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| {{Odd star polygon db|Odd star polygon stat table|p5/2}}
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| [[File:Inverted Pentagram circumscribed.svg|thumb|A pentagram inscribed in a circle;, also known as "[[pentacle]]"]]
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| A '''pentagram''' (sometimes known as a '''pentalpha''' or '''pentangle''' or a '''[[star polygon|star pentagon]]''') is the shape of a [[five-pointed star]] drawn with five straight strokes.
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| In medieval Christian tradition, the pentagram could represent the [[Holy Wounds|five wounds of Jesus]].
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| In the [[Renaissance]] it came to be associated with [[Renaissance magic|magic]] and [[occultism]], and is also found as a magic symbol in the folklore of early modern Germany (''[[Drudenfuss]]'').
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| In modern use, it is sometimes used as representing the [[Seal of Solomon]], and it has religious significance in various [[new religious movements]] (including certain forms of [[Neopaganism]]) as well as in [[occultism]].
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| The word ''pentagram'' comes from the [[Greek language|Greek]] word πεντάγραμμον (''pentagrammon''),<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dpenta%2Fgrammon πεντάγραμμον], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus; a noun form of adjectival πεντάγραμμος (''pentagrammos'') or πεντέγραμμος (''pentegrammos''), a word meaning roughly "five-lined" or "five lines"</ref> from πέντε (''pente''), "five" + γραμμή (''grammē''), "line".<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dpe%2Fnte πέντε], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus; [http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref>
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| The word "[[pentacle]]" is sometimes used synonymously with "pentagram"<ref>this usage is borne out by the Oxford English Dictionary, although that work specifies that a circumscription makes the shape more particularly a pentacle. "Pentacle", ''Oxford English Dictionary'': "Middle French ''pentacle'' talisman [before 1555, in English from 1561], most often in the form of a five-pointed star and its etymon post-classical Latin ''pentaculum '' [...] A pentagram, esp. one enclosed in a circle; a talisman or magical symbol in the shape of or inscribed with a pentagram. Also, in extended use: any similar magical symbol (freq. applied to a hexagram formed by two intersecting or interlaced equilateral triangles)."</ref> The word ''pentalpha'' is a learned modern (17th-century) revival of a post-classical Greek name of the shape.<ref>πένταλϕα
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| "five Alphas", interpreting the shape as five Α shapes overlapping at 72-degree angles.</ref>
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| ==Cultural significance==
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| ===Early history===
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| In early ([[First Dynasty of Ur|Ur I]]) [[Sumerian cuneiform|monumental Sumerian script]], a pentagram glyph served as a logogram for the word ''ub'', meaning "corner, angle, nook; a small room, cavity, hole; pitfall" (this later gave rise to the cuneiform sign UB {{Cuneiform|𒌒}}, composed of five wedges, further reduced to four in Assyrian cuneiform ).
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| The word ''Pentemychos'' ({{lang|grc|πεντέμυχος}} lit. "five corners" or "five recesses")<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dpente%2Fmuxos πεντέμυχος], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> was the title of the [[cosmogony]] of [[Pherecydes of Syros]].<ref>This is a lost book, but its contents are preserved in [[Damascius]], ''De principiis,'' quoted in Kirk and Raven, (1983) [1956], p. 55.</ref>
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| Here, the "five corners" are where the seeds of [[Chronos]] are placed within the Earth in order for the [[cosmos]] to appear.<ref name="Kirk and Raven">"the divine products of [[Chronos]]' seed, when disposed in five recesses, were called πεντέμυχος (Pentemychos)" {{cite book |first1=Geoffrey Stephen |last1=Kirk |first2=John Earle |last2=Raven |first3=Malcolm |last3=Schofield |year=1983 |origyear=1957 |title=The Presocratic Philosophers: A Critical History with a Selection of Texts |edition=2nd edition, illustrated, revised, reprint |publisher=Cambridge University Press |isbn= 978-0-521-27455-5 |pages=51–52, 55 |url=http://books.google.com/books?id=kFpd86J8PLsC&lpg=PP1&pg=PA52 |quote=the only other place in [[Homer]] where [[Ortygia|Ortygie]] {{sic}} is mentioned is ''[[Odyssey]]'' V, 123, where [[Orion (mythology)|Orion]], having been carried off by [[Eos]] [the dawn], is slain... by [[Artemis]]... since solstices would normally be observed at sunrise in summer, and so in the north-east-by-east direction, that is what the phrase might suggest... the dwelling-place of Eos... [[Colchis|''Aia'']]...}}</ref>
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| [[File:Hugieia-pentagram.svg|thumb|A Pythagorean "Hugieia Pentagram"<ref> Allman, G. J., ''Greek Geometry From Thales to Euclid'' (1889), p.26.</ref>]]
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| In [[Neoplatonism]], the pentagram was said to have been used as a symbol or sign of recognition by the [[Pythagoreans]], who called the pentagram {{lang|grc|ὑγιεία ''[[Hygieia|hugieia]]''}} "health"<ref> Allman, G. J., ''Greek Geometry From Thales to Euclid'', part I (1877), in ''Hermathena'' 3.5, pp. [https://archive.org/stream/greekgeometryfro00allmuoft#page/n31/mode/2upp. 183], [https://archive.org/stream/greekgeometryfro00allmuoft#page/n45/mode/2up 197], citing [[Iamblichus]] and the Scholiast on Aristophanes. The pentagram was said to have been so called from Pythagoras himself having written the letters Υ, Γ, Ι, Θ (= /ei/), Α on its vertices.</ref>
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| ===Western symbolism===
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| The pentagram was used in ancient times as a Christian symbol for the five [[sense]]s.<ref>''Christian Symbols Ancient and Modern'', Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, ISBN 0-7135-1960-6.</ref> or of the [[five wounds of Christ]].
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| A Christian use of the pentangle occurs in the 14th-century English poem ''[[Sir Gawain and the Green Knight]]'', in which the symbol decorates the shield of the hero, [[Gawain]]. The [[Gawain poet|unnamed poet]] credits the symbol's origin to King [[Solomon]], and says the symbol is key to understanding the work. The poet explains that each of the five interconnected points represents a virtue tied to a group of five. Gawain is keen in his [[five senses]], dextrous in his five fingers, faithful to the salvation provided through the [[Five Wounds]] of Christ, takes courage from the [[five joys]] that [[Mary, the mother of Jesus|Mary]] had of Jesus, and exemplifies the five virtues of [[knighthood]].<ref>{{cite journal |last1= Morgan |first1= Gerald |year= 1979 |title= The Significance of the Pentangle Symbolism in "Sir Gawain and the Green Knight" |journal= The Modern Language Review |volume= 74 |issue= 4 |pages= 769–790 |publisher= |jstor=3728227 |accessdate=June 28, 2013}}</ref>
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| [[Heinrich Cornelius Agrippa]] and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion. By the mid-19th century a further distinction had developed amongst occultists regarding the pentagram's orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, and was essentially "good". However, the influential writer [[Eliphas Levi]] called it evil whenever the symbol appeared the other way up.
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| *"A reversed pentagram, with two points projecting upwards, is a symbol of evil and attracts sinister forces because it overturns the proper order of things and demonstrates the triumph of matter over spirit. It is the goat of lust attacking the heavens with its horns, a sign execrated by initiates."<ref>{{cite book |title=Transcendental Magic, its Doctrine and Ritual |trans_title=Dogme et rituel de la haute magie |publisher=[[Red Wheel/Weiser/Conari|Weiser]] |location=[[York Beach, Maine|York Beach]] |first=Éliphas |last=Lévi |authorlink=Eliphas Levi |others=Trans. by [[A. E. Waite]] |year=1999 |origyear=1896 (translated), 1854 (first published) |oclc=263626874}}</ref>
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| *"The flaming star, which, when turned upside down, is the {{sic|hierol|gyphic}} sign of the goat of [[Black Magic]], whose head may be drawn in the star, the two horns at the top, the ears to the right and left, the beard at the bottom. It is the sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns."<ref>{{cite book |title=The Key of the Mysteries |trans_title=la Clef des grands mystères suivant Hénoch, Abraham, Hermès Trismégiste et Salomon |publisher=[[Red Wheel/Weiser/Conari|Weiser]] |location=Boston |first=Éliphas |last=Lévi |authorlink=Eliphas Levi |others=Trans. by [[Aleister Crowley]] |page=69 |year=2002 |origyear=1939 (translated), 1859 (first published) |oclc=49053462}}</ref>
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| *"Let us keep the figure of the Five-pointed Star always upright, with the topmost triangle pointing to heaven, for it is the seat of wisdom, and if the figure is reversed, perversion and evil will be the result."<ref>{{cite book |title=Magic, White and Black |publisher=The Path |location=New York |first=Franz |last=Hartmann |authorlink=Franz Hartmann |edition=5th |year=1895 |origyear=1886 |oclc=476635673}}</ref>
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| <center><gallery>
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| Image:Pentagram and human body (Agrippa).jpg|Man inscribed in a pentagram, from [[Heinrich Cornelius Agrippa]]'s ''[[Three Books of Occult Philosophy|De occulta philosophia libri tres]]''. The five signs at the pentagram's [[vertex (geometry)|vertices]] are [[astrological]].
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| Image:Inverted pentacle.PNG|Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.
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| Image:Pentagram (Levi).jpg|The [[occult]]ist and [[Magician (paranormal)|magician]] [[Eliphas Levi]]'s pentagram, which he considered to be a symbol of the [[Macrocosm and microcosm|microcosm]], or human.
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| </gallery></center>
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| The apotropaic use of the pentagram symbol in [[German folklore]] (called ''[[Drudenfuss]]'' in German) is referred to by [[Goethe]] in his ''[[Faust I|Faust]]'' (1808), where a pentagram prevents [[Mephistopheles]] from leaving a room (but did not prevent him from entering by the same way, as the outward pointing corner of the diagram happened to be imperfectly drawn):
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| :''Mephistopheles'':
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| ::I must confess, my stepping o'er
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| ::Thy threshold a slight hindrance doth impede;
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| ::The wizard-foot [Drudenfuss] doth me retain.
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| :''Faust:''
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| ::The pentagram thy peace doth mar?
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| ::To me, thou son of hell, explain,
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| ::How camest thou in, if this thine exit bar?
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| ::Could such a spirit aught ensnare?
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| :''Mephistopheles'':
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| ::Observe it well, it is not drawn with care,
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| ::One of the angles, that which points without,
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| ::Is, as thou seest, not quite closed.
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| ===East Asian symbolism===
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| [[Wu Xing]] ({{zh|c=五行|p=Wǔ Xíng}}) are the five phases, or five elements in Chinese tradition ([[Traditional Chinese medicine|medicine]], [[acupuncture]], [[feng shui]], and [[Taoism]]) They are similar to the ancient [[classical elements|Greek elements]], with more emphasis on their cyclic transformation than on their material aspects. The five phases are: Fire (火 ''huǒ''), Earth (土 ''tǔ''), Metal (金 ''jīn''), Water (水 ''shuǐ''), and Wood (木 ''mù''). They are sometimes{{according to whom|date=December 2013}}<!--the fact that there are five elements in China on its own has nothing to do with the topic "Pentagram". Show us evidence of the actual pentagram shape in Chinese tradition.-->
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| represented as a diagram in the shape of a pentagram.
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| In [[Japanese culture]], the pentagram (五芒星 ''gobōsei'') is a symbol of magical power, associated with the [[onmyoji]] [[Abe no Seimei]]; it is a diagram of the "overcoming cycle" of the five [[Chinese elements]].{{citation needed|date=December 2013}}
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| ===Use in modern occultism===
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| [[Image:Pentagram with one point down (de Guaita).jpg|thumb|right|A goat's head inscribed in a downward-pointing pentagram, from ''La Clef de la Magie Noire'' by Stanislas de Guaita (1897).]]
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| Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists.
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| Following [[Anton LaVey]], and ultimately based on a drawing by French nobleman and occultist Stanislas de Guaita (''La Clef de la Magie Noire'', 1897), the so-called [[Sigil of Baphomet]], a pentagram with two points up inscribed in a double circle with the head of a goat inside the pentagram is the copyrighted logo of the [[Church of Satan]].
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| [[Aleister Crowley]] made use of the pentagram in his [[Thelemic]] system of [[Magick (Aleister Crowley)|magick]]: an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation of [[Lon Milo DuQuette]].<ref>''The Magick of Aleister Crowley'' p 93 and endnote 31 to Chapter Five (p 247).</ref> Crowley contradicted his old comrades in the [[Hermetic Order of the Golden Dawn]], who, following Levi, considered this orientation of the symbol [[evil]] and associated it with the triumph of matter over spirit.
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| ===Use in new religious movements===
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| ====Bahá'í====
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| {{main|Bahá'í symbols}}
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| The five-pointed star is a symbol of the [[Bahá'í Faith]].<ref>[http://reference.bahai.org/en/t/se/DG/dg-141.html Bahá'í Reference Library – Directives from the Guardian, Pages 51–52<!--Bot generated title-->]</ref><ref>[http://bahai-library.com/uhj_nine_pointed_star Nine-Pointed Star, The:History and Symbolism by Universal House of Justice 24 January 1999]</ref> In the Bahá'í Faith, the star is known as the ''Haykal'' ({{lang-ar|"temple"}}), and it was initiated and established by the [[Báb]]. The Báb and [[Bahá'u'lláh]] wrote various works in the form of a pentagram.<ref>[http://altreligion.about.com/od/symbols/ig/Pentagrams/Haykal.htm Haykal - Baha'i Five Pointed Star Symbol]</ref>
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| <center><gallery>
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| Image:Haykal-Bab-2.gif|An unidentified work of the [[Báb]].
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| Image:Haykal2.gif|An unidentified work of the [[Báb]].
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| </gallery></center>
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| ====Latter Day Saint Movement====
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| [[The Church of Jesus Christ of Latter-day Saints]] began using both upright and inverted five-pointed stars in [[Temple (Latter Day Saints)|Temple]] architecture, dating from the [[Nauvoo Illinois Temple]], which was dedicated on 30 April 1846.<ref>See the [http://users.marshall.edu/~brown/nauvoo/nt-parent.html Nauvoo Temple] website discussing its architecture, and particularly the page on [http://users.marshall.edu/~brown/nauvoo/symbols.html Nauvoo Temple exterior symbolism]. Retrieved 16 December 2006.</ref> Other temples decorated with five-pointed stars in both orientations include the [[Salt Lake Temple]], and the [[Logan Utah Temple]]. These symbols come from the symbolism found in [[Book of Revelation|Revelation]] chapter 12, which says, "And there appeared a great wonder in heaven; a woman clothed with the sun, and the moon under her feet, and upon her head a crown of twelve stars."<ref>{{cite web |url=http://www.fairlds.org/pubs/Stars.pdf |title=Inverted Stars on LDS Temples |work=FAIRLDS.org |first=Matthew B |last=Brown |authorlink=Matthew B. Brown |year=2002 |archiveurl=http://web.archive.org/web/20080229073604/http://www.fairlds.org/pubs/Stars.pdf |archivedate=29 February 2008}}</ref>
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| ====Wicca====
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| [[Image:Pentacle 2.svg|thumb|right|A typical Neopagan pentagram (circumscribed).]]
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| [[Wiccan]] publications described ritual practices involving pentagrams since at least the 1980s,<ref>e.g. ''Buckland's Complete Book of Witchcraft'' (1985)</ref> and within [[Gardnerian Wicca|traditional forms]] of Wicca, a pentagram (no circle) with two points up is associated with the Second Degree Initiation.<ref name="Farrars initiation">{{cite book |first=Farrar |last=Stewart |authorlink=Stewart Farrar |title=The Witches' Way }} (1984) Chapter II – Second Degree Initiation</ref>
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| Wiccans began to use the pentagram as a [[Religious symbol|symbol of their faith]] by the 1990s.<ref>Scott Cunningham, ''Living Wicca: A Further Guide for the Solitary Practitioner'', Llewellyn Worldwide, 1993, p. 151.</ref>
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| Because of a perceived association with Satanism and occultism, many United States schools in the late 1990s have sought to prevent students from displaying the pentagram on clothing or jewelry.<ref>[http://www.religioustolerance.org/sch_clot5.htm "Religious Clothing in School"], Robinson, B.A., Ontario Consultants on Religious Tolerance, 20 August 1999, updated 29 April 2005. Retrieved 10 February 2006. {{cite press release | url=http://aclumich.org/modules.php?name=News&file=article&sid=26 | title=ACLU Defends Honor Student Witch Pentacle | publisher=American Civil Liberties Union of Michigan | date=10 February 1999 | accessdate=10 February 2006}} [http://www.post-gazette.com/magazine/20000927witch2.asp "Witches and wardrobes: Boy says he was suspended from school for wearing magical symbol"] Rouvalis, Cristina; Pittsburgh Post-Gazette, 27 September 2000. Retrieved 10 February 2006.</ref>
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| In public schools, such actions by administrators have been determined to be in violation of students' [[First Amendment to the United States Constitution|First Amendment]] right to [[Free Exercise Clause of the First Amendment|free exercise of religion]] in 2000.<ref>
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| {{cite web
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| |url=http://www.firstamendmentcenter.org//news.aspx?id=7234
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| |title=Federal judge upholds Indiana students' right to wear Wiccan symbols
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| |agency=Associated Press
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| |date=1 May 2000
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| |accessdate=21 September 2007}}</ref>
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| The encircled pentagram (referred to as a pentacle by the plaintiffs) was added to the list of 38 approved religious symbols to be placed on the tombstones of fallen service members at [[Arlington National Cemetery]] on 24 April 2007. The decision was made following ten applications from families of fallen soldiers who practiced [[Wicca]]. The government paid the families USD 225,000 to settle their pending lawsuits.<ref>{{cite news |url=http://www.cnn.com/2007/US/04/23/wiccan.grave.ap/index.html |title=Wiccan symbol OK for soldiers' graves |publisher=[[CNN|CNN.com]] |author=[[Associated Press]] |date=23 April 2007 |archiveurl=http://web.archive.org/web/20070426075801/http://www.cnn.com/2007/US/04/23/wiccan.grave.ap/index.html |archivedate=26 April 2007}}</ref><ref>{{cite web |url=http://www.cem.va.gov/hmm/emblems.asp |title=Burial and Memorials: Available Emblems of Belief for Placement on Government Headstones and Markers |publisher=[[United States Department of Veterans Affairs]] |date=3 July 2013 |accessdate=13 January 2014}}</ref>
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| ===Other modern use===
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| *The pentagram is featured on the national flags of [[flag of Morocco|Morocco]] (adopted 1915) and [[flag of Ethiopia|Ethiopia]] (adopted 1996), in both instances in reference to the [[Seal of Solomon]].
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| <center><gallery> | |
| Image:Flag of Morocco.svg|[[Morocco]]'s flag
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| Image:Flag of Ethiopia.svg|[[Ethiopia]]'s flag
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| </gallery></center>
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| *The [[Order of the Eastern Star]], an organization associated with [[Freemasonry]] (established 1850), used to have a point-down pentagram as its symbol, with the five [[isosceles triangle]]s of the points colored red, blue, yellow, white and green (the logo has been changed slightly{{year needed|date=December 2013}} and is now a five-pointed mullet with an inscribed pentagon, but no longer a pentagram)
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| <center><gallery>
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| Image:OrderEasternStar logo from saucer.jpg| [[Order of the Eastern Star]] emblem
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| </gallery></center>
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| ==Geometry==<!--This section is linked from [[Golden ratio]]-->
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| [[Image:Pentagram-phi.svg|thumb|A regular pentagram colored to distinguish its line segments of different lengths. The four lengths are in [[golden ratio]] to one another.]]
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| [[Image:PentagramFractal.PNG|thumb|Fractal pentagram drawn with a [[Euclidean vector|vector]] [[iteration]] program]]
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| The pentagram is the simplest [[regular polygon|regular]] [[star polygon]]. The pentagram contains ten points (the five points of the star, and the five vertices of the inner pentagon) and fifteen line segments. It is represented by the [[Schläfli symbol]] {5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its [[symmetry group]] the [[dihedral group]] of order 10.
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| ===Construction===
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| The pentagram can be constructed by connecting alternate vertices of a [[pentagon]]; see [[Pentagon#Construction of a regular pentagon|details of the construction]]. It can also be constructed as a [[stellation]] of a pentagon, by extending the edges of a pentagon until the lines intersect.
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| ===Golden ratio===
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| The [[golden ratio]], ''φ'' = (1 + √5) / 2 ≈ 1.618, satisfying
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| :<math>\varphi=1+2\sin(\pi/10)=1+2\sin 18^\circ\,</math>
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| :<math>\varphi=1/(2\sin(\pi/10))=1/(2\sin 18^\circ)\,</math>
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| :<math>\varphi=2\cos(\pi/5)=2\cos 36^\circ\,</math>
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| plays an important role in regular pentagons and pentagrams. Each intersection of edges sections the edges in the golden ratio: the ratio of the length of the edge to the longer segment is ''φ'', as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is ''φ''. As the four-color illustration shows:
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| :<math>\frac{\mathrm{red}}{\mathrm{green}} = \frac{\mathrm{green}}{\mathrm{blue}} = \frac{\mathrm{blue}}{\mathrm{magenta}} = \varphi.</math>
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| The pentagram includes ten [[isosceles triangle]]s: five [[acute triangle|acute]] and five [[obtuse triangle|obtuse]] isosceles triangles. In all of them, the ratio of the longer side to the shorter side is ''φ''. The acute triangles are [[Golden triangle (mathematics)|golden triangle]]s. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a [[Golden triangle (mathematics)|golden gnomon]].
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| ===Trigonometric values===
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| {{Further|Exact trigonometric constants#36°: regular pentagon{{!}}Exact trigonometric constants: Pentagon}}
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| :<math>\sin \frac{\pi}{10} = \sin 18^\circ = \frac{\sqrt 5 - 1}{4}=\frac{\varphi-1}{2}=\frac{1}{2\varphi}</math>
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| :<math>\cos \frac{\pi}{10} = \cos 18^\circ = \frac{\sqrt{2(5 + \sqrt 5)}}{4} </math>
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| :
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| :<math>\tan \frac{\pi}{10} = \tan 18^\circ = \frac{\sqrt{5(5 - 2 \sqrt 5)}}{5} </math>
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| :
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| :<math>\cot \frac{\pi}{10} = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5} </math>
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| :
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| :<math>\sin \frac{\pi}{5} = \sin 36^\circ = \frac{\sqrt{2(5 - \sqrt 5)} }{4}</math>
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| :
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| :<math>\cos \frac{\pi}{5} = \cos 36^\circ = \frac{\sqrt 5+1}{4} = \frac{\varphi}{2}</math>
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| :
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| :<math>\tan \frac{\pi}{5} = \tan 36^\circ = \sqrt{5 - 2\sqrt 5} </math>
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| :
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| :<math>\cot \frac{\pi}{5} = \cot 36^\circ = \frac{ \sqrt{5(5 + 2\sqrt 5)}}{5} </math>
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| As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths is ''φ'' times that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle.
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| ===Three-dimensional figures===
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| {{Further|Uniform polyhedron#Icosahedral symmetry{{!}}Uniform polyhedron: Icosahedral symmetry}}
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| Several [[polyhedra]] incorporate pentagrams:
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| <gallery>
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| Image:Pentagrammic prism.png|[[Pentagrammic prism]]
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| Image:Pentagrammic antiprism.png|[[Pentagrammic antiprism]]
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| Image:Pentagrammic crossed antiprism.png|[[Pentagrammic crossed-antiprism|Pentagrammic crossed antiprism]]
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| Image:Small_stellated_dodecahedron.png|[[Small stellated dodecahedron]]
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| Image:Small ditrigonal icosidodecahedron.png|[[Small ditrigonal icosidodecahedron]]
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| Image:Small dodecahemicosahedron.png|[[Small dodecahemicosahedron]]
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| Image:Snub dodecadodecahedron.png|[[Snub dodecadodecahedron]]
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| Image:Sixteenth stellation of icosidodecahedron.png|[[List of Wenninger polyhedron models#Stellations of icosidodecahedron|16th stellation of icosidodecahedron]]
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| </gallery>
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| | |
| ===Higher dimensions===
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| Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures:
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| {| class=wikitable width=720
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| !colspan=2|4D
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| !colspan=2|5D | |
| |-
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| |[[Image:4-simplex t0.svg|180px]]<br>The regular [[5-cell]] (4-[[simplex]]) has 5 vertices and 10 edges.
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| |[[Image:4-simplex t1.svg|180px]]<br>The [[rectified 5-cell]] has 10 vertices and 30 edges.
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| |[[File:5-simplex t1 A4.svg|180px]]<br>The [[rectified 5-simplex]] has 15 vertices, seen in this orthogonal projection as 3 nested pentagrams.
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| |[[File:5-simplex t2 A4.svg|180px]]<br>The [[birectified 5-simplex]] has 20 vertices, seen in this orthogonal projection as 4 overlapping pentagrams.
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| |}
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| All ten 4-dimensional [[Schläfli-Hess polychoron]] have either pentagrammic faces or [[vertex figure]] elements.
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| | |
| <!--very nice, but also original research
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| ==Natural occurrence==
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| {| class=wikitable width=500
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| |align=center|[[Image:Venus pentagram.png|480px]]<br>Successive [[inferior conjunction]]s of Venus repeat very near a 13:8 [[orbital resonance]] (The Earth orbits 8 times for every 13 orbits of Venus), creating a pentagrammic precession sequence.
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| |}
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| -->
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| ==See also==
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| *[[Barnstar]]
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| *[[Command-at-Sea Pin]]
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| *[[Enneagram (geometry)]]
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| *[[Heptagram]]
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| *[[Hexagram]]
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| *[[List of regular polytopes#Two Dimensions 2]]
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| *[[List of symbols]]
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| *[[Lute of Pythagoras]]
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| *[[Mullet (heraldry)]]
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| *[[Icosahedral symmetry#Achiral nonconvex uniform polyhedra|Nonconvex uniform polyhedra with full icosahedral symmetry]] (many show a pattern of pentagrams)
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| *[[Pentachoron]] – the 4-[[simplex]]
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| *[[Pentad]]
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| *[[Pentagram map]] (related concept)
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| *[[Pentalpha]]
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| *[[Petersen graph]]
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| *[[Ptolemy's theorem]]
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| *[[Red star]]
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| *[[Star (glyph)]]
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| *[[Stellation#Stellated polygons|Stellated polygons]]
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| == References ==
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| {{Reflist|2}}
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| | |
| ==Bibliography==
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| *{{cite book |title=Tilings and Patterns |publisher=W. H. Freeman |location=New York |first1=Branko |last1=Grünbaum |author1-link=Branko Grünbaum |first2=Geoffrey Colin |last2=Shephard |author2-link=Geoffrey Colin Shephard |year=1987 |isbn=978-0-7167-1193-3}}
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| *{{cite book |chapter=Polyhedra with Hollow Faces |title=Polytopes: Abstract, Convex and Computational |series=NATO ASI Series C: Mathematical and Physical Sciences |publisher=Springer Netherlands |location=Dordrecht |first1=Branko |last1=Grünbaum |author1-link=Branko Grünbaum |editor1=Bisztriczky, T. |editor2=[[Peter McMullen|McMullen, P.]] |editor3=Schneider, A. |editor4=Weiss, A. Ivić |volume=440 |pages=43–70 |year=1994 |isbn=978-94-010-4398-4 |doi=10.1007/978-94-011-0924-6_3}}
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| "Pentagram" article in ''The Continuum Encyclopedia of Symbols'' Becker, Udo, ed., Garmer, Lance W. translator, New York: Continuum Books, 1994, p. 230.
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| *''Signs and Symbols in Christian Art'' Ferguson, George, Oxford University Press: 1966, p. 59.
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| ==External links==
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| {{Commons category|Pentagrams}}
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| *{{Mathworld |urlname=Pentagram |title=Pentagram}}
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| *[http://www.cs.utk.edu/~mclennan/BA/PP.html The Pythagorean Pentacle] from the Biblioteca Arcana.
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| *[http://jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html In-depth analysis of the Golden Ratio]
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| *[http://freemasonry.bcy.ca/anti-masonry/pentagram.html The pentagram and Freemasonry]
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| {{witchcraft}}
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| {{WiccaandWitchcraft}}
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| {{Polygons}}
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| {{Use dmy dates|date=May 2012}}
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| {{LaVeyan Satanism}}
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| [[Category:Christian symbols]]
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| [[Category:Golden ratio]]
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| [[Category:National symbols of Ethiopia]]
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| [[Category:National symbols of Morocco]]
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| [[Category:Neopaganism]]
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| [[Category:Polygons]]
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| [[Category:Pythagorean symbols]]
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| [[Category:Star symbols]]
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| [[Category:Religious symbols]]
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| [[Category:Magic symbols]]
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