Systems Based on Constraints: A New Kind of Science | Online by Stephen Wolfram [Page 210]

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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"><a href="/nks/chap-5--two-dimensions-and-beyond/">Chapter 5</a></span> <span class="section"><a href="/nks/chap-5--two-dimensions-and-beyond/#sect-5-7--systems-based-on-constraints">Section 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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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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" 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</ul> </div> <div class="xlarge"> <img src="/nks/pages/p210--systems-based-on-constraints/image-xlarge.png" alt="" width="1000" height="1205" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0210a" data-inline-text="" data-area="2000,2420" data-coords="582,1130,1727,1198" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div> </div> <div class="large"> <img src="/nks/pages/p210--systems-based-on-constraints/image-large.png" alt="" width="850" height="1025" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0210a" data-inline-text="" data-area="850,1025" data-coords="248,483,734,508" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div> </div> <div class="medium"> <img src="/nks/pages/p210--systems-based-on-constraints/image-medium.png" alt="" width="750" height="904" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0210a" data-inline-text="" data-area="750,904" data-coords="219,424,650,450" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div> </div> <div class="small"> <img src="/nks/pages/p210--systems-based-on-constraints/image-small.png" alt="" width="650" height="784" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0210a" data-inline-text="" data-area="650,784" data-coords="190,369,564,390" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div> </div><div class="text-content wide-padding"><h2 class="text-section-heading">Systems Based on Constraints</h2><p class="text">In the course of this book we have looked at many different kinds of systems. But in one respect all these systems have ultimately been set up in the same basic way: they are all based on explicit rules that specify how the system evolves from step to step.</p><p class="text">In traditional science, however, it is common to consider systems that are set up in a rather different way: instead of having explicit rules for evolution, the systems are just given constraints to satisfy.</p><p class="text">As a simple example, consider a line of cells in which each cell is colored black or white, and in which the arrangement of colors is subject to the constraint that every cell should have exactly one black and one white neighbor. Knowing only this constraint gives no explicit procedure for working out the color of each cell. And in fact it may at first not be clear that there will be any arrangement of colors that can satisfy the constraint. But it turns out that there is—as shown below.</p><p class="text">And having seen this picture, one might then imagine that there must be many other patterns that would also satisfy the constraint. After all, the constraint is local to neighboring cells, so one might suppose that parts of the pattern sufficiently far apart should always be independent. But in fact this is not true, and instead the system works a bit like a puzzle in which there is only one way to fit in each piece. And in the end it is only the perfectly repetitive pattern shown below that can satisfy the required constraint at every cell.</p><p class="text">Other constraints, however, can allow more freedom. Thus, for example, with the constraint that every cell must have at least one neighbor whose color is different from its own, any of the patterns in the picture at the top of the <a href="/nks/p211--systems-based-on-constraints--webview/">facing page</a> are allowed, as indeed is any pattern that involves no more than two successive cells of the same color.</p><hr class="caphr"><p class="capimg-wrapper"><div class="c2c-image" data-class="capimg" data-file-name="page0210a" data-src="/nks/img/inline/page0210a.png" data-width="794" data-height="27" data-alt="System made of black and white cells" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div></p><p class="captext">A system consisting of a line of black and white cells whose form is defined by the constraint that every cell should have exactly one black and one white neighbor. The pattern shown is the only possible one that satisfies this constraint. The idea of implicitly determining the behavior of a system by giving constraints that it must satisfy is common in traditional science and mathematics.</p></div> <div class="ribbon-wrapper stripe"> </div> <footer id="footer" class="wide-padding"> <section id="related-notes" class="width-full max-width-900 center"> <h2>Notes In the Book Related To This Page:</h2> <div> <div class="thumbnails cf"> <a href="/nks/notes-5-7--the-notion-of-equations-versus-constraints/"><img src="/nks/img/thumbnails/notes-5-7--the-notion-of-equations-versus-constraints--textonly.png" alt="The notion of equations [versus constraints]" title="The notion of equations [versus constraints]" width="76" height="62"></a> <a href="/nks/notes-5-7--linear-and-nonlinear-systems/"><img src="/nks/img/thumbnails/notes-5-7--linear-and-nonlinear-systems--textonly.png" alt="Linear and nonlinear systems" title="Linear and nonlinear systems" width="76" height="62"></a> <a href="/nks/notes-5-7--explanations-based-on-constraints/"><img src="/nks/img/thumbnails/notes-5-7--explanations-based-on-constraints--textonly.png" alt="Explanations based on constraints" title="Explanations based on constraints" width="76" height="62"></a> <a href="/nks/notes-5-7--1d-systems-based-on-constraints/"><img src="/nks/img/thumbnails/notes-5-7--1d-systems-based-on-constraints--textonly.png" alt="1D [systems based on] constraints" title="1D [systems based on] constraints" width="76" height="62"></a> <a href="/nks/notes-5-7--1d-cellular-automata/"><img src="/nks/img/thumbnails/notes-5-7--1d-cellular-automata--textonly.png" alt="1D cellular automata" title="1D cellular automata" width="76" height="62"></a> <a href="/nks/notes-5-7--excluded-blocks-in-dynamical-systems-theory/"><img src="/nks/img/thumbnails/notes-5-7--excluded-blocks-in-dynamical-systems-theory--textonly.png" alt="[Excluded blocks in] dynamical systems theory" title="[Excluded blocks in] dynamical systems theory" width="76" height="62"></a> <a href="/nks/notes-5-7--correspondence-systems-as-constraints/"><img src="/nks/img/thumbnails/notes-5-7--correspondence-systems-as-constraints--textonly.png" alt="Correspondence systems [as constraints]" title="Correspondence systems [as constraints]" width="76" height="62"></a> <a href="/nks/notes-5-7--sequence-equations/"><img src="/nks/img/thumbnails/notes-5-7--sequence-equations--textonly.png" alt="Sequence equations" title="Sequence equations" width="76" height="62"></a> <a href="/nks/notes-5-7--pattern-avoiding-sequences/"><img src="/nks/img/thumbnails/notes-5-7--pattern-avoiding-sequences--textonly.png" alt="Pattern-avoiding sequences" title="Pattern-avoiding sequences" width="76" height="62"></a> </div> <ul class="cols-2 cols-1__600 heirs-width-full__600"> <li><span><a href="/nks/notes-5-7--the-notion-of-equations-versus-constraints/"><span><img src="/nks/img/thumbnails/notes-5-7--the-notion-of-equations-versus-constraints--textonly.png" alt="" width="76" height="62"></span><span>The notion of equations [versus constraints]</span></a></span></li> <li><span><a href="/nks/notes-5-7--linear-and-nonlinear-systems/"><span><img src="/nks/img/thumbnails/notes-5-7--linear-and-nonlinear-systems--textonly.png" alt="" width="76" height="62"></span><span>Linear and nonlinear systems</span></a></span></li> <li><span><a href="/nks/notes-5-7--explanations-based-on-constraints/"><span><img src="/nks/img/thumbnails/notes-5-7--explanations-based-on-constraints--textonly.png" alt="" width="76" height="62"></span><span>Explanations based on constraints</span></a></span></li> <li><span><a href="/nks/notes-5-7--1d-systems-based-on-constraints/"><span><img src="/nks/img/thumbnails/notes-5-7--1d-systems-based-on-constraints--textonly.png" alt="" width="76" height="62"></span><span>1D [systems based on] constraints</span></a></span></li> <li><span><a href="/nks/notes-5-7--1d-cellular-automata/"><span><img src="/nks/img/thumbnails/notes-5-7--1d-cellular-automata--textonly.png" alt="" width="76" height="62"></span><span>1D cellular automata</span></a></span></li> <li><span><a href="/nks/notes-5-7--excluded-blocks-in-dynamical-systems-theory/"><span><img src="/nks/img/thumbnails/notes-5-7--excluded-blocks-in-dynamical-systems-theory--textonly.png" alt="" width="76" height="62"></span><span>[Excluded blocks in] dynamical systems theory</span></a></span></li> <li><span><a href="/nks/notes-5-7--correspondence-systems-as-constraints/"><span><img src="/nks/img/thumbnails/notes-5-7--correspondence-systems-as-constraints--textonly.png" alt="" width="76" height="62"></span><span>Correspondence systems [as constraints]</span></a></span></li> <li><span><a href="/nks/notes-5-7--sequence-equations/"><span><img src="/nks/img/thumbnails/notes-5-7--sequence-equations--textonly.png" alt="" width="76" height="62"></span><span>Sequence equations</span></a></span></li> <li><span><a href="/nks/notes-5-7--pattern-avoiding-sequences/"><span><img src="/nks/img/thumbnails/notes-5-7--pattern-avoiding-sequences--textonly.png" alt="" width="76" height="62"></span><span>Pattern-avoiding sequences</span></a></span></li> <li><span><a href="/nks/notes-5-7--formal-languages-and-constraints/"><span><img src="/nks/img/thumbnails/notes-5-7--formal-languages-and-constraints--textonly.png" alt="" width="76" height="62"></span><span>Formal languages [and constraints]</span></a></span></li> <li><span><a href="/nks/notes-5-7--diophantine-equations/"><span><img src="/nks/img/thumbnails/notes-5-7--diophantine-equations--textonly.png" alt="" width="76" height="62"></span><span>Diophantine equations</span></a></span></li> <li><span><a href="/nks/notes-5-7--constraints-on-formulas/"><span><img src="/nks/img/thumbnails/notes-5-7--constraints-on-formulas--textonly.png" alt="" width="76" height="62"></span><span>Constraints on formulas</span></a></span></li> <li><span><a href="/nks/notes-4-9--existence-and-uniqueness-in-pdes/"><span><img src="/nks/img/thumbnails/notes-4-9--existence-and-uniqueness-in-pdes--textonly.png" alt="" width="76" height="62"></span><span>Existence and uniqueness [in PDEs]</span></a></span></li> <li><span><a href="/nks/notes-12-8--correspondence-systems/"><span><img src="/nks/img/thumbnails/notes-12-8--correspondence-systems--textonly.png" alt="" width="76" height="62"></span><span>Correspondence systems</span></a></span></li> <li><span><a href="/nks/notes-12-8--word-problems/"><span><img src="/nks/img/thumbnails/notes-12-8--word-problems--textonly.png" alt="" width="76" height="62"></span><span>Word problems</span></a></span></li> <li><span><a href="/nks/notes-12-8--sequence-equations/"><span><img src="/nks/img/thumbnails/notes-12-8--sequence-equations--textonly.png" alt="" width="76" height="62"></span><span>Sequence equations</span></a></span></li> </ul> </div> </section> <section id="resources"> <section 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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>   </footer> 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Systems Based on Constraints: A New Kind of Science | Online by Stephen Wolfram [Page 210]