Pentagram
Wikipedia says:
Regular pentagram  

A regular pentagram  
Type  Regular star polygon 
Edges and vertices  5 
Schläfli symbol  {5/2} 
Coxeter diagram  
Symmetry group  Dihedral (D_{5}) 
Internal angle (degrees)  36° 
Dual polygon  self 
Properties  star, cyclic, equilateral, isogonal, isotoxal 
A pentagram (sometimes known as a pentalpha or pentangle or a star pentagon) is the shape of a fivepointed star drawn with five straight strokes.
The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon),^{[1]} from πέντε (pente), "five" + γραμμή (grammē), "line".^{[2]} The word "pentacle" is sometimes used synonymously with "pentagram"^{[3]} The word pentalpha is a learned modern (17thcentury) revival of a postclassical Greek name of the shape.^{[4]}
Contents
Geometry
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.
Construction
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.
Golden ratio
The golden ratio, φ = (1 + √5) / 2 ≈ 1.618, satisfying
 φ = 1 + 2 sin ( π / 10 ) = 1 + 2 sin 18 ∘ {\displaystyle \varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\,}
 φ = 1 / ( 2 sin ( π / 10 ) ) = 1 / ( 2 sin 18 ∘ ) {\displaystyle \varphi =1/(2\sin(\pi /10))=1/(2\sin 18^{\circ })\,}
 φ = 2 cos ( π / 5 ) = 2 cos 36 ∘ {\displaystyle \varphi =2\cos(\pi /5)=2\cos 36^{\circ }\,}
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 fourcolor illustration shows:
 r e d g r e e n = g r e e n b l u e = b l u e m a g e n t a = φ . {\displaystyle {\frac {\mathrm {red} }{\mathrm {green} }}={\frac {\mathrm {green} }{\mathrm {blue} }}={\frac {\mathrm {blue} }{\mathrm {magenta} }}=\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.
Trigonometric values
 sin π 10 = sin 18 ∘ = 5 − 1 4 = φ − 1 2 = 1 2 φ {\displaystyle \sin {\frac {\pi }{10}}=\sin 18^{\circ }={\frac {{\sqrt {5}}1}{4}}={\frac {\varphi 1}{2}}={\frac {1}{2\varphi }}}
 cos π 10 = cos 18 ∘ = 2 ( 5 + 5 ) 4 {\displaystyle \cos {\frac {\pi }{10}}=\cos 18^{\circ }={\frac {\sqrt {2(5+{\sqrt {5}})}}{4}}}
 tan π 10 = tan 18 ∘ = 5 ( 5 − 2 5 ) 5 {\displaystyle \tan {\frac {\pi }{10}}=\tan 18^{\circ }={\frac {\sqrt {5(52{\sqrt {5}})}}{5}}}
 cot π 10 = cot 18 ∘ = 5 + 2 5 {\displaystyle \cot {\frac {\pi }{10}}=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}}
 sin π 5 = sin 36 ∘ = 2 ( 5 − 5 ) 4 {\displaystyle \sin {\frac {\pi }{5}}=\sin 36^{\circ }={\frac {\sqrt {2(5{\sqrt {5}})}}{4}}}
 cos π 5 = cos 36 ∘ = 5 + 1 4 = φ 2 {\displaystyle \cos {\frac {\pi }{5}}=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}}}
 tan π 5 = tan 36 ∘ = 5 − 2 5 {\displaystyle \tan {\frac {\pi }{5}}=\tan 36^{\circ }={\sqrt {52{\sqrt {5}}}}}
 cot π 5 = cot 36 ∘ = 5 ( 5 + 2 5 ) 5 {\displaystyle \cot {\frac {\pi }{5}}=\cot 36^{\circ }={\frac {\sqrt {5(5+2{\sqrt {5}})}}{5}}}
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.
Threedimensional figures
Several polyhedra incorporate pentagrams:
Higher dimensions
Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures:
4D  5D  

The regular 5cell (4simplex) has five vertices and 10 edges.  The rectified 5cell has 10 vertices and 30 edges.  The rectified 5simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams.  The birectified 5simplex has 20 vertices, seen in this orthogonal projection as four overlapping pentagrams. 
All ten 4dimensional Schläfli–Hess 4polytopes have either pentagrammic faces or vertex figure elements.
Pentagram of Venus
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 a approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction.^{[5]} 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. Groups of five intersections of curves, equidistant from the figure's center, have the same geometric relationship.
Cultural significance
Early history
In early (Ur I) monumental Sumerian script, or cuneiform, 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 𒌒, composed of five wedges, further reduced to four in Assyrian cuneiform ).
The word Pentemychos (πεντέμυχος lit. "five corners" or "five recesses")^{[6]} was the title of the cosmogony of Pherecydes of Syros.^{[7]} Here, the "five corners" are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear.^{[8]}
In Neoplatonism, the pentagram was said to have been used as a symbol or sign of recognition by the Pythagoreans, who called the pentagram ὑγιεία hugieia "health"^{[10]}
Western symbolism
The pentagram was used in ancient times as a Christian symbol for the five senses,^{[11]} or of the five wounds of Christ. A Christian use of the pentangle occurs in the 14thcentury 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 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 had of Jesus, and exemplifies the five virtues of knighthood.^{[12]}
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 mid19th 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.
 "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."^{[13]}
 "The flaming star, which, when turned upside down, is the hierolgyphic [sic] 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."^{[14]}
 "Let us keep the figure of the Fivepointed 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."^{[15]}
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.
The occultist and magician Eliphas Levi's pentagram, which he considered to be a symbol of the microcosm, or human.
The apotropaic use of the pentagram symbol in German folklore (called Drudenfuss in German) is referred to by Goethe in his 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):
 Mephistopheles:
 I must confess, my stepping o'er
 Thy threshold a slight hindrance doth impede;
 The wizardfoot [Drudenfuss] doth me retain.
 Faust:
 The pentagram thy peace doth mar?
 To me, thou son of hell, explain,
 How camest thou in, if this thine exit bar?
 Could such a spirit aught ensnare?
 Mephistopheles:
 Observe it well, it is not drawn with care,
 One of the angles, that which points without,
 Is, as thou seest, not quite closed.
East Asian symbolism
Wu Xing (Chinese: 五行; pinyin: Wǔ Xíng) are the five phases, or five elements in Chinese tradition (medicine, acupuncture, feng shui, and Taoism) They are similar to the ancient 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).
Use in modern occultism
Based on Renaissanceera occultism, the pentagram found its way into the symbolism of modern occultists.
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 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.
Aleister Crowley made use of the pentagram in his Thelemic system of magick: an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation of Lon Milo DuQuette.^{[16]} 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.
Use in new religious movements
Bahá'í
The fivepointed star is a symbol of the Bahá'í Faith.^{[17]}^{[18]} In the Bahá'í Faith, the star is known as the Haykal (Arabic: "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.^{[19]}
Latter Day Saints
The Church of Jesus Christ of Latterday Saints began using both upright and inverted fivepointed stars in Temple architecture, dating from the Nauvoo Illinois Temple dedicated on 30 April 1846.^{[20]} Other temples decorated with fivepointed 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."^{[21]}
Wicca
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.^{[22]} In public schools, such actions by administrators were determined to be in violation of students' First Amendment right to free exercise of religion in 2000.^{[23]} 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.^{[24]}^{[25]}
Other religious use
The five pointed star is a symbol of the Serer religion, and a multicolored version is used as symbol of the Druze religion.
Other modern use
 The pentagram is featured on the national flags of Morocco (adopted 1915) and Ethiopia (adopted 1996)
 The Order of the Eastern Star, an organization associated with Freemasonry (established 1850), used to have a pointdown pentagram as its symbol, with the five isosceles triangles of the points colored blue, yellow, white, green, and red (the logo shown here is from Prince Hall Association, the nonPHA logo has the Pentagram inscribed with the cabalistic Word)
Order of the Eastern Star emblem
See also
 Barnstar
 CommandatSea Pin
 Enneagram (geometry)
 Heptagram
 Hexagram
 A particular case of hypotrochoid is a pentagramlike curve
 List of regular polytopes#Stars
 List of symbols
 Lute of Pythagoras
 Medal of Honor
 Mullet (heraldry)
 Nonconvex uniform polyhedra with full icosahedral symmetry (many show a pattern of pentagrams)
 Pentachoron – the 4simplex
 Pentad
 Pentagram map (related concept)
 Pentalpha
 Petersen graph
 Ptolemy's theorem
 Red star
 Star (glyph)
 Stellated polygons
References
 ^ πεντάγραμμον, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus; a noun form of adjectival πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "fivelined" or "five lines"
 ^ πέντε, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus; γραμμή, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus
 ^ 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 fivepointed star and its etymon postclassical 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)."
 ^ πένταλϕα, "five Alphas", interpreting the shape as five Α shapes overlapping at 72degree angles.
 ^ Baez, John (January 4, 2014). "The Pentagram of Venus". Azimuth. Retrieved January 7, 2016.
 ^ πεντέμυχος, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus
 ^ This is a lost book, but its contents are preserved in Damascius, De principiis, quoted in Kirk and Raven, (1983) [1956], p. 55.
 ^ "the divine products of Chronos' seed, when disposed in five recesses, were called πεντέμυχος (Pentemychos)" Kirk, Geoffrey Stephen; Raven, John Earle; Schofield, Malcolm (1983) [1957]. The Presocratic Philosophers: A Critical History with a Selection of Texts (2nd, illustrated, revised, reprint ed.). Cambridge University Press. pp. 51–52, 55. ISBN 9780521274555.
the only other place in Homer where Ortygie [sic] is mentioned is Odyssey V, 123, where 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 northeastbyeast direction, that is what the phrase might suggest... the dwellingplace of Eos... Aia..
 ^ Allman, G. J., Greek Geometry From Thales to Euclid (1889), p.26.
 ^ Allman, G. J., Greek Geometry From Thales to Euclid, part I (1877), in Hermathena 3.5, pp. 183, 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.
 ^ Christian Symbols Ancient and Modern, Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, ISBN 0713519606.
 ^ Morgan, Gerald (1979). "The Significance of the Pentangle Symbolism in "Sir Gawain and the Green Knight"". The Modern Language Review 74 (4): 769–790. JSTOR 3728227.
 ^ Lévi, Éliphas (1999) [1896 (translated), 1854 (first published)]. Transcendental Magic, its Doctrine and Ritual [Dogme et rituel de la haute magie]. Trans. by A. E. Waite. York Beach: Weiser. OCLC 263626874.
 ^ Lévi, Éliphas (2002) [1939 (translated), 1859 (first published)]. The Key of the Mysteries [la Clef des grands mystères suivant Hénoch, Abraham, Hermès Trismégiste et Salomon]. Trans. by Aleister Crowley. Boston: Weiser. p. 69. OCLC 49053462.
 ^ Hartmann, Franz (1895) [1886]. Magic, White and Black (5th ed.). New York: The Path. OCLC 476635673.
 ^ The Magick of Aleister Crowley p 93 and endnote 31 to Chapter Five (p 247).
 ^ Bahá'í Reference Library – Directives from the Guardian, Pages 51–52
 ^ NinePointed Star, The:History and Symbolism by Universal House of Justice 24 January 1999
 ^ Haykal  Baha'i Five Pointed Star Symbol
 ^ See the Nauvoo Temple website discussing its architecture, and particularly the page on Nauvoo Temple exterior symbolism. Retrieved 16 December 2006.
 ^ Brown, Matthew B (2002). "Inverted Stars on LDS Temples" (PDF). FAIRLDS.org. Archived from the original (PDF) on 29 February 2008.
 ^ "Religious Clothing in School", Robinson, B.A., Ontario Consultants on Religious Tolerance, 20 August 1999, updated 29 April 2005. Retrieved 10 February 2006. "ACLU Defends Honor Student Witch Pentacle" (Press release). American Civil Liberties Union of Michigan. 10 February 1999. Retrieved 10 February 2006. "Witches and wardrobes: Boy says he was suspended from school for wearing magical symbol" Rouvalis, Cristina; Pittsburgh PostGazette, 27 September 2000. Retrieved 10 February 2006.
 ^ "Federal judge upholds Indiana students' right to wear Wiccan symbols". Associated Press. 1 May 2000. Retrieved 21 September 2007.
 ^ Associated Press (23 April 2007). "Wiccan symbol OK for soldiers' graves". CNN.com. Archived from the original on 26 April 2007.
 ^ "Burial and Memorials: Available Emblems of Belief for Placement on Government Headstones and Markers". United States Department of Veterans Affairs. 3 July 2013. Retrieved 13 January 2014.
Bibliography
 Grünbaum, Branko; Shephard, Geoffrey Colin (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 9780716711933.
 Grünbaum, Branko (1994). "Polyhedra with Hollow Faces". In Bisztriczky, T.; McMullen, P.; Schneider, A.; Weiss, A. Ivić. Polytopes: Abstract, Convex and Computational. NATO ASI Series C: Mathematical and Physical Sciences 440. Dordrecht: Springer Netherlands. pp. 43–70. doi:10.1007/9789401109246_3. ISBN 9789401043984.
 "Pentagram" article in The Continuum Encyclopedia of Symbols Becker, Udo, ed., Garmer, Lance W. translator, New York: Continuum Books, 1994, p. 230.
 Signs and Symbols in Christian Art Ferguson, George, Oxford University Press: 1966, p. 59.
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 404: Regular starpolytopes Dimension 2)
External links
Wikimedia Commons has media related to Pentagrams. 
 Weisstein, Eric W., "Pentagram", MathWorld.
 The Pythagorean Pentacle from the Biblioteca Arcana.
 Indepth analysis of the Golden Ratio
 The pentagram and Freemasonry


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