Pentagram

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")[7] was the title of the cosmogony of Pherecydes of Syros.[8] Here, the "five corners" are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear.[9]

A Pythagorean "Hugieia Pentagram"[10]

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"[11]

The pentagram was used in ancient times as a Christian symbol for the five senses,[12] 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.[13]

The North rose of Amiens Cathedral

The North rose of Amiens cathedral (built in the 13^th 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.

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 Éliphas Lévi 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."[14]
• "The flaming star, which, when turned upside down, is the heirolgyphic [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."[15]
• "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."[16]

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  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.
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  Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.
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  The occultist and magician Éliphas Lévi'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 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 wizard-foot [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.

Wu Xing's five phases

Wu Xing (Chinese: 五行; pinyin: Wǔ Xíng) are the five phases, or five elements in Taoists Chinese tradition. They are differentiated from the formative ancient Japanese or Greek elements, due to there emphasis on cyclic transformations and change. The five phases are: Fire (火 huǒ), Earth (土 tǔ), Metal (金 jīn), Water (水 shuǐ), and Wood (木 mù). The Wuxing is the fundamental philosophy and doctrine of traditional Chinese Medicine and Acupuncture.[17]

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.[18] 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."[19] 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 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.[20] 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.

The five-pointed star is a symbol of the Baháʼí Faith.[21][22] In the Baháʼí 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.[23][24]

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  Haykal by the Báb written in his own hand.
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  An unidentified work of the Báb.
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  An unidentified work of the Báb.

The Church of Jesus Christ of Latter-day Saints is theorized to have began using both upright and inverted five-pointed stars in Temple architecture, dating from the Nauvoo Illinois Temple dedicated on 30 April 1846.[25] 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."[26]

Typical Neopagan pentagram (circumscribed)

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.[27] 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.[28]

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.[29][30]

Yoonir, symbol of the universe in Serer religion and creation mythology.[31][32]

Representation of the three worlds in Serer cosmogony: the invisible world, the terrestrial world and the nocturnal world.[33]

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.[31][32]

A multicolored version is used as symbol of the Druze religion.

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

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  Morocco's flag
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  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 point down, but in a few it is point up. Grand Chapter officers often have a pentagon inscribed around the star[34](the emblem shown here is from the Prince Hall Association).

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  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.

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  Flag of Haaksbergen

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

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  Flag of Nagasaki

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.

A uniform truncated pentagram t{5/2} produces a doubly-wrapped pentagon with overlapping vertices and edges, {10/2}. A shallower truncation produces an isogonal figure, like this one with equally spaced vertices. A truncated retro-pentagram t{5/3}, or a quasitruncation, produces a decagram, {10/3}.

shallow t{5/2}

t{5/3} = {10/3}

A regular 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

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

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

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

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 four-color illustration shows:

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

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.

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

As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths 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.

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.

Several polyhedra incorporate pentagrams:

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

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

4D		5D
The regular 5-cell (4-simplex) has five vertices and 10 edges.	The rectified 5-cell has 10 vertices and 30 edges.	The rectified 5-simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams.	The birectified 5-simplex has 20 vertices, seen in this orthogonal projection as four overlapping pentagrams.

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