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You can view its source [e]" accesskey="e"><span>View source</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Icosahedron&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div 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vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="vector-more-collapsible-item mw-list-item"><a href="/wiki/Icosahedron"><span>Read</span></a></li><li id="ca-more-viewsource" class="selected vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Icosahedron&action=edit"><span>View source</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Icosahedron&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Icosahedron" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Icosahedron" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Icosahedron&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DIcosahedron%26action%3Dedit"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DIcosahedron%26action%3Dedit"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q188758" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> </div> <div id="contentSub"><div id="mw-content-subtitle">← <a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></div></div> <div id="mw-content-text" class="mw-body-content"><p>You do not have permission to edit this page, for the following reasons: </p> <ul class="permissions-errors"><li class="mw-permissionerror-protectedpagetext"><div class="mw-parser-output"> <div class="mw-parser-output"><style data-mw-deduplicate="TemplateStyles:r1099782930">.mw-parser-output .pptext-whywhat h2{margin-top:1em;border-bottom:0;font-size:130%;font-weight:bold;padding:0.15em}.mw-parser-output .pptext-submit{list-style:none;display:inline;text-align:center}.mw-parser-output .pptext-whywhat{display:flex;flex-wrap:wrap;column-gap:2em}.mw-parser-output .pptext-whywhat>div{flex:1 1 400px}</style><style data-mw-deduplicate="TemplateStyles:r1238441935">.mw-parser-output .fmbox{clear:both;margin:0.2em 0;width:100%;border:1px solid #a2a9b1;background-color:var(--background-color-interactive-subtle,#f8f9fa);box-sizing:border-box;color:var(--color-base,#202122)}.mw-parser-output .fmbox-warning{border:1px solid #bb7070;background-color:#ffdbdb}.mw-parser-output .fmbox-editnotice{background-color:transparent}.mw-parser-output .fmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .fmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .fmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .fmbox .mbox-invalid-type{text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .fmbox-warning{background-color:#300}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .fmbox-warning{background-color:#300}}</style><table id="mw-protectedpagetext" class="plainlinks fmbox fmbox-system" role="presentation" style="border-style:none;"><tbody><tr><td class="mbox-text"><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="plainlinks metadata ambox ambox-protection" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/60px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/80px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div></td><td class="mbox-text" style="font-weight:bold; font-size:130%;"><div class="mbox-text-span"><div style="text-align: center;">This page is currently semi-protected so that only <a href="/wiki/Wikipedia:User_access_levels#Autoconfirmed" title="Wikipedia:User access levels">established</a>, <a href="/wiki/Wikipedia:Why_create_an_account%3F" title="Wikipedia:Why create an account?">registered users</a> can edit it.</div></div></td></tr></tbody></table> <div class="pptext-whywhat"> <div class="pptext-why"> <div class="mw-heading mw-heading2"><h2 id="Why_is_the_page_protected?"><span id="Why_is_the_page_protected.3F"></span>Why is the page protected?</h2></div> <ul> <li>While most articles can be edited by anyone, <a href="/wiki/Wikipedia:Protection_policy#Semi-protection" title="Wikipedia:Protection policy">semi-protection</a> is sometimes necessary to prevent <a href="/wiki/Wikipedia:Vandalism" title="Wikipedia:Vandalism">vandalism</a> to popular pages.</li> <li>The reason for protection can be found in the <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:Log&type=protect&page=Icosahedron">protection log</a>. If there are no relevant entries in the protection log, the page may have been moved after being protected. </li> </ul> </div> <div class="pptext-what"> <div class="mw-heading mw-heading2"><h2 id="What_can_I_do?"><span id="What_can_I_do.3F"></span>What can I do?</h2></div> <ul> <li>If you have a user account, <a href="/wiki/Special:UserLogin" title="Special:UserLogin">log in</a> first. If you do not yet have an account, you may <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:UserLogin&type=signup&campaign=semiprotectednotice">create one</a>; after <a href="/wiki/Wikipedia:User_access_levels#Autoconfirmed_users" title="Wikipedia:User access levels">4 days and 10 edits</a>, you will be able to edit semi-protected pages.</li><li><a href="/wiki/Talk:Icosahedron" title="Talk:Icosahedron">Discuss this page</a> with others.</li> <li>For move-protected pages, see <a href="/wiki/Wikipedia:Requested_moves" title="Wikipedia:Requested moves">requested moves</a>.</li> <li><a href="/wiki/Wikipedia:Requests_for_page_protection#Current_requests_for_reduction_in_protection_level" title="Wikipedia:Requests for page protection">Request that the page's protection level be reduced</a>.</li> <li><a href="/wiki/Help:Introduction" title="Help:Introduction">Find out more about how to get started editing Wikipedia</a>.</li> <li>If you have noticed an error or have a suggestion for a <b>simple, non-controversial change</b>, you can submit an edit request by clicking the button below and following the instructions. An <a href="/wiki/Wikipedia:User_access_levels#Autoconfirmed_users" title="Wikipedia:User access levels">established user</a> may then make the change on your behalf. Please check <a href="/wiki/Talk:Icosahedron" title="Talk:Icosahedron">the talk page</a> first in case the issue is already being discussed.</li> <li class="pptext-submit"><div> <p><span class="plainlinks clickbutton"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Talk%3AIcosahedron&preload=Template%3ASubmit+an+edit+request%2Fpreload&action=edit&section=new&editintro=Template%3AEdit+semi-protected%2Feditintro&preloadtitle=Semi-protected+edit+request+on+27+November+2024&preloadparams%5B%5D=edit+semi-protected&preloadparams%5B%5D=Icosahedron"><span class="mw-ui-button mw-ui-progressive">Submit an edit request</span></a></span> </p> </div></li> <li>If you wrote any text, please save it temporarily to your device until you can edit this page.</li> </ul> </div> </div></td></tr></tbody></table></div> </div></li><li class="mw-permissionerror-blockedtext"> <div id="mw-blocked-text" style="border: 1px solid #AAA; background-color: var(--background-color-warning-subtle, ivory); color: inherit; padding: 1.5em; width: 100%; box-sizing: border-box;"> <div style="text-align: center;"><span style="font-size: 26px;"><span typeof="mw:File"><a href="/wiki/File:Stop_hand_nuvola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/50px-Stop_hand_nuvola.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/75px-Stop_hand_nuvola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/100px-Stop_hand_nuvola.svg.png 2x" data-file-width="240" data-file-height="240" /></a></span><b> This IP address has been <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> from <i>editing</i> Wikipedia.</b></span><br /><span style="font-size: 18px;">This does not affect your ability to <i>read</i> Wikipedia pages.</span></div><div class="paragraphbreak" style="margin-top:0.5em"></div><b>Most people who see this message have done nothing wrong.</b> Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.<div class="paragraphbreak" style="margin-top:0.5em"></div> <p>The IP address or range 8.222.128.0/17 has been <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> by <a href="/wiki/User:L235" title="User:L235">‪L235‬</a> for the following reason(s): </p> <div style="padding:10px; background:var(--background-color-base, white); color:inherit; border:1px #666 solid;"> <div class="user-block colocation-webhost" style="margin-bottom: 0.5em; background-color: #ffefd5; border: 1px solid #AAA; padding: 0.7em;"> <figure class="mw-halign-left" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/40px-Server-multiple.svg.png" decoding="async" width="40" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/60px-Server-multiple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/80px-Server-multiple.svg.png 2x" data-file-width="744" data-file-height="1052" /></span><figcaption></figcaption></figure><b>The <a href="/wiki/IP_address" title="IP address">IP address</a> that you are currently using has been blocked because it is believed to be a <a href="/wiki/Web_hosting_service" title="Web hosting service">web host provider</a> or <a href="/wiki/Colocation_centre" title="Colocation centre">colocation provider</a>.</b> To prevent abuse, <a href="/wiki/Wikipedia:Open_proxies" title="Wikipedia:Open proxies">web hosts and colocation providers may be blocked</a> from editing Wikipedia. <div style="border-top: 1px solid #AAA; clear: both">You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a <a href="/wiki/Wikipedia:Open_proxies" title="Wikipedia:Open proxies">proxy</a> or <a href="/wiki/Virtual_private_network" title="Virtual private network">VPN</a>. <p><b>We recommend that you attempt to use another connection to edit.</b> For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in. </p><p>If you do not have any other way to edit Wikipedia, you will need to <a href="/wiki/Wikipedia:IP_block_exemption#Requesting_and_granting_exemption" title="Wikipedia:IP block exemption">request an IP block exemption</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">How to appeal if you are confident that your connection does not use a colocation provider's IP address:</div><div class="hidden-content mw-collapsible-content" style=""> If you are confident that you are not using a web host, you may <a href="/wiki/Wikipedia:Appealing_a_block" title="Wikipedia:Appealing a block">appeal this block</a> by adding the following text on your <a href="/wiki/Help:Talk_pages" title="Help:Talk pages">talk page</a>: <code>{{<a href="/wiki/Template:Unblock" title="Template:Unblock">unblock</a>|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. <i>Place any further information here.</i> ~~~~}}</code>. <b>You must fill in the blank with your IP address for this block to be investigated.</b> Your IP address can be determined <span class="plainlinks"><b><a class="external text" href="https://en.wikipedia.org/wiki/Wikipedia:Get_my_IP_address?withJS=MediaWiki:Get-my-ip.js">here</a></b></span>. Alternatively, if you wish to keep your IP address private you can use the <a href="/wiki/Wikipedia:Unblock_Ticket_Request_System" title="Wikipedia:Unblock Ticket Request System">unblock ticket request system</a>. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.</div></div> <p><span class="sysop-show" style="font-size: 85%;"><span style="border:#707070 solid 1px;background-color:#ffe0e0;padding:2px"><b>Administrators:</b></span> The <a href="/wiki/Wikipedia:IP_block_exemption" title="Wikipedia:IP block exemption">IP block exemption</a> user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a <a href="/wiki/Wikipedia:CheckUser" title="Wikipedia:CheckUser">CheckUser</a> needs to take a look at the account. This can be requested most easily at <a href="/wiki/Wikipedia:SPI#Quick_CheckUser_requests" class="mw-redirect" title="Wikipedia:SPI">SPI Quick Checkuser Requests</a>. <b>Unblocking</b> an IP or IP range with this template <b>is highly discouraged</b> without at least contacting the blocking administrator.</span> </p> </div></div> </div> <p>This block will expire on 18:23, 24 August 2026. Your current IP address is 8.222.208.146. </p> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>Even when blocked, you will <i>usually</i> still be able to edit your <a href="/wiki/Special:MyTalk" title="Special:MyTalk">user talk page</a>, as well as <a href="/wiki/Wikipedia:Emailing_users" title="Wikipedia:Emailing users">email</a> administrators and other editors. </p> </div> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>For information on how to proceed, please read the <b><a href="/wiki/Wikipedia:Appealing_a_block#Common_questions" title="Wikipedia:Appealing a block">FAQ for blocked users</a></b> and the <a href="/wiki/Wikipedia:Appealing_a_block" title="Wikipedia:Appealing a block">guideline on block appeals</a>. 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The name comes {{ety|grc|''εἴκοσι'' (eíkosi)|twenty||''ἕδρα'' (hédra)|seat}}. The plural can be either "icosahedra" ({{IPAc-en|-|d|r|ə}}) or "icosahedrons". There are infinitely many non-[[similarity (geometry)|similar]] shapes of icosahedra, some of them being more symmetrical than others. The best known is the ([[convex polyhedron|convex]], non-[[stellation|stellated]]) [[regular icosahedron]]—one of the [[Platonic solid]]s—whose faces are 20 [[equilateral triangle]]s. ==Regular icosahedra== {{multiple image |image1 = Icosahedron.png |caption1 = Convex [[regular icosahedron]] |image2 = Great icosahedron.png |caption2 = [[Great icosahedron]] |header = Two kinds of regular icosahedra |total_width = 300px }} There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 [[equilateral triangle]] faces with five meeting at each of its twelve vertices. Both have [[icosahedral symmetry]]. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. ===Convex regular icosahedron=== {{Main|Regular icosahedron}} [[File:golden_rectangle_regular_icosahedron.jpg|thumb|upright|Three interlocking [[golden rectangle]]s inscribed in a con&shy;vex regular icosahedron]] The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular [[Platonic solid]]s, and is represented by its [[Schläfli symbol]] {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex. Its [[dual polyhedron]] is the [[regular dodecahedron]] {5, 3} having three regular pentagonal faces around each vertex. ===Great icosahedron=== {{main|Great icosahedron}} [[File:Icosahedron-spinoza.jpg|alt=A detail of Spinoza monument in Amsterdam|thumb|upright|A detail of [[Baruch Spinoza|Spinoza]] monument in [[Amsterdam]]]] The [[great icosahedron]] is one of the four regular star [[Kepler-Poinsot polyhedra]]. Its [[Schläfli symbol]] is {3, {{sfrac|5|2}}}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a [[pentagram]] rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges. Its [[dual polyhedron]] is the [[great stellated dodecahedron]] {{{sfrac|5|2}}, 3}, having three regular star pentagonal faces around each vertex. ==Stellated icosahedra== [[Stellation]] is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book ''[[The Fifty-Nine Icosahedra]]'', Coxeter et al. enumerated 58 such stellations of the regular icosahedron. Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them. Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such. {{Icosahedron stellations}} ==Pyritohedral symmetry== {| class=wikitable align=right width="320" !bgcolor=#e7dcc3 colspan=2|Pyritohedral and tetrahedral symmetries |- |bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_h|3|node_h|4|node}} (pyritohedral) [[File:Uniform polyhedron-43-h01.svg|24px]]<br />{{CDD|node_h|3|node_h|3|node_h}} (tetrahedral) [[File:Uniform polyhedron-33-s012.svg|24px]] |- |bgcolor=#e7dcc3|[[Schläfli symbol]]||s{3,4}<br />sr{3,3} or <math>s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}</math> |- |bgcolor=#e7dcc3|[[Face (geometry)|Faces]]||20 triangles:<br />8 equilateral<br />12 isosceles |- |bgcolor=#e7dcc3|[[Edge (geometry)|Edges]]||30 (6 short + 24 long) |- |bgcolor=#e7dcc3|[[Vertex (geometry)|Vertices]]||12 |- |bgcolor=#e7dcc3|[[List of spherical symmetry groups#Polyhedral symmetry|Symmetry group]]||[[Pyritohedral symmetry|T<sub>h</sub>]], [4,3<sup>+</sup>], (3*2), order 24 |- |bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||[[Tetrahedral symmetry|T<sub>d</sub>]], [3,3]<sup>+</sup>, (332), order 12 |- |bgcolor=#e7dcc3|[[Dual polyhedron]]||[[Pyritohedron]] |- |bgcolor=#e7dcc3|Properties||[[convex set|convex]] |- !colspan=2|[[File:Pseudoicosahedron flat.png|200px]]<br />[[Net (polyhedron)|Net]] |- | colspan=2| {| width=320px align=right |[[File:Icosahedron in cuboctahedron.png|220px]] |[[File:Icosahedron in cuboctahedron net.png|100px]] |- |colspan=2|A regular icosahedron is topologically identical to a [[cuboctahedron]] with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a [[Kinematics of the cuboctahedron|kinematic transformation between cuboctahedron and icosahedron]]. |} |} A ''regular icosahedron'' can be distorted or marked up as a lower [[pyritohedral]] symmetry,<ref name=Koca>{{cite journal| last1=Koca|first1=Nazife| last2=Al-Mukhaini|first2=Aida| last3=Koca|first3=Mehmet| last4=Al Qanobi|first4=Amal| date=2016-12-01| page=139| title=Symmetry of the Pyritohedron and Lattices| volume=21| doi=10.24200/squjs.vol21iss2pp139-149| journal=Sultan Qaboos University Journal for Science [SQUJS]|issue=2 | doi-access=free}}</ref><ref name=Baez/> and is called a '''snub octahedron''', '''snub tetratetrahedron''', '''snub tetrahedron''', and '''pseudo-icosahedron'''.<ref>{{cite book | last = Kappraff | first = Jay | year = 1991 | title = Connections: The Geometric Bridge Between Art and Science | publisher = World Scientific | edition = 2nd | url = https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475 | page = 475 }}</ref> This can be seen as an [[alternation (geometry)|alternated]] [[truncated octahedron]]. If all the triangles are [[equilateral triangle|equilateral]], the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently. [[Pyritohedral symmetry]] has the symbol (3*2), [3<sup>+</sup>,4], with order 24. [[Tetrahedral symmetry]] has the symbol (332), [3,3]<sup>+</sup>, with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent [[isosceles triangle]]s. These symmetries offer [[Coxeter diagram]]s: {{CDD|node|4|node_h|3|node_h}} and {{CDD|node_h|3|node_h|3|node_h}} respectively, each representing the lower symmetry to the [[regular icosahedron]] {{CDD|node_1|3|node|5|node}}, (*532), [5,3] [[icosahedral symmetry]] of order 120. ===Cartesian coordinates=== [[File:Truncated octahedron internal rectangles.png|150px|thumb|left|Construction from the vertices of a [[truncated octahedron]], showing internal rectangles.]] The [[Cartesian coordinates]] of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the [[truncated octahedron]] with [[alternation (geometry)|alternated]] vertices deleted. This construction is called a ''snub tetrahedron'' in its regular icosahedron form, generated by the same operations carried out starting with the vector (''ϕ'', 1, 0), where ''ϕ'' is the [[golden ratio]].<ref name=Baez>{{cite web|url=http://math.ucr.edu/home/baez/golden.html|title=Fool's Gold|author=John Baez|date=September 11, 2011}}</ref> ===Jessen's icosahedron=== [[File:Jessen's icosahedron.svg|left|thumb|Jessen's icosahedron]] {{main|Jessen's icosahedron}} In Jessen's icosahedron, sometimes called ''Jessen's orthogonal icosahedron'', the 12 isosceles faces are arranged differently so that the figure is non-convex and has [[right angle|right]] [[dihedral angle]]s. It is [[Hilbert's third problem|scissors congruent]] to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube. {{Clear}} ===Cuboctahedron=== [[File:A3-P5-P3.gif|thumb|Progressions between an [[octahedron]], [[pseudoicosahedron]], and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.]] {{main|Kinematics of the cuboctahedron}} A regular icosahedron is topologically identical to a [[cuboctahedron]] with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and [[Orientable double cover|double cover]] octahedron. Cyclical kinematic transformations among the members of this family exist. ==Other icosahedra== [[File:Rhombic icosahedron.png|thumb|[[Rhombic icosahedron]]]] ===Rhombic icosahedron=== {{Main|Rhombic icosahedron}} The [[rhombic icosahedron]] is a [[zonohedron]] made up of 20 congruent rhombs. It can be derived from the [[rhombic triacontahedron]] by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not [[face-transitive]]. ===Pyramid and prism symmetries=== Common icosahedra with pyramid and prism symmetries include: *19-sided [[pyramid]] (plus 1 base = 20). *18-sided [[prism (geometry)|prism]] (plus 2 ends = 20). *9-sided [[antiprism]] (2 sets of 9 sides + 2 ends = 20). *10-sided [[bipyramid]] (2 sets of 10 sides = 20). *10-sided [[trapezohedron]] (2 sets of 10 sides = 20). ===Johnson solids=== Several [[Johnson solid]]s are icosahedra:<ref>[http://mathworld.wolfram.com/Icosahedron.html Icosahedron] on Mathworld.</ref> {| class=wikitable width=720 !J22 !J35 !J36 !J59 !J60 !J92 |- align=center valign=top |[[File:Gyroelongated triangular cupola.png|120px]]<br />[[Gyroelongated triangular cupola]] |[[File:Elongated triangular orthobicupola.png|120px]]<br />[[Elongated triangular orthobicupola]] |[[File:Elongated triangular gyrobicupola.png|120px]]<br />[[Elongated triangular gyrobicupola]] |[[File:Parabiaugmented dodecahedron.png|120px]]<br />[[Parabiaugmented dodecahedron]] |[[File:Metabiaugmented dodecahedron.png|120px]]<br />[[Metabiaugmented dodecahedron]] |[[File:Triangular hebesphenorotunda.png|120px]]<br />[[Triangular hebesphenorotunda]] |- |[[File:Johnson solid 22 net.png|120px]] |[[File:Johnson solid 35 net.png|120px]] |[[File:Johnson solid 36 net.png|120px]] |[[File:Johnson solid 59 net.png|120px]] |[[File:Johnson solid 60 net.png|120px]] |[[File:Johnson solid 92 net.png|120px]] |- align=center valign=top |16 triangles<br />3 squares<br />&nbsp;<br />1 hexagon |8 triangles<br />12 squares |8 triangles<br />12 squares |10 triangles<br />&nbsp;<br />10 pentagons |10 triangles<br />&nbsp;<br />10 pentagons |13 triangles<br />3 squares<br />3 pentagons<br />1 hexagon |} == See also == * [[Truncated icosahedron]] * [[600-cell]] * [[Icosoku]] ==References== {{Commons category}} {{reflist}} {{Polyhedra}} {{Polyhedron navigator}} {{Authority control}} [[Category:Geodesic polyhedra]] 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