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Quantum Phenomena: A New Kind of Science | Online by Stephen Wolfram [Page 537]

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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="chapter"><a href="/nks/chap-9--fundamental-physics/">Chapter 9</a></span> <span class="section"><a href="/nks/chap-9--fundamental-physics/#sect-9-16--quantum-phenomena">Section 16</a></span> <span class="page"><span>Page 537</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"> <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 selected"> <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 selected"> <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" 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Image</a></li> <li class="web-view"><a href="/nks/p537--quantum-phenomena--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/p537--quantum-phenomena/image-xlarge.png" alt="" width="1000" height="1205" class="fill"> </div> <div class="large"> <img src="/nks/pages/p537--quantum-phenomena/image-large.png" alt="" width="850" height="1025" class="fill"> </div> <div class="medium"> <img src="/nks/pages/p537--quantum-phenomena/image-medium.png" alt="" width="750" height="904" class="fill"> </div> <div class="small"> <img src="/nks/pages/p537--quantum-phenomena/image-small.png" alt="" width="650" height="784" class="fill"> </div><div class="text-content wide-padding"><p class="textnoindent">emerges from the same underlying network—or in effect from the structure of space. And indeed even in traditional general relativity one can try avoiding introducing matter explicitly—for example by imagining that everything we call matter is actually made up of pure gravitational energy, or of something like gravitational waves.</p><p class="text">But so far as one can tell, the details of this do not work out—so that at the level of general relativity there is no choice but to introduce matter explicitly. Yet I suspect that this is in effect just a sign of limitations in the Einstein equations and general relativity.</p><p class="text">For while at a large scale these may provide a reasonable description of average behavior in a network, it is almost inevitable that closer to the scale of individual connections they will have to be modified. Yet presumably one can still use the Einstein equations on large scales if one introduces matter with appropriate properties as a way to represent small-scale effects in the network.</p><p class="text">In the <a href="/nks/chap-9--fundamental-physics/#sect-9-14--elementary-particles">previous section</a> I suggested that energy and momentum might in effect be associated with the presence of excess nodes in a network. And this now potentially seems to fit quite well with what we have seen in <a href="/nks/chap-9--fundamental-physics/#sect-9-15--the-phenomenon-of-gravity">this section</a>. For if the underlying rule for a network is going to maintain to a certain approximation the same average number of nodes as flat space, then it follows that wherever there are more nodes corresponding to energy and momentum, this must be balanced by something reducing the number of nodes. But such a reduction is exactly what is needed to correspond to positive curvature of the kind implied by the Einstein equations in the presence of ordinary matter.</p><h2 class="text-section-heading">Quantum Phenomena</h2><p class="text">From our everyday experience with objects that we can see and touch we develop a certain intuition about how things work. But nearly a century ago it became clear that when it comes to things like electrons some of this intuition is no longer correct. Yet there has developed an elaborate mathematical formalism in quantum theory that successfully reproduces much of what is observed. And while some aspects of this</p></div> <div class="ribbon-wrapper stripe"> <div class="ribbon"> <a href="https://www.wolframphysics.org/" target="_blank" class="ribbon-link"> <div class="ribbon-content mini-m"> <span class="icon"><img src="/nks/img/future-icon.svg" height="23" width="23"></span> <span class="title">Wolfram Physics Project <span class="text">— An Approach to the Fundamental Theory of Physics &amp; More</span></span> </div> </a> </div> </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-9-16--history-of-quantum-theory/"><img src="/nks/img/thumbnails/notes-9-16--history-of-quantum-theory--textonly.png" alt="History [of quantum theory]" title="History [of quantum theory]" width="76" height="62"></a> <a href="/nks/notes-9-16--quantum-effects/"><img src="/nks/img/thumbnails/notes-9-16--quantum-effects--textonly.png" alt="Quantum effects" title="Quantum effects" width="76" height="62"></a> <a href="/nks/notes-9-16--reproducing-quantum-phenomena/"><img src="/nks/img/thumbnails/notes-9-16--reproducing-quantum-phenomena--textonly.png" alt="Reproducing quantum phenomena" title="Reproducing quantum phenomena" width="76" height="62"></a> <a href="/nks/notes-9-14--note-for-physicists/"><img src="/nks/img/thumbnails/notes-9-14--note-for-physicists--textonly.png" alt="Note for physicists" title="Note for physicists" width="76" height="62"></a> <a href="/nks/notes-9-15--history-of-gravity-theory/"><img src="/nks/img/thumbnails/notes-9-15--history-of-gravity-theory--textonly.png" alt="History [of gravity theory]" title="History [of gravity theory]" width="76" height="62"></a> <a href="/nks/notes-9-15--differential-geometry/"><img src="/nks/img/thumbnails/notes-9-15--differential-geometry--textonly.png" alt="Differential geometry" title="Differential geometry" width="76" height="62"></a> <a href="/nks/notes-9-15--einstein-equations/"><img src="/nks/img/thumbnails/notes-9-15--einstein-equations--textonly.png" alt="Einstein equations" title="Einstein equations" width="76" height="62"></a> <a href="/nks/notes-9-15--pure-gravity-theory/"><img src="/nks/img/thumbnails/notes-9-15--pure-gravity-theory--textonly.png" alt="Pure gravity [theory]" title="Pure gravity [theory]" width="76" height="62"></a> <a href="/nks/notes-9-15--quantum-gravity/"><img src="/nks/img/thumbnails/notes-9-15--quantum-gravity--textonly.png" alt="Quantum gravity" title="Quantum gravity" width="76" height="62"></a> </div> <ul class="cols-2 cols-1__600 heirs-width-full__600"> <li><span><a href="/nks/notes-9-16--history-of-quantum-theory/"><span><img src="/nks/img/thumbnails/notes-9-16--history-of-quantum-theory--textonly.png" alt="" width="76" height="62"></span><span>History [of quantum theory]</span></a></span></li> <li><span><a href="/nks/notes-9-16--quantum-effects/"><span><img src="/nks/img/thumbnails/notes-9-16--quantum-effects--textonly.png" alt="" width="76" height="62"></span><span>Quantum effects</span></a></span></li> <li><span><a href="/nks/notes-9-16--reproducing-quantum-phenomena/"><span><img src="/nks/img/thumbnails/notes-9-16--reproducing-quantum-phenomena--textonly.png" alt="" width="76" height="62"></span><span>Reproducing quantum phenomena</span></a></span></li> <li><span><a href="/nks/notes-9-14--note-for-physicists/"><span><img src="/nks/img/thumbnails/notes-9-14--note-for-physicists--textonly.png" alt="" width="76" height="62"></span><span>Note for physicists</span></a></span></li> <li><span><a href="/nks/notes-9-15--history-of-gravity-theory/"><span><img src="/nks/img/thumbnails/notes-9-15--history-of-gravity-theory--textonly.png" alt="" width="76" height="62"></span><span>History [of gravity theory]</span></a></span></li> <li><span><a href="/nks/notes-9-15--differential-geometry/"><span><img src="/nks/img/thumbnails/notes-9-15--differential-geometry--textonly.png" alt="" width="76" height="62"></span><span>Differential geometry</span></a></span></li> <li><span><a href="/nks/notes-9-15--einstein-equations/"><span><img src="/nks/img/thumbnails/notes-9-15--einstein-equations--textonly.png" alt="" width="76" height="62"></span><span>Einstein equations</span></a></span></li> <li><span><a href="/nks/notes-9-15--pure-gravity-theory/"><span><img src="/nks/img/thumbnails/notes-9-15--pure-gravity-theory--textonly.png" alt="" width="76" height="62"></span><span>Pure gravity [theory]</span></a></span></li> <li><span><a href="/nks/notes-9-15--quantum-gravity/"><span><img src="/nks/img/thumbnails/notes-9-15--quantum-gravity--textonly.png" alt="" width="76" height="62"></span><span>Quantum gravity</span></a></span></li> <li><span><a href="/nks/notes-9-15--cosmology/"><span><img src="/nks/img/thumbnails/notes-9-15--cosmology--textonly.png" alt="" width="76" height="62"></span><span>Cosmology</span></a></span></li> <li><span><a href="/nks/notes-10-11--reversible-logic/"><span><img src="/nks/img/thumbnails/notes-10-11--reversible-logic--textonly.png" alt="" width="76" height="62"></span><span>Reversible logic</span></a></span></li> <li><span><a href="/nks/notes-12-8--quantum-computers/"><span><img src="/nks/img/thumbnails/notes-12-8--quantum-computers--textonly.png" alt="" width="76" height="62"></span><span>Quantum computers</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 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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> 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