Mechanisms in Programs and Nature: 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 7</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"> <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 selected"> <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-6--starting-from-randomness/"><span>‹</span></a></div> <div id="next"><a href="/nks/chap-8--implications-for-everyday-systems/"><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/p297--universality-of-behavior/">Chapter 7: <span>Mechanisms in Programs and Nature</span></a></h1> <hr> </section> <section class="chapter-content"> <div> <h2 id="sect-7-1--universality-of-behavior" class="section-heading"><a href="/nks/p297--universality-of-behavior/">Section 1: <span>Universality of Behavior</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-7--mechanisms-in-programs-and-nature#sect-7-1--universality-of-behavior</span></span></h2> <div class="pages"> <a href="/nks/p297--universality-of-behavior/"><img src="/nks/pages/p297--universality-of-behavior/image-tiny.png" alt="Page 297" width="70" height="86"></a> <a href="/nks/p298--universality-of-behavior/"><img src="/nks/pages/p298--universality-of-behavior/image-tiny.png" alt="Page 298" width="70" height="86"></a><a href="/nks/sect-7-1--universality-of-behavior--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-7-2--three-mechanisms-for-randomness" class="section-heading"><a href="/nks/p299--three-mechanisms-for-randomness/">Section 2: <span>Three Mechanisms for Randomness</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-7--mechanisms-in-programs-and-nature#sect-7-2--three-mechanisms-for-randomness</span></span></h2> <div class="pages"> <a href="/nks/p299--three-mechanisms-for-randomness/"><img src="/nks/pages/p299--three-mechanisms-for-randomness/image-tiny.png" alt="Page 299" width="70" height="86"></a> <a href="/nks/p300--three-mechanisms-for-randomness/"><img src="/nks/pages/p300--three-mechanisms-for-randomness/image-tiny.png" alt="Page 300" width="70" height="86"></a><a href="/nks/sect-7-2--three-mechanisms-for-randomness--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-7-3--randomness-from-the-environment" class="section-heading"><a href="/nks/p301--randomness-from-the-environment/">Section 3: <span>Randomness from the Environment</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-7--mechanisms-in-programs-and-nature#sect-7-3--randomness-from-the-environment</span></span></h2> <div class="pages"> <a href="/nks/p301--randomness-from-the-environment/"><img src="/nks/pages/p301--randomness-from-the-environment/image-tiny.png" alt="Page 301" width="70" height="86"></a> <a href="/nks/p302--randomness-from-the-environment/"><img src="/nks/pages/p302--randomness-from-the-environment/image-tiny.png" alt="Page 302" width="70" height="86"></a> <a href="/nks/p303--randomness-from-the-environment/"><img src="/nks/pages/p303--randomness-from-the-environment/image-tiny.png" alt="Page 303" width="70" height="86"></a><a href="/nks/sect-7-3--randomness-from-the-environment--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-7-4--chaos-theory-and-randomness-from-initial-conditions" class="section-heading"><a href="/nks/p304--chaos-theory-and-randomness-from-initial-conditions/">Section 4: <span>Chaos Theory and Randomness from Initial Conditions</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-7--mechanisms-in-programs-and-nature#sect-7-4--chaos-theory-and-randomness-from-initial-conditions</span></span></h2> <div class="pages"> <a href="/nks/p304--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p304--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 304" width="70" height="86"></a> <a href="/nks/p305--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p305--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 305" width="70" height="86"></a> <a href="/nks/p306--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p306--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 306" width="70" height="86"></a> <a href="/nks/p307--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p307--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 307" width="70" height="86"></a> <a href="/nks/p308--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p308--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 308" width="70" height="86"></a> <a href="/nks/p309--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p309--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 309" width="70" height="86"></a> <a href="/nks/p310--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p310--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 310" width="70" height="86"></a> <a href="/nks/p311--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p311--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 311" width="70" height="86"></a> <a href="/nks/p312--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p312--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 312" width="70" height="86"></a> <a href="/nks/p313--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p313--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 313" width="70" height="86"></a> <a href="/nks/p314--chaos-theory-and-randomness-from-initial-conditions/"><img src="/nks/pages/p314--chaos-theory-and-randomness-from-initial-conditions/image-tiny.png" alt="Page 314" width="70" height="86"></a><a href="/nks/sect-7-4--chaos-theory-and-randomness-from-initial-conditions--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-7-5--the-intrinsic-generation-of-randomness" class="section-heading"><a href="/nks/p315--the-intrinsic-generation-of-randomness/">Section 5: <span>The Intrinsic Generation of Randomness</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-7--mechanisms-in-programs-and-nature#sect-7-5--the-intrinsic-generation-of-randomness</span></span></h2> <div class="pages"> <a href="/nks/p315--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p315--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 315" width="70" height="86"></a> <a href="/nks/p316--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p316--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 316" width="70" height="86"></a> <a href="/nks/p317--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p317--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 317" width="70" height="86"></a> <a href="/nks/p318--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p318--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 318" width="70" height="86"></a> <a href="/nks/p319--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p319--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 319" width="70" height="86"></a> <a href="/nks/p320--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p320--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 320" width="70" height="86"></a> <a href="/nks/p321--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p321--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 321" width="70" height="86"></a> <a href="/nks/p322--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p322--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 322" width="70" height="86"></a> <a href="/nks/p323--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p323--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 323" width="70" height="86"></a> <a href="/nks/p324--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p324--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 324" width="70" height="86"></a> <a href="/nks/p325--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p325--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 325" width="70" height="86"></a> <a href="/nks/p326--the-intrinsic-generation-of-randomness/"><img src="/nks/pages/p326--the-intrinsic-generation-of-randomness/image-tiny.png" alt="Page 326" width="70" height="86"></a><a href="/nks/sect-7-5--the-intrinsic-generation-of-randomness--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-7-6--the-phenomenon-of-continuity" class="section-heading"><a href="/nks/p327--the-phenomenon-of-continuity/">Section 6: <span>The Phenomenon of Continuity</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-7--mechanisms-in-programs-and-nature#sect-7-6--the-phenomenon-of-continuity</span></span></h2> <div class="pages"> <a href="/nks/p327--the-phenomenon-of-continuity/"><img src="/nks/pages/p327--the-phenomenon-of-continuity/image-tiny.png" alt="Page 327" width="70" height="86"></a> <a href="/nks/p328--the-phenomenon-of-continuity/"><img src="/nks/pages/p328--the-phenomenon-of-continuity/image-tiny.png" alt="Page 328" width="70" height="86"></a> <a href="/nks/p329--the-phenomenon-of-continuity/"><img src="/nks/pages/p329--the-phenomenon-of-continuity/image-tiny.png" alt="Page 329" width="70" height="86"></a> <a href="/nks/p330--the-phenomenon-of-continuity/"><img src="/nks/pages/p330--the-phenomenon-of-continuity/image-tiny.png" alt="Page 330" width="70" height="86"></a> <a href="/nks/p331--the-phenomenon-of-continuity/"><img src="/nks/pages/p331--the-phenomenon-of-continuity/image-tiny.png" alt="Page 331" width="70" height="86"></a> <a href="/nks/p332--the-phenomenon-of-continuity/"><img src="/nks/pages/p332--the-phenomenon-of-continuity/image-tiny.png" alt="Page 332" width="70" height="86"></a> <a href="/nks/p333--the-phenomenon-of-continuity/"><img src="/nks/pages/p333--the-phenomenon-of-continuity/image-tiny.png" alt="Page 333" width="70" height="86"></a> <a href="/nks/p334--the-phenomenon-of-continuity/"><img src="/nks/pages/p334--the-phenomenon-of-continuity/image-tiny.png" alt="Page 334" width="70" height="86"></a> <a href="/nks/p335--the-phenomenon-of-continuity/"><img src="/nks/pages/p335--the-phenomenon-of-continuity/image-tiny.png" alt="Page 335" width="70" height="86"></a> <a href="/nks/p336--the-phenomenon-of-continuity/"><img src="/nks/pages/p336--the-phenomenon-of-continuity/image-tiny.png" alt="Page 336" width="70" height="86"></a><a href="/nks/sect-7-6--the-phenomenon-of-continuity--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-7-7--origins-of-discreteness" class="section-heading"><a href="/nks/p337--origins-of-discreteness/">Section 7: <span>Origins of Discreteness</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-7--mechanisms-in-programs-and-nature#sect-7-7--origins-of-discreteness</span></span></h2> <div class="pages"> <a href="/nks/p337--origins-of-discreteness/"><img src="/nks/pages/p337--origins-of-discreteness/image-tiny.png" alt="Page 337" width="70" height="86"></a> <a href="/nks/p338--origins-of-discreteness/"><img src="/nks/pages/p338--origins-of-discreteness/image-tiny.png" alt="Page 338" width="70" height="86"></a> <a href="/nks/p339--origins-of-discreteness/"><img src="/nks/pages/p339--origins-of-discreteness/image-tiny.png" alt="Page 339" width="70" height="86"></a> <a href="/nks/p340--origins-of-discreteness/"><img src="/nks/pages/p340--origins-of-discreteness/image-tiny.png" alt="Page 340" width="70" height="86"></a> <a href="/nks/p341--origins-of-discreteness/"><img src="/nks/pages/p341--origins-of-discreteness/image-tiny.png" alt="Page 341" width="70" height="86"></a><a href="/nks/sect-7-7--origins-of-discreteness--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-7-8--the-problem-of-satisfying-constraints" class="section-heading"><a href="/nks/p342--the-problem-of-satisfying-constraints/">Section 8: <span>The Problem of Satisfying 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-7--mechanisms-in-programs-and-nature#sect-7-8--the-problem-of-satisfying-constraints</span></span></h2> <div class="pages"> <a href="/nks/p342--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p342--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 342" width="70" height="86"></a> <a href="/nks/p343--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p343--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 343" width="70" height="86"></a> <a href="/nks/p344--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p344--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 344" width="70" height="86"></a> <a href="/nks/p345--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p345--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 345" width="70" height="86"></a> <a href="/nks/p346--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p346--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 346" width="70" height="86"></a> <a href="/nks/p347--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p347--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 347" width="70" height="86"></a> <a href="/nks/p348--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p348--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 348" width="70" height="86"></a> <a href="/nks/p349--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p349--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 349" width="70" height="86"></a> <a href="/nks/p350--the-problem-of-satisfying-constraints/"><img src="/nks/pages/p350--the-problem-of-satisfying-constraints/image-tiny.png" alt="Page 350" width="70" height="86"></a><a href="/nks/sect-7-8--the-problem-of-satisfying-constraints--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-7-9--origins-of-simple-behavior" class="section-heading"><a href="/nks/p351--origins-of-simple-behavior/">Section 9: <span>Origins of Simple Behavior</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline 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href="/nks/p356--origins-of-simple-behavior/"><img src="/nks/pages/p356--origins-of-simple-behavior/image-tiny.png" alt="Page 356" width="70" height="86"></a> <a href="/nks/p357--origins-of-simple-behavior/"><img src="/nks/pages/p357--origins-of-simple-behavior/image-tiny.png" alt="Page 357" width="70" height="86"></a> <a href="/nks/p358--origins-of-simple-behavior/"><img src="/nks/pages/p358--origins-of-simple-behavior/image-tiny.png" alt="Page 358" width="70" height="86"></a> <a href="/nks/p359--origins-of-simple-behavior/"><img src="/nks/pages/p359--origins-of-simple-behavior/image-tiny.png" alt="Page 359" width="70" height="86"></a> <a href="/nks/p360--origins-of-simple-behavior/"><img src="/nks/pages/p360--origins-of-simple-behavior/image-tiny.png" alt="Page 360" width="70" height="86"></a> <a href="/nks/p361--origins-of-simple-behavior/"><img src="/nks/pages/p361--origins-of-simple-behavior/image-tiny.png" alt="Page 361" width="70" height="86"></a><a 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