Note (b) for Why These Discoveries Were Not Made Before: A New Kind of Science

<!doctype html> <html lang="en" class="notes text-version"> <head>  <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <title>Note (b) for Why These Discoveries Were Not Made Before: A New Kind of Science | Online by Stephen Wolfram [Page 880]</title> <meta name="description" content="My work on cellular automata I began serious work on cellular automata in the middle of 1981. I had been thinking for some... – from A New Kind of Science"> <link rel="alternate" hreflang="en" href="https://www.wolframscience.com/nks/index.en.php"> <meta property="og:image" content="http://www.wolframscience.com/share.png"> <meta property="og:url" content="https://www.wolframscience.com/nks/notes-2-3--my-work-on-cellular-automata/"> <meta property="og:type" content="website"> <meta property="og:title" content="Note (b) for Why These Discoveries Were Not Made Before: A New Kind of Science | Online by Stephen Wolfram [Page 880]"> <meta property="og:description" content="My work on cellular automata I began serious work on cellular automata in the middle of 1981. I had been thinking for some... – from A New Kind of Science"> <meta property="og:locale" content="en"> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Note (b) for Why These Discoveries Were Not Made Before: A New Kind of Science | Online by Stephen Wolfram [Page 880]"> <meta name="twitter:description" content="My work on cellular automata I began serious work on cellular automata in the middle of 1981. I had been thinking for some... – from A New Kind of Science"> <meta name="twitter:image:src" content="http://www.wolframscience.com/share.png"> <link rel="canonical" href="https://www.wolframscience.com/nks/notes-2-3--my-work-on-cellular-automata/"> <link rel="stylesheet" href="/common/framework/css/framework.en.css"> <script src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script src="/common/javascript/wal/latest/walLoad.js"></script> <script defer src="/common/js/announcements/script.js"></script> <script src="/common/framework/js/head.en.js"></script> <script src="/common/js/jquery/3.7.1/jquery.min.js"></script> <script src="/common/js/jquery/plugins/migrate/3.5.2/jquery.migrate.min.js"></script> <script src="/common/javascript/analytics.js"></script>  <link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"> <link rel="stylesheet" href="//www.wolframcdn.com/fonts/pt-serif/1.0/all.css"> <link rel="stylesheet" href="//www.wolframcdn.com/fonts/mathematica/1.0/all.css">  <link rel="preconnect" href="https://fonts.googleapis.com"> <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> <link href="https://fonts.googleapis.com/css2?family=Crimson+Text:ital@0;1&display=swap" rel="stylesheet"> <link rel="stylesheet" href="https://use.typekit.net/hod7dyh.css">  <link rel="stylesheet" href="/common/js/clipboard/2.0/clipboard.css"> <link rel="stylesheet" href="/nks/css/c2c.en.css"> <link rel="stylesheet" href="/nks/css/shared.en.css"> <link rel="stylesheet" href="/nks/css/notes.en.css"> </head> <body>    <header id="header"> <div class="table width-full 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="notes"><a href="/nks/notes-for-chapters/">Notes</a></span> <span class="chapter"><a href="/nks/chap-2--the-crucial-experiment--notes/">Chapter 2</a></span> <span class="section"><a href="/nks/sect-2-3--why-these-discoveries-were-not-made-before--notes/">Section 3</a></span> <span class="page"><a href="/nks/p880">Page 880</a></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"> <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 selected"> <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="wide-wrapper"> <div id="previous"><a href="/nks/notes-2-3--the-importance-of-explicitness/"><span>‹</span></a></div> <div id="next"><a href="/nks/notes-2-3--my-papers/"><span>›</span></a></div> <div id="content" class="wide-padding"> <div class="ribbon-wrapper top"> </div> <h1 class="main-heading">Notes</h1> <h2 class="chapter-heading"><a href="/nks/p23--how-do-simple-programs-behave/">Chapter 2: <span>The Crucial Experiment</span></a></h2> <h2 class="section-heading"><a href="/nks/p42--why-these-discoveries-were-not-made-before">Section 3: <span>Why These Discoveries Were Not Made Before</span></a></h2> <hr> <div class="text-content"><p class="notehead">My work on cellular automata</p><p class="notetext">I began serious work on cellular automata in the middle of 1981. I had been thinking for some time about how complicated patterns could arise in natural systems—in apparent violation of the Second Law of Thermodynamics. I had been particularly interested in self-gravitating gases where the basic physics seemed clear, but where complex phenomena like galaxy formation seemed to occur. I had also been interested in neural networks, where there had been fairly simple models developed by <a href="/nks/index/names/k-o/?search=McCulloch,+Warren+S." class="names-index-link">Warren McCulloch</a> and <a href="/nks/index/names/p-s/?search=Pitts,+Walter+H." class="names-index-link">Walter Pitts</a> in the 1940s. I came up with cellular automata as an attempt to capture the essential features of a range of systems, from self-gravitating gases to neural networks. I wanted to find models that had a simple structure like the Ising model in statistical mechanics (studied since the 1920s), but which had definite rules for time evolution and could easily be simulated on a computer. Ironically enough, while cellular automata are good for many things, they turn out to be rather unsuitable for modelling either self-gravitating gases or neural networks. (See page <a href="/nks/notes-9-3--self-gravitating-systems/">1021</a>). But by the time I realized this, it was clear that cellular automata were of great interest for many other purposes.</p><p class="notetext">I did my first major computer experiments on cellular automata late in 1981 (see page <a href="/nks/p19--the-personal-story-of-the-science-in-this-book/">19</a>). Two features initially struck me most. First, that starting from random initial conditions, cellular automata could organize themselves to produce complex patterns. And second, that in cases like rule 90 simple initial conditions led to nested or fractal patterns. During the first half of 1982, I worked hard to analyze the behavior of cellular automata using ideas from statistical mechanics, dynamical systems theory and discrete mathematics. And in June 1982, I finished my first paper on cellular automata, entitled "Statistical Mechanics of Cellular Automata". Published in the journal <em>Reviews of Modern Physics</em> in July 1983, this paper already presents in raw form many of the key ideas that led to the development of the science described in this book. It discusses the fact that by not using traditional mathematical equations, simple models can potentially be made to reproduce complex phenomena, and it mentions some of the consequences of viewing models like cellular automata as computational systems. The paper also contained a small picture of rule 30 started from a single black cell. But at the time, I did not study this picture in detail, and I tacitly assumed that whenever I saw randomness it must come from the random initial conditions that I used. (See page <a href="/nks/p112--how-the-discoveries-in-this-chapter-were-made/">112</a>.)</p><p class="notetext">It was some time in the fall of 1981 that I first found out (at a dinner with some then-young MIT computer scientists) that a version of the systems I had invented had been studied before under the name of "cellular automata". (I had been aware of the Game of Life, but its recreational emphasis had put me off studying it.) Knowing the name cellular automata, I was able to track down quite a number of relevant papers from the 1950s and 1960s. But I found that active research on what had been called cellular automata had more or less petered out (with the slight exception of a group at MIT at that time mainly concerned with building special-purpose hardware for 2D cellular automata). By late 1982 preprints of my paper on cellular automata had created quite a stir, and I got involved in organizing a conference held in March 1983 at Los Alamos to bring together many people newly interested in cellular automata with earlier workers in the field.</p><p class="notetext">As part of preparing for that conference, I decided to use the graphics capabilities of the new workstation computer I had just obtained (a very early unit from Sun Microsystems) to investigate in a systematic way the behavior of a large collection of different cellular automata. And after spending several weeks looking at screen after screen of patterns—and trying to analyze their properties—I came to the conclusion that one could identify in the behavior of cellular automata with random initial conditions just four basic classes, each with its own characteristic features (see page <a href="/nks/p231--four-classes-of-behavior/">231</a>).</p><p class="notetext">In 1982 and early 1983, my efforts to analyze cellular automata were mainly based on ideas from discrete mathematics and dynamical systems theory. In the course of 1983, I also began to make serious use of formal language theory and the theory of computation. But for the most part I concentrated on characterizing behavior obtained from all possible initial conditions. And in fact I still vaguely assumed that if simple initial conditions were used, only fairly simple behavior would be obtained. Several of my papers had actually shown quite detailed pictures where this was not the case. I had noticed them, but they had never been among the examples I had studied in depth, partly for the superficial reason that the rules they involved were not symmetrical, or inevitably led to patterns that were otherwise not convenient for display. I do not know exactly what made me start looking more carefully at simple initial conditions, though I believe that I first systematically generated high-resolution pictures of all the <span class="inlinecode clipboard-inline">k = 2</span>, <span class="inlinecode clipboard-inline">r = 1</span> cellular automata as an exercise for an early laserprinter—probably at the beginning of 1984. And I do know that for example on June 1, 1984 I printed out pictures of rule 30, rule 110 and <span class="inlinecode clipboard-inline">k = 2</span>, <span class="inlinecode clipboard-inline">r = 2</span> totalistic code 10 (see <a href="/nks/notes-2-3--code-10/">note below</a>), took them with me on a flight from New York to London, and a few days later was in Sweden talking about randomness in rule 30 and its potential significance.</p><p class="notetext">A month or so later, writing an article for <em>Scientific American</em>—nominally on the subject of software in science and mathematics—led me to think more carefully about basic issues of computation and modelling, and to describe for the first time the idea of computational irreducibility (see page <a href="/nks/p737--computational-irreducibility/">737</a>). In the fall of 1984 I began to investigate some of the implications of what I had discovered about cellular automata for foundational questions in science. And by early 1985 I had written what I consider to be my two most fundamental (if excessively short) papers from the period: one on undecidability and intractability in theoretical physics, and the other on intrinsic randomness generation and the origins of randomness in physical systems.</p><p class="notetext">In the early summer of 1985 I was doing consulting at a startup company called Thinking Machines Corporation, which had developed a massively parallel computer called the Connection Machine that was fairly well suited to cellular automaton simulation. Partly as an application for this computer I then ended up making a detailed study of rule 30 and its randomness—among other things proposing it as a practical random sequence generator and cryptosystem.</p><p class="notetext">I had always thought that cellular automata could be a way to get at foundational questions in thermodynamics and hydrodynamics. And in mid-1985, partly in an attempt to find uses for the Connection Machine, I devised a practical scheme for doing fluid mechanics with cellular automata (see page <a href="/nks/p378--fluid-flow/">378</a>). Then over the course of that winter and the following spring I analyzed the scheme and worked out its correspondence to the traditional continuum approach.</p><p class="notetext">By 1986, however, I felt that I had answered at least the first round of obvious questions about cellular automata, and it increasingly seemed that it would not be easier to go further with the computational tools available. In June 1986 I organized one last conference on cellular automata—then in August 1986 essentially left the field to begin the development of Mathematica.</p><p class="notetext">Over the years, I have come back to look at cellular automata again and again, and every time I have been amazed and delighted by the richness of the phenomena they exhibit. As I argue in this book, a vast range of systems must in the end show the same basic phenomena. But cellular automata—and especially 1D ones—make the phenomena particularly clear, which is why even after investigating all sorts of other systems 1D cellular automata are still the most common examples that I use in this book.</p></div> </div> <div class="ribbon-wrapper stripe"> </div> <footer id="footer" class="wide-padding"> <section id="related-pages" class="width-full max-width-900 center"> <h2 class="display-ib">Pages Related to this Note:</h2> <span> <a href="/nks/p17--the-personal-story-of-the-science-in-this-book/">17</a>, <a href="/nks/p22--the-personal-story-of-the-science-in-this-book/">22</a>, <a href="/nks/p23--how-do-simple-programs-behave/">23</a>, <a href="/nks/p39--the-need-for-a-new-intuition/">39</a>, <a href="/nks/p44--why-these-discoveries-were-not-made-before/">44</a>, <a href="/nks/p50--why-these-discoveries-were-not-made-before/">50</a>, <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/">108</a>, <a href="/nks/p113--how-the-discoveries-in-this-chapter-were-made/">113</a>, <a href="/nks/p231--four-classes-of-behavior/">231</a>, <a href="/nks/p235--four-classes-of-behavior/">235</a>, <a href="/nks/p378--fluid-flow/">378</a>, <a href="/nks/p381--fluid-flow/">381</a>, <a href="/nks/p737--computational-irreducibility/">737</a> </span> <div> <div class="thumbnails cf"> <a href="/nks/p17--the-personal-story-of-the-science-in-this-book/"><img src="/nks/pages/p17--the-personal-story-of-the-science-in-this-book/image-tiny.png" alt="Page 17" title="Page 17" width="70" height="86"></a> <a href="/nks/p22--the-personal-story-of-the-science-in-this-book/"><img src="/nks/pages/p22--the-personal-story-of-the-science-in-this-book/image-tiny.png" alt="Page 22" title="Page 22" width="70" height="86"></a> <a href="/nks/p23--how-do-simple-programs-behave/"><img src="/nks/pages/p23--how-do-simple-programs-behave/image-tiny.png" alt="Page 23" title="Page 23" width="70" height="86"></a> <a href="/nks/p39--the-need-for-a-new-intuition/"><img src="/nks/pages/p39--the-need-for-a-new-intuition/image-tiny.png" alt="Page 39" title="Page 39" width="70" height="86"></a> <a href="/nks/p44--why-these-discoveries-were-not-made-before/"><img src="/nks/pages/p44--why-these-discoveries-were-not-made-before/image-tiny.png" alt="Page 44" title="Page 44" width="70" height="86"></a> <a href="/nks/p50--why-these-discoveries-were-not-made-before/"><img src="/nks/pages/p50--why-these-discoveries-were-not-made-before/image-tiny.png" alt="Page 50" title="Page 50" width="70" height="86"></a> <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p108--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 108" title="Page 108" width="70" height="86"></a> <a href="/nks/p113--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p113--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 113" title="Page 113" width="70" height="86"></a> <a href="/nks/p231--four-classes-of-behavior/"><img src="/nks/pages/p231--four-classes-of-behavior/image-tiny.png" alt="Page 231" title="Page 231" width="70" height="86"></a> <a href="/nks/p235--four-classes-of-behavior/"><img src="/nks/pages/p235--four-classes-of-behavior/image-tiny.png" alt="Page 235" title="Page 235" width="70" height="86"></a> <a href="/nks/p378--fluid-flow/"><img src="/nks/pages/p378--fluid-flow/image-tiny.png" alt="Page 378" title="Page 378" width="70" height="86"></a> <a href="/nks/p381--fluid-flow/"><img src="/nks/pages/p381--fluid-flow/image-tiny.png" alt="Page 381" title="Page 381" width="70" height="86"></a> <a href="/nks/p737--computational-irreducibility/"><img src="/nks/pages/p737--computational-irreducibility/image-tiny.png" alt="Page 737" title="Page 737" width="70" height="86"></a> </div> </div> </section><br> <section id="related-notes" class="width-full max-width-900 center"> <h2>Other Notes From Page <a href="/nks/p880">880</a>:</h2> <div> <div class="thumbnails cf"> </div> <ul class="cols-2 cols-1__600 heirs-width-full__600 show-thumbnails"> <li><span><a href="/nks/notes-2-3--close-approaches-to-core-discoveries/"><span><img src="/nks/img/thumbnails/notes-2-3--close-approaches-to-core-discoveries--textonly.png" alt="" width="76" height="62"></span><span>Close approaches [to core discoveries]</span></a></span></li> <li><span><a href="/nks/notes-2-3--the-importance-of-explicitness/"><span><img src="/nks/img/thumbnails/notes-2-3--the-importance-of-explicitness--textonly.png" alt="" width="76" height="62"></span><span>The importance of explicitness</span></a></span></li> </ul> </div> </section> <section id="image-sources" class="width-full max-width-900 center hide"> <hr class="even"> <h2>Image Source Notebooks:</h2> <div> <ul class="grid cols-2 width-full cols-1__600 heirs-width-full__600"> </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/sect-2-3--why-these-discoveries-were-not-made-before--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 section</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> <span class="pdf-link mini-t"> <span><img src="/nks/img/icon-pdf.png" alt="" width="10" height="13"></span> <span>Show PDF</span> <span class="tooltip"> <span class="close">✖</span> <a href="https://files.wolframcdn.com/pub/www.wolframscience.com/nks/nks-nts-ch2.pdf" class="chapter">Show Chapter Notes</a><br> <a href="https://files.wolframcdn.com/pub/www.wolframscience.com/nks/nks-notes.pdf" class="all">Show All Notes</a> </span> </span> </div> </div> </footer>  <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Book", "name": "A New Kind of Science", "description": "Starting from a collection of simple computer experiments Stephen Wolfram shows how their unexpected results force a whole new way of looking at the operation of our universe. 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>  <footer id="citation"> <p>From Stephen Wolfram: <em>A New Kind of Science</em> [<a href="/nks/citation/">citation</a>]   <a href="https://creativecommons.org/licenses/by-sa/4.0/" target="_blank"> <svg id="icon-cc-by-sa" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 110 32"> <g transform="translate(76 0)"> <path d="M15.7,0.9c4.2,0,7.7,1.5,10.7,4.4c2.9,2.9,4.4,6.4,4.4,10.6c0,4.2-1.4,7.7-4.3,10.6c-3,3-6.6,4.4-10.7,4.4 c-4.1,0-7.6-1.5-10.6-4.4C2.2,23.5,0.7,20,0.7,15.9c0-4.1,1.5-7.6,4.4-10.6C8.1,2.3,11.6,0.9,15.7,0.9z M15.8,3.6 c-3.4,0-6.3,1.2-8.7,3.6c-2.4,2.5-3.7,5.4-3.7,8.7c0,3.3,1.2,6.2,3.6,8.6c2.4,2.4,5.3,3.6,8.7,3.6c3.3,0,6.2-1.2,8.7-3.7 c2.4-2.3,3.5-5.2,3.5-8.6c0-3.4-1.2-6.3-3.6-8.7C22.1,4.8,19.2,3.6,15.8,3.6z M9.1,13.8c0.3-1.8,1-3.3,2.2-4.3 c1.2-1,2.7-1.5,4.4-1.5c2.4,0,4.2,0.8,5.6,2.3s2.1,3.5,2.1,5.8c0,2.3-0.7,4.2-2.2,5.7c-1.4,1.5-3.3,2.3-5.6,2.3 c-1.7,0-3.2-0.5-4.4-1.5c-1.2-1-2-2.5-2.2-4.3h3.8c0.1,1.8,1.2,2.7,3.3,2.7c1.1,0,1.9-0.5,2.5-1.4c0.6-0.9,1-2.1,1-3.6 c0-1.6-0.3-2.8-0.9-3.6c-0.6-0.8-1.4-1.2-2.5-1.2c-2,0-3.1,0.9-3.4,2.7h1.1l-3,3l-3-3L9.1,13.8L9.1,13.8z" fill="currentColor"></path> </g> <g transform="translate(5 0)"> <path d="M59.4,5.2c-2.9-2.9-6.4-4.3-10.7-4.3c-4.1,0-7.7,1.4-10.6,4.3c-2.9,3-4.4,6.6-4.4,10.7s1.5,7.6,4.4,10.6 c3,2.9,6.5,4.4,10.6,4.4c4.1,0,7.7-1.5,10.7-4.5c2.9-2.8,4.3-6.3,4.3-10.5C63.7,11.7,62.3,8.1,59.4,5.2z M57.5,24.5 c-2.5,2.4-5.4,3.7-8.7,3.7c-3.4,0-6.2-1.2-8.7-3.6c-2.4-2.4-3.6-5.3-3.6-8.6c0-3.3,1.2-6.2,3.7-8.7c2.4-2.4,5.2-3.6,8.7-3.6 c3.4,0,6.3,1.2,8.7,3.6c2.4,2.4,3.6,5.3,3.6,8.7C61.1,19.3,59.9,22.2,57.5,24.5z" fill="currentColor"></path> <path d="M46.7,8.3c0-1.4,0.7-2.1,2.1-2.1s2.1,0.7,2.1,2.1c0,1.4-0.7,2.1-2.1,2.1C47.3,10.4,46.7,9.7,46.7,8.3z" fill="currentColor"></path> <path d="M52.8,18.3h-1.7v7.3h-4.7v-7.3h-1.7c0,0,0-4.1,0-6.1c0-0.6,0.4-1,1-1c2,0,4,0,6,0c0.6,0,1,0.4,1,1 C52.8,14.2,52.8,18.3,52.8,18.3z" fill="currentColor"></path> </g> <g id="cc-logo"> <path d="M15.7,0.9c4.2,0,7.8,1.5,10.7,4.4c1.4,1.4,2.5,3,3.2,4.8c0.7,1.8,1.1,3.7,1.1,5.8c0,2.1-0.4,4-1.1,5.8 c-0.7,1.8-1.8,3.4-3.2,4.8c-1.4,1.4-3.1,2.6-5,3.3c-1.8,0.8-3.8,1.2-5.8,1.2c-2,0-3.9-0.4-5.7-1.1c-1.8-0.8-3.5-1.9-4.9-3.3 c-1.4-1.4-2.5-3.1-3.3-4.9c-0.8-1.8-1.1-3.7-1.1-5.7c0-2,0.4-3.9,1.1-5.7c0.8-1.8,1.9-3.5,3.3-4.9C8,2.3,11.6,0.9,15.7,0.9z M15.8,3.6c-3.4,0-6.3,1.2-8.7,3.6C6,8.3,5,9.7,4.4,11.2c-0.6,1.5-1,3.1-0.9,4.7c0,1.6,0.3,3.2,0.9,4.6C5,22,6,23.4,7.1,24.5 c1.1,1.1,2.5,2.1,4,2.7c1.5,0.6,3,0.9,4.7,0.9c1.6,0,3.2-0.3,4.7-0.9c1.5-0.6,2.9-1.6,4.1-2.7c2.3-2.3,3.5-5.2,3.5-8.6 c0-1.7-0.3-3.2-0.9-4.7c-0.6-1.5-1.5-2.8-2.7-4C22.1,4.8,19.2,3.6,15.8,3.6z M15.6,13.4l-2,1c-0.2-0.4-0.5-0.8-0.8-0.9 c-0.3-0.2-0.6-0.3-0.9-0.3c-1.3,0-2,0.9-2,2.7c0,0.8,0.2,1.4,0.5,1.9c0.3,0.5,0.8,0.7,1.5,0.7c0.9,0,1.5-0.4,1.9-1.3l1.8,0.9 c-0.4,0.7-0.9,1.3-1.6,1.7c-0.7,0.4-1.5,0.6-2.3,0.6c-1.3,0-2.4-0.4-3.2-1.2c-0.8-0.8-1.2-2-1.2-3.4c0-1.4,0.4-2.6,1.2-3.4 c0.8-0.8,1.9-1.3,3.1-1.3C13.4,11.2,14.8,11.9,15.6,13.4L15.6,13.4z M24.2,13.4l-2,1c-0.2-0.4-0.5-0.8-0.8-0.9 c-0.3-0.2-0.6-0.3-0.9-0.3c-1.3,0-2,0.9-2,2.7c0,0.8,0.2,1.4,0.5,1.9c0.3,0.5,0.8,0.7,1.5,0.7c0.9,0,1.5-0.4,1.8-1.3l1.9,0.9 c-0.4,0.7-1,1.3-1.7,1.7c-0.7,0.4-1.5,0.6-2.3,0.6c-1.4,0-2.4-0.4-3.3-1.2c-0.8-0.8-1.2-2-1.2-3.4c0-1.4,0.4-2.6,1.2-3.4 c0.8-0.8,1.9-1.3,3.1-1.3C22.1,11.2,23.4,11.9,24.2,13.4L24.2,13.4z" fill="currentColor"></path> </g> </svg></a> <a href="https://en.wikipedia.org/wiki/Wikipedia:Citing_sources" target="_blank"> <svg id="icon-wiki" xmlns="http://www.w3.org/2000/svg" viewBox="101 0 44 32"> <g transform="translate(0 0)"> <path id="wikipedia" d="M129.6,4.5l0.3,0h0.5l0.2,0h0.4l0.2,0l0.5,0h0.2l0.1,0.1v0.7c0,0.2-0.1,0.3-0.3,0.3c-1.1,0-1.3,0.2-1.7,0.7 c-0.2,0.3-0.6,1-1.1,1.7l-3.7,7l-0.1,0.2l4.6,9.4l0.3,0.1l7.2-17.1c0.3-0.7,0.2-1.2-0.1-1.5c-0.3-0.3-0.6-0.4-1.4-0.5l-0.7,0 c-0.1,0-0.2,0-0.2-0.1c-0.1,0-0.1-0.1-0.1-0.2V4.6l0.1-0.1h8.1l0.1,0.1v0.7c0,0.2-0.1,0.3-0.3,0.3c-1.1,0-1.9,0.3-2.4,0.7 c-0.5,0.4-0.9,1-1.2,1.7c0,0-2.2,5-4.4,10.1l-0.4,0.9c-1.6,3.6-3.2,7.2-4.1,9.2c-0.9,1.7-1.7,1.5-2.5,0c-0.6-1.2-1.6-3.3-2.5-5.5 l-0.7-1.4c-0.4-0.8-0.8-1.7-1.1-2.4c-1.5,3.2-3.6,7.4-4.7,9.4c-1,1.8-1.8,1.5-2.5,0c-2.3-5.4-7-15-9.3-20.3c-0.4-1-0.7-1.6-1-1.9 c-0.3-0.2-0.9-0.4-1.8-0.4c-0.3,0-0.5-0.1-0.5-0.3V4.6l0.1-0.1l0.8,0h1.4l1.4,0h0.4l2.5,0h0.3l1.3,0h0.8l0.1,0.1v0.7 c0,0.2-0.1,0.3-0.4,0.3l-0.9,0c-0.8,0-1.2,0.3-1.2,0.7c0,0.2,0.1,0.5,0.3,1c1.8,4.3,7.9,17.3,7.9,17.2l0.2,0.1l3.9-7.9 c-0.5-1.1-0.8-1.7-0.8-1.7L119,9.6c0,0-0.5-1.1-0.7-1.4c-1.2-2.4-1.2-2.5-2.4-2.6c-0.3,0-0.5-0.1-0.5-0.3V4.6l0.1-0.1h1l2.1,0h3.9 l0.2,0.1v0.7c0,0.2-0.1,0.3-0.4,0.3l-0.5,0.1c-1.3,0.1-1.1,0.6-0.2,2.3l2.6,5.3l2.9-5.7c0.5-1,0.4-1.3,0.2-1.5l0,0 c-0.1-0.1-0.5-0.4-1.3-0.4l-0.3,0c-0.1,0-0.2,0-0.2-0.1c-0.1,0-0.1-0.1-0.1-0.2V4.6l0.1-0.1l1.2,0H129.6z" fill="currentColor"></path> </g> </svg></a> </p> </footer>  </div> </main> <script src="/common/js/c2c/1.0/raw.js"></script> <script src="/common/js/clipboard/2.0/clipboard.js"></script> <script src="/nks/js/shared.en.js"></script> <script src="/nks/js/notes.en.js"></script>  <div id ="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "WebSite", "name": "Stephen Wolfram: A New Kind of Science | Online", "description": "The latest on exploring the computational universe, with free online access to Stephen Wolfram's classic 1,200-page breakthrough book, \"A New Kind of Science\".", "keywords": "A New Kind of Science Online, NKS Online, NKSO", "url": "https://www.wolframscience.com/nks/", "image": "https://www.wolframscience.com/nks/img/a-new-kind-of-science-cover.png", "inLanguage": "en", "creator": { "@type": "Organization", "name": "Wolfram" } } </script> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "WebPage", "name": "Note (b) for Why These Discoveries Were Not Made Before: A New Kind of Science | Online by Stephen Wolfram [Page 880]", "description": "My work on cellular automata I began serious work on cellular automata in the middle of 1981. I had been thinking for some... – from A New Kind of Science", "url": "https://www.wolframscience.com/nks/notes-2-3--my-work-on-cellular-automata/", "image": "http://www.wolframscience.com/share.png", "inLanguage": "en" } </script>  </body> </html>

CINXE.COM

Note (b) for Why These Discoveries Were Not Made Before: A New Kind of Science | Online by Stephen Wolfram [Page 880]