Two Dimensions and Beyond : A New Kind of Science

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height-full"> <div class="heirs-vertical-align-m"> <div class="padding-lr-1"> <h1><a href="/nks/"><img src="/nks/img/a-new-kind-of-science-logo.png" alt="Stephen Wolfram's A New Kind of Science | Online" width="372" height="43" class="fill"></a></h1> </div> <div class="padding-r-1"> <nav id="nav" class="table"> <div class="nav-main heirs-vertical-align-t"> <div class="size-picker"> <i class="icon"></i> <ul> <li class="size-auto checked"><span>auto</span></li> <li class="size-small"><span>small</span></li> <li class="size-medium"><span>medium</span></li> <li class="size-large"><span>large</span></li> <li class="size-x-large"><span>x-large</span></li> </ul> </div> <div class="search"> <div><i class="icon"></i></div> </div> <div class="toc"> <div><a href="/nks/"><i class="icon"></i></a></div> </div> <div class="breadcrumbs"> <div> <span class="chapter"><span>Chapter 5</span></span> </div> </div> </div> <div class="nav-burger"> <div> <div class="hamburger"><i class="icon"></i></div> </div> <div class="burger-nav-l1"> <div id="top-stuff"> <div class="crumb-wrap-supreme"> <div> <span class="toc"><a href="/nks/"><i class="icon"></i></a></span> </div> </div> <div class="size-picker"> <div><i class="icon"></i></div> </div> <ul class="size-pickings"> <li class="size-auto checked"><span>auto</span></li> <li class="size-small"><span>small</span></li> <li class="size-medium"><span>medium</span></li> <li class="size-large"><span>large</span></li> <li class="size-x-large"><span>x-large</span></li> </ul> </div> <ul class="chapters"> <li class="chapter preface"> <a href="/nks/pix--preface/"> <span></span> <span>Preface</span> </a> </li> <li class="chapter chapter-1"> <a href="/nks/chap-1--the-foundations-for-a-new-kind-of-science/"> <span>1</span> <span>The Foundations for a New Kind of Science</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p1--an-outline-of-basic-ideas/"> <span>1</span> <span>An Outline of Basic Ideas</span> </a> </li> <li class="section section-2"> <a href="/nks/p7--relations-to-other-areas/"> <span>2</span> <span>Relations to Other Areas</span> </a> </li> <li class="section section-3"> <a href="/nks/p12--some-past-initiatives/"> <span>3</span> <span>Some Past Initiatives</span> </a> </li> <li class="section section-4"> <a href="/nks/p17--the-personal-story-of-the-science-in-this-book/"> <span>4</span> <span>The Personal Story of the Science in This Book</span> </a> </li> </ul> </li> <li class="chapter chapter-2"> <a href="/nks/chap-2--the-crucial-experiment/"> <span>2</span> <span>The Crucial Experiment</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p23--how-do-simple-programs-behave/"> <span>1</span> <span>How Do Simple Programs Behave?</span> </a> </li> <li class="section section-2"> <a href="/nks/p39--the-need-for-a-new-intuition/"> <span>2</span> <span>The Need for a New Intuition</span> </a> </li> <li class="section section-3"> <a href="/nks/p42--why-these-discoveries-were-not-made-before/"> <span>3</span> <span>Why These Discoveries Were Not Made Before</span> </a> </li> </ul> </li> <li class="chapter chapter-3"> <a href="/nks/chap-3--the-world-of-simple-programs/"> <span>3</span> <span>The World of Simple Programs</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p51--the-search-for-general-features/"> <span>1</span> <span>The Search for General Features</span> </a> </li> <li class="section section-2"> <a href="/nks/p53--more-cellular-automata/"> <span>2</span> <span>More Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p71--mobile-automata/"> <span>3</span> <span>Mobile Automata</span> </a> </li> <li class="section section-4"> <a href="/nks/p78--turing-machines/"> <span>4</span> <span>Turing Machines</span> </a> </li> <li class="section section-5"> <a href="/nks/p82--substitution-systems/"> <span>5</span> <span>Substitution Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p88--sequential-substitution-systems/"> <span>6</span> <span>Sequential Substitution Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p93--tag-systems/"> <span>7</span> <span>Tag Systems</span> </a> </li> <li class="section section-8"> <a href="/nks/p95--cyclic-tag-systems/"> <span>8</span> <span>Cyclic Tag Systems</span> </a> </li> <li class="section section-9"> <a href="/nks/p97--register-machines/"> <span>9</span> <span>Register Machines</span> </a> </li> <li class="section section-10"> <a href="/nks/p102--symbolic-systems/"> <span>10</span> <span>Symbolic Systems</span> </a> </li> <li class="section section-11"> <a href="/nks/p105--some-conclusions/"> <span>11</span> <span>Some Conclusions</span> </a> </li> <li class="section section-12"> <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/"> <span>12</span> <span>How the Discoveries in This Chapter Were Made</span> </a> </li> </ul> </li> <li class="chapter chapter-4"> <a href="/nks/chap-4--systems-based-on-numbers/"> <span>4</span> <span>Systems Based on Numbers</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p115--the-notion-of-numbers/"> <span>1</span> <span>The Notion of Numbers</span> </a> </li> <li class="section section-2"> <a href="/nks/p117--elementary-arithmetic/"> <span>2</span> <span>Elementary Arithmetic</span> </a> </li> <li class="section section-3"> <a href="/nks/p128--recursive-sequences/"> <span>3</span> <span>Recursive Sequences</span> </a> </li> <li class="section section-4"> <a href="/nks/p132--the-sequence-of-primes/"> <span>4</span> <span>The Sequence of Primes</span> </a> </li> <li class="section section-5"> <a href="/nks/p136--mathematical-constants/"> <span>5</span> <span>Mathematical Constants</span> </a> </li> <li class="section section-6"> <a href="/nks/p145--mathematical-functions/"> <span>6</span> <span>Mathematical Functions</span> </a> </li> <li class="section section-7"> <a href="/nks/p149--iterated-maps-and-the-chaos-phenomenon/"> <span>7</span> <span>Iterated Maps and the Chaos Phenomenon</span> </a> </li> <li class="section section-8"> <a href="/nks/p155--continuous-cellular-automata/"> <span>8</span> <span>Continuous Cellular Automata</span> </a> </li> <li class="section section-9"> <a href="/nks/p161--partial-differential-equations/"> <span>9</span> <span>Partial Differential Equations</span> </a> </li> <li class="section section-10"> <a href="/nks/p167--continuous-versus-discrete-systems/"> <span>10</span> <span>Continuous Versus Discrete Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-5 selected"> <a href="/nks/chap-5--two-dimensions-and-beyond/"> <span>5</span> <span>Two Dimensions and Beyond</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p169--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p170--cellular-automata/"> <span>2</span> <span>Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p184--turing-machines/"> <span>3</span> <span>Turing Machines</span> </a> </li> <li class="section section-4"> <a href="/nks/p187--substitution-systems-and-fractals/"> <span>4</span> <span>Substitution Systems and Fractals</span> </a> </li> <li class="section section-5"> <a href="/nks/p193--network-systems/"> <span>5</span> <span>Network Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p204--multiway-systems/"> <span>6</span> <span>Multiway Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p210--systems-based-on-constraints/"> <span>7</span> <span>Systems Based on Constraints</span> </a> </li> </ul> </li> <li class="chapter chapter-6"> <a href="/nks/chap-6--starting-from-randomness/"> <span>6</span> <span>Starting from Randomness</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p223--the-emergence-of-order/"> <span>1</span> <span>The Emergence of Order</span> </a> </li> <li class="section section-2"> <a href="/nks/p231--four-classes-of-behavior/"> <span>2</span> <span>Four Classes of Behavior</span> </a> </li> <li class="section section-3"> <a href="/nks/p250--sensitivity-to-initial-conditions/"> <span>3</span> <span>Sensitivity to Initial Conditions</span> </a> </li> <li class="section section-4"> <a href="/nks/p255--systems-of-limited-size-and-class-2-behavior/"> <span>4</span> <span>Systems of Limited Size and Class 2 Behavior</span> </a> </li> <li class="section section-5"> <a href="/nks/p261--randomness-in-class-3-systems/"> <span>5</span> <span>Randomness in Class 3 Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p266--special-initial-conditions/"> <span>6</span> <span>Special Initial Conditions</span> </a> </li> <li class="section section-7"> <a href="/nks/p275--the-notion-of-attractors/"> <span>7</span> <span>The Notion of Attractors</span> </a> </li> <li class="section section-8"> <a href="/nks/p281--structures-in-class-4-systems/"> <span>8</span> <span>Structures in Class 4 Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-7"> <a href="/nks/chap-7--mechanisms-in-programs-and-nature/"> <span>7</span> <span>Mechanisms in Programs and Nature</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p297--universality-of-behavior/"> <span>1</span> <span>Universality of Behavior</span> </a> </li> <li class="section section-2"> <a href="/nks/p299--three-mechanisms-for-randomness/"> <span>2</span> <span>Three Mechanisms for Randomness</span> </a> </li> <li class="section section-3"> <a href="/nks/p301--randomness-from-the-environment/"> <span>3</span> <span>Randomness from the Environment</span> </a> </li> <li class="section section-4"> <a href="/nks/p304--chaos-theory-and-randomness-from-initial-conditions/"> <span>4</span> <span>Chaos Theory and Randomness from Initial Conditions</span> </a> </li> <li class="section section-5"> <a href="/nks/p315--the-intrinsic-generation-of-randomness/"> <span>5</span> <span>The Intrinsic Generation of Randomness</span> </a> </li> <li class="section section-6"> <a href="/nks/p327--the-phenomenon-of-continuity/"> <span>6</span> <span>The Phenomenon of Continuity</span> </a> </li> <li class="section section-7"> <a href="/nks/p337--origins-of-discreteness/"> <span>7</span> <span>Origins of Discreteness</span> </a> </li> <li class="section section-8"> <a href="/nks/p342--the-problem-of-satisfying-constraints/"> <span>8</span> <span>The Problem of Satisfying Constraints</span> </a> </li> <li class="section section-9"> <a href="/nks/p351--origins-of-simple-behavior/"> <span>9</span> <span>Origins of Simple Behavior</span> </a> </li> </ul> </li> <li class="chapter chapter-8"> <a href="/nks/chap-8--implications-for-everyday-systems/"> <span>8</span> <span>Implications for Everyday Systems</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p363--issues-of-modelling/"> <span>1</span> <span>Issues of Modelling</span> </a> </li> <li class="section section-2"> <a href="/nks/p369--the-growth-of-crystals/"> <span>2</span> <span>The Growth of Crystals</span> </a> </li> <li class="section section-3"> <a href="/nks/p374--the-breaking-of-materials/"> <span>3</span> <span>The Breaking of Materials</span> </a> </li> <li class="section section-4"> <a href="/nks/p376--fluid-flow/"> <span>4</span> <span>Fluid Flow</span> </a> </li> <li class="section section-5"> <a href="/nks/p383--fundamental-issues-in-biology/"> <span>5</span> <span>Fundamental Issues in Biology</span> </a> </li> <li class="section section-6"> <a href="/nks/p400--growth-of-plants-and-animals/"> <span>6</span> <span>Growth of Plants and Animals</span> </a> </li> <li class="section section-7"> <a href="/nks/p422--biological-pigmentation-patterns/"> <span>7</span> <span>Biological Pigmentation Patterns</span> </a> </li> <li class="section section-8"> <a href="/nks/p429--financial-systems/"> <span>8</span> <span>Financial Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-9"> <a href="/nks/chap-9--fundamental-physics/"> <span>9</span> <span>Fundamental Physics</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p433--the-problems-of-physics/"> <span>1</span> <span>The Problems of Physics</span> </a> </li> <li class="section section-2"> <a href="/nks/p435--the-notion-of-reversibility/"> <span>2</span> <span>The Notion of Reversibility</span> </a> </li> <li class="section section-3"> <a href="/nks/p441--irreversibility-and-the-second-law-of-thermodynamics/"> <span>3</span> <span>Irreversibility and the Second Law of Thermodynamics</span> </a> </li> <li class="section section-4"> <a href="/nks/p458--conserved-quantities-and-continuum-phenomena/"> <span>4</span> <span>Conserved Quantities and Continuum Phenomena</span> </a> </li> <li class="section section-5"> <a href="/nks/p465--ultimate-models-for-the-universe/"> <span>5</span> <span>Ultimate Models for the Universe</span> </a> </li> <li class="section section-6"> <a href="/nks/p472--the-nature-of-space/"> <span>6</span> <span>The Nature of Space</span> </a> </li> <li class="section section-7"> <a href="/nks/p475--space-as-a-network/"> <span>7</span> <span>Space as a Network</span> </a> </li> <li class="section section-8"> <a href="/nks/p481--the-relationship-of-space-and-time/"> <span>8</span> <span>The Relationship of Space and Time</span> </a> </li> <li class="section section-9"> <a href="/nks/p486--time-and-causal-networks/"> <span>9</span> <span>Time and Causal Networks</span> </a> </li> <li class="section section-10"> <a href="/nks/p497--the-sequencing-of-events-in-the-universe/"> <span>10</span> <span>The Sequencing of Events in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p504--uniqueness-and-branching-in-time/"> <span>11</span> <span>Uniqueness and Branching in Time</span> </a> </li> <li class="section section-12"> <a href="/nks/p508--evolution-of-networks/"> <span>12</span> <span>Evolution of Networks</span> </a> </li> <li class="section section-13"> <a href="/nks/p516--space-time-and-relativity/"> <span>13</span> <span>Space, Time and Relativity</span> </a> </li> <li class="section section-14"> <a href="/nks/p525--elementary-particles/"> <span>14</span> <span>Elementary Particles</span> </a> </li> <li class="section section-15"> <a href="/nks/p530--the-phenomenon-of-gravity/"> <span>15</span> <span>The Phenomenon of Gravity</span> </a> </li> <li class="section section-16"> <a href="/nks/p537--quantum-phenomena/"> <span>16</span> <span>Quantum Phenomena</span> </a> </li> </ul> </li> <li class="chapter chapter-10"> <a href="/nks/chap-10--processes-of-perception-and-analysis/"> <span>10</span> <span>Processes of Perception and Analysis</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p547--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p548--what-perception-and-analysis-do/"> <span>2</span> <span>What Perception and Analysis Do</span> </a> </li> <li class="section section-3"> <a href="/nks/p552--defining-the-notion-of-randomness/"> <span>3</span> <span>Defining the Notion of Randomness</span> </a> </li> <li class="section section-4"> <a href="/nks/p557--defining-complexity/"> <span>4</span> <span>Defining Complexity</span> </a> </li> <li class="section section-5"> <a href="/nks/p560--data-compression/"> <span>5</span> <span>Data Compression</span> </a> </li> <li class="section section-6"> <a href="/nks/p572--irreversible-data-compression/"> <span>6</span> <span>Irreversible Data Compression</span> </a> </li> <li class="section section-7"> <a href="/nks/p577--visual-perception/"> <span>7</span> <span>Visual Perception</span> </a> </li> <li class="section section-8"> <a href="/nks/p585--auditory-perception/"> <span>8</span> <span>Auditory Perception</span> </a> </li> <li class="section section-9"> <a href="/nks/p588--statistical-analysis/"> <span>9</span> <span>Statistical Analysis</span> </a> </li> <li class="section section-10"> <a href="/nks/p598--cryptography-and-cryptanalysis/"> <span>10</span> <span>Cryptography and Cryptanalysis</span> </a> </li> <li class="section section-11"> <a href="/nks/p606--traditional-mathematics-and-mathematical-formulas/"> <span>11</span> <span>Traditional Mathematics and Mathematical Formulas</span> </a> </li> <li class="section section-12"> <a href="/nks/p620--human-thinking/"> <span>12</span> <span>Human Thinking</span> </a> </li> <li class="section section-13"> <a href="/nks/p632--higher-forms-of-perception-and-analysis/"> <span>13</span> <span>Higher Forms of Perception and Analysis</span> </a> </li> </ul> </li> <li class="chapter chapter-11"> <a href="/nks/chap-11--the-notion-of-computation/"> <span>11</span> <span>The Notion of Computation</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p637--computation-as-a-framework/"> <span>1</span> <span>Computation as a Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p638--computations-in-cellular-automata/"> <span>2</span> <span>Computations in Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p642--the-phenomenon-of-universality/"> <span>3</span> <span>The Phenomenon of Universality</span> </a> </li> <li class="section section-4"> <a href="/nks/p644--a-universal-cellular-automaton/"> <span>4</span> <span>A Universal Cellular Automaton</span> </a> </li> <li class="section section-5"> <a href="/nks/p656--emulating-other-systems-with-cellular-automata/"> <span>5</span> <span>Emulating Other Systems with Cellular Automata</span> </a> </li> <li class="section section-6"> <a href="/nks/p664--emulating-cellular-automata-with-other-systems/"> <span>6</span> <span>Emulating Cellular Automata with Other Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p674--implications-of-universality/"> <span>7</span> <span>Implications of Universality</span> </a> </li> <li class="section section-8"> <a href="/nks/p675--the-rule-110-cellular-automaton/"> <span>8</span> <span>The Rule 110 Cellular Automaton</span> </a> </li> <li class="section section-9"> <a href="/nks/p690--the-significance-of-universality-in-rule-110/"> <span>9</span> <span>The Significance of Universality in Rule 110</span> </a> </li> <li class="section section-10"> <a href="/nks/p691--class-4-behavior-and-universality/"> <span>10</span> <span>Class 4 Behavior and Universality</span> </a> </li> <li class="section section-11"> <a href="/nks/p694--the-threshold-of-universality-in-cellular-automata/"> <span>11</span> <span>The Threshold of Universality in Cellular Automata</span> </a> </li> <li class="section section-12"> <a href="/nks/p706--universality-in-turing-machines-and-other-systems/"> <span>12</span> <span>Universality in Turing Machines and Other Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-12"> <a href="/nks/chap-12--the-principle-of-computational-equivalence/"> <span>12</span> <span>The Principle of Computational Equivalence</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p715--basic-framework/"> <span>1</span> <span>Basic Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p716--outline-of-the-principle/"> <span>2</span> <span>Outline of the Principle</span> </a> </li> <li class="section section-3"> <a href="/nks/p719--the-content-of-the-principle/"> <span>3</span> <span>The Content of the Principle</span> </a> </li> <li class="section section-4"> <a href="/nks/p726--the-validity-of-the-principle/"> <span>4</span> <span>The Validity of the Principle</span> </a> </li> <li class="section section-5"> <a href="/nks/p735--explaining-the-phenomenon-of-complexity/"> <span>5</span> <span>Explaining the Phenomenon of Complexity</span> </a> </li> <li class="section section-6"> <a href="/nks/p737--computational-irreducibility/"> <span>6</span> <span>Computational Irreducibility</span> </a> </li> <li class="section section-7"> <a href="/nks/p750--the-phenomenon-of-free-will/"> <span>7</span> <span>The Phenomenon of Free Will</span> </a> </li> <li class="section section-8"> <a href="/nks/p753--undecidability-and-intractability/"> <span>8</span> <span>Undecidability and Intractability</span> </a> </li> <li class="section section-9"> <a href="/nks/p772--implications-for-mathematics-and-its-foundations/"> <span>9</span> <span>Implications for Mathematics and Its Foundations</span> </a> </li> <li class="section section-10"> <a href="/nks/p822--intelligence-in-the-universe/"> <span>10</span> <span>Intelligence in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p840--implications-for-technology/"> <span>11</span> <span>Implications for Technology</span> </a> </li> <li class="section section-12"> <a href="/nks/p844--historical-perspectives/"> <span>12</span> <span>Historical Perspectives</span> </a> </li> </ul> </li> <li class="chapter notes"> <a href="/nks/notes-for-chapters/"> <span></span> <span>Notes</span> </a> </li> </ul> <div class="search"> </div> </div> </div> </nav> </div> </div> </div> <div id="search"> <div class="text-align-c__600"> <div class="inline-grid cols-3 cols-1__600"> <form id="search-index-form" action="/nks/index/"><div><label for="search-index-input">Look up in index</label><input type="search" id="search-index-input" name="search-index-input"></div></form> <form id="search-all-form" action="/nks/search/"><div><label for="search-all-input">Text search</label><input type="search" id="search-all-input" name="search-all-input"></div></form> <form id="search-jump-form"><div><label for="search-jump-input">Jump to page</label><input type="search" id="search-jump-input" name="search-jump-input"></div></form> </div> </div> </div> </header>  <main id="main"> <div class="thin-wrapper"> <div id="previous"><a href="/nks/chap-4--systems-based-on-numbers/"><span>‹</span></a></div> <div id="next"><a href="/nks/chap-6--starting-from-randomness/"><span>›</span></a></div> <div id="content" class="thin-padding"> <div class="ribbon-wrapper top"> </div> <section class="chapter-title"> <h1 class="chapter-heading"><a href="/nks/p169--introduction/">Chapter 5: <span>Two Dimensions and Beyond</span></a></h1> <hr> </section> <section class="chapter-content"> <div> <h2 id="sect-5-1--introduction" class="section-heading"><a href="/nks/p169--introduction/">Section 1: <span>Introduction</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-1--introduction</span></span></h2> <div class="pages"> <a href="/nks/p169--introduction/"><img src="/nks/pages/p169--introduction/image-tiny.png" alt="Page 169" width="70" height="86"></a><a href="/nks/sect-5-1--introduction--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-5-2--cellular-automata" class="section-heading"><a href="/nks/p170--cellular-automata/">Section 2: <span>Cellular Automata</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-2--cellular-automata</span></span></h2> <div class="pages"> <a href="/nks/p170--cellular-automata/"><img src="/nks/pages/p170--cellular-automata/image-tiny.png" alt="Page 170" width="70" height="86"></a> <a href="/nks/p171--cellular-automata/"><img src="/nks/pages/p171--cellular-automata/image-tiny.png" alt="Page 171" width="70" height="86"></a> <a href="/nks/p172--cellular-automata/"><img src="/nks/pages/p172--cellular-automata/image-tiny.png" alt="Page 172" width="70" height="86"></a> <a href="/nks/p173--cellular-automata/"><img src="/nks/pages/p173--cellular-automata/image-tiny.png" alt="Page 173" width="70" height="86"></a> <a href="/nks/p174--cellular-automata/"><img src="/nks/pages/p174--cellular-automata/image-tiny.png" alt="Page 174" width="70" height="86"></a> <a href="/nks/p175--cellular-automata/"><img src="/nks/pages/p175--cellular-automata/image-tiny.png" alt="Page 175" width="70" height="86"></a> <a href="/nks/p176--cellular-automata/"><img src="/nks/pages/p176--cellular-automata/image-tiny.png" alt="Page 176" width="70" height="86"></a> <a href="/nks/p177--cellular-automata/"><img src="/nks/pages/p177--cellular-automata/image-tiny.png" alt="Page 177" width="70" height="86"></a> <a href="/nks/p178--cellular-automata/"><img src="/nks/pages/p178--cellular-automata/image-tiny.png" alt="Page 178" width="70" height="86"></a> <a href="/nks/p179--cellular-automata/"><img src="/nks/pages/p179--cellular-automata/image-tiny.png" alt="Page 179" width="70" height="86"></a> <a href="/nks/p180--cellular-automata/"><img src="/nks/pages/p180--cellular-automata/image-tiny.png" alt="Page 180" width="70" height="86"></a> <a href="/nks/p181--cellular-automata/"><img src="/nks/pages/p181--cellular-automata/image-tiny.png" alt="Page 181" width="70" height="86"></a> <a href="/nks/p182--cellular-automata/"><img src="/nks/pages/p182--cellular-automata/image-tiny.png" alt="Page 182" width="70" height="86"></a> <a href="/nks/p183--cellular-automata/"><img src="/nks/pages/p183--cellular-automata/image-tiny.png" alt="Page 183" width="70" height="86"></a><a href="/nks/sect-5-2--cellular-automata--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-5-3--turing-machines" class="section-heading"><a href="/nks/p184--turing-machines/">Section 3: <span>Turing Machines</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-3--turing-machines</span></span></h2> <div class="pages"> <a href="/nks/p184--turing-machines/"><img src="/nks/pages/p184--turing-machines/image-tiny.png" alt="Page 184" width="70" height="86"></a> <a href="/nks/p185--turing-machines/"><img src="/nks/pages/p185--turing-machines/image-tiny.png" alt="Page 185" width="70" height="86"></a> <a href="/nks/p186--turing-machines/"><img src="/nks/pages/p186--turing-machines/image-tiny.png" alt="Page 186" width="70" height="86"></a><a href="/nks/sect-5-3--turing-machines--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-5-4--substitution-systems-and-fractals" class="section-heading"><a href="/nks/p187--substitution-systems-and-fractals/">Section 4: <span>Substitution Systems and Fractals</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-4--substitution-systems-and-fractals</span></span></h2> <div class="pages"> <a href="/nks/p187--substitution-systems-and-fractals/"><img src="/nks/pages/p187--substitution-systems-and-fractals/image-tiny.png" alt="Page 187" width="70" height="86"></a> <a href="/nks/p188--substitution-systems-and-fractals/"><img src="/nks/pages/p188--substitution-systems-and-fractals/image-tiny.png" alt="Page 188" width="70" height="86"></a> <a href="/nks/p189--substitution-systems-and-fractals/"><img src="/nks/pages/p189--substitution-systems-and-fractals/image-tiny.png" alt="Page 189" width="70" height="86"></a> <a href="/nks/p190--substitution-systems-and-fractals/"><img src="/nks/pages/p190--substitution-systems-and-fractals/image-tiny.png" alt="Page 190" width="70" height="86"></a> <a href="/nks/p191--substitution-systems-and-fractals/"><img src="/nks/pages/p191--substitution-systems-and-fractals/image-tiny.png" alt="Page 191" width="70" height="86"></a> <a href="/nks/p192--substitution-systems-and-fractals/"><img src="/nks/pages/p192--substitution-systems-and-fractals/image-tiny.png" alt="Page 192" width="70" height="86"></a><a href="/nks/sect-5-4--substitution-systems-and-fractals--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-5-5--network-systems" class="section-heading"><a href="/nks/p193--network-systems/">Section 5: <span>Network Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-5--network-systems</span></span></h2> <div class="pages"> <a href="/nks/p193--network-systems/"><img src="/nks/pages/p193--network-systems/image-tiny.png" alt="Page 193" width="70" height="86"></a> <a href="/nks/p194--network-systems/"><img src="/nks/pages/p194--network-systems/image-tiny.png" alt="Page 194" width="70" height="86"></a> <a href="/nks/p195--network-systems/"><img src="/nks/pages/p195--network-systems/image-tiny.png" alt="Page 195" width="70" height="86"></a> <a href="/nks/p196--network-systems/"><img src="/nks/pages/p196--network-systems/image-tiny.png" alt="Page 196" width="70" height="86"></a> <a href="/nks/p197--network-systems/"><img src="/nks/pages/p197--network-systems/image-tiny.png" alt="Page 197" width="70" height="86"></a> <a href="/nks/p198--network-systems/"><img src="/nks/pages/p198--network-systems/image-tiny.png" alt="Page 198" width="70" height="86"></a> <a href="/nks/p199--network-systems/"><img src="/nks/pages/p199--network-systems/image-tiny.png" alt="Page 199" width="70" height="86"></a> <a href="/nks/p200--network-systems/"><img src="/nks/pages/p200--network-systems/image-tiny.png" alt="Page 200" width="70" height="86"></a> <a href="/nks/p201--network-systems/"><img src="/nks/pages/p201--network-systems/image-tiny.png" alt="Page 201" width="70" height="86"></a> <a href="/nks/p202--network-systems/"><img src="/nks/pages/p202--network-systems/image-tiny.png" alt="Page 202" width="70" height="86"></a> <a href="/nks/p203--network-systems/"><img src="/nks/pages/p203--network-systems/image-tiny.png" alt="Page 203" width="70" height="86"></a><a href="/nks/sect-5-5--network-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-5-6--multiway-systems" class="section-heading"><a href="/nks/p204--multiway-systems/">Section 6: <span>Multiway Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-6--multiway-systems</span></span></h2> <div class="pages"> <a href="/nks/p204--multiway-systems/"><img src="/nks/pages/p204--multiway-systems/image-tiny.png" alt="Page 204" width="70" height="86"></a> <a href="/nks/p205--multiway-systems/"><img src="/nks/pages/p205--multiway-systems/image-tiny.png" alt="Page 205" width="70" height="86"></a> <a href="/nks/p206--multiway-systems/"><img src="/nks/pages/p206--multiway-systems/image-tiny.png" alt="Page 206" width="70" height="86"></a> <a href="/nks/p207--multiway-systems/"><img src="/nks/pages/p207--multiway-systems/image-tiny.png" alt="Page 207" width="70" height="86"></a> <a href="/nks/p208--multiway-systems/"><img src="/nks/pages/p208--multiway-systems/image-tiny.png" alt="Page 208" width="70" height="86"></a> <a href="/nks/p209--multiway-systems/"><img src="/nks/pages/p209--multiway-systems/image-tiny.png" alt="Page 209" width="70" height="86"></a><a href="/nks/sect-5-6--multiway-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-5-7--systems-based-on-constraints" class="section-heading"><a href="/nks/p210--systems-based-on-constraints/">Section 7: <span>Systems Based on Constraints</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-5--two-dimensions-and-beyond#sect-5-7--systems-based-on-constraints</span></span></h2> <div class="pages"> <a href="/nks/p210--systems-based-on-constraints/"><img src="/nks/pages/p210--systems-based-on-constraints/image-tiny.png" alt="Page 210" width="70" height="86"></a> <a href="/nks/p211--systems-based-on-constraints/"><img src="/nks/pages/p211--systems-based-on-constraints/image-tiny.png" alt="Page 211" width="70" height="86"></a> <a href="/nks/p212--systems-based-on-constraints/"><img src="/nks/pages/p212--systems-based-on-constraints/image-tiny.png" alt="Page 212" width="70" height="86"></a> <a href="/nks/p213--systems-based-on-constraints/"><img src="/nks/pages/p213--systems-based-on-constraints/image-tiny.png" alt="Page 213" width="70" height="86"></a> <a href="/nks/p214--systems-based-on-constraints/"><img src="/nks/pages/p214--systems-based-on-constraints/image-tiny.png" alt="Page 214" width="70" height="86"></a> <a href="/nks/p215--systems-based-on-constraints/"><img src="/nks/pages/p215--systems-based-on-constraints/image-tiny.png" alt="Page 215" width="70" height="86"></a> <a href="/nks/p216--systems-based-on-constraints/"><img src="/nks/pages/p216--systems-based-on-constraints/image-tiny.png" alt="Page 216" width="70" height="86"></a> <a href="/nks/p217--systems-based-on-constraints/"><img src="/nks/pages/p217--systems-based-on-constraints/image-tiny.png" alt="Page 217" width="70" height="86"></a> <a href="/nks/p218--systems-based-on-constraints/"><img src="/nks/pages/p218--systems-based-on-constraints/image-tiny.png" alt="Page 218" width="70" height="86"></a> <a href="/nks/p219--systems-based-on-constraints/"><img src="/nks/pages/p219--systems-based-on-constraints/image-tiny.png" alt="Page 219" width="70" height="86"></a> <a href="/nks/p220--systems-based-on-constraints/"><img src="/nks/pages/p220--systems-based-on-constraints/image-tiny.png" alt="Page 220" width="70" height="86"></a> <a href="/nks/p221--systems-based-on-constraints/"><img src="/nks/pages/p221--systems-based-on-constraints/image-tiny.png" alt="Page 221" width="70" height="86"></a><a href="/nks/sect-5-7--systems-based-on-constraints--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> </section> <section id="related-resources" class="width-full max-width-900" style="display: none;"> <hr class="even"> <h2>External Resources Related to this Page:</h2> <div> <ul class="grid cols-2 width-full heirs-width-1-2 cols-1__600 heirs-width-full__600"> </ul> </div> </section>  <footer id="inline-footer"> <div class="heirs-display-ib heirs-margin-bt-1-2__600 heirs-display-b__600"> <div> <a href="/nks/chap-5--two-dimensions-and-beyond--notes/" class="view-notes mini-t"> <span><img src="/nks/img/icon-notes.png" alt="" width="8" height="8"></span> <span>All notes for this chapter</span> </a> </div> <div> <a href="/nks/programs/" class="programs-link mini-t"> <span><img src="/nks/img/icon-download.png" alt="" width="11" height="13"></span> <span>Download programs</span> </a> </div> <div> <a 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Wolfram uses his approach to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems, to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe.", "author": { "@type": "Person", "name": "Stephen Wolfram" }, "publisher": { "@type": "Corporation", "name": "Wolfram Media, Inc.", "url": "http://www.wolfram-media.com/" }, "isbn": "1-57955-008-8", "datePublished": "2002", "numberOfPages": "1197", "bookFormat": "print, online, iPad", "genre": "Science, Cellular Automata, Computation, Biology, Physics", "audience": "Scientists, Researchers, Science Educators, Biologists, Physicists, Mathematicians, Programmers", "discussionUrl": "http://community.wolfram.com/content?curTag=wolfram%20science", "inLanguage": "en" } </script>   </div> <div 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