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Regular pentagram

A pentagram
Type Star polygon
Edge and vertices 5
Schläfli symbol {5/2}
Coxeter–Dynkin diagram
Symmetry group D5 (Order 10)
Internal angle
(degree)
36°


A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), a noun form of πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "five-lined" or "five lines".

Pentagrams were used symbolically in ancient Greece and Babylonia, and are used today as a symbol of faith by many Wiccans, akin to the use of the cross by Christians and the Star of David by Jews. The pentagram has magical associations, and many people who practice Neopagan faiths wear jewelry incorporating the symbol. Christians once more commonly used the pentagram to represent the five wounds of Jesus, and it also has associations within Freemasonry.

The word "pentacle" is sometimes used synonymously with "pentagram", and this usage is borne out by the Oxford English Dictionary, although that work specifies that a circumscription makes the shape more particularly a pentacle. Wiccans and Neo-pagans often make use of this more specific definition for a pentagram enclosed in a circle.

Early history

Sumer

The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC.The Sumerian pentagrams served as pictograms for the word "UB," meaning "corner, angle, nook; a small room, cavity, hole; pitfall," suggesting something very similar to the pentemychos (see below on the Pythagorean use for what pentemychos means). In René Labat's index system of Sumerian hieroglyphs/pictograms it is shown with two points up.In the Babylonian context, the edges of the pentagram were probably orientations: forward, backward, left, right, and "above". These directions also had an astrological meaning, representing the five planets Jupiter, Mercury, Mars and Saturn, and Venus as the "Queen of Heaven" (Ishtar) above.

Pythagoreans

The Pythagoreans called the pentagram ὑγιεία Hygieia ("health"; also the Greek goddess of health, Hygieia), and saw in the pentagram a mathematical perfection (see Geometry section below).

The five vertices were also used by the medieval neo-pythagoreans (whom one could argue were not pythagoreans at all) to represent the five classical elements:

  • ὕδωρ, hydor, water
  • γαῖα, gaia earth
  • εἱλή, heile, heat (fire)
  • ἰδέα, idea or ἱερόν, Hieron "a divine thing"
  • ἀήρ, aer, air


The vertices were labeled in the letters of υ-γ-ε-ι-α. The ordering (clockwise or counter-clockwise) and starting vertex varied.

The ancient Pythagorean pentagram was drawn with two points up and represented the doctrine of Pentemychos. Pentemychos means "five recesses" or "five chambers", also known as the pentagonas — the five-angle, and was the title of a work written by Pythagoras's teacher and friend Pherecydes of Syros. It was also the "place" where the first pre-cosmic offspring had to be put in order for the ordered cosmos to appear.

European occultism

Heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, keeping the Pythagorean attributions of elements to the five points. 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.
"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 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."
"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."


Image: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.Image:Inverted pentacle.PNG|Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.Image:Pentagram (Levi).jpg|The occultist and magician Eliphas Levi's pentagram, which he considered to be a symbol of the microcosm, or human.


Religious symbolism

Christianity

The pentagram is used as a Christian symbol for the five senses, and if the letters S, A, L, V, and S are inscribed in the points, it can be taken as a symbol of health (from Latin salus).

Medieval Christians believed it to symbolise the five wounds of Christ. The pentagram was believed to protect against witches and demons.

The pentagram figured in a heavily symbolic Arthurian romance: it appears on the shield of Sir Gawain in the 14th century poem Sir Gawain and the Green Knight. As the poet explains, the five points of the star each have five meanings: they represent the five senses, the five fingers, the five wounds of Christ, the five joys that Mary had of Jesus (the Annunciation, the Nativity, the Resurrection, the Ascension, and the Assumption), and the five virtues of knighthood which Gawain hopes to embody: noble generosity, fellowship, purity, courtesy, and compassion.

Most Christians, probably due to their misinterpretation of symbols used by ceremonial magicians, came to associate it with Satanism and subsequently rejected the symbol sometime in the twentieth century.

Mormonism

The Church of Jesus Christ of Latter-day Saints has used pentagrams and five-pointed stars in Temple architecture, specifically the Nauvoo Illinois Temple and the Salt Lake Temple. These symbols derived from traditional morning star pentagrams that are no longer commonly used in mainstream Christianity.

Judaism

The pentagram was the official seal of the city of Jerusalemmarker for a time. Due to the similarity of the star shapes, it is occasionally confused with the Star of David by those unfamiliar with the symbols.

Neopaganism

A typical Neopagan pentagram (circumscribed).
Many Neopagans, especially Wiccans, use the pentagram as a symbol of faith similar to the Christian cross or the Jewish Star of David. It is not, however, a universal symbol for Neopaganism, and is rarely used by Reconstructionists. Its religious symbolism is commonly explained by reference to the neo-Pythagorean understanding that the five vertices of the pentagram represent the four elements with the addition of Spirit as the uppermost point. As a representation of the elements, the pentagram is involved in the Wiccan practice of summoning the elemental spirits of the four directions at the beginning of a ritual.

The outer circle of the circumscribed pentagram is sometimes interpreted as binding the elements together or bringing them into harmony with each other. The Neopagan pentagram is generally displayed with one point up, partly because of the "inverted" goat's head pentagram's association with Satanism; however, within traditional forms of Wicca a pentagram with two points up is associated with the Second Degree Initiation and in this context has no relation to Satanism.

Because of a perceived association with Satanism and also because of negative societal attitudes towards Neopagan religions and the "occult", many United States schools have sought to prevent students from displaying the pentagram on clothing or jewelry. In public schools, such actions by administrators have been determined to be in violation of students' First Amendment right to free exercise of religion.

Satanism

Satanists use a pentagram with two points up, often inscribed in a double circle, with the head of a goat inside the pentagram. This is referred to as the Sigil of Baphomet. They use it much the same way as the Pythagoreans, as Tartaros literally translates from Greek as a "Pit" or "Void" in Christian terminology (the word is used as such in the Bible, referring to the place where the fallen angels are fettered). The Pythagorean Greek letters are most often replaced by the Hebrew letters לויתן forming the name Leviathan. Less esoteric LaVeyan Satanists use it as a sign of rebellion or religious identification, the three downward points symbolising rejection of the holy Trinity.

Thelema

Aleister Crowley also made use of the pentagram and 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. 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.

Bahá'í Faith

The pentagram is the official symbol of the Bahá'í Faith. In the Bahá'í Faith, the pentagram is known as the Haykal ( ), and it was initiated and established by the Báb. Both Báb and Bahá'u'lláh wrote various works in the form of a pentagram.

Image:Haykal-Bab-2.gif|An unidentified work of the Báb.Image:Haykal2.gif|An unidentified work of the Báb.


Taoism

Representation of the Chinese five elements (Wǔ Xíng)
Taoism conceived of a five element system which governed the natural world which they called the Wu Xing(Chinese:五行/Wǔ Xíng). Unlike the Greek system of four elements being Earth, Air, Fire, and Water, the Chinese system involved Fire(火/huǒ), Earth(土/tǔ), Metal(金/jīn), Water(水/shuǐ), and Wood(木/mù). This five element system was normally depicted as a pentagram(Chinese:五星/Wǔ Xīng) ringed by a circle. The circle was used to describe the generative cycle where wood feeds fire, fire creates earth (ash), earth bears metal, metal carries water, and water nourishes wood. The pentagram describes the destructive or overcoming system where Wood parts Earth, Earth absorbs Water, Water quenches Fire, Fire melts Metal, Metal chops Wood, or the alternative destructive system: Wood absorbs Water, Water rusts Metal, Metal breaks up Earth, Earth smothers Fire, Fire burns Wood. This system informs traditional Chinese medicine (TCM) even today. As the Wu Xing is of great antiquity and the silk road had been open to Europe since before the Roman Empire it is quite likely that this medical system was imported to Europe as a misunderstood and exotic practice involving spirits (Shen).

Political symbolism

Flags

While a solid five-pointed star is found on many flags, the pentagram is relatively rare. It appears on three national flags, those of Ethiopiamarker, Ghanamarker and Moroccomarker and in some coats of arms.

According to Ivan Sache, on the Moroccanmarker flags, the pentagram represents the link between God and the nation. It is also possible that both flags use the pentagram as a symbol of King Solomon (see Seal of Solomon), the archetypal wise king of Jewish, Christian and Muslim lore. The Ghana pentagram refers to the hope of Ghana being the Black Star and a shining example to the continent.

Image:Flag of Morocco.svg|Moroccomarker's flagImage:Flag of Ethiopia.svg|Ethiopiamarker's flag


Other organizations

Order of the Eastern Star

The Order of the Eastern Star, a fraternal organization associated with Freemasonry, has employed a point-down pentagram as its symbol, with the five isosceles triangles of the points colored red, blue, yellow, white and green. This is an older form of the order's emblem and it is now more commonly depicted with the central pentagon rotated 36° so that it is no longer strictly a pentagram.

In literature

In the medieval romance of Sir Gawain and the Green Knight, the pentagram on Gawain's shield is given a Christian interpretation (see above).

In Goethe's Faust, the pentagram prevents Mephistopheles from leaving a room.

Mephistopheles:
:I must confess, my stepping o'er
:Thy threshold a slight hindrance doth impede;
:The wizard-foot doth me retain.


Faust:
:The pentagram thy peace doth mar?
:To me, thou son of hell, explain,
:How earnest thou in, if this thine exit bar?
:Could such a spirit aught ensnare?


In H. P. Lovecraft's Cthulhu Mythos stories, the version of The Elder Sign devised by August Derleth is a warped pentagram with a flaming eye or pillar of flame in the center. It was first described in Derleth's novel, The Lurker at the Threshold. (This was, however, different from the symbol that Lovecraft himself had envisaged.)

In Dan Brown's novel The Da Vinci Code, the pentagram represents Venus, based on the successive inferior conjunctions of Venus against the Zodiac.

In Japanesemarker 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. As a predominantly non-Christian country, with a different set of associations attached to the symbol, there is no social stigma associated with it.

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
\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 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 2 intersecting edges (a side of the pentagon in the pentagram's center) is φ. 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 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 triangle. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon.

Trigonometric values

:See Exact trigonometric constants: Pentagon


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


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


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


\cot \frac{\pi}{10} = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5}


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


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


\tan \frac{\pi}{5} = \tan 36^\circ = \sqrt{5 - 2\sqrt 5}


\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.

Three-dimensional figures

See Uniform polyhedron: Icosahedral symmetry
Several polyhedra incorporate pentagrams:

Image:Pentagrammic prism.png|Pentagrammic prismImage:Pentagrammic antiprism.png|Pentagrammic antiprismImage:Pentagrammic crossed antiprism.png|Pentagrammic crossed antiprismImage:Small_stellated_dodecahedron.png|Small stellated dodecahedronImage:Small ditrigonal icosidodecahedron.png|Small ditrigonal icosidodecahedronImage:Small dodecahemicosahedron.png|Small dodecahemicosahedronImage:Snub dodecadodecahedron.png|Snub dodecadodecahedronImage:Sixteenth stellation of icosidodecahedron.png|16th stellation of icosidodecahedron

Higher dimensions

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


The regular 5-cell (4-simplex) has 5 vertices and 10 edges

Rectified 5-cell has 10 vertices and 30 edges


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

In Astronomy and nature


Successive inferior conjunctions 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.


See also





Notes

  1. "Pentagram" article in The Continuum Encyclopedia of Symbols Becker, Udo, ed., Garmer, Lance W. translator, New York: Continuum Books, 1994, p. 230.
  2. Signs and Symbols in Christian Art Ferguson, George, Oxford University Press: 1966, p. 59.
  3. Order of the Eastern Star
  4. "Pentacle", Oxford English Dictionary.
  5. Labat, René. Manuel d'épigraphie akkadienne: Signes, Syllabaire, Idéogrammes. The pentagram is symbol number 306 in this system.
  6. http://merlinravensong2.tripod.com/the-pentagram.html
  7. This is a lost book, but its contents are preserved in Damascius, De principiis, quoted in Kirk and Raven, The Pre-Socratic Philosophers, Cambridge Univ. Press, 1956, p. 55.
  8. Found at http://www.scribd.com/doc/9668236/The-Key-of-the-Mysteries-by-Levi, page 69.
  9. Christian Symbols Ancient and Modern, Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, ISBN 0-7135-1960-6.
  10. http://www.mediade.si/media/symbolic.meaning.of.the.pentagram.extract.pdf
  11. Pentagram, pentacle
  12. Christian Symbols and How To Use Them, Knapp, Justina; Milwaukee: The Bruce Publishing Company, 1955. Plate LXV, Plate LV (Imprimatur, Jos. F. Busch, Bishop of St. Cloud)
  13. See the Nauvoo Temple website discussing its architecture, and particularly the page on Nauvoo Temple exterior symbolism. Retrieved 13 December 2006.
  14. Pentagram, pentacle
  15. Chapter II - Second Degree Initiation
  16. "Religious Clothing in School", Robinson, B.A., Ontario Consultants on Religious Tolerance, 20 August, 1999, updated 29 April, 2005. accessed 10 February, 2006.
  17. "ACLU Defends Honor Student Witch Pentacle" press release, American Civil Liberties Union of Michigan, 10 February, 1999. accessed 10 February, 2006.
  18. "Witches and wardrobes: Boy says he was suspended from school for wearing magical symbol" Rouvalis, Cristina; Pittsburgh Post-Gazette, September 27, 2000. accessed 10 February, 2006.
  19. The Magick of Aleister Crowley p 93 and endnote 31 to Chapter Five (p 247).
  20. Bahá'í Reference Library - Directives from the Guardian, Pages 51-52
  21. Moroccan flag on Flagspot.net accessed on 10 February, 2006.


References

  • Grünbaum, B. and G. C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
  • Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43-70.


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