Pentagram

Early pentagrams have been found on Sumerian pottery from Ur c. 3500 BCE, and the five-pointed star was at various times the symbol of Ishtar or Marduk.{{cite book |last=Budge |first=Sir E. A. Wallis |title=Amulets and Talismans |date=1968 |pages=433}}{{cite web |last=Scott |first=Dustin Jon |date=2006 |title=History of the Pentagram |url=https://djonscott.com/pent-hist.html |archive-url=https://web.archive.org/web/20210427111819/https://djonscott.com/pent-hist.html |url-status=usurped |archive-date=27 April 2021 |access-date=18 May 2021 |website= |publisher= |quote=}}File:Hugieia-pentagram.svg

File:Right-handed interlaced pentagram.svg often bears the left-handed version.]]

File:Japanese Crest Abe no Seimei Hann.svg's pentagram mon represents the Wu Xing.]]

Pentagram symbols from about 5,000 years ago were found in the Liangzhu culture of China.{{Cite web |url=https://fashion.chinadaily.com.cn/a/201909/23/WS5d885fd5a31099ab995e19ea.html |title=距今5000年！良渚文物中發現最古老五角星圖案 |author=馬愛平 |date= September 23, 2019 |publisher=China Daily |language=Chinese |accessdate= |quote= }}{{cite news |last1=Chen |first1=Minzhen |title=Chinese antiques give new insight into history of pentagram |url=http://www.csstoday.com/Item/12413.aspx |access-date=1 December 2024 |work=Chinese Social Sciences Today |publisher=Social Sciences in China Press |date=2024}} A pentagram appeared in a Chinese text on music theory from the Warring States period ({{circa|475}}{{snd}}221 BC) as a diagram of the mathematical relations between the five notes in a particular Chinese musical scale.

The pentagram was known to the ancient Greeks, with a depiction on a vase possibly dating back to the 7th century BCE.Coxeter, H.S.M.; Regular Polytopes, 3rd edn, Dover, 1973, p. 114. Pythagoreanism originated in the 6th century BCE and used the pentagram as a symbol of mutual recognition, of wellbeing, and to recognize good deeds and charity.Ball, W. W. Rouse and Coxeter, H. S. M.; Mathematical Recreations and Essays, 13th Edn., Dover, 1987, p. 176.

From around 300–150 BCE the pentagram stood as the symbol of Jerusalem, marked by the 5 Hebrew letters ירשלם spelling its name.{{Cite web|url=https://aboutjewishpeople.com/star-of-david-vs-pentagram/|title=Star of David vs. Pentagram: Everything You Need to Know|date=17 July 2020}}

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 }} {{lang|grc-Latn|hugieia}} "health".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.{{cite book |last1=Agrippa von Nettesheim |first1=Heinrich Cornelius |title=De occulta philosophia libri tres |date=1533 |location=Cologne |pages=160, 163, 276-277 |url=https://www.loc.gov/resource/rbc0001.2009gen12345/?sp=280 |access-date=28 November 2024 |quote={{lang|la|eius generis sunt signaculum Constantino ostensum, quod crucem plerique appelabant, Latinis literis inscriptum, 'in hoc vince', et alterum revelatum Antiocho cognomine Soteris, in figura pentagoni, quod sanitatem edicit, nam resolutum in literas, edicit vocabulum ὑγίεα, id est, 'sanitas', in quorum signorum fidutia et virtute, uterque regum insignem contra hostes victoriam reportauit. Sic Iudas qui ob eam rem postea cognominatus est Machabaeus, cum Iudaeis pugnaturus contra Antiochum Eupatorem, nobile illud signaculum מׄכׄבׄיׄ ad angelum accepit,}}}}

In Serer religion and Serer cosmogony, the pentagram, called Yoonir (or "Yooniir" in Serer) is the symbol of the Universe amongst the Serer people.Gravrand, Henry, La civilisation sereer, vol. II : Pangool, Nouvelles éditions africaines, Dakar, 1990, pp. 20, {{ISBN|2-7236-1055-1}}Madiya, Clémentine Faïk-Nzuji, Canadian Museum of Civilization, Canadian Centre for Folk Culture Studies. "International Centre for African Language, Literature and Tradition", (Louvain, Belgium), pp. 27, 155, {{ISBN|0-660-15965-1}} In Serer religion and Serer primordial time, the peak of the Star represents the Serer supreme deity and creator, Roog. The other four points represent the cardinal points of the Universe. The crossing of the lines ("bottom left" and "top right" and "top left and bottom right") pinpoints the axis of the Universe, that all energies pass. The top point is "the point of departure and conclusion, the origin and the end".Madiya, Clémentine Faïk-Nzuji, Canadian Museum of Civilization, Canadian Centre for Folk Culture Studies, "International Centre for African Language, Literature and Tradition", (Louvain, Belgium), pp. 27, 155, {{ISBN|0-660-15965-1}} Yoonir also represents "good fortune and destiny" – in a Serer religious sense, and in an enthno nationlistic sense–following centuries of their religious and ethnic persecution, it also represents the Serer people,Gravrand, Henry, La civilisation sereer, vol. II : Pangool, Nouvelles éditions africaines, Dakar, 1990, p. 21, {{ISBN|2-7236-1055-1}} an ethnoreligious group and nation–today, found in Senegal, Gambia, and Mauritania.[https://www.jstor.org/stable/1161299 "Charisma and Ethnicity in Political Context: A Case Study in the Establishment of a Senegalese Religious Clientele"], Leonardo A. Villalón, Journal of the International African Institute, Vol. 63, No. 1 (1993), p. 95, Cambridge University Press on behalf of the International African InstituteVillalón, Leonardo A., Islamic Society and State Power in Senegal: Disciples and Citizens in Fatick, p. 62, Cambridge University Press (2006), {{ISBN|9780521032322}}Bulletin de la Société de géographie, Volume 26. Société de Géographie (1855), pp. 35 - 36. [https://books.google.co.uk/books?id=hJ9bQf7f-RoC&pg=PA35] (retrieved 24 March 2025).Maury, Alfred, Rapports à la Soc. de géogr, Volume 1. (1855). p. 25 [https://books.google.co.uk/books?id=Ujg-AAAAcAAJ&pg=PA25] (retrieved 24 March 2025)Marty, Paul, L'Islám en Mauritanie et au Sénégal. E. Leroux (1916), p. 49 The Serer have a detailed pictorial representation of the Universe, representing the three worlds in Serer primordial time: the invisible world, the terrestrial world, and the nocturnal world.Gravrand, Henry, La civilisation sereer, vol. II : Pangool, Nouvelles éditions africaines, Dakar, 1990, p. 216, {{ISBN|2-7236-1055-1}}

The pentagram was used in ancient times as a Christian symbol for the five senses,Christian Symbols Ancient and Modern, Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, {{isbn|0-7135-1960-6}}. or of the five wounds of Christ. The pentagram plays an important symbolic role in the 14th-century English poem Sir Gawain and the Green Knight, in which the symbol decorates the shield of the hero, Gawain. The unnamed poet credits the symbol's origin to King Solomon, and explains that each of the five interconnected points represents a virtue tied to a group of five: Gawain is perfect in his five senses and five fingers, faithful to the Five Wounds of Christ, takes courage from the five joys that Mary had of Jesus, and exemplifies the five virtues of knighthood,{{cite journal |last1=Morgan |first1=Gerald |date=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 |doi=10.2307/3728227 |jstor=3728227}} which are generosity, friendship, chastity, chivalry, and piety.Sir Gawain and the Green Knight, lines 619–665

File:Amiens Rose Nord 1.jpg]]

The North rose of Amiens Cathedral (built in the 13th century) exhibits a pentagram-based motif. Some sources interpret the unusual downward-pointing star as symbolizing the Holy Spirit descending on people.

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. Agrippa depicts the human body inscribed in an 'upright' (point-up) pentagram and another with its hands in rotated pentagrams, among numerous other geometrical figures, in the section on 'the proportions and harmonious measures of the human body', and an 'inverted' (point-down) version of the Pythagorean 'hygeia' pentagram in the section on 'characters, received only by revelation, which no other kind of reasoning can discover', alongside variations of the Chi-Rho and the Hebrew word Makabi.

'Of this type are the signet shown to Constantine, which most people called a cross, inscribed in Latin letters, 'in this conquer', and another revealed to Antiochus who was surnamed Soteris, in the shape of a pentagon, which issued health, for resolved into letters, it issued the word ὑγίεα, that is, 'health', in the confidence and virtue of which signs, each of the kings won a notable victory against their enemies. Thus Judas, who for this reason was afterwards known as Maccabeus, was about to fight with the Jews against Antiochus Eupatorus, and received that noble seal מׄכׄבׄיׄ from the angel'.

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 but controversial writer Éliphas Lévi, known for believing that magic was a real science, had called it evil whenever the symbol appeared the other way up:

• "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."{{cite book |title=Transcendental Magic, its Doctrine and Ritual |trans-title=Dogme et rituel de la haute magie |publisher=Weiser |location=York Beach |first=Éliphas |last=Lévi |author-link=Éliphas Lévi |others=Trans. by A. E. Waite |date=1999 |orig-year=1896 (translated), 1854 (first published) |oclc=263626874}}
• "The flaming star, which, when turned upside down, is the {{sic|heirol|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 a sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns."{{cite book |title=The Key of the Mysteries |trans-title=la Clef des grands mystères suivant Hénoch, Abraham, Hermès Trismégiste et Salomon |publisher=Weiser |location=Boston |first=Éliphas |last=Lévi |author-link=Éliphas Lévi |others=Trans. by Aleister Crowley |page=69 |date=2002 |orig-year=1939 (translated), 1859 (first published) |oclc=49053462}}
• "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."{{cite book |title=Magic, White and Black |publisher=The Path |location=New York |first=Franz |last=Hartmann |author-link=Franz Hartmann |edition=5th |date=1895 |orig-year=1886 |oclc=476635673}}

File:Pentagram and human body (Agrippa).jpg|Man inscribed in a pentagram, from Heinrich Cornelius Agrippa's De occulta philosophia libri tres. The five signs at the pentagram's vertices are astrological.

File:Inverted pentacle.PNG|Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.

File:Pentagram (Levi).jpg|The occultist and magician Éliphas Lévi's pentagram, which he considered to be a symbol of the microcosm, or human

{{Star polygons}}

The apotropaic (protective) use in German folklore of the pentagram symbol (called Drudenfuss in German) is referred to by Goethe in 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):

{{blockquote|

Mephistopheles:

:I must confess, I'm prevented though

:By a little thing that hinders me,

:The Druid's-foot on your doorsill–

Faust:

:The Pentagram gives you pain?

:Then tell me, you Son of Hell,

:If that's the case, how did you gain

:Entry? Are spirits like you cheated?

Mephistopheles:

:Look carefully! It's not completed:

:One angle, if you inspect it closely

:Has, as you see, been left a little open.{{Cite web|title=Goethe, Johann Wolfgang von (1749–1832) - Faust, Part I: Scenes I to III|url=https://www.poetryintranslation.com/PITBR/German/FaustIScenesItoIII.php|access-date=2021-05-25|website=www.poetryintranslation.com}}}}

Also protective is the use in Icelandic folklore of a gestured or carved rather than painted pentagram (called {{lang|is|smèrhnút}} in Icelandic), according to 19th century folklorist Jón Árnason:{{cite book|first=Jón |last=Árnason |author-link=Jón Árnason (author)|date=1862 |language=is|chapter={{lang|is|Töfrabrogð}} [Magic trick]|title=Íslenzkar Þjoðsögur og Æfintýri |trans-title=Icelandic Folktales and Legends |volume=1 |location=Leipzig |publisher=J. C. Hinrich's Bookstore |page=432 |url=https://baekur.is/bok/d1007e0b-6463-47a4-9e8d-ce48e82042e7/1/472/Islenzkar_thjodsogur_og#page/n471/mode/1up |quote={{lang|is|Smèr það, er verður af tilberaspýunni, er kallað tilberasmèr; er það útlits sem annað smèr; en gjöri maður krossmark yfir því, eða risti á það kross, eða mynd þá, er smèrhnútur heitir,^* springur það alt í smámola og verður eins og draflakyrníngur, svo ekki sèst eptir af því, nema agnir einar, eða það hjaðnar niður sem froða. Þykir það því varlegra, ef manni er boðið óhrjálegt smèr að borða, eða í gjöld, að gjóra annaðhvort þetta mark á það, því tilberasmèr þolir hvorki krossmark né smjörhnút. / * Smèrhnútur er svo í lögun: 15px}}}}

:A butter that comes from the fake vomit is called a fake butter; it looks like any other butter; but if one makes a sign of a cross over it, or carves a cross on it, or a figure called a buttermilk-knot,^* it all explodes into small pieces and becomes like a grain of dross, so that nothing remains of it, except only particles, or it subsides like foam. Therefore it seems more prudent, if a person is offered a horrible butter to eat, or as a fee,In the Middle Ages, butter was used for payment, e.g. rent. See:
• {{cite book |last=Sexton |first=Regina |chapter=The Role and Function of Butter in the Diet of the Monk and Penitent in Early Medieval Ireland |title=The Fat of the Land: Proceedings of the Oxford Symposium on Food and Cooking 2002 |editor-last=Walker |editor-first=Harlan |location=Bristol |publisher=Footwork |date=2003 |pages=253–269 |chapter-url=https://books.google.com/books?id=ZJ5qBQAAQBAJ&pg=PT259}} to make either mark on it, because a fake butter cannot withstand either a cross mark or a butter-knot.

:* The butter-knot is shaped like this: 15px

Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists. Its major use is a continuation of the ancient Babylonian use of the pentagram as an apotropaic charm to protect against evil forces.{{cite book |url=https://books.google.com/books?id=zwAlAAAAMAAJ |title=The Pentagram as Medical Symbol: An Iconological Study |publisher=Hes & De Graaf |first=Jan |last=Schouten |page=18 |date=1968 |isbn=978-90-6004-166-6}} Éliphas Lévi claimed that "The Pentagram expresses the mind's domination over the elements and it is by this sign that we bind the demons of the air, the spirits of fire, the spectres of water, and the ghosts of earth."{{cite book |url=https://books.google.com/books?id=jX9BAAAAYAAJ&pg=PA136 |title=The Mysteries of Magic: A Digest of the Writings of Eliphas Lévi |publisher=George Redway |location=London |first=Arthur Edward |last=Waite |pages=136 |date=1886}} In this spirit, the Hermetic Order of the Golden Dawn developed the use of the pentagram in the lesser banishing ritual of the pentagram, which is still used to this day by those who practice Golden Dawn-type magic.

Aleister Crowley made use of the pentagram in the system of magick used in Thelema: an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation of Lon Milo DuQuette.{{cite book |url=https://books.google.com/books?id=-07yFC31j6wC&pg=PA93 |title=The Magick of Aleister Crowley: A Handbook of the Rituals of Thelema |publisher=Weiser Books |first=Lon Milo |last=DuQuette |pages=93, 247 |date=2003 |isbn=978-1-57863-299-2}} 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.

{{Main|Baháʼí symbols}}

File:Bab-Star-Tablet-Haykal.jpg written in his own hand]]

The five-pointed star is a symbol of the Baháʼí Faith.{{Cite web|url=https://reference.bahai.org/en/t/se/DG/dg-141.html.utf8|title=Bahá'í Reference Library - Directives from the Guardian, Pages 51-52|website=reference.bahai.org}}{{Cite web|url=https://bahai-library.com/uhj_nine_pointed_star|title=The Nine-Pointed Star|website=bahai-library.com}} In the Baháʼí Faith, the star is known as the Haykal ({{langx|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.Moojan Momen (2019). [https://blogs.bl.uk/asian-and-african/2019/10/the-star-tablet-of-the-bab.html The Star Tablet of the Bab]. British Library Blog.Bayat, Mohamad Ghasem (2001). [https://bahai-library.com/bayat_surat_haykal An Introduction to the Súratu'l-Haykal (Discourse of The Temple)] in Lights of Irfan, Book 2.

The Church of Jesus Christ of Latter-day Saints is theorized to have begun using both upright and inverted five-pointed stars in Temple architecture, dating from the Nauvoo Illinois Temple dedicated on 30 April 1846.See the [http://users.marshall.edu/~brown/nauvoo/nt-parent.html Nauvoo Temple] {{Webarchive|url=https://web.archive.org/web/20200517215640/http://users.marshall.edu/~brown/nauvoo/nt-parent.html |date=17 May 2020 }} 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. Other temples decorated with five-pointed stars in both orientations include the Salt Lake Temple and the Logan Utah Temple. These usages come from the symbolism found in Revelation chapter 12: "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."{{cite web |url=http://www.fairlds.org/pubs/Stars.pdf |title=Inverted Stars on LDS Temples |work=FAIRLDS.org |first=Matthew B |last=Brown |author-link=Matthew B. Brown |date=2002 |archive-url=https://web.archive.org/web/20080229073604/http://www.fairlds.org/pubs/Stars.pdf |archive-date=29 February 2008}}

{{multiple image |align=right |direction=horizontal |total_width=300

|image1=Pentacle (fixed width).svg |caption1=Typical Neopagan pentagram (circumscribed)

|image2=USVA headstone emb-37.svg |caption2=USVA headstone emblem 37

}}

Because of a perceived association with Satanism and occultism, many United States schools in the late 1990s sought to prevent students from displaying the pentagram on clothing or jewelry.[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 |access-date=10 February 2006 |url-status=bot: unknown |archive-url=https://web.archive.org/web/20031108123215/http://aclumich.org/modules.php?name=News&file=article&sid=26 |archive-date=8 November 2003}} [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. In public schools, such actions by administrators were determined in 2000 to be in violation of students' First Amendment right to free exercise of religion.{{cite web |url=http://www.firstamendmentcenter.org/federal-judge-upholds-indiana-students-right-to-wear-wiccan-symbols |title=Federal judge upholds Indiana students' right to wear Wiccan symbols |agency=Associated Press |date=1 May 2000 |access-date=21 September 2007 |url-status=dead |archive-url=https://swap.stanford.edu/20140330195413/http://www.firstamendmentcenter.org/federal-judge-upholds-indiana-students-right-to-wear-wiccan-symbols |archive-date=30 March 2014}}

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 {{US$|225,000}} to settle their pending lawsuits.{{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.com |agency=Associated Press |date=23 April 2007 |archive-url=https://web.archive.org/web/20070426075801/http://www.cnn.com/2007/US/04/23/wiccan.grave.ap/index.html |archive-date=26 April 2007}}{{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 |access-date=13 January 2014}}

{{multiple image

| align = right

| direction = horizontal

| total_width = 300

| image1 = Pentagram4.svg

| caption1 = The inverted pentagram is the most notable and widespread symbol of Satanism.

| image2 = Baphomet Pentagram.svg

| caption2 = The goat head of Baphomet forms an inverted pentagram, as depicted in the Sigil of Baphomet

}}

The inverted pentagram is broadly used in Satanism, sometimes depicted with the goat's head of Baphomet, as popularized by the Church of Satan since 1968. LaVeyan Satanists pair the goat head with Hebrew letters at the five points of the pentagram to form the Sigil of Baphomet. The Baphomet sigil was adapted for the Joy of Satan Ministries logo, using cuneiform characters at the five points of the pentagram, reflecting the shape's earliest use in Sumeria. The inverted pentagram also appears in The Satanic Temple logo, with an alternative depiction of Baphomet's head. Other depictions of the Satanic goat's head resemble the inverted pentagram without its explicit outline.

The five-pointed star is a symbol of the Serer religion and the Serer people of West Africa. Called Yoonir in their language, it symbolizes the universe in the Serer creation myth, and also represents the star Sirius.{{Cite book |title=La civilisation Sereer, Volume II: Pangool |series=Nouvelles éditions Africaines du Sénégal |location=Dakar, Senegal |language=fr |first=Henry |last=Gravrand |author-link=Henry Gravrand |year=1990 |isbn=2-7236-1055-1 |page=20}}{{cite book |title=Tracing Memory: A Glossary of Graphic Signs and Symbols in African Art and Culture |series=Mercury series, no. 71 |publisher=Canadian Museum of Civilization |location=Hull, Québec |first=Clémentine Faïk-Nzuji |last=Madiya |author-link=Clémentine Nzuji |pages=27, 155 |date=1996 |isbn=0-660-15965-1}}

• The pentagram is featured on the national flags of Morocco (adopted 1915) and Ethiopia (adopted 1996 and readopted 2009)

Image:Flag of Morocco (large stroke).svg|Morocco's flag

Image:Flag of Ethiopia.svg|Ethiopia's flag

• The Order of the Eastern Star, an organization (established 1850) associated with Freemasonry, uses a pentagram as its symbol, with the five isosceles triangles of the points colored blue, yellow, white, green, and red. In most Grand Chapters the pentagram is used pointing down, but in a few, it is pointing up. Grand Chapter officers often have a pentagon inscribed around the starRitual of the Order of the Eastern Star, 1976(the emblem shown here is from the Prince Hall Association).

Image:OrderEasternStar logo from saucer.jpg| Order of the Eastern Star emblem

• A pentagram is featured on the flag of the Dutch city of Haaksbergen, as well on its coat of arms.

Image:Flag of Haaksbergen.svg| Flag of Haaksbergen

• A pentagram is featured on the flag of the Japanese city of Nagasaki, as well on its emblem.

Image:Flag of Nagasaki, Nagasaki.svg| Flag of Nagasaki

==Geometry==

File:Stellamerlata.gifs drawn with MSWLogo (in Tartapelago{{cite web|url=http://www.pietrocola.eu/maecla/tartapelago/frattali/index.htm |title=Tartapelago. Exposure of fractals|first=Giorgio|last=Pietrocola |date=2005 |website=Maecla }}) ]]

File:Pentagram Angles Notation.svg

The pentagram is the simplest 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.

It can be seen as a net of a pentagonal pyramid although with isosceles triangles.

The pentagram can be constructed by connecting alternate vertices of a pentagon; see 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.

Image:Pentagram-phi.svg to one another.]]

File:golden_triangle_pentagram.svg in a regular pentagram.]]

The golden ratio, φ = (1 + {{radic|5}}) / 2 ≈ 1.618, satisfying:

:$\backslash varphi=1+2\backslash sin(\backslash pi/10)=1+2\backslash sin\; 18^\backslash circ\backslash ,$

:$\backslash varphi=1/(2\backslash sin(\backslash pi/10))=1/(2\backslash sin\; 18^\backslash circ)\backslash ,$

:$\backslash varphi=2\backslash cos(\backslash pi/5)=2\backslash cos\; 36^\backslash circ\backslash ,$

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:

:$\backslash frac\{\backslash mathrm\{red\}\}\{\backslash mathrm\{green\}\}\; =\; \backslash frac\{\backslash mathrm\{green\}\}\{\backslash mathrm\{blue\}\}\; =\; \backslash frac\{\backslash mathrm\{blue\}\}\{\backslash mathrm\{magenta\}\}\; =\; \backslash varphi.$

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon.

{{main|Exact trigonometric values}}

:$$

\begin{align}

\sin \frac{\pi}{10} & = \sin 18^\circ = \frac{\sqrt 5 - 1}{4}=\frac{\varphi-1}{2}=\frac{1}{2\varphi} \\[5pt]

\cos \frac{\pi}{10} & = \cos 18^\circ = \frac{\sqrt{2(5 + \sqrt 5)}}{4} \\[5pt]

\tan \frac{\pi}{10} & = \tan 18^\circ = \frac{\sqrt{5(5 - 2 \sqrt 5)}}{5} \\[5pt]

\cot \frac{\pi}{10} & = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5} \\[5pt]

\sin \frac{\pi}{5} & = \sin 36^\circ = \frac{\sqrt{2(5 - \sqrt 5)} }{4} \\[5pt]

\cos \frac{\pi}{5} & = \cos 36^\circ = \frac{\sqrt 5+1}{4} = \frac{\varphi}{2} \\[5pt]

\tan \frac{\pi}{5} & = \tan 36^\circ = \sqrt{5 - 2\sqrt 5} \\[5pt]

\cot \frac{\pi}{5} & = \cot 36^\circ = \frac{ \sqrt{5(5 + 2\sqrt 5)}}{5}

\end{align}

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.

{{Further| Pentagramma mirificum}}

A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss.

{{Further|Uniform polyhedron#Icosahedral symmetry{{!}}Uniform polyhedron: Icosahedral symmetry}}

Several polyhedra incorporate pentagrams:

Image:Pentagrammic prism.png|Pentagrammic prism

Image:Pentagrammic antiprism.png|Pentagrammic antiprism

Image:Pentagrammic crossed antiprism.png|Pentagrammic crossed-antiprism

Image:Small_stellated_dodecahedron.png|Small stellated dodecahedron

Image:Great_stellated_dodecahedron.png|Great stellated dodecahedron

Image:Small ditrigonal icosidodecahedron.png|Small ditrigonal icosidodeca­hedron

Image:Dodecadodecahedron.png|Dodecadodecahedron

Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures:

All ten 4-dimensional Schläfli–Hess 4-polytopes have either pentagrammic faces or vertex figure elements.

File:Venus geocentric orbit curve simplified Line (pentagram).svg

The pentagram of Venus is the apparent path of the planet Venus as observed from Earth. Successive inferior conjunctions of Venus repeat with an orbital resonance of approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction.{{cite web |url=https://johncarlosbaez.wordpress.com/2014/01/04/the-pentagram-of-venus |title=The Pentagram of Venus |first=John |last=Baez |author-link=John C. Baez |work=Azimuth |date=4 January 2014 |access-date=7 January 2016 |url-status=dead |archive-url=https://web.archive.org/web/20151214005503/https://johncarlosbaez.wordpress.com/2014/01/04/the-pentagram-of-venus/ |archive-date=14 December 2015}} The tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram, and each group of five intersections equidistant from the figure's center have the same geometric relationship.

{{see also|Unicode input}}

The pentagram has these Unicode code points that enable them to be included in documents:

• {{unichar|26E4|pentagram}}
• {{unichar|26E5|right-handed interlaced pentagram}}
• {{unichar|26E6|left-handed interlaced pentagram}}
• {{unichar|26E7|inverted pentagram}}

{{div col|colwidth=25em}}

• {{Annotated link |Abe no Seimei}}
• {{Annotated link |Christian symbolism}}
• {{Annotated link |Command at Sea insignia}}
• {{Annotated link |Enneagram (geometry)}}
• {{Annotated link |Five-pointed star}}
• {{Annotated link |Heptagram}}
• {{Annotated link |Hexagram}}
• {{Annotated link |Lute of Pythagoras}}
• {{Annotated link |Medal of Honor}}
• Pentachoron – the 4-simplex
• {{Annotated link |Pentagram map}}
• {{Annotated link |Pentalpha}}
• {{Annotated link |Petersen graph}}
• {{Annotated link |Ptolemy's theorem}}
• {{Annotated link |Seal of Solomon}}
• {{Annotated link |Star polygons in art and culture}}
• {{Annotated link |Star (heraldry)}}
• {{Annotated link |Stellation#Stellating polygons|Stellated polygons}}

{{div col end}}

{{Reflist}}

• {{cite encyclopedia |title=Pentagram |encyclopedia=The Continuum Encyclopedia of Symbols |publisher=Continuum Books |location=New York City |first=Udo |last=Becker |translator-first=Lance W. |translator-last=Garmer |date=1994 |isbn=978-0-8264-0644-6 |url=https://archive.org/details/continuumencyclo00beck/page/230 |pages=230ff |ref=none}}
• {{cite book |chapter=Chapter 26, Higher Still: Regular Star-Polytopes |title=The Symmetries of Things |publisher=A. K. Peters |location=Wellesley, Massachusetts |first1=John Horton |last1=Conway |author1-link=John Horton Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss |page=404 |date=April 2008 |isbn=978-1-56881-220-5 |ref=none}}
• {{cite book |title=Signs and Symbols in Christian Art |publisher=Oxford University Press |location=New York City |first=George Wells |last=Ferguson |page=59 |date=1966 |orig-year=1954 |oclc=65081051 |ref=none}}
• {{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 |date=1987 |isbn=978-0-7167-1193-3 |url=https://archive.org/details/isbn_0716711931 |ref=none}}
• {{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=McMullen, P. |editor2-link=Peter McMullen |editor3=Schneider, A. |editor4=Weiss, A. Ivić |volume=440 |pages=43–70 |date=1994 |isbn=978-94-010-4398-4 |doi=10.1007/978-94-011-0924-6_3 |ref=none}}

{{Wikiquote}}

{{Commons category|Pentagrams}}

• {{MathWorld |urlname=Pentagram |title=Pentagram}}
• [http://www.cs.utk.edu/~mclennan/BA/PP.html The Pythagorean Pentacle] from the Biblioteca Arcana.

{{Polygons}}

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{{Thelema series}}

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