CINXE.COM

<!doctype html> <html lang="en" class="geometry mathworldcontributors recreationalmathematics"> <head> <title>Cube -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Cube" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content=" The cube, illustrated above together with a wireframe version and a net that can be used for its construction, is the Platonic solid composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron with Maeder index 6 (Maeder 1997), Wenninger index 3 (Wenninger 1989), Coxeter index 18 (Coxeter et al. 1954), and Har'El index 11 (Har'El 1993). It is described by the Schläfli symbol {4,3} and Wythoff symbol 3|24. ..." /> <meta name="description" content=" The cube, illustrated above together with a wireframe version and a net that can be used for its construction, is the Platonic solid composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron with Maeder index 6 (Maeder 1997), Wenninger index 3 (Wenninger 1989), Coxeter index 18 (Coxeter et al. 1954), and Har'El index 11 (Har'El 1993). It is described by the Schläfli symbol {4,3} and Wythoff symbol 3|24. ..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2001-12-30" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-02-04" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-06-01" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-10-26" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-11-22" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-12-08" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-12-14" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-10-06" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2007-10-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2019-01-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2022-04-12" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2023-08-18" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2023-08-25" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2023-09-10" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2024-01-04" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Cubes" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Hexahedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Platonic Solids" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Canonical Polyhedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Isohedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Parallelohedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Rhombohedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Space-Filling Polyhedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Stereohedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Trapezohedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Solid Geometry:Polyhedra:Zonohedra" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Recreational Mathematics:Folding:Origami" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Recreational Mathematics:Mathematical Records" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Recreational Mathematics:Mathematics in the Arts:Mathematics in Architecture" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Recreational Mathematics:Mathematical Art:Polyhedron Nets" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Cantrell" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Herrstrom, Emily" /> <meta name="DC.Subject" scheme="MSC_2000" content="51M04" /> <meta name="DC.Subject" scheme="MSC_2000" content="52B" /> <meta name="DC.Rights" content="Copyright 1999-2025 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement." /> <meta name="DC.Format" scheme="IMT" content="text/html" /> <meta name="DC.Identifier" scheme="URI" content="https://mathworld.wolfram.com/Cube.html" /> <meta name="DC.Language" scheme="RFC3066" content="en" /> <meta name="DC.Publisher" content="Wolfram Research, Inc." /> <meta name="DC.Relation.IsPartOf" scheme="URI" content="https://mathworld.wolfram.com/" /> <meta name="DC.Type" scheme="DCMIType" content="Text" /> <meta name="Last-Modified" content="2024-01-04" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_Cube.png"> <meta property="og:url" content="https://mathworld.wolfram.com/Cube.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Cube -- from Wolfram MathWorld"> <meta property="og:description" content=" The cube, illustrated above together with a wireframe version and a net that can be used for its construction, is the Platonic solid composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron with Maeder index 6 (Maeder 1997), Wenninger index 3 (Wenninger 1989), Coxeter index 18 (Coxeter et al. 1954), and Har'El index 11 (Har'El 1993). It is described by the Schläfli symbol {4,3} and Wythoff symbol 3|24. ..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Cube -- from Wolfram MathWorld"> <meta name="twitter:description" content=" The cube, illustrated above together with a wireframe version and a net that can be used for its construction, is the Platonic solid composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron with Maeder index 6 (Maeder 1997), Wenninger index 3 (Wenninger 1989), Coxeter index 18 (Coxeter et al. 1954), and Har'El index 11 (Har'El 1993). It is described by the Schläfli symbol {4,3} and Wythoff symbol 3|24. ..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_Cube.png"> <link rel="canonical" href="https://mathworld.wolfram.com/Cube.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="topics"> <main id="entry"> <div class="wrapper"> <section id="container"> <header class="text-align-c"> <div id="header-dropdown-menu"> <img src="/images/header/menu-icon.png" width="18" height="12" id="menu-icon"> <span class="display-n__600">TOPICS</span> </div> <svg version="1.1" id="logo" width="490" height="30" class="hide__600" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 480.7 30" style="enable-background:new 0 0 480.7 30;" xml:space="preserve"> <g> <g> <g> <polygon fill="#0095AA" points="144.2,17.1 137.8,4 133.2,4 133.2,28.5 136.5,28.5 136.5,8.6 143,21.8 143,21.9 145.2,21.9 151.7,8.6 151.7,28.5 155,28.5 155,4 150.6,4"/> <path fill="#0095AA" d="M170.8,10.8c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.3,0.8c-1.2,0.6-2.1,1.4-2.6,2.4l0,0.1l2.6,1.6l0-0.1 c0.3-0.5,0.8-1,1.4-1.4c0.7-0.4,1.6-0.6,2.8-0.6c1.4,0,2.5,0.3,3.2,0.8c0.8,0.5,1.1,1.4,1.1,2.8v0.8l-4.5,0.4 c-1.8,0.2-3.2,0.6-4.1,1.1c-1,0.5-1.7,1.2-2.2,2c-0.5,0.8-0.8,1.9-0.8,3.2c0,1.7,0.5,3.1,1.5,4.2c1,1.1,2.4,1.6,4.1,1.6 c1.2,0,2.3-0.3,3.3-0.8c0.9-0.5,1.8-1.2,2.7-1.9v2.2h3.2V15.9C172.6,13.5,172,11.8,170.8,10.8z M169.4,19.2v4.4 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6c-0.4,0.2-0.9,0.2-1.3,0.2c-1.1,0-1.9-0.2-2.4-0.7 c-0.5-0.5-0.8-1.2-0.8-2.3c0-1.1,0.4-2,1.2-2.6c0.8-0.6,2.2-1,4.2-1.2L169.4,19.2z"/> <path fill="#0095AA" d="M184.9,25.9c-0.7,0.3-1.6,0.5-2.4,0.5c-1,0-1.6-0.3-1.9-0.9c-0.3-0.6-0.4-1.7-0.4-3.2v-9.7h4.7V9.9h-4.7V4.7 h-3.2v5.1h-2.6v2.7h2.6v10.3c0,2.1,0.4,3.7,1.2,4.7c0.8,1,2.2,1.5,4.1,1.5c1.2,0,2.3-0.2,3.5-0.6l0.1,0L184.9,25.9L184.9,25.9z" /> <path fill="#0095AA" d="M198.1,9.3c-1.2,0-2.3,0.3-3.5,1c-1.1,0.6-2,1.3-2.9,2.1V2.1h-3.2v26.4h3.2V15.4c0.9-0.8,1.9-1.5,2.9-2.1 c1-0.6,1.9-0.9,2.7-0.9c0.8,0,1.3,0.1,1.7,0.4c0.4,0.3,0.7,0.7,0.8,1.2c0.2,0.6,0.3,1.7,0.3,3.5v11h3.2V16.1 c0-2.4-0.4-4.2-1.3-5.2C201.2,9.9,199.9,9.3,198.1,9.3z"/> <polygon fill="#0095AA" points="225.2,23.5 220.6,4 220.6,4 216.8,4 212.2,23.3 207.8,4 207.8,4 204.3,4 210.1,28.4 210.1,28.5 214.1,28.5 218.6,9.3 223.1,28.4 223.2,28.5 227.2,28.5 233.1,4 229.7,4"/> <path fill="#0095AA" d="M264.2,9.6c-1.2,0-2.2,0.3-3.2,1c-0.9,0.6-1.7,1.3-2.5,2.1V9.9h-3.2v18.7h3.2V15.7c1.1-1.1,2.1-1.8,2.9-2.3 c0.8-0.4,1.7-0.6,2.6-0.6c0.6,0,1.1,0.1,1.4,0.2l0.1,0l0.7-3.1l-0.1,0C265.6,9.7,264.9,9.6,264.2,9.6z"/> <rect x="269" y="2.1" fill="#0095AA" width="3.2" height="26.4"/> <path fill="#0095AA" d="M287.6,2.1v9.2c-1.5-1.3-3.2-1.9-5-1.9c-2.3,0-4.2,0.9-5.5,2.7c-1.3,1.8-2,4.3-2,7.3c0,3.1,0.6,5.5,1.8,7.2 c1.2,1.7,3,2.5,5.2,2.5c1.1,0,2.2-0.2,3.2-0.7c0.9-0.4,1.7-0.9,2.4-1.6v1.7h3.2V2.1H287.6z M287.6,14.2v9.7 c-0.6,0.6-1.4,1.1-2.2,1.6c-0.9,0.5-1.8,0.7-2.7,0.7c-2.9,0-4.3-2.3-4.3-6.9c0-2.4,0.4-4.2,1.3-5.4c0.9-1.1,2-1.7,3.3-1.7 C284.5,12.2,286,12.9,287.6,14.2z"/> </g> <g> <polygon fill="#666666" points="29,3.3 25.4,3.3 21,22.5 16.5,3.4 16.5,3.3 12.5,3.3 8,22.4 3.7,3.3 0,3.3 5.8,28.1 5.8,28.2 10,28.2 14.4,9.3 18.8,28.1 18.8,28.2 23.1,28.2 29,3.5"/> <path fill="#666666" d="M37,8.7c-2.7,0-4.8,0.8-6.2,2.5c-1.4,1.7-2.1,4.2-2.1,7.6c0,3.3,0.7,5.8,2.1,7.5c1.4,1.7,3.5,2.5,6.2,2.5 c2.7,0,4.8-0.9,6.2-2.6c1.4-1.7,2.1-4.2,2.1-7.5c0-3.3-0.7-5.8-2.1-7.5C41.8,9.5,39.7,8.7,37,8.7z M40.5,24 c-0.8,1.1-1.9,1.7-3.6,1.7c-1.7,0-2.9-0.5-3.7-1.7c-0.8-1.1-1.2-2.9-1.2-5.3c0-2.5,0.4-4.3,1.2-5.4c0.8-1.1,2-1.6,3.6-1.6 c1.6,0,2.8,0.5,3.5,1.6c0.8,1.1,1.2,2.9,1.2,5.4C41.7,21.2,41.3,22.9,40.5,24z"/> <rect x="48.7" y="1.4" fill="#666666" width="3.4" height="26.8"/> <path fill="#666666" d="M67,1.6c-1.1-0.5-2.3-0.7-3.6-0.7c-2.2,0-3.7,0.6-4.6,1.7c-0.9,1.1-1.3,2.8-1.3,5.1v1.6h-2.7v3h2.7v16.1h3.4 V12.2h4.1v-3h-4.1V7.6c0-1.5,0.2-2.5,0.6-3c0.4-0.5,1-0.7,2-0.7c0.4,0,0.8,0.1,1.3,0.2c0.5,0.1,0.9,0.3,1.2,0.4l0.2,0.1l1-3 L67,1.6z"/> <path fill="#666666" d="M76.3,8.9c-1.2,0-2.3,0.4-3.2,1c-0.8,0.6-1.6,1.2-2.3,2V9.2h-3.4v19h3.4v-13c1.1-1.1,2-1.8,2.8-2.2 c0.8-0.4,1.7-0.6,2.5-0.6c0.6,0,1,0.1,1.3,0.2l0.2,0.1l0.7-3.3l-0.1-0.1C77.7,9.1,77.1,8.9,76.3,8.9z"/> <path fill="#666666" d="M92.3,10.2c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.4,0.8c-1.2,0.6-2.1,1.4-2.7,2.5l-0.1,0.2l2.8,1.7 l0.1-0.2c0.3-0.5,0.7-0.9,1.4-1.3c0.6-0.4,1.6-0.6,2.8-0.6c1.3,0,2.4,0.3,3.2,0.7c0.7,0.5,1.1,1.4,1.1,2.8v0.7l-4.4,0.4 c-1.8,0.2-3.2,0.6-4.2,1.1c-1,0.5-1.7,1.2-2.2,2.1c-0.5,0.9-0.8,2-0.8,3.3c0,1.7,0.5,3.2,1.5,4.3c1,1.1,2.4,1.7,4.2,1.7 c1.2,0,2.3-0.3,3.3-0.8c0.8-0.5,1.7-1.1,2.5-1.8v2.1h3.4V15.4C94.1,12.9,93.5,11.2,92.3,10.2z M85.8,25.6c-1,0-1.8-0.2-2.3-0.7 c-0.5-0.4-0.8-1.2-0.8-2.2c0-1.1,0.4-1.9,1.1-2.5c0.8-0.6,2.2-1,4.1-1.1l2.7-0.3V23c-0.4,0.3-0.7,0.6-1.1,0.9 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6C86.7,25.5,86.2,25.6,85.8,25.6z"/> <path fill="#666666" d="M122,11.6c-0.3-0.9-0.9-1.7-1.6-2.2c-0.7-0.5-1.7-0.8-2.8-0.8c-1.3,0-2.4,0.4-3.4,1.1 c-0.9,0.6-1.8,1.4-2.7,2.3c-0.3-1.1-0.8-2-1.5-2.5c-0.8-0.6-1.8-0.9-3.1-0.9c-1.2,0-2.3,0.3-3.3,1c-0.8,0.6-1.6,1.2-2.4,2V9.2 h-3.4v19h3.4V15c0.9-0.9,1.8-1.7,2.7-2.2c0.8-0.5,1.6-0.8,2.2-0.8c0.7,0,1.2,0.1,1.5,0.3c0.3,0.2,0.5,0.6,0.6,1.2 c0.1,0.6,0.2,1.8,0.2,3.5v11.2h3.4V15c1.2-1.1,2.2-1.9,2.9-2.3c0.7-0.4,1.4-0.6,2-0.6c0.7,0,1.2,0.1,1.5,0.3 c0.3,0.2,0.5,0.6,0.7,1.1c0.1,0.6,0.2,1.8,0.2,3.6v11.2h3.4V15.6C122.5,13.9,122.3,12.5,122,11.6z"/> </g> <path fill="#0095AA" d="M242.6,8.3c-5.8,0-10.5,4.7-10.5,10.5s4.7,10.5,10.5,10.5c5.8,0,10.5-4.7,10.5-10.5S248.4,8.3,242.6,8.3z M235.7,23.9c0-0.5,0.1-1,0.2-1.6c1.1,0.9,2.3,1.7,3.7,2.4l-0.6,1.4C237.8,25.5,236.6,24.7,235.7,23.9z M238.8,26.5l-0.2,0.5 c-0.9-0.4-1.8-1-2.5-1.6c-0.2-0.2-0.3-0.5-0.3-0.8C236.6,25.4,237.7,26,238.8,26.5z M235.1,23.4c-0.8-0.8-1.4-1.6-1.6-2.4 c-0.1-0.5-0.1-1.1-0.1-1.6c0.4,0.8,1.1,1.7,2,2.5C235.3,22.4,235.2,23,235.1,23.4z M235.2,24.6L235.2,24.6 c-0.3-0.4-0.6-0.8-0.8-1.2c0.2,0.2,0.5,0.5,0.7,0.7C235.2,24.3,235.2,24.4,235.2,24.6z M233.4,17.5c0-0.1,0-0.1,0-0.2 c0.1-0.7,0.4-1.5,0.7-2.2c0-0.1,0-0.1,0.1-0.2c0,0.3,0.1,0.6,0.2,0.9c-0.3,0.4-0.5,0.8-0.7,1.2C233.5,17.1,233.5,17.3,233.4,17.5z M233.6,18.5c0.1-0.4,0.2-0.8,0.4-1.2c0.1-0.3,0.3-0.6,0.5-0.9c0.4,0.9,1.1,1.9,2.1,2.8c-0.2,0.4-0.4,0.8-0.6,1.2 c-0.1,0.3-0.3,0.6-0.4,0.9C234.6,20.4,234,19.4,233.6,18.5z M250.7,15.2c-0.1,0.4-0.3,0.8-0.8,1c-0.3-0.8-0.8-1.6-1.3-2.4 c0.2-0.1,0.3-0.3,0.4-0.5C249.7,14,250.3,14.6,250.7,15.2z M249.2,12.4c0.4,0.2,0.7,0.5,1,0.8c0.3,0.4,0.4,0.7,0.5,1 c0,0,0,0.1,0,0.1c-0.4-0.5-0.9-1-1.4-1.5C249.2,12.7,249.2,12.6,249.2,12.4C249.2,12.4,249.2,12.4,249.2,12.4z M251.8,17.8 c-0.1,0.6-0.5,1.1-1.1,1.5c-0.1-0.8-0.3-1.7-0.6-2.5c0.4-0.2,0.7-0.6,0.9-0.9C251.4,16.5,251.7,17.1,251.8,17.8z M251.3,15.2 C251.3,15.2,251.3,15.2,251.3,15.2C251.3,15.2,251.3,15.2,251.3,15.2C251.3,15.2,251.3,15.2,251.3,15.2z M239.2,16.5 c0.6-0.7,1.2-1.3,1.8-1.9c0.7,0.6,1.5,1.1,2.5,1.6l-1.1,2.4C241.2,18,240.1,17.3,239.2,16.5z M242.2,19.1l-1.1,2.4 c-1.4-0.7-2.7-1.5-3.7-2.4c0.4-0.7,0.9-1.4,1.5-2.1C239.8,17.8,241,18.5,242.2,19.1z M244,16.4c0.8,0.3,1.5,0.6,2.3,0.7 c0.2,0,0.4,0.1,0.6,0.1c0,0.9-0.1,1.8-0.2,2.6c-0.3,0-0.6-0.1-0.9-0.1c-1-0.2-2-0.5-3-0.9C243.3,18,243.7,17.2,244,16.4z M246.7,14.2c0.1,0.8,0.2,1.6,0.3,2.4c-0.2,0-0.4-0.1-0.5-0.1c-0.7-0.1-1.4-0.4-2.1-0.7c0.3-0.7,0.6-1.5,0.9-2.1 c0.4,0.2,0.8,0.3,1.2,0.4C246.5,14.2,246.6,14.2,246.7,14.2z M246.6,12.2c0.5,0.4,0.9,0.9,1.3,1.4c-0.2,0.1-0.5,0.1-0.8,0.1 C247,13.1,246.8,12.6,246.6,12.2z M245,13.1c-0.3-0.2-0.7-0.4-1-0.6c0.5-0.3,1.1-0.6,1.6-0.7L245,13.1z M244.7,13.6l-1,2.1 c-0.8-0.4-1.6-0.9-2.3-1.4c0.7-0.6,1.3-1,2-1.5C243.9,13.1,244.3,13.4,244.7,13.6z M241.1,13.9c-0.6-0.6-1-1.1-1.2-1.7 c0.8-0.3,1.7-0.5,2.6-0.7c0.1,0.3,0.3,0.6,0.6,0.9C242.5,12.8,241.8,13.3,241.1,13.9z M242.3,11c-0.9,0.1-1.8,0.3-2.6,0.6 c-0.1-0.5,0-0.9,0.2-1.3c0.8-0.1,1.6-0.1,2.4,0C242.3,10.6,242.3,10.8,242.3,11z M240.7,14.3c-0.7,0.6-1.3,1.2-1.9,1.9 c-0.8-0.8-1.4-1.6-1.7-2.4c0.7-0.5,1.5-0.9,2.3-1.3C239.6,13,240.1,13.7,240.7,14.3z M239.2,11.9c-0.8,0.3-1.6,0.8-2.3,1.2 c-0.2-0.7,0-1.4,0.4-1.9c0.6-0.3,1.3-0.6,2-0.8C239.1,10.9,239.1,11.4,239.2,11.9z M238.4,16.6c-0.6,0.7-1.1,1.4-1.5,2.1 c-1-0.9-1.7-1.9-2-2.9c0.5-0.6,1-1.2,1.7-1.8C236.9,14.9,237.6,15.8,238.4,16.6z M237,19.6c1.1,0.9,2.4,1.8,3.8,2.4l-1,2.1 c-1.4-0.7-2.7-1.5-3.8-2.4c0.1-0.3,0.3-0.7,0.4-1.1C236.7,20.3,236.8,20,237,19.6z M240.2,24.9c1.2,0.5,2.4,0.9,3.5,1.1 c0.3,0.1,0.6,0.1,0.9,0.1c-0.3,0.5-0.7,0.9-1.1,1.2c-0.3,0-0.7-0.1-1-0.1c-1-0.2-2-0.5-3-0.9C239.7,25.9,239.9,25.5,240.2,24.9z M243.7,27.9c0.3,0,0.6,0,0.9,0c-0.4,0.1-0.9,0.2-1.3,0.2C243.5,28.1,243.6,28,243.7,27.9z M244.3,27.4c0.3-0.3,0.7-0.7,1-1.2 c1.1,0.1,2.2,0,3-0.2c-0.4,0.4-0.8,0.7-1.2,1C246.3,27.3,245.4,27.5,244.3,27.4z M244.9,25.7c-0.4,0-0.7-0.1-1.1-0.2 c-1.1-0.2-2.3-0.6-3.4-1.1c0.1-0.3,0.3-0.6,0.5-1l0.5-1.1c1.2,0.5,2.4,0.9,3.5,1.1c0.3,0.1,0.7,0.1,1,0.2 c-0.1,0.4-0.3,0.7-0.4,1.1C245.3,25,245.1,25.3,244.9,25.7z M245.1,22.8c-1.1-0.2-2.3-0.6-3.4-1.1c0.4-0.8,0.7-1.6,1.1-2.4 c1,0.4,2.1,0.8,3.1,1c0.3,0.1,0.6,0.1,0.9,0.1c-0.1,0.9-0.3,1.7-0.6,2.5C245.8,22.9,245.4,22.9,245.1,22.8z M247.3,20.5 c1.1,0.1,2.2,0,3-0.4c0,0.8,0,1.7-0.2,2.4c-0.9,0.4-2,0.5-3.4,0.5C247,22.2,247.1,21.4,247.3,20.5z M247.4,19.9 c0.1-0.9,0.2-1.8,0.1-2.6c0.8,0,1.6,0,2.2-0.3c0.3,0.8,0.5,1.7,0.5,2.5C249.5,19.9,248.5,20,247.4,19.9z M249.5,16.5 c-0.5,0.2-1.2,0.3-2,0.2c0-0.9-0.1-1.7-0.3-2.5c0.4,0,0.7,0,1-0.1C248.8,14.8,249.2,15.6,249.5,16.5z M248.4,13.4 c-0.3-0.4-0.7-0.8-1.1-1.2c0.5,0.3,0.9,0.5,1.3,0.9C248.6,13.2,248.5,13.3,248.4,13.4z M247.4,11.7c0.4,0.1,0.7,0.2,1,0.3 c0.1,0.2,0.2,0.3,0.2,0.5C248.2,12.2,247.8,11.9,247.4,11.7z M246.7,11.1c0.2-0.1,0.4-0.2,0.5-0.2c0.2,0.1,0.3,0.2,0.5,0.4 C247.4,11.2,247.1,11.2,246.7,11.1z M246.2,10.9c0-0.2-0.1-0.5-0.1-0.7c0.2,0.1,0.5,0.2,0.7,0.3C246.6,10.7,246.4,10.8,246.2,10.9 z M246.6,13.6c0,0-0.1,0-0.1,0c-0.3-0.1-0.7-0.2-1-0.3c0.2-0.5,0.4-1,0.6-1.4C246.2,12.4,246.4,13,246.6,13.6z M244.8,10 c0.1,0,0.2,0,0.3,0.1c0.1,0,0.2,0,0.3,0.1c0.1,0.1,0.2,0.3,0.2,0.6C245.4,10.4,245.1,10.2,244.8,10z M245,11.4 c-0.5,0.2-0.9,0.4-1.4,0.7c-0.2-0.2-0.4-0.5-0.5-0.7C243.8,11.4,244.4,11.4,245,11.4z M244.6,11c-0.6,0-1.1,0-1.7,0 c0-0.2,0-0.3,0.1-0.5C243.5,10.6,244.1,10.8,244.6,11z M243.4,10.1c0.1-0.1,0.3-0.1,0.4-0.1c0.3,0.1,0.6,0.3,0.9,0.5 C244.3,10.4,243.9,10.2,243.4,10.1z M242.7,10c-0.5-0.1-1-0.1-1.5-0.1c-0.2,0-0.4,0-0.5,0c0.3-0.2,0.7-0.3,1.1-0.4 c0.5,0,0.9,0,1.4,0.2C243,9.8,242.8,9.9,242.7,10z M236.4,13.5c-0.6,0.5-1.2,1.1-1.7,1.7c-0.1-0.6-0.1-1.1,0.1-1.5 c0.4-0.7,1-1.3,1.6-1.8C236.2,12.4,236.2,12.9,236.4,13.5z M239.3,26.8c1,0.4,2,0.8,3.1,0.9c0.2,0,0.3,0.1,0.5,0.1 c-0.3,0.1-0.5,0.2-0.8,0.2c-0.3,0-0.5-0.1-0.8-0.1c-0.7-0.1-1.5-0.4-2.2-0.7C239.2,27.1,239.2,27,239.3,26.8z M245.6,25.7 c0.2-0.3,0.3-0.6,0.5-0.9c0.2-0.4,0.3-0.8,0.5-1.2c1.3,0.1,2.4,0,3.4-0.4c-0.1,0.3-0.2,0.6-0.3,0.9c-0.2,0.4-0.4,0.8-0.6,1.1 C248,25.6,246.9,25.8,245.6,25.7z M250,24.3c0.2-0.4,0.3-0.8,0.5-1.3c0.2-0.1,0.5-0.3,0.7-0.4c-0.3,0.7-0.7,1.3-1.1,1.9 c-0.1,0.1-0.1,0.1-0.2,0.2C249.9,24.5,250,24.4,250,24.3z M250.7,22.3c0.1-0.8,0.2-1.5,0.1-2.3c0.5-0.3,0.9-0.6,1.2-1 c0,0.1,0,0.2,0,0.3c0,0.6-0.1,1.3-0.3,1.8C251.5,21.5,251.2,21.9,250.7,22.3z"/> </g> <g> <path fill="#0095AA" d="M297.9,7.6h4.5v0.8h-3.5V11h2.9v0.8h-2.9v3.5h-1V7.6z"/> <path fill="#0095AA" d="M303.7,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4H308l-1.9-3.3h-1.4v3.3h-1V7.6z M306,11.2 c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H306z"/> <path fill="#0095AA" d="M309.9,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C311.2,15.5,309.9,13.9,309.9,11.4z M315.5,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C314.6,14.6,315.5,13.3,315.5,11.4z"/> <path fill="#0095AA" d="M318.1,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4H321l-1.4-4L319,8.7h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M330.7,8.4h-2.3V7.6h5.7v0.8h-2.3v6.9h-1V8.4z"/> <path fill="#0095AA" d="M335.4,7.6h1v3.2h3.6V7.6h1v7.7h-1v-3.6h-3.6v3.6h-1V7.6z"/> <path fill="#0095AA" d="M343.1,7.6h4.5v0.8h-3.5v2.4h2.9v0.8h-2.9v2.8h3.6v0.8h-4.6V7.6z"/> <path fill="#0095AA" d="M351.7,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4h-0.7l-1.4-4l-0.6-1.8h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M361.8,7.6h1.1l2.6,7.7h-1.1l-0.7-2.3H361l-0.7,2.3h-1L361.8,7.6z M361.2,12.2h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L361.2,12.2z"/> <path fill="#0095AA" d="M366.7,7.6h1v3.9h0l3.2-3.9h1.1l-2.4,2.9l2.8,4.8h-1.1l-2.3-4l-1.3,1.6v2.4h-1V7.6z"/> <path fill="#0095AA" d="M373,7.6h4.5v0.8H374v2.4h2.9v0.8H374v2.8h3.6v0.8H373V7.6z"/> <path fill="#0095AA" d="M379.2,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4h-1.1l-1.9-3.3h-1.4v3.3h-1V7.6z M381.5,11.2c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H381.5z"/> <path fill="#0095AA" d="M385.4,14.3l0.6-0.6c0.6,0.6,1.4,0.9,2.2,0.9c1,0,1.6-0.5,1.6-1.3c0-0.8-0.6-1-1.3-1.4l-1.1-0.5 c-0.7-0.3-1.6-0.8-1.6-2c0-1.2,1-2.1,2.4-2.1c0.9,0,1.7,0.4,2.3,0.9L390,9c-0.5-0.4-1.1-0.7-1.7-0.7c-0.9,0-1.4,0.4-1.4,1.1 c0,0.7,0.7,1,1.3,1.3l1.1,0.5c0.9,0.4,1.6,0.9,1.6,2.1c0,1.2-1,2.2-2.6,2.2C387,15.5,386.1,15,385.4,14.3z"/> <path fill="#0095AA" d="M393.7,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C395.1,15.5,393.7,13.9,393.7,11.4z M399.3,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C398.4,14.6,399.3,13.3,399.3,11.4z"/> <path fill="#0095AA" d="M402,7.6h4.5v0.8H403V11h2.9v0.8H403v3.5h-1V7.6z"/> <path fill="#0095AA" d="M410.1,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4H413l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M420.2,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L420.2,7.6z M420,12.2h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L420,12.2z"/> <path fill="#0095AA" d="M425.6,8.8h-2.2V7.6h5.8v1.2H427v6.5h-1.4V8.8z"/> <path fill="#0095AA" d="M430,7.6h1.4v3.1h3.1V7.6h1.4v7.7h-1.4v-3.4h-3.1v3.4H430V7.6z"/> <path fill="#0095AA" d="M437.4,7.6h4.7v1.2h-3.3v1.9h2.8v1.2h-2.8v2.2h3.4v1.2h-4.8V7.6z"/> <path fill="#0095AA" d="M443.3,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4h-0.9l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M453.3,7.6h1.7l2.5,7.7H456l-0.6-2.1h-2.6l-0.6,2.1h-1.4L453.3,7.6z M453.2,12.2h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L453.2,12.2z"/> <path fill="#0095AA" d="M459,8.8h-2.2V7.6h5.8v1.2h-2.2v6.5H459V8.8z"/> <path fill="#0095AA" d="M463.7,7.6h1.4v7.7h-1.4V7.6z"/> <path fill="#0095AA" d="M466.4,11.5c0-2.5,1.6-4,3.5-4c1,0,1.7,0.4,2.2,1l-0.7,0.9c-0.4-0.4-0.9-0.6-1.5-0.6c-1.2,0-2.1,1-2.1,2.8 c0,1.7,0.8,2.8,2.1,2.8c0.7,0,1.2-0.3,1.6-0.7l0.8,0.8c-0.6,0.7-1.5,1.1-2.4,1.1C467.9,15.5,466.4,14,466.4,11.5z"/> <path fill="#0095AA" d="M474.8,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L474.8,7.6z M474.7,12.2h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L474.7,12.2z"/> <path fill="#0095AA" d="M299.5,20.7h1.1l2.6,7.7h-1.1l-0.7-2.3h-2.8l-0.7,2.3h-1L299.5,20.7z M298.9,25.3h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L298.9,25.3z"/> <path fill="#0095AA" d="M304.1,20.7h1.1l2.8,4.8c0.3,0.5,0.6,1.1,0.8,1.7h0c-0.1-0.8-0.1-1.7-0.1-2.5v-4h0.9v7.7h-1.1l-2.8-4.8 c-0.3-0.5-0.6-1.1-0.8-1.7h0c0.1,0.8,0.1,1.6,0.1,2.4v4.1h-0.9V20.7z"/> <path fill="#0095AA" d="M311.4,20.7h1.9c2.4,0,3.7,1.4,3.7,3.8c0,2.5-1.3,3.9-3.6,3.9h-2V20.7z M313.3,27.6c1.8,0,2.7-1.1,2.7-3.1 c0-1.9-0.9-3-2.7-3h-0.9v6.1H313.3z"/> <path fill="#0095AA" d="M320.4,20.6l1.4,0l0.6,3.9c0.1,0.8,0.2,1.6,0.3,2.5l0,0c0.2-0.9,0.3-1.7,0.5-2.5l1-3.9l1.3,0l0.9,3.9 c0.2,0.8,0.3,1.6,0.5,2.5l0,0c0.1-0.9,0.2-1.7,0.4-2.5l0.7-3.9l1.3,0l-1.6,7.7l-1.8,0l-0.8-4.1c-0.1-0.6-0.2-1.2-0.3-1.9l0,0 c-0.1,0.7-0.2,1.2-0.3,1.9l-0.9,4l-1.8,0L320.4,20.6z"/> <path fill="#0095AA" d="M329.8,24.5c0-2.5,1.5-3.9,3.5-3.9c2,0,3.4,1.5,3.3,4c0,2.5-1.5,4-3.5,4C331.2,28.5,329.8,27,329.8,24.5z M335.2,24.6c0-1.7-0.7-2.7-1.9-2.8c-1.2,0-2,1-2,2.7c0,1.7,0.7,2.8,1.9,2.9C334.4,27.4,335.2,26.3,335.2,24.6z"/> <path fill="#0095AA" d="M337.8,20.7l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L337.8,20.7z"/> <path fill="#0095AA" d="M343.4,20.7l4.7,0.1l0,1.2l-3.3-0.1l0,2.2l2.8,0l0,1.2l-2.8,0l-0.1,3.2l-1.4,0L343.4,20.7z"/> <path fill="#0095AA" d="M349,20.7l2.6,0c1.6,0,2.8,0.6,2.8,2.3c0,1.2-0.6,1.9-1.6,2.2l1.8,3.2l-1.6,0l-1.6-3l-1.1,0l-0.1,3l-1.4,0 L349,20.7z M351.4,24.3c1,0,1.6-0.4,1.6-1.3c0-0.9-0.5-1.2-1.6-1.2l-1.1,0l0,2.5L351.4,24.3z"/> <path fill="#0095AA" d="M357.4,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L357.4,20.7z M357.2,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L357.2,25.2z"/> <path fill="#0095AA" d="M362.5,20.6l1.7,0l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5l0,0c0.2-0.6,0.3-1,0.5-1.5l1.4-3.8l1.7,0l-0.1,7.7l-1.3,0 l0.1-3.5c0-0.7,0.1-1.8,0.3-2.5l0,0l-0.6,1.9l-1.3,3.4l-0.9,0l-1.2-3.5l-0.6-1.9l0,0c0.1,0.7,0.2,1.8,0.2,2.5l-0.1,3.5l-1.3,0 L362.5,20.6z"/> <path fill="#0095AA" d="M370.8,19.1l0.9,0L371.5,30l-0.9,0L370.8,19.1z"/> <path fill="#0095AA" d="M375.1,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L375.1,20.7z M374.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L374.8,25.2z"/> <path fill="#0095AA" d="M379.9,20.7l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L379.9,20.7z"/> <path fill="#0095AA" d="M385.6,20.7l2.4,0c1.7,0,3,0.7,3,2.4c0,1.7-1.3,2.5-3,2.4l-1.1,0l0,2.8l-1.4,0L385.6,20.7z M387.9,24.5 c1.1,0,1.7-0.4,1.7-1.4c0-0.9-0.6-1.3-1.7-1.3l-0.9,0l0,2.7L387.9,24.5z"/> <path fill="#0095AA" d="M392,20.6l1.4,0l-0.1,3.1l3.1,0.1l0.1-3.1l1.4,0l-0.1,7.7l-1.4,0l0.1-3.4l-3.1-0.1l-0.1,3.4l-1.4,0L392,20.6z "/> <path fill="#0095AA" d="M401,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L401,20.7z M400.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L400.8,25.2z"/> </g> </g> <a href="https://www.wolfram.com/mathematica/"><rect x="409" y="5.9" style="fill:#ffffff00;" width="70" height="11.6"/></a> <a href="https://wolframalpha.com/"><rect x="319.6" y="18.2" style="fill:#ffffff00;" width="86.9" height="11.6"/></a> <a href="/"><rect y="0.1" style="fill:#ffffff00;" width="292.4" height="29.9"/></a> </svg> <svg version="1.1" id="logo-600" class="hide show__600" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 480.7 30" style="enable-background:new 0 0 480.7 30;" xml:space="preserve"> <g> <g> <g> <polygon fill="#0095AA" points="145.2,17.1 138.8,3.9 134.2,3.9 134.2,28.5 137.5,28.5 137.5,8.6 144,21.8 144,21.9 146.2,21.9 152.7,8.6 152.7,28.5 156,28.5 156,3.9 151.6,3.9"/> <path fill="#0095AA" d="M171.8,10.8c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.3,0.8c-1.2,0.6-2.1,1.4-2.6,2.4l0,0.1l2.6,1.6l0-0.1 c0.3-0.5,0.8-1,1.4-1.4c0.7-0.4,1.6-0.6,2.8-0.6c1.4,0,2.5,0.3,3.2,0.8c0.8,0.5,1.1,1.4,1.1,2.8v0.8l-4.5,0.4 c-1.8,0.2-3.2,0.6-4.1,1.1c-1,0.5-1.7,1.2-2.2,2c-0.5,0.8-0.8,1.9-0.8,3.2c0,1.7,0.5,3.1,1.5,4.2c1,1.1,2.4,1.6,4.1,1.6 c1.2,0,2.3-0.3,3.3-0.8c0.9-0.5,1.8-1.2,2.7-1.9v2.2h3.2V15.9C173.6,13.5,173,11.8,171.8,10.8z M170.4,19.1v4.4 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6c-0.4,0.2-0.9,0.2-1.3,0.2c-1.1,0-1.9-0.2-2.4-0.7 c-0.5-0.5-0.8-1.2-0.8-2.3c0-1.1,0.4-2,1.2-2.6c0.8-0.6,2.2-1,4.2-1.2L170.4,19.1z"/> <path fill="#0095AA" d="M185.9,25.8c-0.7,0.3-1.6,0.5-2.4,0.5c-1,0-1.6-0.3-1.9-0.9c-0.3-0.6-0.4-1.7-0.4-3.2v-9.7h4.7V9.8h-4.7V4.7 h-3.2v5.1h-2.6v2.7h2.6v10.3c0,2.1,0.4,3.7,1.2,4.7c0.8,1,2.2,1.5,4.1,1.5c1.2,0,2.3-0.2,3.5-0.6l0.1,0L185.9,25.8L185.9,25.8z" /> <path fill="#0095AA" d="M199.1,9.3c-1.2,0-2.3,0.3-3.5,1c-1.1,0.6-2,1.3-2.9,2.1V2.1h-3.2v26.4h3.2V15.4c0.9-0.8,1.9-1.5,2.9-2.1 c1-0.6,1.9-0.9,2.7-0.9c0.8,0,1.3,0.1,1.7,0.4c0.4,0.3,0.7,0.7,0.8,1.2c0.2,0.6,0.3,1.7,0.3,3.5v11h3.2V16.1 c0-2.4-0.4-4.2-1.3-5.2C202.2,9.8,200.9,9.3,199.1,9.3z"/> <polygon fill="#0095AA" points="226.2,23.5 221.6,4 221.6,3.9 217.8,3.9 213.2,23.3 208.8,4 208.8,3.9 205.3,3.9 211.1,28.4 211.1,28.5 215.1,28.5 219.6,9.3 224.1,28.4 224.1,28.5 228.2,28.5 234.1,3.9 230.7,3.9 "/> <path fill="#0095AA" d="M265.2,9.6c-1.2,0-2.2,0.3-3.2,1c-0.9,0.6-1.7,1.3-2.5,2.1V9.8h-3.2v18.7h3.2V15.7c1.1-1.1,2.1-1.8,2.9-2.3 c0.8-0.4,1.7-0.6,2.6-0.6c0.6,0,1.1,0.1,1.4,0.2l0.1,0l0.7-3.1l-0.1,0C266.6,9.7,265.9,9.6,265.2,9.6z"/> <rect x="270" y="2.1" fill="#0095AA" width="3.2" height="26.4"/> <path fill="#0095AA" d="M288.6,2.1v9.2c-1.5-1.3-3.2-1.9-5-1.9c-2.3,0-4.2,0.9-5.5,2.7c-1.3,1.8-2,4.3-2,7.3c0,3.1,0.6,5.5,1.8,7.2 c1.2,1.7,3,2.5,5.2,2.5c1.1,0,2.2-0.2,3.2-0.7c0.9-0.4,1.7-0.9,2.4-1.6v1.7h3.2V2.1H288.6z M288.6,14.1v9.7 c-0.6,0.6-1.4,1.1-2.2,1.6c-0.9,0.5-1.8,0.7-2.7,0.7c-2.9,0-4.3-2.3-4.3-6.9c0-2.4,0.4-4.2,1.3-5.4c0.9-1.1,2-1.7,3.3-1.7 C285.5,12.2,287,12.8,288.6,14.1z"/> </g> <g> <polygon fill="#666666" points="30,3.2 26.4,3.2 22,22.5 17.5,3.4 17.5,3.2 13.5,3.2 9,22.4 4.7,3.2 1,3.2 6.8,28 6.8,28.2 11,28.2 15.4,9.2 19.8,28 19.8,28.2 24,28.2 30,3.5"/> <path fill="#666666" d="M38,8.7c-2.7,0-4.8,0.8-6.2,2.5c-1.4,1.7-2.1,4.2-2.1,7.6c0,3.3,0.7,5.8,2.1,7.5c1.4,1.7,3.5,2.5,6.2,2.5 c2.7,0,4.8-0.9,6.2-2.6c1.4-1.7,2.1-4.2,2.1-7.5c0-3.3-0.7-5.8-2.1-7.5C42.8,9.5,40.7,8.7,38,8.7z M41.5,24 c-0.8,1.1-1.9,1.7-3.6,1.7c-1.7,0-2.9-0.5-3.7-1.7c-0.8-1.1-1.2-2.9-1.2-5.3c0-2.5,0.4-4.3,1.2-5.4c0.8-1.1,2-1.6,3.6-1.6 c1.6,0,2.8,0.5,3.5,1.6c0.8,1.1,1.2,2.9,1.2,5.4C42.7,21.1,42.3,22.9,41.5,24z"/> <rect x="49.7" y="1.4" fill="#666666" width="3.4" height="26.8"/> <path fill="#666666" d="M67.9,1.5c-1.1-0.5-2.3-0.7-3.6-0.7c-2.2,0-3.7,0.6-4.6,1.7c-0.9,1.1-1.3,2.8-1.3,5.1v1.6h-2.7v3h2.7v16.1 h3.4V12.2h4.1v-3h-4.1V7.6c0-1.5,0.2-2.5,0.6-3c0.4-0.5,1-0.7,2-0.7c0.4,0,0.8,0.1,1.3,0.2c0.5,0.1,0.9,0.3,1.2,0.4l0.2,0.1l1-3 L67.9,1.5z"/> <path fill="#666666" d="M77.3,8.9c-1.2,0-2.3,0.4-3.2,1c-0.8,0.6-1.6,1.2-2.3,2V9.2h-3.4v19h3.4v-13c1.1-1.1,2-1.8,2.8-2.2 c0.8-0.4,1.7-0.6,2.5-0.6c0.6,0,1,0.1,1.3,0.2l0.2,0.1l0.7-3.3l-0.1-0.1C78.7,9,78.1,8.9,77.3,8.9z"/> <path fill="#666666" d="M93.3,10.2c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.4,0.8c-1.2,0.6-2.1,1.4-2.7,2.5l-0.1,0.2l2.8,1.7 l0.1-0.2c0.3-0.5,0.7-0.9,1.4-1.3c0.6-0.4,1.6-0.6,2.8-0.6c1.3,0,2.4,0.3,3.2,0.7c0.7,0.5,1.1,1.4,1.1,2.8v0.7l-4.4,0.4 c-1.8,0.2-3.2,0.6-4.2,1.1c-1,0.5-1.7,1.2-2.2,2.1c-0.5,0.9-0.8,2-0.8,3.3c0,1.7,0.5,3.2,1.5,4.3c1,1.1,2.4,1.7,4.2,1.7 c1.2,0,2.3-0.3,3.3-0.8c0.8-0.5,1.7-1.1,2.5-1.8v2.1h3.4V15.4C95.1,12.9,94.5,11.2,93.3,10.2z M86.8,25.6c-1,0-1.8-0.2-2.3-0.7 c-0.5-0.4-0.8-1.2-0.8-2.2c0-1.1,0.4-1.9,1.1-2.5c0.8-0.6,2.2-1,4.1-1.1l2.7-0.3V23c-0.4,0.3-0.7,0.6-1.1,0.9 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6C87.7,25.5,87.2,25.6,86.8,25.6z"/> <path fill="#666666" d="M123,11.6c-0.3-0.9-0.9-1.7-1.6-2.2c-0.7-0.5-1.7-0.8-2.8-0.8c-1.3,0-2.4,0.4-3.4,1.1 c-0.9,0.6-1.8,1.4-2.7,2.3c-0.3-1.1-0.8-2-1.5-2.5c-0.8-0.6-1.8-0.9-3.1-0.9c-1.2,0-2.3,0.3-3.3,1c-0.8,0.6-1.6,1.2-2.4,2V9.2 h-3.4v19h3.4V14.9c0.9-0.9,1.8-1.7,2.7-2.2c0.8-0.5,1.6-0.8,2.2-0.8c0.7,0,1.2,0.1,1.5,0.3c0.3,0.2,0.5,0.6,0.6,1.2 c0.1,0.6,0.2,1.8,0.2,3.5v11.2h3.4V14.9c1.2-1.1,2.2-1.9,2.9-2.3c0.7-0.4,1.4-0.6,2-0.6c0.7,0,1.2,0.1,1.5,0.3 c0.3,0.2,0.5,0.6,0.7,1.1c0.1,0.6,0.2,1.8,0.2,3.6v11.2h3.4V15.6C123.5,13.9,123.3,12.5,123,11.6z"/> </g> <path fill="#0095AA" d="M243.6,8.3c-5.8,0-10.5,4.7-10.5,10.5s4.7,10.5,10.5,10.5c5.8,0,10.5-4.7,10.5-10.5S249.4,8.3,243.6,8.3z M236.7,23.9c0-0.5,0.1-1,0.2-1.6c1.1,0.9,2.3,1.7,3.7,2.4L240,26C238.8,25.5,237.6,24.7,236.7,23.9z M239.8,26.5l-0.2,0.5 c-0.9-0.4-1.8-1-2.5-1.6c-0.2-0.2-0.3-0.5-0.3-0.8C237.6,25.4,238.7,26,239.8,26.5z M236.1,23.4c-0.8-0.8-1.4-1.6-1.6-2.4 c-0.1-0.5-0.1-1.1-0.1-1.6c0.4,0.8,1.1,1.7,2,2.5C236.3,22.4,236.2,22.9,236.1,23.4z M236.2,24.5L236.2,24.5 c-0.3-0.4-0.6-0.8-0.8-1.2c0.2,0.2,0.5,0.5,0.7,0.7C236.2,24.3,236.2,24.4,236.2,24.5z M234.4,17.4c0-0.1,0-0.1,0-0.2 c0.1-0.7,0.4-1.5,0.7-2.2c0-0.1,0-0.1,0.1-0.2c0,0.3,0.1,0.6,0.2,0.9c-0.3,0.4-0.5,0.8-0.7,1.2C234.5,17.1,234.5,17.3,234.4,17.4z M234.6,18.4c0.1-0.4,0.2-0.8,0.4-1.2c0.1-0.3,0.3-0.6,0.5-0.9c0.4,0.9,1.1,1.9,2.1,2.8c-0.2,0.4-0.4,0.8-0.6,1.2 c-0.1,0.3-0.3,0.6-0.4,0.9C235.6,20.4,235,19.4,234.6,18.4z M251.7,15.2c-0.1,0.4-0.3,0.8-0.8,1c-0.3-0.8-0.8-1.6-1.3-2.4 c0.2-0.1,0.3-0.3,0.4-0.5C250.7,13.9,251.3,14.5,251.7,15.2z M250.2,12.3c0.4,0.2,0.7,0.5,1,0.8c0.3,0.4,0.4,0.7,0.5,1 c0,0,0,0.1,0,0.1c-0.4-0.5-0.9-1-1.4-1.5C250.2,12.7,250.2,12.5,250.2,12.3C250.2,12.4,250.2,12.3,250.2,12.3z M252.8,17.8 c-0.1,0.6-0.5,1.1-1.1,1.5c-0.1-0.8-0.3-1.7-0.6-2.5c0.4-0.2,0.7-0.6,0.9-0.9C252.4,16.5,252.7,17.1,252.8,17.8z M252.3,15.2 C252.3,15.2,252.3,15.2,252.3,15.2C252.3,15.2,252.3,15.2,252.3,15.2C252.3,15.2,252.3,15.2,252.3,15.2z M240.2,16.5 c0.6-0.7,1.2-1.3,1.8-1.9c0.7,0.6,1.5,1.1,2.5,1.6l-1.1,2.4C242.2,18,241.1,17.3,240.2,16.5z M243.2,19.1l-1.1,2.4 c-1.4-0.7-2.7-1.5-3.7-2.4c0.4-0.7,0.9-1.4,1.5-2.1C240.8,17.8,242,18.5,243.2,19.1z M245,16.4c0.8,0.3,1.5,0.6,2.3,0.7 c0.2,0,0.4,0.1,0.6,0.1c0,0.9-0.1,1.8-0.2,2.6c-0.3,0-0.6-0.1-0.9-0.1c-1-0.2-2-0.5-3-0.9C244.3,18,244.7,17.2,245,16.4z M247.7,14.2c0.1,0.8,0.2,1.6,0.3,2.4c-0.2,0-0.4-0.1-0.5-0.1c-0.7-0.1-1.4-0.4-2.1-0.7c0.3-0.7,0.6-1.5,0.9-2.1 c0.4,0.2,0.8,0.3,1.2,0.4C247.5,14.1,247.6,14.2,247.7,14.2z M247.6,12.2c0.5,0.4,0.9,0.9,1.3,1.4c-0.2,0.1-0.5,0.1-0.8,0.1 C248,13.1,247.8,12.6,247.6,12.2z M246,13.1c-0.3-0.2-0.7-0.4-1-0.6c0.5-0.3,1.1-0.6,1.6-0.7L246,13.1z M245.7,13.6l-1,2.1 c-0.8-0.4-1.6-0.9-2.3-1.4c0.7-0.6,1.3-1,2-1.5C244.9,13.1,245.3,13.3,245.7,13.6z M242.1,13.9c-0.6-0.6-1-1.1-1.2-1.7 c0.8-0.3,1.7-0.5,2.6-0.7c0.1,0.3,0.3,0.6,0.6,0.9C243.5,12.8,242.8,13.3,242.1,13.9z M243.3,11c-0.9,0.1-1.8,0.3-2.6,0.6 c-0.1-0.5,0-0.9,0.2-1.3c0.8-0.1,1.6-0.1,2.4,0C243.3,10.6,243.3,10.8,243.3,11z M241.7,14.2c-0.7,0.6-1.3,1.2-1.9,1.9 c-0.8-0.8-1.4-1.6-1.7-2.4c0.7-0.5,1.5-0.9,2.3-1.3C240.6,13,241.1,13.6,241.7,14.2z M240.2,11.9c-0.8,0.3-1.6,0.8-2.3,1.2 c-0.2-0.7,0-1.4,0.4-1.9c0.6-0.3,1.3-0.6,2-0.8C240.1,10.9,240.1,11.4,240.2,11.9z M239.4,16.6c-0.6,0.7-1.1,1.4-1.5,2.1 c-1-0.9-1.7-1.9-2-2.9c0.5-0.6,1-1.2,1.7-1.8C237.9,14.9,238.6,15.8,239.4,16.6z M238,19.6c1.1,0.9,2.4,1.8,3.8,2.4l-1,2.1 c-1.4-0.7-2.7-1.5-3.8-2.4c0.1-0.3,0.3-0.7,0.4-1.1C237.7,20.3,237.8,20,238,19.6z M241.2,24.9c1.2,0.5,2.4,0.9,3.5,1.1 c0.3,0.1,0.6,0.1,0.9,0.1c-0.3,0.5-0.7,0.9-1.1,1.2c-0.3,0-0.7-0.1-1-0.1c-1-0.2-2-0.5-3-0.9C240.7,25.9,240.9,25.5,241.2,24.9z M244.7,27.9c0.3,0,0.6,0,0.9,0c-0.4,0.1-0.9,0.2-1.3,0.2C244.5,28,244.6,28,244.7,27.9z M245.3,27.4c0.3-0.3,0.7-0.7,1-1.2 c1.1,0.1,2.2,0,3-0.2c-0.4,0.4-0.8,0.7-1.2,1C247.3,27.3,246.4,27.4,245.3,27.4z M245.9,25.6c-0.4,0-0.7-0.1-1.1-0.2 c-1.1-0.2-2.3-0.6-3.4-1.1c0.1-0.3,0.3-0.6,0.5-1l0.5-1.1c1.2,0.5,2.4,0.9,3.5,1.1c0.3,0.1,0.7,0.1,1,0.2 c-0.1,0.4-0.3,0.7-0.4,1.1C246.3,25,246.1,25.3,245.9,25.6z M246.1,22.8c-1.1-0.2-2.3-0.6-3.4-1.1c0.4-0.8,0.7-1.6,1.1-2.4 c1,0.4,2.1,0.8,3.1,1c0.3,0.1,0.6,0.1,0.9,0.1c-0.1,0.9-0.3,1.7-0.6,2.5C246.8,22.9,246.4,22.9,246.1,22.8z M248.3,20.5 c1.1,0.1,2.2,0,3-0.4c0,0.8,0,1.7-0.2,2.4c-0.9,0.4-2,0.5-3.4,0.5C248,22.2,248.1,21.4,248.3,20.5z M248.4,19.9 c0.1-0.9,0.2-1.8,0.1-2.6c0.8,0,1.6,0,2.2-0.3c0.3,0.8,0.5,1.7,0.5,2.5C250.5,19.8,249.5,20,248.4,19.9z M250.5,16.4 c-0.5,0.2-1.2,0.3-2,0.2c0-0.9-0.1-1.7-0.3-2.5c0.4,0,0.7,0,1-0.1C249.8,14.8,250.2,15.6,250.5,16.4z M249.4,13.4 c-0.3-0.4-0.7-0.8-1.1-1.2c0.5,0.3,0.9,0.5,1.3,0.9C249.6,13.2,249.5,13.3,249.4,13.4z M248.4,11.7c0.4,0.1,0.7,0.2,1,0.3 c0.1,0.2,0.2,0.3,0.2,0.5C249.2,12.1,248.8,11.9,248.4,11.7z M247.7,11.1c0.2-0.1,0.4-0.2,0.5-0.2c0.2,0.1,0.3,0.2,0.5,0.4 C248.4,11.2,248.1,11.2,247.7,11.1z M247.2,10.9c0-0.2-0.1-0.5-0.1-0.7c0.2,0.1,0.5,0.2,0.7,0.3C247.6,10.7,247.4,10.8,247.2,10.9 z M247.6,13.6c0,0-0.1,0-0.1,0c-0.3-0.1-0.7-0.2-1-0.3c0.2-0.5,0.4-1,0.6-1.4C247.2,12.4,247.4,13,247.6,13.6z M245.8,10 c0.1,0,0.2,0,0.3,0.1c0.1,0,0.2,0,0.3,0.1c0.1,0.1,0.2,0.3,0.2,0.6C246.4,10.4,246.1,10.2,245.8,10z M246,11.4 c-0.5,0.2-0.9,0.4-1.4,0.7c-0.2-0.2-0.4-0.5-0.5-0.7C244.7,11.4,245.4,11.4,246,11.4z M245.6,10.9c-0.6,0-1.1,0-1.7,0 c0-0.2,0-0.3,0.1-0.5C244.5,10.6,245.1,10.7,245.6,10.9z M244.4,10.1c0.1-0.1,0.3-0.1,0.4-0.1c0.3,0.1,0.6,0.3,0.9,0.5 C245.3,10.3,244.9,10.2,244.4,10.1z M243.7,9.9c-0.5-0.1-1-0.1-1.5-0.1c-0.2,0-0.4,0-0.5,0c0.3-0.2,0.7-0.3,1.1-0.4 c0.5,0,0.9,0,1.4,0.2C243.9,9.7,243.8,9.8,243.7,9.9z M237.4,13.5c-0.6,0.5-1.2,1.1-1.7,1.7c-0.1-0.6-0.1-1.1,0.1-1.5 c0.4-0.7,1-1.3,1.6-1.8C237.2,12.4,237.2,12.9,237.4,13.5z M240.3,26.8c1,0.4,2,0.8,3.1,0.9c0.2,0,0.3,0.1,0.5,0.1 c-0.3,0.1-0.5,0.2-0.8,0.2c-0.3,0-0.5-0.1-0.8-0.1c-0.7-0.1-1.5-0.4-2.2-0.7C240.1,27.1,240.2,27,240.3,26.8z M246.6,25.7 c0.2-0.3,0.3-0.6,0.5-0.9c0.2-0.4,0.3-0.8,0.5-1.2c1.3,0.1,2.4,0,3.4-0.4c-0.1,0.3-0.2,0.6-0.3,0.9c-0.2,0.4-0.4,0.8-0.6,1.1 C249,25.6,247.9,25.8,246.6,25.7z M251,24.3c0.2-0.4,0.3-0.8,0.5-1.3c0.2-0.1,0.5-0.3,0.7-0.4c-0.3,0.7-0.7,1.3-1.1,1.9 c-0.1,0.1-0.1,0.1-0.2,0.2C250.9,24.5,251,24.4,251,24.3z M251.7,22.2c0.1-0.8,0.2-1.5,0.1-2.3c0.5-0.3,0.9-0.6,1.2-1 c0,0.1,0,0.2,0,0.3c0,0.6-0.1,1.3-0.3,1.8C252.5,21.5,252.2,21.9,251.7,22.2z"/> </g> <g> <path fill="#0095AA" d="M298.9,7.6h4.5v0.8h-3.5V11h2.9v0.8h-2.9v3.5h-1V7.6z"/> <path fill="#0095AA" d="M304.7,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4H309l-1.9-3.3h-1.4v3.3h-1V7.6z M307,11.2 c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H307z"/> <path fill="#0095AA" d="M310.9,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C312.2,15.4,310.9,13.9,310.9,11.4z M316.5,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C315.6,14.6,316.5,13.3,316.5,11.4z"/> <path fill="#0095AA" d="M319.1,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4H322l-1.4-4L320,8.7h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M331.7,8.4h-2.3V7.6h5.7v0.8h-2.3v6.9h-1V8.4z"/> <path fill="#0095AA" d="M336.4,7.6h1v3.2h3.6V7.6h1v7.7h-1v-3.6h-3.6v3.6h-1V7.6z"/> <path fill="#0095AA" d="M344.1,7.6h4.5v0.8h-3.5v2.4h2.9v0.8h-2.9v2.8h3.6v0.8h-4.6V7.6z"/> <path fill="#0095AA" d="M352.7,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4h-0.7l-1.4-4l-0.6-1.8h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M362.8,7.6h1.1l2.6,7.7h-1.1l-0.7-2.3H362l-0.7,2.3h-1L362.8,7.6z M362.2,12.1h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L362.2,12.1z"/> <path fill="#0095AA" d="M367.7,7.6h1v3.9h0l3.2-3.9h1.1l-2.4,2.9l2.8,4.8h-1.1l-2.3-4l-1.3,1.6v2.4h-1V7.6z"/> <path fill="#0095AA" d="M374,7.6h4.5v0.8H375v2.4h2.9v0.8H375v2.8h3.6v0.8H374V7.6z"/> <path fill="#0095AA" d="M380.2,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4h-1.1l-1.9-3.3h-1.4v3.3h-1V7.6z M382.5,11.2c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H382.5z"/> <path fill="#0095AA" d="M386.4,14.3l0.6-0.6c0.6,0.6,1.4,0.9,2.2,0.9c1,0,1.6-0.5,1.6-1.3c0-0.8-0.6-1-1.3-1.4l-1.1-0.5 c-0.7-0.3-1.6-0.8-1.6-2c0-1.2,1-2.1,2.4-2.1c0.9,0,1.7,0.4,2.3,0.9L391,9c-0.5-0.4-1.1-0.7-1.7-0.7c-0.9,0-1.4,0.4-1.4,1.1 c0,0.7,0.7,1,1.3,1.3l1.1,0.5c0.9,0.4,1.6,0.9,1.6,2.1c0,1.2-1,2.2-2.6,2.2C388,15.4,387.1,15,386.4,14.3z"/> <path fill="#0095AA" d="M394.7,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C396.1,15.4,394.7,13.9,394.7,11.4z M400.3,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C399.4,14.6,400.3,13.3,400.3,11.4z"/> <path fill="#0095AA" d="M403,7.6h4.5v0.8H404V11h2.9v0.8H404v3.5h-1V7.6z"/> <path fill="#0095AA" d="M411.1,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4H414l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M421.2,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L421.2,7.6z M421,12.1h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L421,12.1z"/> <path fill="#0095AA" d="M426.6,8.8h-2.2V7.6h5.8v1.2H428v6.5h-1.4V8.8z"/> <path fill="#0095AA" d="M431,7.6h1.4v3.1h3.1V7.6h1.4v7.7h-1.4v-3.4h-3.1v3.4H431V7.6z"/> <path fill="#0095AA" d="M438.4,7.6h4.7v1.2h-3.3v1.9h2.8v1.2h-2.8v2.2h3.4v1.2h-4.8V7.6z"/> <path fill="#0095AA" d="M444.3,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4h-0.9l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M454.3,7.6h1.7l2.5,7.7H457l-0.6-2.1h-2.6l-0.6,2.1h-1.4L454.3,7.6z M454.2,12.1h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L454.2,12.1z"/> <path fill="#0095AA" d="M460,8.8h-2.2V7.6h5.8v1.2h-2.2v6.5H460V8.8z"/> <path fill="#0095AA" d="M464.7,7.6h1.4v7.7h-1.4V7.6z"/> <path fill="#0095AA" d="M467.4,11.5c0-2.5,1.6-4,3.5-4c1,0,1.7,0.4,2.2,1l-0.7,0.9c-0.4-0.4-0.9-0.6-1.5-0.6c-1.2,0-2.1,1-2.1,2.8 c0,1.7,0.8,2.8,2.1,2.8c0.7,0,1.2-0.3,1.6-0.7l0.8,0.8c-0.6,0.7-1.5,1.1-2.4,1.1C468.9,15.4,467.4,14,467.4,11.5z"/> <path fill="#0095AA" d="M475.8,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L475.8,7.6z M475.7,12.1h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L475.7,12.1z"/> <path fill="#0095AA" d="M300.5,20.7h1.1l2.6,7.7h-1.1l-0.7-2.3h-2.8l-0.7,2.3h-1L300.5,20.7z M299.8,25.2h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L299.8,25.2z"/> <path fill="#0095AA" d="M305.1,20.7h1.1l2.8,4.8c0.3,0.5,0.6,1.1,0.8,1.7h0c-0.1-0.8-0.1-1.7-0.1-2.5v-4h0.9v7.7h-1.1l-2.8-4.8 c-0.3-0.5-0.6-1.1-0.8-1.7h0c0.1,0.8,0.1,1.6,0.1,2.4v4.1h-0.9V20.7z"/> <path fill="#0095AA" d="M312.4,20.7h1.9c2.4,0,3.7,1.4,3.7,3.8c0,2.5-1.3,3.9-3.6,3.9h-2V20.7z M314.3,27.6c1.8,0,2.7-1.1,2.7-3.1 c0-1.9-0.9-3-2.7-3h-0.9v6.1H314.3z"/> <path fill="#0095AA" d="M321.4,20.6l1.4,0l0.6,3.9c0.1,0.8,0.2,1.6,0.3,2.5l0,0c0.2-0.9,0.3-1.7,0.5-2.5l1-3.9l1.3,0l0.9,3.9 c0.2,0.8,0.3,1.6,0.5,2.5l0,0c0.1-0.9,0.2-1.7,0.4-2.5l0.7-3.9l1.3,0l-1.6,7.7l-1.8,0l-0.8-4.1c-0.1-0.6-0.2-1.2-0.3-1.9l0,0 c-0.1,0.7-0.2,1.2-0.3,1.9l-0.9,4l-1.8,0L321.4,20.6z"/> <path fill="#0095AA" d="M330.8,24.4c0-2.5,1.5-3.9,3.5-3.9c2,0,3.4,1.5,3.3,4c0,2.5-1.5,4-3.5,4C332.2,28.5,330.8,26.9,330.8,24.4z M336.2,24.5c0-1.7-0.7-2.7-1.9-2.8c-1.2,0-2,1-2,2.7c0,1.7,0.7,2.8,1.9,2.9C335.4,27.3,336.2,26.3,336.2,24.5z"/> <path fill="#0095AA" d="M338.8,20.6l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L338.8,20.6z"/> <path fill="#0095AA" d="M344.4,20.6l4.7,0.1l0,1.2l-3.3-0.1l0,2.2l2.8,0l0,1.2l-2.8,0l-0.1,3.2l-1.4,0L344.4,20.6z"/> <path fill="#0095AA" d="M350,20.6l2.6,0c1.6,0,2.8,0.6,2.8,2.3c0,1.2-0.6,1.9-1.6,2.2l1.8,3.2l-1.6,0l-1.6-3l-1.1,0l-0.1,3l-1.4,0 L350,20.6z M352.4,24.3c1,0,1.6-0.4,1.6-1.3c0-0.9-0.5-1.2-1.6-1.2l-1.1,0l0,2.5L352.4,24.3z"/> <path fill="#0095AA" d="M358.4,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L358.4,20.7z M358.2,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L358.2,25.2z"/> <path fill="#0095AA" d="M363.5,20.6l1.7,0l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5l0,0c0.2-0.6,0.3-1,0.5-1.5l1.4-3.8l1.7,0l-0.1,7.7l-1.3,0 l0.1-3.5c0-0.7,0.1-1.8,0.3-2.5l0,0l-0.6,1.9l-1.3,3.4l-0.9,0l-1.2-3.5l-0.6-1.9l0,0c0.1,0.7,0.2,1.8,0.2,2.5l-0.1,3.5l-1.3,0 L363.5,20.6z"/> <path fill="#0095AA" d="M371.8,19l0.9,0L372.5,30l-0.9,0L371.8,19z"/> <path fill="#0095AA" d="M376.1,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L376.1,20.7z M375.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L375.8,25.2z"/> <path fill="#0095AA" d="M380.9,20.6l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L380.9,20.6z"/> <path fill="#0095AA" d="M386.6,20.6l2.4,0c1.7,0,3,0.7,3,2.4c0,1.7-1.3,2.5-3,2.4l-1.1,0l0,2.8l-1.4,0L386.6,20.6z M388.9,24.5 c1.1,0,1.7-0.4,1.7-1.4c0-0.9-0.6-1.3-1.7-1.3l-0.9,0l0,2.7L388.9,24.5z"/> <path fill="#0095AA" d="M393,20.6l1.4,0l-0.1,3.1l3.1,0.1l0.1-3.1l1.4,0l-0.1,7.7l-1.4,0l0.1-3.4l-3.1-0.1l-0.1,3.4l-1.4,0L393,20.6z "/> <path fill="#0095AA" d="M402,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L402,20.7z M401.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L401.8,25.2z"/> </g> </g> <a href="https://www.wolfram.com/mathematica/"><rect x="296.2" y="0.1" style="fill:#ffffff00;" width="183.8" height="16"/></a> <a href="https://wolframalpha.com/"><rect x="296.2" y="16.4" style="fill:#ffffff00;" width="123.4" height="13.6"/></a> <a href="/"><rect x="1" y="0.1" style="fill:#ffffff00;" width="292.4" height="29.9"/></a> </svg> <form method="get" action="/search/" name="search" id="search" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchField"> <img src="/images/header/search-icon.png" width="18" height="18" class="search-btn" title="Search" alt="Search"> <img src="/images/header/menu-close.png" width="16" height="16" id="search-close" class="close-btn" title="Close" alt="Close"> </form> </header> <section class="left-side"> <img src="/images/sidebar/menu-close.png" width="16" height="16" id="dropdown-topics-menu-close" class="close-btn"> <form method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Cubes.html">Cubes</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Hexahedra.html">Hexahedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/PlatonicSolids.html">Platonic Solids</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/CanonicalPolyhedra.html">Canonical Polyhedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Isohedra.html">Isohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Parallelohedra.html">Parallelohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Rhombohedra.html">Rhombohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Space-FillingPolyhedra.html">Space-Filling Polyhedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Stereohedra.html">Stereohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Trapezohedra.html">Trapezohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Zonohedra.html">Zonohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/Folding.html">Folding</a> </li> <li> <a href="/topics/Origami.html">Origami</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicalRecords.html">Mathematical Records</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicsintheArts.html">Mathematics in the Arts</a> </li> <li> <a href="/topics/MathematicsinArchitecture.html">Mathematics in Architecture</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicalArt.html">Mathematical Art</a> </li> <li> <a href="/topics/PolyhedronNets.html">Polyhedron Nets</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Cantrell.html">Cantrell</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/HerrstromEmily.html">Herrstrom, Emily</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>   <h1>Cube</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/Polyhedra/Cube.nb" download="Cube.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="798.84" src="images/eps-svg/CubeSolidWireframeNet_1000.svg" class="" alt="CubeSolidWireframeNet" /> </div> <p> <div class="make-your-own"> <h2>Make Your Own Cube</h2><a href="/pdf/Cube.pdf" target="_blank"> <div class="downloadbutton" style="background-image: url('../images/buttons/Cube-net.png');"> <span>Print & fold</span><img src="../images/buttons/download-arrow.svg" /> </div> </a><a href="/stl/Cube.stl" target="_blank"> <div class="downloadbutton threeD" style="background-image: url('../images/buttons/Cube-3D.png');"> <span>Print in 3D</span><img src="../images/buttons/download-arrow.svg" /> </div> </a> </div> </p> <p> The cube, illustrated above together with a wireframe version and a <a href="/Net.html">net</a> that can be used for its construction, is the <a href="/PlatonicSolid.html">Platonic solid</a> composed of six <a href="/Square.html">square</a> faces that meet each other at <a href="/RightAngle.html">right angles</a> and has eight vertices and 12 edges. It is also the <a href="/UniformPolyhedron.html">uniform polyhedron</a> with Maeder index 6 (Maeder 1997), Wenninger index 3 (Wenninger 1989), Coxeter index 18 (Coxeter <i>et al. </i>1954), and Har'El index 11 (Har'El 1993). It is described by the <a href="/SchlaefliSymbol.html">Schläfli symbol</a> <img src="/images/equations/Cube/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="{4,3}" /> and <a href="/WythoffSymbol.html">Wythoff symbol</a> <img src="/images/equations/Cube/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="24" alt="3|24" />. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="489.822" src="images/eps-svg/CubeProjections_600.svg" class="" alt="CubeProjections" /> </div> <p> Three symmetric projections of the cube are illustrated above. </p> <p> The cube is the unique regular convex <a href="/Hexahedron.html">hexahedron</a>. The topologically distinct <a href="/PentagonalWedge.html">pentagonal wedge</a> is the only other convex <a href="/Hexahedron.html">hexahedron</a> that shares the same number of vertices, edges, and faces as the cube (though of course with different face shapes; the pentagonal wedge consists of triangles, 2 quadrilaterals, and 2 pentagons). </p> <p> The cube is implemented in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> as <tt><a href="http://reference.wolfram.com/language/ref/Cube.html">Cube</a></tt>[] or <tt><a href="http://reference.wolfram.com/language/ref/UniformPolyhedron.html">UniformPolyhedron</a></tt>[<tt>"Cube"</tt>]. Precomputed properties are available as <tt><a href="http://reference.wolfram.com/language/ref/PolyhedronData.html">PolyhedronData</a></tt>[<tt>"Cube"</tt>, <i>prop</i>]. </p> <p> The cube is a <a href="/Space-FillingPolyhedron.html">space-filling polyhedron</a> and therefore has <a href="/DehnInvariant.html">Dehn invariant</a> 0. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="684.72" src="images/eps-svg/CubeConvexHulls_1000.svg" class="" alt="CubeConvexHulls" /> </div> <p> It is the <a href="/ConvexHull.html">convex hull</a> of the <a href="/Endodocahedron.html">endodocahedron</a> and <a href="/StellaOctangula.html">stella octangula</a>. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="489.595" src="images/eps-svg/CubeNets_950.svg" class="" alt="CubeNets" /> </div> <p> There are a total of 11 distinct <a href="/Net.html">nets</a> for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch), illustrated above, the same number as the <a href="/Octahedron.html">octahedron</a>. Questions of <a href="/PolyhedronColoring.html">polyhedron coloring</a> of the cube can be addressed using the <a href="/PolyaEnumerationTheorem.html">Pólya enumeration theorem</a>. </p> <p> A cube with unit edge lengths is called a <a href="/UnitCube.html">unit cube</a>. </p> <p> The <a href="/SurfaceArea.html">surface area</a> and <a href="/Volume.html">volume</a> of a cube with edge length <img src="/images/equations/Cube/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" /> are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline4.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="S" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline5.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline6.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="28" height="20" alt="6a^2" /></td><td align="right" width="10"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline7.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="V" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline8.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline9.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="20" height="20" alt="a^3." /></td><td align="right" width="10"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> Because the <a href="/Volume.html">volume</a> of a cube of edge length <img src="/images/equations/Cube/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" /> is given by <img src="/images/equations/Cube/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="a^3" />, a number <a href="/OftheForm.html">of the form</a> <img src="/images/equations/Cube/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="a^3" /> is called a <a href="/CubicNumber.html">cubic number</a> (or sometimes simply "a cube"). Similarly, the operation of taking a number to the third <a href="/Power.html">power</a> is called <a href="/Cubed.html">cubing</a>. </p> <p> A <a href="/UnitCube.html">unit cube</a> has <a href="/Inradius.html">inradius</a>, <a href="/Midradius.html">midradius</a>, and <a href="/Circumradius.html">circumradius</a> of </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline13.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="7" height="20" alt="r" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline14.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline15.svg" class="displayformula" style="max-width:100%;max-height:100%;" width="12" height="26" border="0" alt="1/2" /></td><td align="right" width="10"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline16.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="rho" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline17.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline18.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="43" height="27" alt="1/2sqrt(2)" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline19.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="R" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline20.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline21.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="47" height="27" alt="1/2sqrt(3)." /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr> </table> </div> <p> The cube has a <a href="/DihedralAngle.html">dihedral angle</a> of </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/Cube/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="57" height="26" alt=" alpha=1/2pi. " /></td><td align="right" width="3"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr> </table> </div> <p> In terms of the <a href="/Inradius.html">inradius</a> <img src="/images/equations/Cube/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="r" /> of a cube, its surface area <img src="/images/equations/Cube/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="S" /> and volume <img src="/images/equations/Cube/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="V" /> are given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline25.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="S" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline26.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline27.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="34" height="20" alt="24r^2" /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/Cube/Inline28.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="V" /></td><td align="center" width="14"><img src="/images/equations/Cube/Inline29.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/Cube/Inline30.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="8r^3," /></td><td align="right" width="10"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr> </table> </div> <p> so the volume, inradius, and surface area are related by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/Cube/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="64" height="39" alt=" (dV)/(dr)=S, " /></td><td align="right" width="3"> <div id="eqn9" class="eqnum"> (9) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/Cube/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="h=r" /> is the <a href="/HarmonicParameter.html">harmonic parameter</a> (Dorff and Hall 2003, Fjelstad and Ginchev 2003). </p> <div class="center-image"> <img src="/images/gifs/cube-o.jpg" class="" style="max-width:100%;max-height:100%;" width="300" alt="Origami cube" /> </div> <p> The illustration above shows an <a href="/Origami.html">origami</a> cube constructed from a single sheet of paper (Kasahara and Takahama 1987, pp. 58-59). </p> <p> Sodium chloride (NaCl; common table salt) naturally forms cubic crystals. </p> <div class="center-image"> <img src="/images/gifs/atomium.jpg" class="" style="max-width:100%;max-height:100%;" width="257" alt="Atomium" /> </div> <p> The world's largest cube is the Atomium, a structure built for the 1958 Brussels World's Fair, illustrated above (© 2006 Art Creation (ASBL); Artists Rights Society (ARS), New York; SABAM, Belgium). The Atomium is 334.6 feet high, and the nine <a href="/Sphere.html">spheres</a> at the vertices and center have diameters of 59.0 feet. The distance between the spheres along the edge of the cube is 95.1 feet, and the diameter of the tubes connecting the spheres is 9.8 feet. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="500.274" src="images/eps-svg/CubeAndDual_1000.svg" class="" alt="CubeAndDual" /> </div> <p> The <a href="/DualPolyhedron.html">dual polyhedron</a> of a <a href="/UnitCube.html">unit cube</a> is an <a href="/Octahedron.html">octahedron</a> with edge lengths <img src="/images/equations/Cube/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" />. </p> <p> The cube has the <a href="/OctahedralGroup.html">octahedral group</a> <img src="/images/equations/Cube/Inline33.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="O_h" /> of symmetries, and is an <a href="/EquilateralZonohedron.html">equilateral zonohedron</a> and a <a href="/Rhombohedron.html">rhombohedron</a>. It has 13 axes of symmetry: <img src="/images/equations/Cube/Inline34.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="6C_2" /> (axes joining midpoints of opposite edges), <img src="/images/equations/Cube/Inline35.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="22" alt="4C_3" /> (space diagonals), and <img src="/images/equations/Cube/Inline36.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="3C_4" /> (axes joining opposite face centroids). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="456.48" src="images/eps-svg/CubicalGraph_900.svg" class="" alt="CubicalGraph" /> </div> <p> The connectivity of the vertices of the cube is given by the <a href="/CubicalGraph.html">cubical graph</a>. </p> <p> Using so-called "wallet hinges," a ring of six cubes can be rotated continuously (Wells 1975; Wells 1991, pp. 218-219). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="535.255" src="images/eps-svg/CubeCutByPlanes_1100.svg" class="" alt="CubeCutByPlanes" /> </div> <p> The illustrations above show the cross sections obtained by cutting a <a href="/UnitCube.html">unit cube</a> centered at the origin with various planes. The following table summarizes the metrical properties of these slices. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">cutting plane</td><td align="left">face shape</td><td align="left">edge lengths</td><td align="left"><a href="/SurfaceArea.html">surface area</a></td><td align="left"><a href="/Volume.html">volume</a> of pieces</td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline37.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="z=0" /></td><td align="left"><a href="/Square.html">square</a></td><td align="left">1</td><td align="left">1</td><td align="left"><img src="/images/equations/Cube/Inline38.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" />, <img src="/images/equations/Cube/Inline39.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" /></td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline40.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="66" height="21" alt="x+z=0" /></td><td align="left"><a href="/Rectangle.html">rectangle</a></td><td align="left">1, <img src="/images/equations/Cube/Inline41.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" /></td><td align="left"><img src="/images/equations/Cube/Inline42.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" /></td><td align="left"><img src="/images/equations/Cube/Inline43.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" />, <img src="/images/equations/Cube/Inline44.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" /></td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline45.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="94" height="21" alt="x+y+z=0" /></td><td align="left"><a href="/Hexagon.html">hexagon</a></td><td align="left"><img src="/images/equations/Cube/Inline46.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="28" alt="1/2sqrt(2)" /></td><td align="left"><img src="/images/equations/Cube/Inline47.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="28" alt="3/4sqrt(3)" /></td><td align="left"><img src="/images/equations/Cube/Inline48.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" />, <img src="/images/equations/Cube/Inline49.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" /></td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline50.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="125" height="27" alt="x+y+z-1/2=0" /></td><td align="left"><a href="/EquilateralTriangle.html">equilateral triangle</a></td><td align="left"><img src="/images/equations/Cube/Inline51.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" /></td><td align="left"><img src="/images/equations/Cube/Inline52.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="28" alt="1/2sqrt(3)" /></td><td align="left"><img src="/images/equations/Cube/Inline53.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/6" />, <img src="/images/equations/Cube/Inline54.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="26" alt="5/6" /></td></tr> </table> </div> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="368.037" src="images/eps-svg/CubeHexagon_600.svg" class="" alt="CubeHexagon" /> </div> <p> As shown above, a <a href="/Plane.html">plane</a> passing through the <a href="/Midpoint.html">midpoints</a> of opposite edges (perpendicular to a <img src="/images/equations/Cube/Inline55.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="22" alt="C_3" /> axis) cuts the cube in a regular <a href="/Hexagon.html">hexagonal</a> <a href="/CrossSection.html">cross section</a> (Gardner 1960; Steinhaus 1999, p. 170; Kasahara 1988, p. 118; Cundy and Rollett 1989, p. 157; Holden 1991, pp. 22-23). Since there are four such axes, there are four possible <a href="/Hexagon.html">hexagonal</a> <a href="/CrossSection.html">cross sections</a>. If the vertices of the cube are <img src="/images/equations/Cube/Inline56.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="92" height="21" alt="(+/-1,+/-1+/-1)" />, then the vertices of the inscribed <a href="/Hexagon.html">hexagon</a> are <img src="/images/equations/Cube/Inline57.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="81" height="21" alt="(0,-1,-1)" />, <img src="/images/equations/Cube/Inline58.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="69" height="21" alt="(1,0,-1)" />, <img src="/images/equations/Cube/Inline59.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="(1,1,0)" />, <img src="/images/equations/Cube/Inline60.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="(0,1,1)" />, <img src="/images/equations/Cube/Inline61.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="69" height="21" alt="(-1,0,1)" />, and <img src="/images/equations/Cube/Inline62.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="81" height="21" alt="(-1,-1,0)" />. A <a href="/Hexagon.html">hexagon</a> is also obtained when the cube is viewed from above a corner along the extension of a space diagonal (Steinhaus 1999, p. 170). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="508.69" src="images/eps-svg/CubePlaneCuttingArea_1000.svg" class="" alt="CubePlaneCuttingArea" /> </div> <p> The maximal cross sectional area that can be obtained by cutting a unit cube with a plane passing through its center is <img src="/images/equations/Cube/Inline63.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" />, corresponding to a rectangular section intersecting the cube in two diagonally opposite edges and along two opposite face diagonals. The area obtained as a function of normal to the plane <img src="/images/equations/Cube/Inline64.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="(a,b,1)" /> is illustrated above (Hidekazu). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="203.831" src="images/eps-svg/CubeSpinning_800.svg" class="" alt="CubeSpinning" /> </div> <p> A <a href="/Hyperboloid.html">hyperboloid</a> of one sheet is obtained as the envelope of a cube rotated about a space diagonal (Steinhaus 1999, pp. 171-172; Kabai 2002, p. 11). The resulting volume for a cube with edge length <img src="/images/equations/Cube/Inline65.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" /> is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/Cube/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="106" height="27" alt=" V=1/3sqrt(3)pia^3 " /></td><td align="right" width="3"> <div id="eqn10" class="eqnum"> (10) </div> </td></tr> </table> </div> <p> (Cardot and Wolinski 2004). </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="544.043" src="images/eps-svg/CubeSolidofRevolution_950.svg" class="" alt="CubeSolidofRevolution" /> </div> <p> More generally, consider the <a href="/SolidofRevolution.html">solid of revolution</a> obtained for revolution axis passing through the center and the point <img src="/images/equations/Cube/Inline66.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="(x,y,1)" />, several examples of which are shown above. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="570.6" src="images/eps-svg/CubeSolidofRevolutionPlots_900.svg" class="" alt="CubeSolidofRevolutionPlots" /> </div> <p> As shown by Hidekazu, the solid with maximum volume is obtained for parameters of approximately <img src="/images/equations/Cube/Inline67.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="215" height="21" alt="(a,b)=(0.529307,0.237593)" />. This corresponds to the rightmost plot above. </p> <div class="table-responsive-noborders"> <table align="center" border="0"> <tr style=""><td align="center"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="146.066" src="images/eps-svg/cubeoct1_500.svg" class="" alt="cubeoct1" /> </div> </td><td align="center"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="162.193" src="images/eps-svg/cubeoct2_500.svg" class="" alt="cubeoct2" /> </div> </td></tr> </table> </div> <p> The centers of the faces of an <a href="/Octahedron.html">octahedron</a> form a cube, and the centers of the faces of a cube form an <a href="/Octahedron.html">octahedron</a> (Steinhaus 1999, pp. 194-195). The largest <a href="/Square.html">square</a> which will fit inside a cube of edge length <img src="/images/equations/Cube/Inline68.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" /> has each corner a distance 1/4 from a corner of a cube. The resulting <a href="/Square.html">square</a> has edge length <img src="/images/equations/Cube/Inline69.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="78" height="27" alt="3sqrt(2)a/4" />, and the cube containing that edge is called <a href="/PrinceRupertsCube.html">Prince Rupert's cube</a>. </p> <div class="table-responsive-noborders"> <table align="center" border="0"> <tr style=""><td align="center"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="131.555" src="images/eps-svg/StellaOctangula_400.svg" class="" alt="StellaOctangula" /> </div> </td><td align="center"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="129.946" src="images/eps-svg/StellaOctangulaCube_380.svg" class="" alt="StellaOctangulaCube" /> </div> </td><td align="center"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="111.185" src="images/eps-svg/RhombicDodecahedronCube_400.svg" class="" alt="RhombicDodecahedronCube" /> </div> </td></tr> </table> </div> <p> The solid formed by the faces having the edges of the <a href="/StellaOctangula.html">stella octangula</a> (left figure) as <a href="/PolygonDiagonal.html">polygon diagonals</a> is a cube (right figure; Ball and Coxeter 1987). Affixing a <a href="/SquarePyramid.html">square pyramid</a> of height 1/2 on each face of a cube having unit edge length results in a <a href="/RhombicDodecahedron.html">rhombic dodecahedron</a> (Brückner 1900, p. 130; Steinhaus 1999, p. 185). </p> <p> Since its eight faces are mutually perpendicular or parallel, the cube cannot be <a href="/Stellation.html">stellated</a>. </p> <p> The cube can be constructed by <a href="/Augmentation.html">augmentation</a> of a unit edge-length <a href="/Tetrahedron.html">tetrahedron</a> by a pyramid with height <img src="/images/equations/Cube/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="28" alt="1/6sqrt(6)" />. The following table gives polyhedra which can be constructed by <a href="/Augmentation.html">augmentation</a> of a <i>cube</i> by pyramids of given heights <img src="/images/equations/Cube/Inline71.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="h" />. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left"><img src="/images/equations/Cube/Inline72.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="h" /></td><td align="left">result</td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline73.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/6" /></td><td align="left"><a href="/TetrakisHexahedron.html">tetrakis hexahedron</a></td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline74.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="1/2" /></td><td align="left"><a href="/RhombicDodecahedron.html">rhombic dodecahedron</a></td></tr><tr style=""><td align="left"><img src="/images/equations/Cube/Inline75.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="28" alt="1/2sqrt(2)" /></td><td align="left">star <a href="/EquilateralPolygon.html">equilateral</a> 24-<a href="/Deltahedron.html">deltahedron</a></td></tr> </table> </div> <p> The <a href="/PolyhedronVertex.html">polyhedron vertices</a> of a cube of edge length 2 with face-centered axes are given by <img src="/images/equations/Cube/Inline76.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="96" height="21" alt="(+/-1,+/-1,+/-1)" />. If the cube is oriented with a space diagonal along the <a href="/z-Axis.html"><i>z</i>-axis</a>, the coordinates are (0, 0, <img src="/images/equations/Cube/Inline77.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(3)" />), (0, <img src="/images/equations/Cube/Inline78.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="62" height="25" alt="2sqrt(2/3)" />, <img src="/images/equations/Cube/Inline79.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="52" height="27" alt="1/sqrt(3)" />), (<img src="/images/equations/Cube/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" />, <img src="/images/equations/Cube/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="25" alt="sqrt(2/3)" />, <img src="/images/equations/Cube/Inline82.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="64" height="27" alt="-1/sqrt(3)" />), (<img src="/images/equations/Cube/Inline83.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="sqrt(2)" />, <img src="/images/equations/Cube/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="25" alt="-sqrt(2/3)" />, <img src="/images/equations/Cube/Inline85.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="52" height="27" alt="1/sqrt(3)" />), (0, <img src="/images/equations/Cube/Inline86.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="74" height="25" alt="-2sqrt(2/3)" />, <img src="/images/equations/Cube/Inline87.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="64" height="27" alt="-1/sqrt(3)" />), (<img src="/images/equations/Cube/Inline88.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="23" alt="-sqrt(2)" />, <img src="/images/equations/Cube/Inline89.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="25" alt="-sqrt(2/3)" />, <img src="/images/equations/Cube/Inline90.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="52" height="27" alt="1/sqrt(3)" />), (<img src="/images/equations/Cube/Inline91.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="23" alt="-sqrt(2)" />, <img src="/images/equations/Cube/Inline92.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="25" alt="sqrt(2/3)" />, <img src="/images/equations/Cube/Inline93.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="64" height="27" alt="-1/sqrt(3)" />), and the negatives of these vectors. A <a href="/Faceting.html">faceted</a> version is the <a href="/GreatCubicuboctahedron.html">great cubicuboctahedron</a>. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AugmentedTruncatedCube.html">Augmented Truncated Cube</a>, <a href="/BiaugmentedTruncatedCube.html">Biaugmented Truncated Cube</a>, <a href="/BidiakisCube.html">Bidiakis Cube</a>, <a href="/BislitCube.html">Bislit Cube</a>, <a href="/BrowkinsTheorem.html">Browkin's Theorem</a>, <a href="/CubeDissection.html">Cube Dissection</a>, <a href="/CubeDovetailingProblem.html">Cube Dovetailing Problem</a>, <a href="/CubeDuplication.html">Cube Duplication</a>, <a href="/CubicNumber.html">Cubic Number</a>, <a href="/CubicalGraph.html">Cubical Graph</a>, <a href="/Cuboid.html">Cuboid</a>, <a href="/GoursatsSurface.html">Goursat's Surface</a>, <a href="/HadwigerProblem.html">Hadwiger Problem</a>, <a href="/Hypercube.html">Hypercube</a>, <a href="/KellersConjecture.html">Keller's Conjecture</a>, <a href="/PentagonalWedge.html">Pentagonal Wedge</a>, <a href="/PlatonicSolid.html">Platonic Solid</a>, <a href="/PolyhedronColoring.html">Polyhedron Coloring</a>, <a href="/PrinceRupertsCube.html">Prince Rupert's Cube</a>, <a href="/Prism.html">Prism</a>, <a href="/RubiksCube.html">Rubik's Cube</a>, <a href="/SomaCube.html">Soma Cube</a>, <a href="/StellaOctangula.html">Stella Octangula</a>, <a href="/Tesseract.html">Tesseract</a>, <a href="/UnitCube.html">Unit Cube</a> <a href="/classroom/Cube.html" class="explore-classroom">Explore this topic in the MathWorld classroom</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/entities/geometry/cube/23/fz/wz/"> cube </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=1.05+*+12%2C000">1.05 * 12,000</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=circle%2C+diameter%3D10">circle, diameter=10</a></li> </ul> </div> </div>   <h2>References</h2><cite>Atomium. "Atomium: The Most Astonishing Building in the World." <a href="http://www.atomium.be/">http://www.atomium.be/</a>.</cite><cite>Ball, W. W. R. and Coxeter, H. S. M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/ericstreasuretro">Mathematical Recreations and Essays, 13th ed.</a></i> New York: Dover, 1987.</cite><cite>Beyer, W. H. (Ed.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/1584882913/ref=nosim/ericstreasuretro">CRC Standard Mathematical Tables, 28th ed.</a></i> Boca Raton, FL: CRC Press, pp. 127 and 228, 1987.</cite><cite>Brückner, M. <i>Vielecke under Vielflache.</i> Leipzig, Germany: Teubner, 1900.</cite><cite>Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <img src="/images/equations/Cube/Inline94.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="28" height="21" alt="<=4" />." <i>Disc. Math.</i> <b>186</b>, 69-94, 1998.</cite><cite>Cardot C. and Wolinski F. "Récréations scientifiques." <i>La jaune et la rouge</i>, No. 594, 41-46, 2004.</cite><cite>Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." <i>Phil. Trans. Roy. Soc. London Ser. A</i> <b>246</b>, 401-450, 1954.</cite><cite>Cundy, H. and Rollett, A. "Cube. <img src="/images/equations/Cube/Inline95.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="4^3" />" and "Hexagonal Section of a Cube." §3.5.2 and 3.15.1 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/ericstreasuretro">Mathematical Models, 3rd ed.</a></i> Stradbroke, England: Tarquin Pub., pp. 85 and 157, 1989.</cite><cite>Davie, T. "The Cube (Hexahedron)." <a href="http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/cube.html">http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/cube.html</a>.</cite><cite>Dorff, M. and Hall, L. "Solids in <img src="/images/equations/Cube/Inline96.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="R^n" /> Whose Area is the Derivative of the Volume." <i>College Math. J.</i> <b>34</b>, 350-358, 2003.</cite><cite>Eppstein, D. "Rectilinear Geometry." <a href="http://www.ics.uci.edu/~eppstein/junkyard/rect.html">http://www.ics.uci.edu/~eppstein/junkyard/rect.html</a>.</cite><cite>Fischer, G. (Ed.). Plate 2 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/3528089911/ref=nosim/ericstreasuretro">Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband.</a></i> Braunschweig, Germany: Vieweg, p. 3, 1986.</cite><cite>Fjelstad, P. and Ginchev, I. "Volume, Surface Area, and the Harmonic Mean." <i>Math. Mag.</i> <b>76</b>, 126-129, 2003.</cite><cite>Gardner, M. "Mathematical Games: More about the Shapes That Can Be Made with Complex Dominoes." <i>Sci. Amer.</i> <b>203</b>, 186-198, Nov. 1960.</cite><cite>Geometry Technologies. "Cube." <a href="http://www.scienceu.com/geometry/facts/solids/cube.html">http://www.scienceu.com/geometry/facts/solids/cube.html</a>.</cite><cite>Har'El, Z. "Uniform Solution for Uniform Polyhedra." <i>Geometriae Dedicata</i> <b>47</b>, 57-110, 1993.</cite><cite>Harris, J. W. and Stocker, H. "Cube" and "Cube (Hexahedron)." §4.2.4 and 4.4.3 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387947469/ref=nosim/ericstreasuretro">Handbook of Mathematics and Computational Science.</a></i> New York: Springer-Verlag, pp. 97-98 and 100, 1998.</cite><cite><a href="/contribute/updated_hyperlink.html#Cube"><img src="/images/entries/contribute_link.gif" width="88" height="24" border="0" alt="Update a link" /></a>Hidekazu, T. "Research on a Cube." <a href="http://www.biwako.ne.jp/~hidekazu/materials/cubee.htm">http://www.biwako.ne.jp/~hidekazu/materials/cubee.htm</a></cite><cite>Holden, A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486268519/ref=nosim/ericstreasuretro">Shapes, Space, and Symmetry.</a></i> New York: Dover, 1991.</cite><cite>Kabai, S. <i>Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica.</i> Püspökladány, Hungary: Uniconstant, p. 231, 2002.</cite><cite>Kasahara, K. "Cube A--Bisecting I," "Cube B--Bisecting II," "Cube C--Bisecting Horizontally," "Cube D--Bisecting on the Diagonal," "Cube E--Bisecting III," "Making a Cube from a Cube with a Single Cut," and "Module Cube." <i><a href="http://www.amazon.com/exec/obidos/ASIN/4817090014/ref=nosim/ericstreasuretro">Origami Omnibus: Paper-Folding for Everyone.</a></i> Tokyo: Japan Publications, pp. 104-108, 112, and 118-120, and 208, 1988.</cite><cite>Kasahara, K. and Takahama, T. <i><a href="http://www.amazon.com/exec/obidos/ASIN/4817090022/ref=nosim/ericstreasuretro">Origami for the Connoisseur.</a></i> Tokyo: Japan Publications, 1987.</cite><cite>Kern, W. F. and Bland, J. R. "Cube." §9 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0007FQY5S/ref=nosim/ericstreasuretro">Solid Mensuration with Proofs, 2nd ed.</a></i> New York: Wiley, pp. 19-20, 1948.</cite><cite>Maeder, R. E. "06: Cube." 1997. <a href="https://www.mathconsult.ch/static/unipoly/06.html">https://www.mathconsult.ch/static/unipoly/06.html</a>.</cite><cite>Malkevitch, J. "Nets: A Tool for Representing Polyhedra in Two Dimensions." <a href="http://www.ams.org/new-in-math/cover/nets.html">http://www.ams.org/new-in-math/cover/nets.html</a>.</cite><cite>Malkevitch, J. "Unfolding Polyhedra." <a href="http://www.york.cuny.edu/~malk/unfolding.html">http://www.york.cuny.edu/~malk/unfolding.html</a>.</cite><cite>Steinhaus, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486409147/ref=nosim/ericstreasuretro">Mathematical Snapshots, 3rd ed.</a></i> New York: Dover, 1999.</cite><cite>Turney, P. D. "Unfolding the Tesseract." <i>J. Recr. Math.</i> <b>17</b>, No. 1, 1-16, 1984-85.</cite><cite>Wells, D. "Puzzle Page." <i>Games and Puzzles.</i> Sep. 1975.</cite><cite>Wells, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/ericstreasuretro">The Penguin Dictionary of Curious and Interesting Geometry.</a></i> London: Penguin, pp. 41-42 and 218-219, 1991.</cite><cite>Wenninger, M. J. "The Hexahedron (Cube)." Model 3 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521098599/ref=nosim/ericstreasuretro">Polyhedron Models.</a></i> Cambridge, England: Cambridge University Press, p. 16, 1989.</cite>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Cube." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/Cube.html">https://mathworld.wolfram.com/Cube.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Cubes.html">Cubes</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Hexahedra.html">Hexahedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/PlatonicSolids.html">Platonic Solids</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/CanonicalPolyhedra.html">Canonical Polyhedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Isohedra.html">Isohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Parallelohedra.html">Parallelohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Rhombohedra.html">Rhombohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Space-FillingPolyhedra.html">Space-Filling Polyhedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Stereohedra.html">Stereohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Trapezohedra.html">Trapezohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/SolidGeometry.html">Solid Geometry</a> </li> <li> <a href="/topics/Polyhedra.html">Polyhedra</a> </li> <li> <a href="/topics/Zonohedra.html">Zonohedra</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/Folding.html">Folding</a> </li> <li> <a href="/topics/Origami.html">Origami</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicalRecords.html">Mathematical Records</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicsintheArts.html">Mathematics in the Arts</a> </li> <li> <a href="/topics/MathematicsinArchitecture.html">Mathematics in Architecture</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicalArt.html">Mathematical Art</a> </li> <li> <a href="/topics/PolyhedronNets.html">Polyhedron Nets</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Cantrell.html">Cantrell</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/HerrstromEmily.html">Herrstrom, Emily</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,247 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Wed Mar 5 2025</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2025 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>