The Growth of Crystals: A New Kind of Science | Online by Stephen Wolfram [Page 369]

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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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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 selected"> <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="wide-wrapper"> <div id="previous"><a href="/nks/p368--issues-of-modelling/"><span>‹</span></a></div> <div id="next"><a href="/nks/p370--the-growth-of-crystals/"><span>›</span></a></div> <div id="content" class="ribbon-top"> <div class="ribbon-wrapper top"> </div> <div class="side-tabs"> <ul> <li class="page-img"><a href="/nks/p369--the-growth-of-crystals/">Page Image</a></li> <li class="web-view"><a href="/nks/p369--the-growth-of-crystals--webview/">eBook View</a></li> <li class="notes"><a href="#related-notes">Notes</a></li> <li class="resources hide"><a href="#resources">Resources</a></li> </ul> </div> <div class="xlarge"> <img src="/nks/pages/p369--the-growth-of-crystals/image-xlarge.png" alt="" width="1000" height="1205" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369a" data-inline-text="" data-area="2000,2420" data-coords="293,1788,1329,1952" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369b" data-inline-text="" data-area="2000,2420" data-coords="292,1978,1329,2146" 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/p369--the-growth-of-crystals/image-large.png" alt="" width="850" height="1025" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369a" data-inline-text="" data-area="850,1025" data-coords="117,755,569,831" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369b" data-inline-text="" data-area="850,1025" data-coords="118,842,566,912" 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/p369--the-growth-of-crystals/image-medium.png" alt="" width="750" height="904" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369a" data-inline-text="" data-area="750,904" data-coords="108,668,498,733" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369b" data-inline-text="" data-area="750,904" data-coords="110,743,499,803" 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/p369--the-growth-of-crystals/image-small.png" alt="" width="650" height="784" class="fill"><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369a" data-inline-text="" data-area="650,784" data-coords="93,579,433,636" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"></div><div class="c2c-template c2c-xlarge" data-hyperlink='null' data-file-name="page0369b" data-inline-text="" data-area="650,784" data-coords="92,643,430,695" 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"><p class="textnoindent">fifty or so years, it has almost never been possible to demonstrate that results obtained from such approximations even correctly reproduce what the original mathematical equations would imply.</p><p class="text">Models based on simple programs, however, suffer from no such problems. For essentially all of them involve only discrete elements which can be handled quite directly on a practical computer. And this means that it becomes straightforward in principle—and often highly efficient in practice—to work out at least the basic consequences of such models.</p><p class="text">Many of the models that I discuss in <a href="/nks/chap-8--implications-for-everyday-systems/">this chapter</a> are actually based on some of the very simplest kinds of programs that I consider anywhere in this book. But as we shall see, even these models appear quite sufficient to capture the behavior of a remarkably wide range of systems from nature and elsewhere—establishing beyond any doubt, I believe, the practical value of thinking in terms of simple programs.</p><h2 class="text-section-heading">The Growth of Crystals</h2><p class="text">At a microscopic level crystals consist of regular arrays of atoms laid out much like the cells in a cellular automaton. A crystal forms when a liquid or gas is cooled below its freezing point. Crystals always start from a seed—often a foreign object such as a grain of dust—and then grow by progressively adding more atoms to their surface.</p><p class="text">As an idealization of this process, one can consider a cellular automaton in which black cells represent regions of solid and white cells represent regions of liquid or gas. If one assumes that any cell which is adjacent to a black cell will itself become black on the next step, then one gets the patterns of growth shown below.</p><hr class="caphr"><p class="capimg-wrapper"><div class="c2c-image" data-class="capimg" data-file-name="page0369a" data-src="/nks/img/inline/page0369a.png" data-width="794" data-height="125" data-alt="Cellular automata with rules" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"><img class="capimg" src="/nks/img/inline/page0369a.png" width="794" height="125"></div><div class="c2c-image" data-class="capimg" data-file-name="page0369b" data-src="/nks/img/inline/page0369b.png" data-width="794" data-height="117" data-alt="Cellular automata with rules 2" data-code="show" data-graphics="show" data-png-1200="show" data-png-900="show" data-png-600="show" data-pdf="show"><img class="capimg" src="/nks/img/inline/page0369b.png" width="794" height="117"></div></p><p class="captext">Cellular automata with rules that specify that a cell should become black if any of its neighbors are already black. The patterns produced have a simple faceted form that reflects directly the structure of the underlying lattice of cells.</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-8-1--consequences-of-models/"><img src="/nks/img/thumbnails/notes-8-1--consequences-of-models--textonly.png" alt="Consequences of models" title="Consequences of models" width="76" height="62"></a> <a href="/nks/notes-8-2--nucleation-of-crystals/"><img src="/nks/img/thumbnails/notes-8-2--nucleation-of-crystals--textonly.png" alt="Nucleation [of crystals]" title="Nucleation [of crystals]" width="76" height="62"></a> <a href="/nks/notes-8-2--implementation-of-hexagonal-cellular-automata/"><img src="/nks/img/thumbnails/notes-8-2--implementation-of-hexagonal-cellular-automata--textonly.png" alt="Implementation [of hexagonal cellular automata]" title="Implementation [of hexagonal cellular automata]" width="76" height="62"></a> <a href="/nks/notes-8-2--history-of-crystal-growth/"><img src="/nks/img/thumbnails/notes-8-2--history-of-crystal-growth--textonly.png" alt="History of crystal growth" title="History of crystal growth" width="76" height="62"></a> <a href="/nks/notes-8-2--models-of-crystal-growth/"><img src="/nks/img/thumbnails/notes-8-2--models-of-crystal-growth--textonly.png" alt="Models of crystal growth" title="Models of crystal growth" width="76" height="62"></a> <a href="/nks/notes-8-2--boiling/"><img src="/nks/img/thumbnails/notes-8-2--boiling--textonly.png" alt="Boiling" title="Boiling" width="76" height="62"></a> <a href="/nks/notes-8-1--history-of-modelling/"><img src="/nks/img/thumbnails/notes-8-1--history-of-modelling--textonly.png" alt="History of modelling" title="History of modelling" width="76" height="62"></a> <a href="/nks/notes-1-1--history-of-programs-and-nature/"><img src="/nks/img/thumbnails/notes-1-1--history-of-programs-and-nature--textonly.png" alt="History of programs and nature" title="History of programs and nature" width="76" height="62"></a> </div> <ul class="cols-2 cols-1__600 heirs-width-full__600"> <li><span><a href="/nks/notes-8-1--consequences-of-models/"><span><img src="/nks/img/thumbnails/notes-8-1--consequences-of-models--textonly.png" alt="" width="76" height="62"></span><span>Consequences of models</span></a></span></li> <li><span><a href="/nks/notes-8-2--nucleation-of-crystals/"><span><img src="/nks/img/thumbnails/notes-8-2--nucleation-of-crystals--textonly.png" alt="" width="76" height="62"></span><span>Nucleation [of crystals]</span></a></span></li> <li><span><a href="/nks/notes-8-2--implementation-of-hexagonal-cellular-automata/"><span><img src="/nks/img/thumbnails/notes-8-2--implementation-of-hexagonal-cellular-automata--textonly.png" alt="" width="76" height="62"></span><span>Implementation [of hexagonal cellular automata]</span></a></span></li> <li><span><a href="/nks/notes-8-2--history-of-crystal-growth/"><span><img src="/nks/img/thumbnails/notes-8-2--history-of-crystal-growth--textonly.png" alt="" width="76" height="62"></span><span>History of crystal growth</span></a></span></li> <li><span><a href="/nks/notes-8-2--models-of-crystal-growth/"><span><img src="/nks/img/thumbnails/notes-8-2--models-of-crystal-growth--textonly.png" alt="" width="76" height="62"></span><span>Models of crystal growth</span></a></span></li> <li><span><a href="/nks/notes-8-2--boiling/"><span><img src="/nks/img/thumbnails/notes-8-2--boiling--textonly.png" alt="" width="76" height="62"></span><span>Boiling</span></a></span></li> <li><span><a href="/nks/notes-8-1--history-of-modelling/"><span><img src="/nks/img/thumbnails/notes-8-1--history-of-modelling--textonly.png" alt="" width="76" height="62"></span><span>History of modelling</span></a></span></li> <li><span><a href="/nks/notes-1-1--history-of-programs-and-nature/"><span><img src="/nks/img/thumbnails/notes-1-1--history-of-programs-and-nature--textonly.png" alt="" width="76" height="62"></span><span>History of programs and nature</span></a></span></li> </ul> </div> </section> <section id="resources"> <section id="exportable-images" class="width-full max-width-900 center hide"> <hr class="even"> <h2>Exportable Images for This Page:</h2> <div> <ul class="grid width-full"> </ul> </div> </section> <section id="related-resources" class="width-full max-width-900 hide"> <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> </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-8-2--the-growth-of-crystals--notes/" class="view-notes mini-t"> <span><img 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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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The Growth of Crystals: A New Kind of Science | Online by Stephen Wolfram [Page 369]