Pentagram

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. 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 pentagram has long been associated with the planet Venus, and the worship of the goddess Venus, or her equivalent. It is also associated with the Roman word lucifer, which was a term used for Venus as the Morning Star, associated with the bringer of light and knowledge. It is most likely to have originated from the observations of prehistoric astronomers. When viewed from Earth, successive inferior conjunctions of Venus plot a nearly perfect pentagram shape around the zodiac every eight years.

The word "pentacle" is sometimes used synonymously with "pentagram," although their technical usages are different, and their etymologies may be unrelated.

File:Venus pentagram.png

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.

Early history

Sumer

The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 B.C.E. 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".^{[citation needed]} 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.^{[citation needed]}

Pythagoreans

The Pythagoreans called the pentagram ύγιεια 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:

• ύδωρ, hydor, water
• γαια, gaia earth
• ίδέα, idea or ίερόν, Hieron "a divine thing"
• έιλή, heile, heat (fire)
• άήρ, aer, air

The vertices were labeled in the letters of υ-γ-ι-ει-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. The pentemychos is in Tartaros, also known as "The Gates of Hell".^{[citation needed]}

In very early Greek thought, Tartaros (or Chaos, according to Hesiod) was the primordial Darkness from which the cosmos is born. While it was locked away after the emergence and ordering of the cosmos, it still continued to have an influence. In fact, it was known as "the subduer of both gods and men" (Homer), and it was from this that the world got its "psyche" (soul) and its "daimon." The Boundless Darkness held influence through Mychos or Krater. Apart from being the gateway from "there" to "here" it was also a way in the opposite direction, from "here" to "there," as is evident in the many tales about how Greek heroes, philosophers and mystics descended through Krater to Tartaros/Hades (the distinction between the two was very optional back then) in quest for Wisdom. The Underworld as the source of wisdom was the rule.

Tartaros was also later seen as the "chthonic realm" where all the enemies of the cosmic order were locked away, also called the "prison-house" of Zeus. It was said to lay outside of the aither over which Zeus had lordship; what we today would call space, back then called "Zeus' defense-wall," yet it was also beneath the earth. Plato (in Cratylus) said that the aither had a penetrating power that permeates the whole world, and he found it both inside and outside of our bodies. The pentemychos is outside, or in-side, of the aither.

In the play Medea by Euripides, the sorceress Medea calls upon Hecate with the words, "By that dread queen whom I revere before all others and have chosen to share my task, by Hecate who dwells within my inmost chamber, not one of them shall wound my heart and rue it not." Note that she speaks of the Heart. The inmost chamber is the Mychos. Normally, Hecate and Persephone are portrayed solely as the rulers of the Underworld. In Medea, however, Hecate is called the Lady of Tartaros, Phulada (Guardian), Propulaia (Before the Gates), Kleidophoros (Key-bearer) and Kleidoukhos (Key-holder, Priestess). This Underworld of the Greeks and Pythagoreans is also the "inmost chamber" and the Core of Inner Being.

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 other way up was considered evil.

"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."

"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."

• Pentagram and human body (Agrippa).jpg

  Man inscribed in a pentagram, from Heinrich Cornelius Agrippa's Libri tres de occulta philosophia. The five signs at the pentagram's vertices are astrological.

• Inverted pentacle.PNG

  Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.

• 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

File:Bolzani XP Pentacle.JPG

Christ as a Pentagram, from Valeriano Bolzani's Hieroglyphica (Basel, 1556)

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).^{[citation needed]}

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.

Probably due to misinterpretation of symbols used by ceremonial magicians, it later became associated with Satanism and subsequently rejected by most of Christianity sometime in the twentieth century.

Mormonism

The Church of Jesus Christ of Latter-day Saints has traditionally used pentagrams and five-pointed stars in Temple architecture, particularly 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 Jerusalem 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.

Satanism

A goat's head inscribed in a pentagram, from La Clef de la Magie Noire by the Rosicrucian Stanislas de Guaita (1897).

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.

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.

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 (Arabic: "temple"), and it was initiated and established by the Báb. Both Báb and Bahá'u'lláh wrote various works in the form of a pentagram.

• Haykal-Bab-2.gif

  An unidentified work of the Báb.

• Haykal2.gif

  An unidentified work of the Báb.

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, not the triumph over matter which was considered evil as taught by the Hermetic Order of the Golden Dawn.

Samael Aun Weor

Samael Aun Weor used the Pentagram to represent man's Atman, or Internal Christ. When a man's limbs are outstretched thus that his feet are planted on the ground while his head is situated atop his body it creates the omnipotent symbol of the pentagram. Through the Mantra "Klim, Krishna, Govindaya, Gopijana, Vallebayah, Swahah" one's inner being is said to be awakened and come to the initiate's aid. Aun Weor stated that no demon could resist the power of this mantra, since one's Logos cannot be overcome by a demon of any stature.

In contrast to representing one's Logos, the inverted pentagram represents one's Umbral Guardian, the malignant antithesis of the divine father. When the pentagram's inferior rays point upwards, it represents Satan. This symbol is therefore shown above as the goat of the Witches' Sabbath, which serves as a call to the vast columns of demons.

Political symbolism

Flags

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

According to Ivan Sache, on the Moroccan 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.

• 

  Morocco's flag

• 

  Ethiopia's flag

Other organizations

File:OrderEasternStar logo from saucer.jpg

Order of the Eastern Star emblem

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 Japanese culture, the pentagram (五芒星 gobōsei) is a symbol of magical power, associated with the onmyoji Abe no Seimei; it is a diagram of the "overcoming cycle" of the five Chinese elements. 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

File:Complete graph K5.svg

Complete graph K₅

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

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

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 triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon.

Fractal pentagram drawn with a vector iteration program

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:

• Pentagrammic prism.png

  Pentagrammic prism

• Pentagrammic antiprism.png

  Pentagrammic antiprism

• Pentagrammic crossed antiprism.png

  Pentagrammic crossed antiprism

• 

  Small stellated dodecahedron

• Small ditrigonal icosidodecahedron.png

  Small ditrigonal icosidodecahedron

• Small dodecahemicosahedron.png

  Small dodecahemicosahedron

• Snub dodecadodecahedron.png

  Snub dodecadodecahedron

• Sixteenth stellation of icosidodecahedron.png

  16th stellation of icosidodecahedron

References

ISBN links support NWE through referral fees

• 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. Edited by T. Bisztriczky et al. Kluwer Academic, 1994. p. 43-70.

External links

All links retrieved July 1, 2008.

• Eric W. Weisstein. Pentagram. MathWorld
• The Pythagorean Pentacle from the Biblioteca Arcana.
• In-depth analysis of the Golden Ratio
• The pentagram and Freemasonry

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