# Notes

# Chapter 5: Two Dimensions and Beyond

## Section 1: Introductionhttps://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]

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]

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]

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]

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]

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]

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]