CINXE.COM
MathWorld Terms and Conditions of Use
<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>MathWorld Terms and Conditions of Use</title> <!-- gl-head-includes.en.html --> <link rel="icon" href="/favicon.ico" sizes="48x48"> <link rel="icon" href="/favicon.svg" sizes="any" type="image/svg+xml"> <link rel="apple-touch-icon" href="/apple-touch-icon.png"> <link rel="manifest" href="/manifest.webmanifest"> <meta prefix="og: http://ogp.me/ns#" property="og:image" content="http://www.wolfram.com/share.png"> <meta prefix="og: http://ogp.me/ns#" property="og:image:secure_url" content="https://www.wolfram.com/share.png"> <meta prefix="og: http://ogp.me/ns#" property="og:image:width" content="1200"> <meta prefix="og: http://ogp.me/ns#" property="og:image:height" content="1200"> <link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"> <link rel="stylesheet" href="/common/css/gl-styles.en.css"> <script src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script src="/common/javascript/wal/latest/walLoad.js"></script> <script defer src="/common/js/announcements/script.js"></script> <script src="/common/js/jquery/3.7.1/jquery.min.js"></script> <script src="/common/js/jquery/plugins/migrate/3.5.2/jquery.migrate.min.js"></script> <script src="/common/javascript/gl-head-scripts.en.js"></script> <script src="/common/javascript/analytics.js"></script> <!--/gl-head-includes.en.html --> <link rel="stylesheet" href="/common/css/normalize/current/normalize.min.css"> <link rel="stylesheet" type="text/css" href="/legal/css/legal.css"> <script type="text/javascript" language="Javascript"> $(document).ready(function(){ $('.termsArrow').toggleClass('keyarrowUp'); $('.termsLinks').toggle(); $(".selector").hover(function () { if ($(this).hasClass("sidebarLinks")) { $(this).removeClass("sidebarLinks"); $(this).addClass("sidebarLinksHover"); } }, function () { if ($(this).hasClass("sidebarLinksHover")) { $(this).removeClass("sidebarLinksHover"); $(this).addClass("sidebarLinks"); } } ); $(".selector").click(function () { if ($(this).children().hasClass("keyarrowUp")) { $(this).children().removeClass("keyarrowUp"); $(this).children().addClass("keyarrowDown"); } else { $(this).children().removeClass("keyarrowDown"); $(this).children().addClass("keyarrowUp"); } $(this).siblings().filter(".linkUL").slideToggle("fast"); }); }); </script> </head> <body> <!-- gl-header.en.html --> <div id="gl-outer-wrapper" class="gl-en cf"> <header id="_header"> <div id="_header-grid" class="_page-width"> <h1 id="_header-logo"> <a href="https://www.wolfram.com/?source=nav" data-walid="GUIHeader"><svg width="177" height="37"><use href="#_top-logo"></use><title>WOLFRAM</title></svg></a> </h1> <nav id="_nav-center"> <div id="_nav-products" class="_each-link"> <a href="https://www.wolfram.com/products/?source=nav" data-walid="GUIHeader" class="_label"><span>Products</span></a> <div class="_dropdown"> <ul class="_dropdown-menu"> <li><a href="https://www.wolfram.com/wolfram-one/?source=nav" data-walid="GUIHeader">Wolfram|One</a> <li><a href="https://www.wolfram.com/mathematica/?source=nav" data-walid="GUIHeader">Mathematica</a> <li><a href="https://www.wolfram.com/wolfram-alpha-notebook-edition/?source=nav" data-walid="GUIHeader">Wolfram|Alpha Notebook Edition</a> <li><a href="https://www.wolfram.com/system-modeler/?source=nav" data-walid="GUIHeader">System Modeler</a> <li><a href="https://www.wolfram.com/products/?source=nav" data-walid="GUIHeader">All Products</a> </ul> </div> </div> <div id="_nav-consulting" class="_each-link"> <a href="https://www.wolfram.com/solutions/?source=nav" data-walid="GUIHeader" class="_label"><span>Consulting & Solutions</span></a> <div class="_dropdown"> <ul class="_dropdown-menu"> <li><a href="https://www.wolframconsulting.com/?source=nav" data-walid="GUIHeader">Wolfram Consulting</a> <li><a href="https://www.wolfram.com/solutions/?source=nav" data-walid="GUIHeader">Industry Solutions</a> <li><a href="https://www.wolfram.com/education/?source=nav" data-walid="GUIHeader">Solutions for Education</a> </ul> </div> </div> <div id="_nav-learning" class="_each-link"> <a href="https://www.wolfram.com/resources/?source=nav" data-walid="GUIHeader" class="_label"><span>Learning & Support</span></a> <div class="_dropdown"> <ul class="_dropdown-menu"> <li><a href="https://www.wolfram.com/wolfram-u/?source=nav" data-walid="GUIHeader">Wolfram U Courses</a> <li><a href="https://www.wolfram.com/language/?source=nav#learning-resources" data-walid="GUIHeader">Wolfram Language Resources</a> <li><a href="https://community.wolfram.com/?source=nav" data-walid="GUIHeader">Wolfram Community</a> <li><a href="https://support.wolfram.com/?source=nav" data-walid="GUIHeader">Support FAQs</a> <li><a href="https://www.wolfram.com/support/contact/?source=nav" data-walid="GUIHeader">Contact Support</a> </ul> </div> </div> <div id="_nav-company" class="_each-link"> <a href="https://www.wolfram.com/company/?source=nav" data-walid="GUIHeader" class="_label"><span>Company</span></a> <div class="_dropdown"> <ul class="_dropdown-menu"> <li><a href="https://www.wolfram.com/company/?source=nav" data-walid="GUIHeader">About Wolfram</a> <li><a href="https://www.wolfram.com/company/careers/?source=nav" data-walid="GUIHeader">Careers</a> <li><a href="https://events.wolfram.com/?source=nav" data-walid="GUIHeader">Events</a> <li><a href="https://education.wolfram.com/programs/?source=nav" data-walid="GUIHeader">Educational Programs</a> <li><a href="https://www.wolfram.com/resources/?source=nav#mobile-apps" data-walid="GUIHeader">All Sites and Resources</a> </ul> </div> </div> <div id="_nav-alpha" class="_each-link"> <a href="https://www.wolframalpha.com/?source=nav" data-walid="GUIHeader" class="_label" target="_blank"><span>Wolfram|Alpha<svg width="9" height="9"><use href="#_top-wa"></use></svg></span></a> </div> </nav> <nav id="_nav-right"> <div id="_nav-cloud" class="_each-link"> <a href="https://www.wolframcloud.com/?source=nav" data-walid="GUIHeader" class="_label" target="_blank"><span><svg width="32" height="24"><use href="#_top-cloud"></use><title>Wolfram Cloud</title></svg></span></a> </div> <div id="_nav-user" class="_each-link"> <a href="https://account.wolfram.com/login/oauth2/sign-in/?source=nav" data-walid="GUIHeader" class="_label" target="_blank"><span><svg width="22" height="25"><use href="#_top-user"></use><title>Your Account</title></svg></span></a> <div class="_dropdown"> <ul class="_dropdown-menu"> <li><a href="https://account.wolfram.com/?source=nav" data-walid="GUIHeader" target="_blank">Your Account</a> <li><a href="https://user.wolfram.com/?source=nav" data-walid="GUIHeader" target="_blank">User Portal</a> </ul> </div> </div> <div id="_nav-search" class="_each-link"> <a href="https://search.wolfram.com/?source=nav" data-walid="GUIHeader" class="_label" target="_blank"><span><svg width="24" height="24"><use href="#_top-search"></use><title>Search</title></svg></span></a> <div class="_dropdown"> <form id="_search-form" action="//search.wolfram.com/" method="get" accept-charset="utf-8" class="no-bfc _page-width"> <div id="_search-wrapper"> <label for="_search-input"><svg width="24" height="24"><use href="#_search-magnifier"></use></svg></label> <button type="button" class="close"><svg width="14" height="14"><use href="#_search-x"></use><title>Close</title></svg></button> <span><input type="search" name="query" placeholder="Search across Wolfram sites" autocomplete="off" id="_search-input"></span> <input type="hidden" name="source" value="GUIHeader" class="cf"> </div> </form> </div> </div> </nav> </div> </header> <noscript id="_noscript" class="remove__ready"> <div> <div><svg class="_noscript-warning"><use href="#_noscript-warning"></use></svg></div> <div>Enable JavaScript to interact with content and submit forms on Wolfram websites. <a href="http://www.enable-javascript.com/" target="_blank" data-walid="GUIFooter" class="chevron-after">Learn how</a></div> </div> </noscript> <div style="height: 0; width: 0; position: absolute; visibility: hidden; top: -999999999999px;"> <svg xmlns="http://www.w3.org/2000/svg"> <g id="_top-icons"> <symbol id="_top-logo" viewBox="0 0 183 38.92"> <g> <path fill="#fff" d="M42.62,9.63h4.87l2.25,12.52,3.33-12.52h4.23l3.96,12.79,2.34-12.79h4.6l-3.96,18.83h-5.32l-3.51-11.71-3.6,11.71h-5.59l-3.6-18.83Z"/> <path fill="#fff" d="M183,28.46h-4.87l-2.25-12.61-3.33,12.61h-4.23l-3.96-12.79-2.34,12.79h-4.6l3.96-18.83h5.32l3.51,11.71,3.6-11.71h5.59l3.6,18.83Z"/> <path fill="#fff" d="M78.39,9.27c6.13,0,9.28,3.69,9.28,9.73s-3.24,9.73-9.28,9.73-9.28-3.78-9.28-9.73c0-6.04,3.15-9.73,9.28-9.73Zm0,15.77c2.97,0,4.23-2.88,4.23-6.04s-1.17-6.04-4.23-6.04-4.23,2.7-4.23,6.04c.09,3.15,1.26,6.04,4.23,6.04Z"/> <path fill="#fff" d="M90.46,9.63h4.96v15.04h7.66v3.69h-12.61V9.63Z"/> <path fill="#fff" d="M105.15,9.63h12.43v3.51h-7.48v4.14h7.12v3.51h-7.12v7.57h-4.96s0-18.74,0-18.74Z"/> <path fill="#fff" d="M120.2,9.63h8.83c3.33,0,5.95,1.53,5.95,4.77,0,2.52-1.17,4.14-3.6,4.77h0c1.62,.45,1.71,1.89,2.61,4.14l1.8,5.14h-5.14l-1.08-3.42c-.99-3.24-1.62-4.05-3.51-4.05h-1.08v7.39h-4.96V9.63h.18Zm4.87,7.84h1.26c1.62,0,3.6-.09,3.6-2.16,0-1.71-1.71-2.07-3.6-2.07h-1.26v4.23Z"/> <path fill="#fff" d="M143.35,9.63h6.04l7.12,18.83h-5.41l-1.35-4.05h-6.94l-1.44,4.05h-5.05l7.03-18.83Zm3.06,3.87h0l-2.43,7.39h4.69l-2.25-7.39Z"/> </g> <path fill="#d10" d="M31.81,19.46l5.41-6.04-7.93-1.71,.81-8.02-7.39,3.24L18.65,0l-4.05,6.94L7.12,3.69l.81,8.02L0,13.42l5.41,6.04L0,25.5l7.93,1.71-.81,8.02,7.39-3.24,4.05,6.94,4.05-6.94,7.39,3.24-.81-8.02,7.93-1.71-5.32-6.04Zm1.35,3.87l-3.51-1.17-2.43-3.15,3.15,1.17,2.79,3.15Zm-11.53,7.48l-2.25,3.78v-3.96l2.34-3.33-.09,3.51Zm-5.23-19.73l-3.78-1.35-2.16-2.97,3.96,1.71,1.98,2.61Zm6.49-2.52l3.96-1.71-2.16,2.97-3.78,1.35,1.98-2.61Zm4.78,3.51l-1.89,2.7,.09-4.14,2.25-3.06-.45,4.5Zm-9.01,17.03l-2.97-4.32,2.97-4.05,2.97,4.05-2.97,4.32Zm-5.59-12.34l-.18-5.23,4.96,1.8v5.14l-4.78-1.71Zm6.31-3.51l4.96-1.8-.18,5.32-4.78,1.62v-5.14Zm-9.82-1.17l-.45-4.41,2.25,3.06,.09,4.14-1.89-2.79Zm7.84,7.75l-2.97,4.05-5.05-1.53,3.24-4.14,4.78,1.62Zm-1.89,7.48l2.34,3.33v3.96l-2.25-3.78-.09-3.51Zm4.32-7.48l4.78-1.62,3.24,4.14-5.05,1.53-2.97-4.05Zm14.42-5.41l-3.69,4.14-4.42-1.62,2.61-3.78,5.5,1.26Zm-12.79-6.58l-2.79,3.6-2.79-3.6,2.79-4.86,2.79,4.86ZM2.97,14.32l5.5-1.17,2.61,3.78-4.42,1.62-3.69-4.23Zm3.96,5.77l3.15-1.17-2.52,3.24-3.6,1.26,2.97-3.33Zm-2.61,4.77l3.69-1.26,3.96,1.17-3.24,.99-4.42-.9Zm5.14,2.34l4.51-1.44,.09,4.77-5.14,2.25,.54-5.59Zm13.7,3.33l.09-4.77,4.51,1.44,.54,5.59-5.14-2.25Zm5.32-4.68l-3.33-1.08,3.96-1.17,3.69,1.26-4.32,.99Z"/> </symbol> <symbol id="_top-wa" viewBox="0 0 9 9"> <path fill="currentColor" d="M2.91,0c-.34,0-.56,.23-.56,.56,0,.39,.22,.62,.56,.62H6.99L.17,7.99c-.22,.23-.22,.56,0,.79,.28,.28,.61,.28,.84,.06L7.83,2.03V6.14c0,.34,.22,.56,.56,.56,.39,0,.61-.23,.61-.56V0H2.91Z"/> </symbol> <symbol id="_top-cloud" viewBox="0 0 31 24"> <path fill="currentColor" d="M16.26,24c-1.19,0-2.38-.33-3.58-1.09-.43-.33-.87-.76-1.19-1.09-.11,0-.33,.11-.43,.11-3.36,.44-6.18-1.09-6.61-3.71v-.44c-2.71-.98-4.44-3.38-4.44-6.11,0-3.6,3.03-6.33,6.83-6.66,.22-1.64,1.08-2.84,2.28-3.71C10.51,.21,12.47-.33,14.42,.21c1.08,.22,2.06,.65,3.03,1.2,1.41-.44,3.03-.44,4.66,0,1.84,.76,3.25,2.18,4.12,4.15,2.71,.33,4.77,2.4,4.77,5.13,0,1.2-.43,2.4-1.3,3.27,.43,.98,.76,1.96,.43,3.06-.65,2.95-4.23,4.69-8.13,3.82h-.22c-.98,1.64-2.38,2.62-4.12,2.95-.43,.11-.98,.22-1.41,.22Zm-3.9-4.47l.33,.44c.33,.44,.76,.87,1.19,1.2,.98,.76,2.17,.98,3.36,.76,1.3-.22,2.6-1.2,3.25-2.62l.33-.65,.76,.11c.22,.11,.43,.11,.76,.22,2.82,.65,5.64-.44,5.96-2.29,.22-.87-.22-1.75-.65-2.18l-.43-.76,.65-.44c.76-.65,1.19-1.64,1.19-2.4,0-1.85-1.63-3.27-3.58-3.27h-.76l-.11-.55c-.43-2.07-1.73-3.27-3.25-3.93-1.19-.44-2.6-.33-3.79,.11l-.43,.22-.43-.33c-.76-.65-1.73-1.09-2.71-1.2-1.41-.22-2.93,0-4.01,.87-.87,.76-1.41,1.85-1.63,3.16l-.11,.87h-1.19c-2.93,0-5.31,2.07-5.31,4.8,0,2.07,1.63,4.04,3.9,4.47l.87,.22-.22,.87c-.11,.22-.11,.44,0,.65,.22,1.42,2.28,2.4,4.44,2.07,.33,0,.65-.11,.87-.22l.76-.22Z"/> <path fill="currentColor" d="M15.61,19.92l-1.75-3-3.08,1.34,.31-3.41-3.39-.72,2.26-2.59-2.26-2.48,3.39-.72-.31-3.41,3.08,1.34,1.75-3,1.75,3,3.08-1.34-.31,3.41,3.39,.72-2.26,2.59,2.26,2.59-3.39,.72,.31,3.41-3.08-1.34-1.75,2.9Zm.41-3.72v.93l.62-.93v-.83l-.62,.83Zm-1.54-.1l.62,.93v-.83l-.62-.83v.72Zm3.08-.21l1.75,.72-.21-1.86-1.54-.41v1.55Zm-5.55-1.14l-.21,1.86,1.75-.72v-1.66l-1.54,.52Zm2.47-1.03l.92,1.45,.92-1.34-.92-1.45-.92,1.34Zm4.32,0l.72,.21,1.13-.21-.82-.31-1.03,.31Zm-8.32,0l1.13,.21,.72-.21-1.03-.31-.82,.31Zm5.86-1.97l.92,1.34,1.64-.41-1.03-1.34-1.54,.41Zm-4.21,.83l1.64,.41,.92-1.34-1.64-.62-.92,1.55Zm7.91-.1l.72,.21-.72-.83-.72-.21,.72,.83Zm-9.04-.52l-.72,.72,.82-.21,.62-.72-.72,.21Zm7.91-1.55l1.54,.62,1.23-1.34-1.95-.31-.82,1.03Zm-9.35-.83l1.23,1.34,1.54-.62-.92-1.14-1.85,.41Zm6.47-.52v1.76l1.64-.62v-1.66l-1.64,.52Zm-2.57,1.24l1.64,.62v-1.86l-1.64-.62v1.86Zm-1.34-1.86l.41,.52v-1.03l-.41-.72v1.24Zm6.58-.41v1.03l.41-.62,.1-1.14-.51,.72Zm-4.01-1.24l.92,1.14,.82-1.14-.92-1.66-.82,1.66Zm-1.44,.72l.92,.31-.31-.41-.92-.52,.31,.62Zm4.11-.31l-.31,.41,.92-.31,.51-.52-1.13,.41Z"/> </symbol> <symbol id="_top-user" viewBox="0 0 21 24"> <path fill="currentColor" d="M17.77,14.77c-.58-.23-1.17-.58-1.87-.69-.93-.35-1.4-.46-1.4-.69,.12-.58,.47-1.27,.7-1.73,.82-1.38,1.28-2.88,1.4-4.5,0-1.96-.7-3.81-1.98-5.31C13.57,.7,12.05,.01,10.42,.01c-1.87-.12-3.62,.69-4.79,2.08s-1.75,3.23-1.52,4.96c.12,1.73,.82,3.46,1.75,4.84,.35,.35,.47,.92,.58,1.38,0,.23-.35,.35-1.4,.69l-1.75,.69C1.08,15.81,.38,18.46,.03,20.42c-.12,.92,.12,1.85,.82,2.54,.58,.69,1.52,1.04,2.33,1.04h14.59c.93,0,1.87-.46,2.45-1.15s.93-1.61,.7-2.65c-.35-2.65-1.4-4.61-3.15-5.42Zm.82,6.69c-.12,.23-.35,.35-.58,.35H3.3c-.23,0-.47-.12-.58-.35-.23-.23-.23-.46-.23-.81,.12-1.27,.7-3.34,1.98-4.04,.47-.23,.93-.46,1.52-.58,1.28-.46,2.92-1.04,2.92-3v-.12c-.12-.81-.47-1.73-.93-2.42-.82-1.15-1.28-2.54-1.4-4.04-.12-1.15,.23-2.31,.93-3.11,.82-.92,1.98-1.27,3.15-1.27,.93,0,1.75,.46,2.33,1.15,.82,.92,1.4,2.19,1.4,3.46-.12,1.27-.47,2.54-1.17,3.69-.47,.81-.82,1.61-1.05,2.54v.23c0,1.96,1.75,2.54,3.04,3,.47,.12,1.05,.35,1.52,.58,1.4,.69,1.87,2.77,2.1,3.92,0,.35-.12,.69-.23,.81Z"/> </symbol> <symbol id="_top-search" viewBox="0 0 24 24"> <path fill="currentColor" d="M23.66,21.85l-6.46-6.45c3.23-4.18,2.51-10.15-1.67-13.49C11.34-1.19,5.24-.47,2.01,3.71S-.5,13.85,3.69,17.2c3.47,2.63,8.25,2.63,11.72,0l6.46,6.45c.48,.48,1.2,.48,1.67,0,.6-.6,.6-1.31,.12-1.79ZM2.49,9.56c0-3.94,3.23-7.16,7.17-7.16s7.17,3.22,7.17,7.16-3.23,7.16-7.17,7.16-7.17-3.1-7.17-7.16Z"/> </symbol> </g> <g id="_search-icons"> <symbol id="_search-magnifier" viewBox="0 0 24 24"> <path fill="#b5b5b5" d="M23.66,21.85l-6.46-6.45c3.23-4.18,2.51-10.15-1.67-13.49C11.34-1.19,5.24-.47,2.01,3.71S-.5,13.85,3.69,17.2c3.47,2.63,8.25,2.63,11.72,0l6.46,6.45c.48,.48,1.2,.48,1.67,0,.6-.6,.6-1.31,.12-1.79ZM2.49,9.56c0-3.94,3.23-7.16,7.17-7.16s7.17,3.22,7.17,7.16-3.23,7.16-7.17,7.16-7.17-3.1-7.17-7.16Z"/> </symbol> <symbol id="_search-x" viewBox="0 0 14 14"> <path fill="#5e5e5e" d="M1.9,13.9l-1.4-1.5L5.6,7L0.5,1.7l1.4-1.5L7,5.5l5.1-5.4l1.4,1.5L8.4,7l5.1,5.4l-1.4,1.5L7,8.5L1.9,13.9z"/> </symbol> </g> <g id="_noscript-icons"> <symbol id="_noscript-warning" viewBox="0 0 28.21 25"> <path d="M27.91,21.7L16.01,1.1c-.84-1.47-2.96-1.47-3.81,0L.3,21.7c-.85,1.47,.21,3.3,1.9,3.3H26c1.69,0,2.75-1.83,1.91-3.3ZM15.75,7.44l-.46,9.42h-2.37l-.48-9.42h3.31Zm-1.65,14.25c-1.12,0-1.87-.81-1.87-1.89s.77-1.89,1.87-1.89,1.83,.79,1.85,1.89c0,1.08-.73,1.89-1.85,1.89Z" fill="#f47821"/> <polygon points="15.77 7.47 15.31 16.89 12.94 16.89 12.46 7.47 15.77 7.47" fill="#fff"/> <path d="M14.1,21.69c-1.12,0-1.87-.81-1.87-1.89s.77-1.89,1.87-1.89,1.83,.79,1.85,1.89c0,1.08-.73,1.89-1.85,1.89Z" fill="#fff"/> </symbol> </g> </svg> </div> <div id="gl-inner-wrapper"> <!--/gl-header.en.html --> <div id="wrapper"> <div class="leftcolumn"> <div class="header"> <h1>LEGAL INFORMATION</h1> </div> <h1><em>MathWorld</em> Terms and Conditions of Use</h1> <p>Access to <i>MathWorld</i> is granted on the following terms:</p> <p>Readers are encouraged to use the contents of these pages for education and enjoyment, but these pages may not be <a href="http://mathworld.wolfram.com/about/faq.html#buy">copied</a>, <a href="http://mathworld.wolfram.com/about/faq.html#mirror">mirrored</a>, redistributed, printed, or reproduced in bulk without permission. <a href="http://mathworld.wolfram.com/about/faq.html#copyright">Reproduction for commercial purposes is not permitted</a>, nor is use of robots to create archival copies.</p> <p><a href="http://mathworld.wolfram.com/about/faq.html#permissions">Limited use of material for academic research and teaching is permitted</a>, providing <a href="http://mathworld.wolfram.com/about/faq.html#citation">conventional citation standards</a> are followed. Access to the <i>MathWorld</i> website will be <a href="http://mathworld.wolfram.com/about/faq.html#access">blocked</a> to subnets violating these terms. Indiscriminant use of prefetching results in bandwidth degredation for other <i>MathWorld</i> users and may violate the above proscription against bulk downloading. Since excessive use of prefetching has resulted in extensive imposition of blocking, <i>MathWorld</i> now denies access to prefetching software and browsers. To re-enable access, you can simply turn off prefetching in your browser.</p> <p><a href="http://mathworld.wolfram.com/contact/contribute.html">Contributions are welcome</a>. All contributions will be reviewed for correctness and suitability, and may be edited for content and/or presentation. Material should be specific, concise, verifiable, and preferably useful and interesting. Every attempt will be made to recognize contributors for their assistance and to identify contributed entries on the <i>MathWorld</i> website.</p> <p>When making a submission, unless you request otherwise, you are granting non-exclusive permission to include contributed material (or edited versions thereof) in <i>MathWorld</i>, as well as in any derivative works in any medium, including but not limited to book form or CD-ROM as further indicated in the <a href="http://mathworld.wolfram.com/permission_form.html">permission form</a>. Please see the <a href="http://mathworld.wolfram.com/about/faq.html#contribute">FAQ on contributing</a> for additional details about the formatting and submission of contributed material.</p> <p>Wolfram takes privacy and data protection seriously, and is committed to safeguarding data you share with us. To learn more, please see our <a href="/legal/privacy/wolfram/">Privacy Policy</a>.</p> </div> <div class="rightcolumn"> <div class="sidebarTop"><a href="/legal/">Legal</a></div> <div class="sidebar"> <ul> <li class="selector sidebarLinks">Privacy Policies<span class="privacyArrow keyarrowDown keyarrowUp"></span></li> <li class="linkUL privacyLinks"><div class="legalLinks"> <ul class="sublist"> <li><a href="/legal/privacy/wolfram/">Wolfram</a></li> <li><a href="/legal/privacy/wolfram/">Wolfram|Alpha</a></li> <li><a href="/legal/privacy/data-retention.html">Data Retention</a></li> <li><a href="/legal/privacy/ccpa-addendum/">CCPA Addendum</a></li> </ul> </div></li> </ul> <ul> <li class="selector sidebarLinks">Terms of Use<span class="termsArrow keyarrowDown keyarrowUp"></span></li> <li class="linkUL termsLinks"><div class="legalLinks"> <ul class="sublist"> <li><a href="/legal/terms/wolfram/">Wolfram</a></li> <li><a href="/legal/terms/wolfram-research.html">Wolfram Research</a></li> <li><a href="http://www.wolframalpha.com/termsofuse/">Wolfram|Alpha</a></li> <li><a href="https://www.wolframalpha.com/termsofuse/pro/">Wolfram|Alpha Pro</a></li> <li><a href="https://products.wolframalpha.com/api/commercial-termsofuse">Wolfram|Alpha API</a></li> <li><a href="/legal/terms/wolfram-id-terms.html">Wolfram ID</a></li> <li><a href="/legal/terms/wolfram-engine.html">Wolfram Engine</a></li> <li><a href="/notebooks/licensing-options.html">Wolfram Notebook</a></li> <li><a href="/legal/terms/wolfram-enterprise-private-cloud.html">Wolfram Enterprise Private Cloud</a></li> <li><a href="/legal/terms/wolfram-application-server.html">Wolfram Application Server</a></li> <li><a href="/legal/terms/wolfram-programming-lab.html">Wolfram Programming Lab</a></li> <li><a href="http://community.wolfram.com/tou">Wolfram Community</a></li> <li><a href="/legal/terms/connected-devices-project.html">Wolfram Connected Devices Project</a></li> <li><a href="/legal/terms/wolfram-demonstrations-project.html">Wolfram Demonstrations Project</a></li> <li><a href="/legal/terms/wolfram-blockchain-labs-liveminting/">Liveminting</a></li> <li><a href="/legal/terms/mathworld.html">Mathworld</a></li> <li><a href="/legal/terms/wolfram-tones.html">WolframTones</a></li> <li><a href="/legal/terms/wolfram-science.html">Wolfram Science</a></li> <li><a href="/legal/terms/screensavers-wallpapers.html">Wolfram Research Screensavers and Wallpapers</a></li> <li><a href="/legal/terms/wolfram-id-terms.html">Wolfram User Portal</a></li> <li><a href="/legal/terms/wolfram-chat.html">Wolfram Chat</a></li> </ul> </div></li> </ul> <ul> <li class="selector sidebarLinks">License Agreements<span class="licenseArrow keyarrowDown keyarrowUp"></span></li> <li class="linkUL licenseLinks"><div class="legalLinks"> <ul class="sublist"> <li><a href="/legal/agreements/wolfram-mathematica/">Wolfram Mathematica</a></li> <li><a href="/legal/agreements/wolfram-system-modeler/">Wolfram System Modeler</a></li> <li><a href="/legal/agreements/wolfram-system-modeler-library/">Wolfram System Modeler Library</a></li> <li><a href="/legal/agreements/wolfram-finance-platform/">Wolfram Finance Platform</a></li> <li><a href="/legal/agreements/wolfram-alpha-notebook-edition/">Wolfram|Alpha Notebook Edition</a></li> <li><a href="/legal/agreements/wolfram-mathematica-raspberry-pi/">Wolfram Raspberry Pi Bundle</a></li> <li><a href="/legal/agreements/wolfram-player/">Wolfram Player</a></li> <li><a href="/legal/agreements/wolfram-webmathematica/">webMathematica Professional</a></li> <li><a href="/legal/agreements/wolfram-webmathematica-amateur/">webMathematica Amateur</a></li> <li><a href="/legal/agreements/wolfram-tones.html">WolframTones</a></li> <li><a href="/legal/agreements/mathlink.html">MathLink</a></li> <li><a href="/legal/agreements/jlink.html">J/Link</a></li> <li><a href="/legal/agreements/wolframalpha-toolbar.html">Wolfram|Alpha Toolbar</a></li> <li><a href="/legal/agreements/wolfram-app/">Wolfram Application Pack</a></li> <li><a href="/legal/third-party-license-agreements/">Other</a></li> </ul> </div></li> </ul> <ul> <li class="selector sidebarLinks">Publishing and Submissions<span class="licenseArrow keyarrowDown keyarrowUp"></span></li> <li class="linkUL licenseLinks"><div class="legalLinks"> <ul class="sublist"> <li><a href="/legal/submission-policies/wolfram-challenges.html">Wolfram Challenges</a></li> <li><a href="/legal/submission-policies/wolfram-competitions/">Wolfram Competitions</a></li> <li><a href="https://datarepository.wolframcloud.com/submission-agreement">Wolfram Data Repository</a></li> <li><a href="http://demonstrations.wolfram.com/submissionpolicy.html">Wolfram Demonstrations</a></li> <li><a href="https://datarepository.wolframcloud.com/publisher-agreement">Wolfram Resource System</a></li> </ul> </div></li> </ul> <ul> <li class="selector sidebarLinks">Terms of Service<span class="serviceArrow keyarrowDown keyarrowUp"></span></li> <li class="linkUL serviceLinks"><div class="legalLinks"> <ul class="sublist"> <li><a href="/legal/terms-of-service/enterprise-support.html">Enterprise Support</a></li> <li><a href="/legal/terms-of-service/premier-support.html">Premier Support</a></li> <li><a href="/legal/terms-of-service/personal-license-support.html">Personal License Support</a></li> </ul> </div></li> </ul> <ul> <li class="selector sidebarLinks">Accessibility<span class="accessArrow keyarrowDown keyarrowUp"></span></li> <li class="linkUL accessLinks"><div class="legalLinks"> <ul class="sublist"> <li><a href="/legal/accessibility/wolframalpha">Wolfram|Alpha</a></li> <li><a href="/legal/accessibility/wolfram-mathematica/">Wolfram Mathematica and Wolfram Player</a></li> <li><a href="/legal/accessibility/wolfram-system-modeler/">Wolfram System Modeler</a></li> </ul> </div></li> </ul> </div> <div><img src="/legal/images/sidebarBottom.gif"></div> </div> </div> <!-- gl-footer.en.html --> <div id="gl-footer-difference"><span><!-- --></span></div> <div id="gl-footer" class="cf"> <div id="gl-footer-menu" class="cf"> <div class="gl-col gl-col-1"> <ul> <li>Products</li> <li><a href="https://www.wolfram.com/wolfram-one/?source=footer">Wolfram|One</a></li> <li><a href="https://www.wolfram.com/mathematica/?source=footer">Mathematica</a></li> <li><a href="https://www.wolfram.com/wolfram-alpha-notebook-edition/?source=footer">Wolfram|Alpha Notebook Edition</a></li> <li><a href="http://www.wolframalpha.com/pro/?source=footer">Wolfram|Alpha Pro</a></li> <li><a href="https://www.wolfram.com/products/?source=footer#mobile-apps">Mobile Apps</a></li> <li><a href="https://www.wolfram.com/finance-platform/?source=footer">Finance Platform</a></li> <li><a href="https://www.wolfram.com/system-modeler/?source=footer">System Modeler</a></li> <li><a href="https://www.wolfram.com/player/?source=footer">Wolfram Player</a></li> <li><a href="https://www.wolfram.com/engine/?source=footer">Wolfram Engine</a></li> <li><a href="https://www.wolfram.com/wolframscript/?source=footer">WolframScript</a></li> <li><a href="https://www.wolfram.com/products/workbench/?source=footer">Wolfram Workbench</a></li> <li><a href="https://www.wolfram.com/group-organization-licensing/?source=footer">Volume & Site Licensing</a></li> <li><a href="https://www.wolfram.com/enterprise-private-cloud/?source=footer">Enterprise Private Cloud</a></li> <li><a href="https://www.wolfram.com/application-server/?source=footer">Application Server</a></li> <li class="gl-all"><a href="https://www.wolfram.com/products/?source=footer">View all...</a></li> </ul> </div> <div class="gl-col gl-col-2"> <ul> <li>Services</li> <li><a href="https://www.wolfram.com/technical-consulting/?source=footer">Technical Consulting</a></li> <li><a href="http://www.wolframsolutions.com/?source=footer">Corporate Consulting</a></li> </ul> <ul> <li>For Customers</li> <li><a href="https://www.wolfram.com/get-products-services/?source=footer">Online Store</a></li> <li><a href="https://user.wolfram.com/portal/ProductRegistration?source=footer">Product Registration</a></li> <li><a href="https://user.wolfram.com/portal/login.html?source=footer">Product Downloads</a></li> <li><a href="https://user.wolfram.com/portal/login.html?source=footer">Service Plans Benefits</a></li> <li><a href="http://user.wolfram.com/portal/?source=footer">User Portal</a></li> <li><a href="https://account.wolfram.com/?source=footer">Your Account</a></li> </ul> <ul> <li>Support</li> <li><a href="https://support.wolfram.com/?source=footer">Support FAQ</a></li> <li><a href="https://www.wolfram.com/support/contact/email/?source=footer">Customer Service</a></li> <li><a href="https://www.wolfram.com/support/contact/?source=footer">Contact Support</a></li> </ul> </div> <div class="gl-col gl-col-3"> <ul> <li>Learning</li> <li><a href="http://reference.wolfram.com/language/?source=footer">Wolfram Language Documentation</a></li> <li><a href="https://www.wolfram.com/language/elementary-introduction/?source=footer">Wolfram Language Introductory Book</a></li> <li><a href="https://www.wolfram.com/start/?source=footer">Get Started with Wolfram</a></li> <li><a href="https://www.wolfram.com/language/fast-introduction-for-programmers/?source=footer">Fast Introduction for Programmers</a></li> <li><a href="https://www.wolfram.com/language/fast-introduction-for-math-students/?source=footer">Fast Introduction for Math Students</a></li> <li><a href="https://events.wolfram.com/?source=footer">Webinars & Training</a></li> <li><a href="https://www.wolfram.com/wolfram-u/?source=footer">Wolfram U</a></li> <li><a href="https://education.wolfram.com/programs/?source=footer">Summer Programs</a></li> <li><a href="https://www.wolfram.com/broadcast/?source=footer">Videos</a></li> <li><a href="https://www.wolfram.com/books/?source=footer">Books</a></li> </ul> </div> <div class="gl-col gl-col-4"> <ul> <li>Public Resources</li> <li><a href="http://www.wolframalpha.com/?source=footer">Wolfram|Alpha</a></li> <li><a href="http://demonstrations.wolfram.com/?source=footer">Demonstrations Project</a></li> <li><a href="https://resources.wolframcloud.com/?source=footer">Resource System</a></li> <li><a href="http://devices.wolfram.com/?source=footer">Connected Devices Project</a></li> <li><a href="https://datadrop.wolframcloud.com/?source=footer">Wolfram Data Drop</a></li> <li><a href="https://www.wolfram.com/raspberry-pi/?source=footer">Wolfram + Raspberry Pi</a></li> <li><a href="http://www.wolframscience.com/?source=footer">Wolfram Science</a></li> <li><a href="http://www.computerbasedmath.org/?source=footer">Computer-Based Math</a></li> <li><a href="http://mathworld.wolfram.com/?source=footer" target="_blank">MathWorld</a></li> <li><a href="https://www.wolfram.com/hackathons/?source=footer">Hackathons</a></li> <li><a href="https://www.wolfram.com/resources/computational-thinking/?source=footer">Computational Thinking</a></li> <li class="gl-all"><a href="https://www.wolfram.com/resources/?source=footer">View all...</a></li> </ul> </div> <div class="gl-col gl-col-5"> <ul> <li>Company</li> <li><a href="http://company.wolfram.com/events/?source=footer">Events</a></li> <li><a href="https://www.wolfram.com/company/?source=footer">About Wolfram</a></li> <li><a href="https://www.wolfram.com/company/careers/?source=footer">Careers</a></li> <li><a href="https://www.wolfram.com/company/contact/?source=footer">Contact</a></li> </ul> <ul> <li>Connect</li> <li><a href="http://community.wolfram.com/?source=footer">Wolfram Community</a></li> <li><a href="http://blog.wolfram.com/?source=footer">Wolfram Blog</a></li> <li><a href="https://www.wolfram.com/newsletters/">Newsletter</a></li> <li><a href="https://www.wolfram.com/connect/?source=footer"><i class="gl-icon gl-r7-c1"></i></a></li> </ul> </div> </div><!--/#gl-footer-menu --> <div id="gl-footer-copyright" class="cf"> <div class="gl-col1"> <span>© <span class="gl-year">2024</span> <a href="https://www.wolfram.com/?source=footer">Wolfram</a>. All rights reserved.</span> </div> <div class="gl-col2 cf"> <ul class="cf"> <li><a href="https://www.wolfram.com/legal/?source=footer">Legal</a> & <a href="https://www.wolfram.com/legal/privacy/wolfram/?source=footer">Privacy Policy</a></li> <li><a href="https://www.wolfram.com/site-map/?source=footer">Site Map</a></li> <li><a href="http://www.wolframalpha.com/?source=footer">WolframAlpha.com</a></li> <li><a href="https://www.wolframcloud.com/?source=footer">WolframCloud.com</a></li> </ul> </div> </div><!--/#gl-footer-copyright --> </div><!--/#gl-footer --> <!--/gl-footer.en.html --> <!-- gl-footer-includes.en.html --> </div><!--/#gl-inner-wrapper --> </div><!--/#gl-outer-wrapper --> <span id="gl-footer-lang-message"><i class="gl-icon gl-r7-c2"></i></span> <script src="/common/javascript/gl-footer-scripts.en.js"></script> <div id ="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> <!--/gl-footer-includes.en.html --> </body> </html>