Pentagram

This article is about the shape with intersecting lines. For the shape showing only the outline, seeFive-pointed star.For other uses, seePentagram (disambiguation).

Pentagram

Apentagram(sometimes known as a pent Alpha, pentangle, or star pentagon) is a regular five-pointedstar polygon,formed from the diagonalline segmentsof a convex (or simple, or non-self-intersecting)regular pentagon.Drawing a circle around the five points creates a similar symbol referred to as thepentacle,which is used widely byWiccansand inpaganism,or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle.

The wordpentagramcomes from theGreekword πεντάγραμμον (pentagrammon),from πέντε (pente), "five" + γραμμή (grammē), "line". The word pentagram refers to just the star and the wordpentaclerefers to the star within a circle, although there is some overlap in usage.The wordpent Alphais a 17th-century revival of a post-classical Greek name of the shape.

History

Early history

Early pentagrams have been found on Sumerian pottery from Ur c. 3500BCE,and the five-pointed star was at various times the symbol ofIshtarorMarduk.

A Pythagorean "Hugieia Pentagram"

A right-handed interlaced pentagram, popular with Wiccans and some other neo-pagans. TheFlag of Moroccooften bearsthe left-handed version.

Pentagram symbols from about 5,000 years ago were found in theLiangzhu cultureofChina.

The pentagram was known to theancient Greeks,with a depiction on a vase possibly dating back to the 7th century BCE.Pythagoreanismoriginated in the 6th century BCE and used the pentagram as a symbol of mutual recognition, of wellbeing, and to recognize good deeds and charity.

From around 300–150 BCE the pentagram stood as the symbol of Jerusalem, marked by the 5 Hebrew letters ירשלם spelling its name.

InNeoplatonism,the pentagram was said to have been used as a symbol or sign of recognition by thePythagoreans,who called the pentagramὑγιείαhugieia"health"

Western symbolism

Middle Ages

The pentagram was used in ancient times as a Christian symbol for the fivesenses,or of thefive wounds of Christ.The pentagram plays an important symbolic role in the 14th-century English poemSir Gawain and the Green Knight,in which the symbol decorates the shield of the hero,Gawain.Theunnamed poetcredits the symbol's origin to KingSolomon,and explains that each of the five interconnected points represents a virtue tied to a group of five: Gawain is perfect in hisfive sensesand five fingers, faithful to the Five Wounds of Christ, takes courage from thefive joysthatMaryhad of Jesus, and exemplifies the five virtues ofknighthood,which are generosity, friendship, chastity, chivalry, and piety.

The North rose ofAmiens Cathedral

The North rose ofAmiens 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.

Renaissance

Heinrich Cornelius Agrippaand others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion.

Romanticism

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 writerÉliphas 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."
• "The flaming star, which, when turned upside down, is the heirolgyphic [sic] sign of the goat ofblack 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. "
• "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."

• 
  Man inscribed in a pentagram, fromHeinrich Cornelius Agrippa'sDe occulta philosophia libri tres.The five signs at the pentagram'sverticesareastrological.
• 
  Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.
• 
  TheoccultistandmagicianÉliphas Lévi's pentagram, which he considered to be a symbol of themicrocosm,or human

Theapotropaic(protective) use inGerman folkloreof the pentagram symbol (calledDrudenfussin German) is referred to byGoetheinFaust(1808), where a pentagram preventsMephistophelesfrom 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, 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.

Also protective is the use inIcelandic folkloreof a gestured or carved rather than painted pentagram (calledsmèrhnútin Icelandic), according to 19th century folkloristJón Árnason:

A butter that comes from the fake vomit is called a fake butter; it looks like something else; 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,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:

Uses in modern occultism

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 anapotropaic charmto protect against evil forces.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."In this spirit, theHermetic Order of the Golden Dawndeveloped the use of the pentagram in thelesser banishing ritual of the pentagram,which is still used to this day by those who practice Golden Dawn-type magic.

Aleister Crowleymade use of the pentagram in the system ofmagickused inThelema:an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation ofLon Milo DuQuette.Crowley contradicted his old comrades in theHermetic Order of the Golden Dawn,who, following Levi, considered this orientation of the symboleviland associated it with the triumph of matter over spirit.

Use in new religious movements

Baháʼí Faith

Main article:Baháʼí symbols

Haykal by theBábwritten in his own hand

The five-pointed star is a symbol of theBaháʼí Faith.In the Baháʼí Faith, the star is known as theHaykal(Arabic:"temple"), and it was initiated and established by theBáb.The Báb andBahá'u'lláhwrote various works in the form of a pentagram.

The Church of Jesus Christ of Latter-day Saints

The Church of Jesus Christ of Latter-day Saintsis theorized to have begun using both upright and inverted five-pointed stars inTemplearchitecture, dating from theNauvoo Illinois Templededicated on 30 April 1846.Other temples decorated with five-pointed stars in both orientations include theSalt Lake Templeand theLogan Utah Temple.These usages come from the symbolism found inRevelationchapter 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."

Wicca

Typical Neopagan pentagram (circumscribed)

USVAheadstone 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.In public schools, such actions by administrators were determined in 2000 to be in violation of students'First Amendmentright tofree exercise of religion.

The encircled pentagram (referred to as apentacleby the plaintiffs) was added to the list of 38 approved religious symbols to be placed on the tombstones of fallen service members atArlington National Cemeteryon 24 April 2007. The decision was made following ten applications from families of fallen soldiers who practicedWicca.The government paid the familiesUS$225,000to settle their pending lawsuits.

Other religious use

Satanism

The inverted pentagram is the most notable and widespread symbol ofSatanism.

The goat head ofBaphometforms an inverted pentagram, as depicted in theSigil of Baphomet

The inverted pentagram is broadly used inSatanism,sometimes depicted with thegoat's head ofBaphomet,as popularized by theChurch of Satansince 1968.LaVeyan Satanistspair the goat head with Hebrew letters at the five points of the pentagram to form theSigil of Baphomet.The Baphomet sigil was adapted for theJoy of Satan Ministrieslogo, usingcuneiformcharacters at the five points of the pentagram, reflecting the shape's earliest use inSumeria.The inverted pentagram also appears inThe Satanic Templelogo, with an alternate depiction of Baphomet's head. Other depictions of the Satanic goat's head resemble the inverted pentagram without its explicit outline.

Serer religion

The five-pointed star is a symbol of theSerer religionand theSerer peopleof West Africa. CalledYoonirintheir language,it symbolizes the universe in theSerer creation myth,and also represents the starSirius.

Other modern use

• The pentagram is featured on the national flags ofMorocco(adopted 1915) andEthiopia(adopted 1996 and readopted 2009)

• 
  Morocco's flag
• 
  Ethiopia's flag

• TheOrder of the Eastern Star,an organization (established 1850) associated withFreemasonry,uses a pentagram as its symbol, with the fiveisosceles trianglesof 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 star(the emblem shown here is from the Prince Hall Association).

• 
  Order of the Eastern Staremblem

• A pentagram is featured on the flag of theDutchcity ofHaaksbergen,as well on its coat of arms.

• 
  Flag of Haaksbergen

• A pentagram is featured on the flag of theJapanesecity ofNagasaki,as well on its emblem.

• 
  Flag of Nagasaki

Geometry

Koch snowflakesdrawn withMSWLogo(in Tartapelago)

The pentagram is the simplestregularstar 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 theSchläfli symbol{5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as itssymmetry groupthedihedral groupof order 10.

It can be seen as a net of apentagonal pyramidalthough with isosceles triangles.

Construction

The pentagram can be constructed by connecting alternate vertices of apentagon;seedetails of the construction.It can also be constructed as astellationof a pentagon, by extending the edges of a pentagon until the lines intersect.

Golden ratio

A regular pentagram colored to distinguish its line segments of different lengths. The four lengths are ingolden ratioto one another.

Thegolden ratio,φ= (1 +√5) / 2 ≈ 1.618, satisfying:

$\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\,$

$\varphi =1/(2\sin(\pi /10))=1/(2\sin 18^{\circ })\,$

$\varphi =2\cos(\pi /5)=2\cos 36^{\circ }\,$

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 isφ,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) isφ.As the four-color illustration shows:

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

The pentagram includes tenisosceles triangles:fiveacuteand fiveobtuseisosceles triangles. In all of them, the ratio of the longer side to the shorter side isφ.The acute triangles aregolden triangles.The obtuse isosceles triangle highlighted via the colored lines in the illustration is agolden gnomon.

Trigonometric values

Main article:Exact trigonometric values

${\begin{aligned}\sin {\frac {\pi }{10}}&=\sin 18^{\circ }={\frac {{\sqrt {5}}-1}{4}}={\frac {\varphi -1}{2}}={\frac {1}{2\varphi }}\\[5pt]\cos {\frac {\pi }{10}}&=\cos 18^{\circ }={\frac {\sqrt {2(5+{\sqrt {5}})}}{4}}\\[5pt]\tan {\frac {\pi }{10}}&=\tan 18^{\circ }={\frac {\sqrt {5(5-2{\sqrt {5}})}}{5}}\\[5pt]\cot {\frac {\pi }{10}}&=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}\\[5pt]\sin {\frac {\pi }{5}}&=\sin 36^{\circ }={\frac {\sqrt {2(5-{\sqrt {5}})}}{4}}\\[5pt]\cos {\frac {\pi }{5}}&=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}}\\[5pt]\tan {\frac {\pi }{5}}&=\tan 36^{\circ }={\sqrt {5-2{\sqrt {5}}}}\\[5pt]\cot {\frac {\pi }{5}}&=\cot 36^{\circ }={\frac {\sqrt {5(5+2{\sqrt {5}})}}{5}}\end{aligned}}$

As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths isφtimes that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle.

Spherical pentagram

Further information:Pentagramma mirificum

A pentagram can be drawn as astar polygonon a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described byJohn Napierin his 1614 bookMirifici 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 byCarl Friedrich Gauss.

Three-dimensional figures

Further information:Uniform polyhedron: Icosahedral symmetry

Severalpolyhedraincorporate pentagrams:

• 
  Pentagrammic prism
• 
  Pentagrammic antiprism
• 
  Pentagrammic crossed-antiprism
• 
  Small stellated dodecahedron
• 
  Great stellated dodecahedron
• 
  Small ditrigonal icosidodeca­hedron
• 
  Dodecadodecahedron

Higher dimensions

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

4D		5D
The regular5-cell(4-simplex) has five vertices and 10 edges.	Therectified 5-cellhas 10 vertices and 30 edges.	Therectified 5-simplexhas 15 vertices, seen in this orthogonal projection as three nested pentagrams.	Thebirectified 5-simplexhas 20 vertices, seen in this orthogonal projection as four overlapping pentagrams.

All ten 4-dimensionalSchläfli–Hess 4-polytopeshave either pentagrammic faces orvertex figureelements.

Pentagram of Venus

The pentagram of Venus

The pentagram of Venus is the apparent path of theplanetVenusas observed fromEarth.Successiveinferior conjunctionsof Venus repeat with anorbital resonanceof approximately 13:8—that is, Venus orbits theSunapproximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction.The tips of the five loops at the center of the figure have the same geometric relationship to one another as the fivevertices,or points, of a pentagram, and each group of fiveintersectionsequidistantfrom the figure's center have the same geometric relationship.

In computer systems

See also:Unicode input

The pentagram has theseUnicodecode points that enable them to be included in documents:

• U+26E4⛤PENTAGRAM
• U+26E5⛥RIGHT-HANDED INTERLACED PENTAGRAM
• U+26E6⛦LEFT-HANDED INTERLACED PENTAGRAM
• U+26E7⛧INVERTED PENTAGRAM

See also

• Abe no Seimei– Japanese mystic
• Christian symbolism– Use of symbols, including archetypes, acts, artwork or events, by Christianity
• Command at Sea insignia– Badge in the US military
• Enneagram (geometry)– Nine-pointed star polygon
• Five-pointed star– Common ideogram in modern culture
• Heptagram– Star polygon
• Hexagram– Six-pointed star polygon
• Lute of Pythagoras– Self-similar geometric figure
• Medal of Honor– Highest award in the United States Armed Forces
• Pentachoron– the 4-simplex
• Pentagram map– Discrete dynamical system on the moduli space of polygons in the projective plane
• Pent Alpha– Puzzle involving stones and a pentagram
• Petersen graph– Cubic graph with 10 vertices and 15 edges
• Ptolemy's theorem– Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle
• Seal of Solomon– Signet ring attributed to the Israelite king Solomon
• Star polygons in art and culture– Polygons as symbolic elements
• Star (heraldry)– Heraldic symbol
• Stellated polygons– Extending the elements of a polytope to form a new figure

Further reading

• Becker, Udo (1994)."Pentagram".The Continuum Encyclopedia of Symbols.Translated by Garmer, Lance W. New York City: Continuum Books. pp. 230ff.ISBN978-0-8264-0644-6.
• Conway, John Horton;Burgiel, Heidi; Goodman-Strauss, Chaim (April 2008). "Chapter 26, Higher Still: Regular Star-Polytopes".The Symmetries of Things.Wellesley, Massachusetts: A. K. Peters. p. 404.ISBN978-1-56881-220-5.
• Ferguson, George Wells (1966) [1954].Signs and Symbols in Christian Art.New York City: Oxford University Press. p. 59.OCLC65081051.
• Grünbaum, Branko;Shephard, Geoffrey Colin(1987).Tilings and Patterns.New York: W. H. Freeman.ISBN978-0-7167-1193-3.
• Grünbaum, Branko(1994). "Polyhedra with Hollow Faces". In Bisztriczky, T.;McMullen, P.;Schneider, A.; Weiss, A. Ivić (eds.).Polytopes: Abstract, Convex and Computational.NATO ASI Series C: Mathematical and Physical Sciences. Vol. 440. Dordrecht: Springer Netherlands. pp. 43–70.doi:10.1007/978-94-011-0924-6_3.ISBN978-94-010-4398-4.

External links

• Weisstein, Eric W."Pentagram".MathWorld.
• The Pythagorean Pentaclefrom the Biblioteca Arcana.