary-conditions-in-cellular-automata--textonly.png "Implementing boundary conditions [in cellular automata]")
Maximum periods [in cellular automata]
Additive cellular automata [in finite regions]
Section 5: Randomness in Class 3 Systemshttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-5--randomness-in-class-3-systems
![Rule 22 [with simple initial conditions]](/nks/img/thumbnails/notes-6-5--rule-22-with-simple-initial-conditions--textonly.png "Rule 22 [with simple initial conditions]")
![Rule 225 [with simple initial conditions]](/nks/img/thumbnails/notes-6-5--rule-225-with-simple-initial-conditions--textonly.png "Rule 225 [with simple initial conditions]")
![Rule 94 [with simple initial conditions]](/nks/img/thumbnails/notes-6-5--rule-94-with-simple-initial-conditions--textonly.png "Rule 94 [with simple initial conditions]")
![Rule 218 [with simple initial conditions]](/nks/img/thumbnails/notes-6-5--rule-218-with-simple-initial-conditions--textonly.png "Rule 218 [with simple initial conditions]")
![Additive [cellular automaton] rules](/nks/img/thumbnails/notes-6-5--additive-cellular-automaton-rules--textonly.png "Additive [cellular automaton] rules")
![Generalized additivity [in cellular automata]](/nks/img/thumbnails/notes-6-5--generalized-additivity-in-cellular-automata--textonly.png "Generalized additivity [in cellular automata]")
![Probabilistic estimates [of cellular automaton properties]](/nks/img/thumbnails/notes-6-5--probabilistic-estimates-of-cellular-automaton-properties--textonly.png "Probabilistic estimates [of cellular automaton properties]")
![Densities in other [cellular automaton] rules](/nks/img/thumbnails/notes-6-5--densities-in-other-cellular-automaton-rules--textonly.png "Densities in other [cellular automaton] rules")
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]
Section 6: Special Initial Conditionshttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-6--special-initial-conditions
![Repeating blocks [in cellular automata]](/nks/img/thumbnails/notes-6-6--repeating-blocks-in-cellular-automata--textonly.png "Repeating blocks [in cellular automata]")
![Localized structures [in cellular automata]](/nks/img/thumbnails/notes-6-6--localized-structures-in-cellular-automata--textonly.png "Localized structures [in cellular automata]")
![[Repetition in] 2D cellular automata](/nks/img/thumbnails/notes-6-6--repetition-in-2d-cellular-automata--textonly.png "[Repetition in] 2D cellular automata")
![[Repetition in] systems based on numbers](/nks/img/thumbnails/notes-6-6--repetition-in-systems-based-on-numbers--textonly.png "[Repetition in] systems based on numbers")
![Self-similarity of additive [cellular automaton] rules](/nks/img/thumbnails/notes-6-6--self-similarity-of-additive-cellular-automaton-rules--textonly.png "Self-similarity of additive [cellular automaton] rules")
![Fractal dimensions [of additive cellular automata]](/nks/img/thumbnails/notes-6-6--fractal-dimensions-of-additive-cellular-automata--textonly.png "Fractal dimensions [of additive cellular automata]")
![General associative [cellular automaton] rules](/nks/img/thumbnails/notes-6-6--general-associative-cellular-automaton-rules--textonly.png "General associative [cellular automaton] rules")
![Uniqueness of patterns [in cellular automata]](/nks/img/thumbnails/notes-6-6--uniqueness-of-patterns-in-cellular-automata--textonly.png "Uniqueness of patterns [in cellular automata]")
![Nested initial conditions [in cellular automata]](/nks/img/thumbnails/notes-6-6--nested-initial-conditions-in-cellular-automata--textonly.png "Nested initial conditions [in cellular automata]")
Repeating blocks [in cellular automata]
Localized structures [in cellular automata]
[Repetition in] 2D cellular automata
[Repetition in] systems based on numbers
Self-similarity of additive [cellular automaton] rules
Fractal dimensions [of additive cellular automata]
General associative [cellular automaton] rules
Section 7: The Notion of Attractorshttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-7--the-notion-of-attractors
![[Attractors in] discrete systems](/nks/img/thumbnails/notes-6-7--attractors-in-discrete-systems--textonly.png "[Attractors in] discrete systems")
![Implementation [of cellular automaton state networks]](/nks/img/thumbnails/notes-6-7--implementation-of-cellular-automaton-state-networks--textonly.png "Implementation [of cellular automaton state networks]")
![Generating functions [for regular languages]](/nks/img/thumbnails/notes-6-7--generating-functions-for-regular-languages--textonly.png "Generating functions [for regular languages]")
![History [of finite automata]](/nks/img/thumbnails/notes-6-7--history-of-finite-automata--textonly.png "History [of finite automata]")
![[Cellular automaton state] network properties](/nks/img/thumbnails/notes-6-7--cellular-automaton-state-network-properties--textonly.png "[Cellular automaton state] network properties")
![Excluded blocks [in cellular automaton evolution]](/nks/img/thumbnails/notes-6-7--excluded-blocks-in-cellular-automaton-evolution--textonly.png "Excluded blocks [in cellular automaton evolution]")
![Entropies and dimensions [in cellular automata]](/nks/img/thumbnails/notes-6-7--entropies-and-dimensions-in-cellular-automata--textonly.png "Entropies and dimensions [in cellular automata]")
![2D generalizations [of entropies]](/nks/img/thumbnails/notes-6-7--2d-generalizations-of-entropies--textonly.png "2D generalizations [of entropies]")
![Surjectivity and injectivity [of cellular automaton maps]](/nks/img/thumbnails/notes-6-7--surjectivity-and-injectivity-of-cellular-automaton-maps--textonly.png "Surjectivity and injectivity [of cellular automaton maps]")
![Temporal sequences [in cellular automata]](/nks/img/thumbnails/notes-6-7--temporal-sequences-in-cellular-automata--textonly.png "Temporal sequences [in cellular automata]")
![Spacetime patches [in cellular automata]](/nks/img/thumbnails/notes-6-7--spacetime-patches-in-cellular-automata--textonly.png "Spacetime patches [in cellular automata]")
![History [of dynamical systems approaches]](/nks/img/thumbnails/notes-6-7--history-of-dynamical-systems-approaches--textonly.png "History [of dynamical systems approaches]")
![[State networks for] systems of limited size](/nks/img/thumbnails/notes-6-7--state-networks-for-systems-of-limited-size--textonly.png "[State networks for] systems of limited size")
![Symmetries [and state networks]](/nks/img/thumbnails/notes-6-7--symmetries-and-state-networks--textonly.png "Symmetries [and state networks]")
![[State networks for] shift rules](/nks/img/thumbnails/notes-6-7--state-networks-for-shift-rules--textonly.png "[State networks for] shift rules")
![[State networks for] additive rules](/nks/img/thumbnails/notes-6-7--state-networks-for-additive-rules--textonly.png "[State networks for] additive rules")
![[Properties of] random networks](/nks/img/thumbnails/notes-6-7--properties-of-random-networks--textonly.png "[Properties of] random networks")
[Attractors in] discrete systems
Implementation [of cellular automaton state networks]
Generating functions [for regular languages]
[Cellular automaton state] network properties
Excluded blocks [in cellular automaton evolution]
Entropies and dimensions [in cellular automata]
2D generalizations [of entropies]
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
[State networks for] systems of limited size
Symmetries [and state networks]
[State networks for] shift rules
Section 8: Structures in Class 4 Systemshttps://www.wolframscience.com/nks/chap-6--starting-from-randomness--notes#sect-6-8--structures-in-class-4-systems
![Survival data [for initial conditions in code 20]](/nks/img/thumbnails/notes-6-8--survival-data-for-initial-conditions-in-code-20--textonly.png "Survival data [for initial conditions in code 20]")
![Background [in rule 110]](/nks/img/thumbnails/notes-6-8--background-in-rule-110--textonly.png "Background [in rule 110]")
![Structures [in rule 110]](/nks/img/thumbnails/notes-6-8--structures-in-rule-110--textonly.png "Structures [in rule 110]")
![Glider gun [in rule 110]](/nks/img/thumbnails/notes-6-8--glider-gun-in-rule-110--textonly.png "Glider gun [in rule 110]")
![Collisions [in rule 110]](/nks/img/thumbnails/notes-6-8--collisions-in-rule-110--textonly.png "Collisions [in rule 110]")
![[Structures in] the Game of Life](/nks/img/thumbnails/notes-6-8--structures-in-the-game-of-life--textonly.png "[Structures in] the Game of Life")
![[Structures in] other 2D cellular automata](/nks/img/thumbnails/notes-6-8--structures-in-other-2d-cellular-automata--textonly.png "[Structures in] other 2D cellular automata")
Survival data [for initial conditions in code 20]
[Structures in] the Game of Life