Pentagram

For other uses, see Pentagram (disambiguation).

Regular pentagram
A regular pentagram
Type	Regular polygon
Edges and vertices	5
Schläfli symbol	{5/2}
Coxeter diagram
Symmetry group	Dihedral (D5)
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 five-pointed star drawn with five straight strokes.

The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line". The word "pentacle" is sometimes used synonymously with "pentagram" The word pentalpha is a learned modern (17th-century) revival of a post-classical Greek name of the shape.

Geometry

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

Fractal pentagram drawn with a vector iteration program

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 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:

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

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

Further information: 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

Further information: Uniform polyhedron: Icosahedral symmetry

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

• 

  Great stellated dodecahedron

• 

  Small ditrigonal icosidodeca­hedron

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  Dodecadodecahedron

• 

  Great icosi­dodeca­hedron

• 

  Snub dodecadodecahedron

• 

  Small snub icosicosidodeca­hedron

• 

  Small retrosnub icosicosidodeca­hedron

• 

  Great retrosnub icosidodeca­hedron

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  Great dirhombicosidodeca­hedron

• 

  16th stellation of icosidodeca­hedron

Higher dimensions

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

4D		5D
The regular 5-cell (4-simplex) has 5 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 3 nested pentagrams.	The birectified 5-simplex has 20 vertices, seen in this orthogonal projection as 4 overlapping pentagrams.

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

Pentagram of Venus

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

A Pythagorean "Hugieia Pentagram"

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"

Western symbolism

The pentagram was used in ancient times as a Christian symbol for the five senses, or of the five wounds of Christ. A Christian use of the pentangle occurs 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 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.

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 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."

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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 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 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.

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

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

Based on Renaissance-era 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. 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

Main article: Bahá'í symbols

The five-pointed star is a symbol of the Bahá'í Faith. 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.

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  An unidentified work of the Báb.

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  An unidentified work of the Báb.

Latter Day Saint Movement

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

Wicca

A typical Neopagan pentagram (circumscribed).

Because of a perceived association with Satanism and occultism, many United States schools in the late 1990s 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 in 2000. 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.

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)

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  Morocco's flag

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  Ethiopia's flag

• The Order of the Eastern Star, an organization associated with Freemasonry (established 1850), used to have a point-down 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)

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  Order of the Eastern Star emblem

See also

• Barnstar
• Command-at-Sea Pin
• Enneagram (geometry)
• Heptagram
• Hexagram
• A particular case of hypotrochoid is a pentagram-like 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 4-simplex
• Pentad
• Pentagram map (related concept)
• Pentalpha
• Petersen graph
• Ptolemy's theorem
• Red star
• Star (glyph)
• Stellated polygons

Bibliography

• "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 Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

External links

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