Notes
Chapter 5: Two Dimensions and Beyond
Section 1: Introduction
https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-1--introduction
Section 2: Cellular Automatahttps://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-2--cellular-automata
![Implementation [of 2D cellular automata]](/nks/img/thumbnails/notes-5-2--implementation-of-2d-cellular-automata--textonly.png "Implementation [of 2D cellular automata]")
![General rules [for multidimensional cellular automata]](/nks/img/thumbnails/notes-5-2--general-rules-for-multidimensional-cellular-automata--textonly.png "General rules [for multidimensional cellular automata]")
![Numbers of possible [2D cellular automaton] rules](/nks/img/thumbnails/notes-5-2--numbers-of-possible-2d-cellular-automaton-rules--textonly.png "Numbers of possible [2D cellular automaton] rules")
![Symmetric 5-neighbor [2D cellular automaton] rules](/nks/img/thumbnails/notes-5-2--symmetric-5-neighbor-2d-cellular-automaton-rules--textonly.png "Symmetric 5-neighbor [2D cellular automaton] rules")
![Growth [2D cellular automaton] rules](/nks/img/thumbnails/notes-5-2--growth-2d-cellular-automaton-rules--textonly.png "Growth [2D cellular automaton] rules")
![History [of 2D cellular automata]](/nks/img/thumbnails/notes-5-2--history-of-2d-cellular-automata--textonly.png "History [of 2D cellular automata]")
![Limiting shapes [in 2D cellular automata]](/nks/img/thumbnails/notes-5-2--limiting-shapes-in-2d-cellular-automata--textonly.png "Limiting shapes [in 2D cellular automata]")
![Projections from 3D [cellular automata]](/nks/img/thumbnails/notes-5-2--projections-from-3d-cellular-automata--textonly.png "Projections from 3D [cellular automata]")
![Other geometries [for cellular automata]](/nks/img/thumbnails/notes-5-2--other-geometries-for-cellular-automata--textonly.png "Other geometries [for cellular automata]")
![Networks [as basis for cellular automata]](/nks/img/thumbnails/notes-5-2--networks-as-basis-for-cellular-automata--textonly.png "Networks [as basis for cellular automata]")
Implementation [of 2D cellular automata]
General rules [for multidimensional cellular automata]
Numbers of possible [2D cellular automaton] rules
Symmetric 5-neighbor [2D cellular automaton] rules
Growth [2D cellular automaton] rules
History [of 2D cellular automata]
Limiting shapes [in 2D cellular automata]
Projections from 3D [cellular automata]
Section 3: Turing Machineshttps://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-3--turing-machines
![Implementation [of 2D Turing machines]](/nks/img/thumbnails/notes-5-3--implementation-of-2d-turing-machines--textonly.png "Implementation [of 2D Turing machines]")
![History [of 2D Turing machines]](/nks/img/thumbnails/notes-5-3--history-of-2d-turing-machines--textonly.png "History [of 2D Turing machines]")
![Visualization [of 2D Turing machines]](/nks/img/thumbnails/notes-5-3--visualization-of-2d-turing-machines--textonly.png "Visualization [of 2D Turing machines]")
![Rules [for 2D Turing machines] based on turning](/nks/img/thumbnails/notes-5-3--rules-for-2d-turing-machines-based-on-turning--textonly.png "Rules [for 2D Turing machines] based on turning")
Implementation [of 2D Turing machines]
History [of 2D Turing machines]
Visualization [of 2D Turing machines]
Section 4: Substitution Systems and Fractalshttps://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-4--substitution-systems-and-fractals
![Implementation [of 2D substitution systems]](/nks/img/thumbnails/notes-5-4--implementation-of-2d-substitution-systems--textonly.png "Implementation [of 2D substitution systems]")
![Connection [of 2D substitution systems] with digit sequences](/nks/img/thumbnails/notes-5-4--connection-of-2d-substitution-systems-with-digit-sequences--textonly.png "Connection [of 2D substitution systems] with digit sequences")
![[2D substitution systems with] non-white backgrounds](/nks/img/thumbnails/notes-5-4--2d-substitution-systems-with-non-white-backgrounds--textonly.png "[2D substitution systems with] non-white backgrounds")
![Higher-dimensional generalizations [of substitution systems]](/nks/img/thumbnails/notes-5-4--higher-dimensional-generalizations-of-substitution-systems--textonly.png "Higher-dimensional generalizations [of substitution systems]")
![[Substitution systems based on] other shapes](/nks/img/thumbnails/notes-5-4--substitution-systems-based-on-other-shapes--textonly.png "[Substitution systems based on] other shapes")
![Implementation [of geometric substitution systems]](/nks/img/thumbnails/notes-5-4--implementation-of-geometric-substitution-systems--textonly.png "Implementation [of geometric substitution systems]")
![Connection [of geometric substitution systems] with digit sequences](/nks/img/thumbnails/notes-5-4--connection-of-geometric-substitution-systems-with-digit-sequences--textonly.png "Connection [of geometric substitution systems] with digit sequences")
![Visualization [of geometric substitution systems]](/nks/img/thumbnails/notes-5-4--visualization-of-geometric-substitution-systems--textonly.png "Visualization [of geometric substitution systems]")
![Neighbor-dependent [2D] substitution systems](/nks/img/thumbnails/notes-5-4--neighbor-dependent-2d-substitution-systems--textonly.png "Neighbor-dependent [2D] substitution systems")
Implementation [of 2D substitution systems]
Connection [of 2D substitution systems] with digit sequences
[2D substitution systems with] non-white backgrounds
Higher-dimensional generalizations [of substitution systems]
[Substitution systems based on] other shapes
Implementation [of geometric substitution systems]
Connection [of geometric substitution systems] with digit sequences
Visualization [of geometric substitution systems]
Section 5: Network Systemshttps://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-5--network-systems
![Implementation [of network systems]](/nks/img/thumbnails/notes-5-5--implementation-of-network-systems--textonly.png "Implementation [of network systems]")
![Rule structure [for network systems]](/nks/img/thumbnails/notes-5-5--rule-structure-for-network-systems--textonly.png "Rule structure [for network systems]")
![[Systems based on] undirected networks](/nks/img/thumbnails/notes-5-5--systems-based-on-undirected-networks--textonly.png "[Systems based on] undirected networks")
![Computer science [and network systems]](/nks/img/thumbnails/notes-5-5--computer-science-and-network-systems--textonly.png "Computer science [and network systems]")
![Properties [of network systems]](/nks/img/thumbnails/notes-5-5--properties-of-network-systems--textonly.png "Properties [of network systems]")
![Implementation [of network cellular automata]](/nks/img/thumbnails/notes-5-5--implementation-of-network-cellular-automata--textonly.png "Implementation [of network cellular automata]")
Implementation [of network systems]
Rule structure [for network systems]
[Systems based on] undirected networks
Computer science [and network systems]
Properties [of network systems]
Section 6: Multiway Systemshttps://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-6--multiway-systems
![Implementation [of multiway systems]](/nks/img/thumbnails/notes-5-6--implementation-of-multiway-systems--textonly.png "Implementation [of multiway systems]")
![General properties [of multiway systems]](/nks/img/thumbnails/notes-5-6--general-properties-of-multiway-systems--textonly.png "General properties [of multiway systems]")
![Properties [of multiway system example]](/nks/img/thumbnails/notes-5-6--properties-of-multiway-system-example--textonly.png "Properties [of multiway system example]")
![Frequency of behavior [in multiway systems]](/nks/img/thumbnails/notes-5-6--frequency-of-behavior-in-multiway-systems--textonly.png "Frequency of behavior [in multiway systems]")
![History [of multiway systems]](/nks/img/thumbnails/notes-5-6--history-of-multiway-systems--textonly.png "History [of multiway systems]")
![Semigroups and groups [and multiway systems]](/nks/img/thumbnails/notes-5-6--semigroups-and-groups-and-multiway-systems--textonly.png "Semigroups and groups [and multiway systems]")
![Formal languages [and multiway systems]](/nks/img/thumbnails/notes-5-6--formal-languages-and-multiway-systems--textonly.png "Formal languages [and multiway systems]")
![Limited size versions [of multiway systems]](/nks/img/thumbnails/notes-5-6--limited-size-versions-of-multiway-systems--textonly.png "Limited size versions [of multiway systems]")
Implementation [of multiway systems]
General properties [of multiway systems]
Properties [of multiway system example]
Frequency of behavior [in multiway systems]
Semigroups and groups [and multiway systems]
Formal languages [and multiway systems]
Multidimensional multiway systems
Limited size versions [of multiway systems]
Section 7: Systems Based on Constraintshttps://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond--notes#sect-5-7--systems-based-on-constraints
![The notion of equations [versus constraints]](/nks/img/thumbnails/notes-5-7--the-notion-of-equations-versus-constraints--textonly.png "The notion of equations [versus constraints]")
![1D [systems based on] constraints](/nks/img/thumbnails/notes-5-7--1d-systems-based-on-constraints--textonly.png "1D [systems based on] constraints")
![[Excluded blocks in] dynamical systems theory](/nks/img/thumbnails/notes-5-7--excluded-blocks-in-dynamical-systems-theory--textonly.png "[Excluded blocks in] dynamical systems theory")
![2D [systems based on] constraints](/nks/img/thumbnails/notes-5-7--2d-systems-based-on-constraints--textonly.png "2D [systems based on] constraints")
![Numbering scheme [for 2D constraints]](/nks/img/thumbnails/notes-5-7--numbering-scheme-for-2d-constraints--textonly.png "Numbering scheme [for 2D constraints]")
![Identifying the 171 patterns [that satisfy 2D constraints]](/nks/img/thumbnails/notes-5-7--identifying-the-171-patterns-that-satisfy-2d-constraints--textonly.png "Identifying the 171 patterns [that satisfy 2D constraints]")
![Checking [tilings with] constraints](/nks/img/thumbnails/notes-5-7--checking-tilings-with-constraints--textonly.png "Checking [tilings with] constraints")
![Representing repetitive [2D] patterns](/nks/img/thumbnails/notes-5-7--representing-repetitive-2d-patterns--textonly.png "Representing repetitive [2D] patterns")
![Searching for patterns [that satisfy constraints]](/nks/img/thumbnails/notes-5-7--searching-for-patterns-that-satisfy-constraints--textonly.png "Searching for patterns [that satisfy constraints]")
![Undecidability [for 2D constraints]](/nks/img/thumbnails/notes-5-7--undecidability-for-2d-constraints--textonly.png "Undecidability [for 2D constraints]")
![NP completeness [for 2D constraints]](/nks/img/thumbnails/notes-5-7--np-completeness-for-2d-constraints--textonly.png "NP completeness [for 2D constraints]")
![Non-periodic pattern [forced by 2D constraint]](/nks/img/thumbnails/notes-5-7--non-periodic-pattern-forced-by-2d-constraint--textonly.png "Non-periodic pattern [forced by 2D constraint]")
![Other types of [2D] constraints](/nks/img/thumbnails/notes-5-7--other-types-of-2d-constraints--textonly.png "Other types of [2D] constraints")
![Forcing nested [2D] patterns](/nks/img/thumbnails/notes-5-7--forcing-nested-2d-patterns--textonly.png "Forcing nested [2D] patterns")
![Non-computable [2D] patterns](/nks/img/thumbnails/notes-5-7--non-computable-2d-patterns--textonly.png "Non-computable [2D] patterns")
![Tiling [problems]](/nks/img/thumbnails/notes-5-7--tiling-problems--textonly.png "Tiling [problems]")
![Correspondence systems [as constraints]](/nks/img/thumbnails/notes-5-7--correspondence-systems-as-constraints--textonly.png "Correspondence systems [as constraints]")
![Formal languages [and constraints]](/nks/img/thumbnails/notes-5-7--formal-languages-and-constraints--textonly.png "Formal languages [and constraints]")
The notion of equations [versus constraints]
Explanations based on constraints
1D [systems based on] constraints
[Excluded blocks in] dynamical systems theory
2D [systems based on] constraints
Numbering scheme [for 2D constraints]
Identifying the 171 patterns [that satisfy 2D constraints]
Checking [tilings with] constraints
Representing repetitive [2D] patterns
Searching for patterns [that satisfy constraints]
Undecidability [for 2D constraints]
NP completeness [for 2D constraints]
Non-periodic pattern [forced by 2D constraint]
Other types of [2D] constraints
Relation to 2D cellular automata
Relation to 1D cellular automata
Correspondence systems [as constraints]
Formal languages [and constraints]