Notes
Chapter 7: Mechanisms in Programs and Nature
Section 1: Universality of Behavior
https://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-1--universality-of-behavior
![History [of universality of behavior]](/nks/img/thumbnails/notes-7-1--history-of-universality-of-behavior--textonly.png "History [of universality of behavior]")
Section 2: Three Mechanisms for Randomnesshttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-2--three-mechanisms-for-randomness
![Definition [of randomness]](/nks/img/thumbnails/notes-7-2--definition-of-randomness--textonly.png "Definition [of randomness]")
![History [of randomness]](/nks/img/thumbnails/notes-7-2--history-of-randomness--textonly.png "History [of randomness]")
Section 3: Randomness from the Environmenthttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-3--randomness-from-the-environment
![Power spectra [of random processes]](/nks/img/thumbnails/notes-7-3--power-spectra-of-random-processes--textonly.png "Power spectra [of random processes]")
Section 4: Chaos Theory and Randomness from Initial Conditionshttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-4--chaos-theory-and-randomness-from-initial-conditions
![Spinning and tossing [as sources of randomness]](/nks/img/thumbnails/notes-7-4--spinning-and-tossing-as-sources-of-randomness--textonly.png "Spinning and tossing [as sources of randomness]")
![Instability [and chaos theory]](/nks/img/thumbnails/notes-7-4--instability-and-chaos-theory--textonly.png "Instability [and chaos theory]")
![Simple case [of three-body problem]](/nks/img/thumbnails/notes-7-4--simple-case-of-three-body-problem--textonly.png "Simple case [of three-body problem]")
Spinning and tossing [as sources of randomness]
Information content of initial conditions
Instability [and chaos theory]
Section 5: The Intrinsic Generation of Randomnesshttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-5--the-intrinsic-generation-of-randomness
![Cellular automata [as randomness generators]](/nks/img/thumbnails/notes-7-5--cellular-automata-as-randomness-generators--textonly.png "Cellular automata [as randomness generators]")
![[Generating sequences with] unequal probabilities](/nks/img/thumbnails/notes-7-5--generating-sequences-with-unequal-probabilities--textonly.png "[Generating sequences with] unequal probabilities")
![Probabilistic rules [for cellular automata]](/nks/img/thumbnails/notes-7-5--probabilistic-rules-for-cellular-automata--textonly.png "Probabilistic rules [for cellular automata]")
Cellular automata [as randomness generators]
[Generating sequences with] unequal probabilities
Sources of repeatable randomness
Section 6: The Phenomenon of Continuityhttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-6--the-phenomenon-of-continuity
![History [of Central Limit Theorem]](/nks/img/thumbnails/notes-7-6--history-of-central-limit-theorem--textonly.png "History [of Central Limit Theorem]")
![Related results [to Central Limit Theorem]](/nks/img/thumbnails/notes-7-6--related-results-to-central-limit-theorem--textonly.png "Related results [to Central Limit Theorem]")
![Self-avoiding [random] walks](/nks/img/thumbnails/notes-7-6--self-avoiding-random-walks--textonly.png "Self-avoiding [random] walks")
![Implementation [of basic aggregation model]](/nks/img/thumbnails/notes-7-6--implementation-of-basic-aggregation-model--textonly.png "Implementation [of basic aggregation model]")
![[Circularity in] code 746](/nks/img/thumbnails/notes-7-6--circularity-in-code-746--textonly.png "[Circularity in] code 746")
![[Circularity in] other rules](/nks/img/thumbnails/notes-7-6--circularity-in-other-rules--textonly.png "[Circularity in] other rules")
![Isotropy [in lattice systems]](/nks/img/thumbnails/notes-7-6--isotropy-in-lattice-systems--textonly.png "Isotropy [in lattice systems]")
![Domains [in code 976]](/nks/img/thumbnails/notes-7-6--domains-in-code-976--textonly.png "Domains [in code 976]")
Discreteness in computer programs
History [of Central Limit Theorem]
Related results [to Central Limit Theorem]
Implementation [of basic aggregation model]
Generalized aggregation models
Section 7: Origins of Discretenesshttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-7--origins-of-discreteness
![1D [discrete] transitions [in cellular automata]](/nks/img/thumbnails/notes-7-7--1d-discrete-transitions-in-cellular-automata--textonly.png "1D [discrete] transitions [in cellular automata]")
![2D [discrete] transitions [in cellular automata]](/nks/img/thumbnails/notes-7-7--2d-discrete-transitions-in-cellular-automata--textonly.png "2D [discrete] transitions [in cellular automata]")
1D [discrete] transitions [in cellular automata]
2D [discrete] transitions [in cellular automata]
Section 8: The Problem of Satisfying Constraintshttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-8--the-problem-of-satisfying-constraints
![NP completeness [and constraint satisfaction]](/nks/img/thumbnails/notes-7-8--np-completeness-and-constraint-satisfaction--textonly.png "NP completeness [and constraint satisfaction]")
![Distribution [of constraint satisfaction]](/nks/img/thumbnails/notes-7-8--distribution-of-constraint-satisfaction--textonly.png "Distribution [of constraint satisfaction]")
![Implementation [of constraint satisfaction]](/nks/img/thumbnails/notes-7-8--implementation-of-constraint-satisfaction--textonly.png "Implementation [of constraint satisfaction]")
![Iterative improvement [of constraint satisfaction]](/nks/img/thumbnails/notes-7-8--iterative-improvement-of-constraint-satisfaction--textonly.png "Iterative improvement [of constraint satisfaction]")
![Gradient descent [in constraint satisfaction]](/nks/img/thumbnails/notes-7-8--gradient-descent-in-constraint-satisfaction--textonly.png "Gradient descent [in constraint satisfaction]")
![Biologically motivated schemes [for combinatorial optimization]](/nks/img/thumbnails/notes-7-8--biologically-motivated-schemes-for-combinatorial-optimization--textonly.png "Biologically motivated schemes [for combinatorial optimization]")
![History [of combinatorial optimization]](/nks/img/thumbnails/notes-7-8--history-of-combinatorial-optimization--textonly.png "History [of combinatorial optimization]")
![[Invariance examples in] 2D cellular automata](/nks/img/thumbnails/notes-7-8--invariance-examples-in-2d-cellular-automata--textonly.png "[Invariance examples in] 2D cellular automata")
![[Sphere packings in] higher dimensions](/nks/img/thumbnails/notes-7-8--sphere-packings-in-higher-dimensions--textonly.png "[Sphere packings in] higher dimensions")
NP completeness [and constraint satisfaction]
Distribution [of constraint satisfaction]
Implementation [of constraint satisfaction]
Iterative improvement [of constraint satisfaction]
Gradient descent [in constraint satisfaction]
Biologically motivated schemes [for combinatorial optimization]
History [of combinatorial optimization]
[Invariance examples in] 2D cellular automata
Section 9: Origins of Simple Behaviorhttps://www.wolframscience.com/nks/chap-7--mechanisms-in-programs-and-nature--notes#sect-7-9--origins-of-simple-behavior
![Previous approaches [to complexity]](/nks/img/thumbnails/notes-7-9--previous-approaches-to-complexity--textonly.png "Previous approaches [to complexity]")
![[Nesting in] phase transitions](/nks/img/thumbnails/notes-7-9--nesting-in-phase-transitions--textonly.png "[Nesting in] phase transitions")
![[Nesting in] random walks](/nks/img/thumbnails/notes-7-9--nesting-in-random-walks--textonly.png "[Nesting in] random walks")
![[Overall] structure of algorithms](/nks/img/thumbnails/notes-7-9--overall-structure-of-algorithms--textonly.png "[Overall] structure of algorithms")
Previous approaches [to complexity]
Repetition in continuous systems
[Nesting in] phase transitions