# Pentagram

The pentagram is a figure formed using five line segments of identical length, arranged to produce five identical points surrounding a pentagram.

A pentagram, a five sided, transparent star, often within a circle, is one of the oldest markings known to humankind. Dating back to Europe as far as 8000 years ago, the pentagram is a symbol fraught with mystery, intrigue, and meaning. It has been the symbol of various religions and nations, from Christianity and Islam to Morocco and Ethiopia to ancient Jerusalem. In modern times, the pentagram has been used by Wiccans and distorted further for the use of Satanists (though one errs to presume a connection between the two). The five points of the pentagram usually have five different meanings; for example, for the Wiccans, the five points represent earth, sky, fire, water, and Spirit, with Spirit occupying the top and most important point. However, these values change from culture to culture.

The pentagram has been used for a wide variety of purposes over the course of human history. Pentagons have been used both to call forth evil and to ward it off.

A standard pentagram, enclosed within a circle.

## History

The pentagram is one of the oldest markings known to humankind, apparently discovered by astronomical research in the Tigris-Euphrates region of the Middle East as far back as 6000 B.C.E. [1] Isolated pentagrams have been found in Israel, in layers dating to 4000 B.C.E.[1] It then shows up among the Sumerians, with the five points believed by scholars to represent either the four corners of the earth and "the vault of heaven," or the five visible planets of the night sky: Jupiter, Mercury, Mars, Saturn, and Venus (with Venus a representative of the Queen of Heaven). Most scholars tend to dismiss the first theory as far fetched, but it is difficult to ascertain exactly what the pentagram meant to ancient peoples due to the lack of thorough documentation. In fact, there is no clear evidence on how the pentagram was used, especially after Sumer, until around 400 B.C.E. and the rise of Pythagorean mysticism.[1]

### Pythagoreans

The Pythagoreans called the pentagram, ύγιεια (Hygieia) ("health;" also the Greek goddess of health, Hygieia), and saw in the pentagram a mathematical perfection which would later come to be known as the Golden ratio. The Pythagoreans, named so after Pythagoras (fl 580-500, B.C.E.), a mathematician who encouraged his followers to seek out truth and knowledge, were driven underground, and used the pentagram to identify themselves to each other, signing letters and communications with it.[2] During this time, the pentagram represented the five points of a human being: Two feet, two hands, and one head, although this seems to underestimate the knowledge of the Pythagoreans, as they were almost undoubtedly aware of its mathematical properties.[3]

What is known with a good amount of certainty, however, is that the pentagram was the main image in the logotype, or official seal of the city of Jerusalem during the period of 300-150 B.C.E.[3]

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.[4]

### European occultism

Heinrich Cornelius Agrippa, among others, perpetuated the popularity of the pentagram as a magical symbol, maintaining an attribution of elements (earth, fire, air, water) to the five points. By the mid-nineteenth 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." Conversely, a pentagram with two points up was considered evil. At other times also, especially during the Middle Ages, it came to represent devil worship.

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.[5]

However, in Nordic countries (such as Norway and Sweden), the pentagram was used to ward off trolls and evil, in general, and was drawn on doors and walls.

## 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.

An illustration of the Golden ratio, specifically applied to a pentagram.

### Construction

The pentagram can be constructed by connecting alternate vertices of a pentagon. 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. In mathematics, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. Thus, the golden ratio is approximately 1.6180339887. 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:

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.
$\frac{\mathrm{red}}{\mathrm{green}} = \frac{\mathrm{green}}{\mathrm{blue}} = \frac{\mathrm{blue}}{\mathrm{magenta}} = \varphi .$

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

$\sin \frac{\pi}{10} = \sin 18^\circ = \frac{\sqrt 5 - 1}{4}=\frac{\varphi-1}{2}=\frac{1}{2\varphi}$
$\cos \frac{\pi}{10} = \cos 18^\circ = \frac{\sqrt{2(5 + \sqrt 5)}}{4}$
$\tan \frac{\pi}{10} = \tan 18^\circ = \frac{\sqrt{5(5 - 2 \sqrt 5)}}{5}$
$\cot \frac{\pi}{10} = \cot 18^\circ = \sqrt{5 + 2 \sqrt 5}$
$\sin \frac{\pi}{5} = \sin 36^\circ = \frac{\sqrt{2(5 - \sqrt 5)} }{4}$
$\cos \frac{\pi}{5} = \cos 36^\circ = \frac{\sqrt 5+1}{4} = \frac{\varphi}{2}$
$\tan \frac{\pi}{5} = \tan 36^\circ = \sqrt{5 - 2\sqrt 5}$
$\cot \frac{\pi}{5} = \cot 36^\circ = \frac{ \sqrt{5(5 + 2\sqrt 5)}}{5}$

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.

## Religious symbolism

### Christianity

Connections between the pentagram and Christianity are many.[3] It adorned jewelry, amulets, and battle attire of early Christians, especially before the cross was introduced. This was not only because the pentagram was associated with the five wounds of Christ, but also because it could be drawn in a single stroke, through one continuous movement of a pen, representing beginning and end (Alpha and Omega) as one.[3]

Some also theorize that the pentagram was an expression of an early, secret Gnostic heresy, found hidden here and there throughout Christianity's history, a symbol of Isis/Venus as the "secret goddess," or female principle.[3] This symbolism commonly shows up in the Arthurian Grail romances, which many see as Gnostic and in kabbalistic teachings disguised as knightly quests and their tales.[3] For example, a pentagram appears on the shield of Sir Gawain in the fourteenth century poem, "Sir Gawain and the Green Knight."

The pentagram was embodied as a symbol of this feminine principle by the five petaled rose, found in many gothic cathedral ornamentations—they are truly subtle, not quite secret pentagrams.[3]

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

### Mormonism

Church of Jesus Christ of Latter-day Saints has traditionally used pentagrams and five-pointed stars in Temple architecture, particularly the Nauvoo Illinois Temple[6] and the Salt Lake Temple. These symbols derived from traditional morning star pentagrams that are no longer commonly used in mainstream Christianity.[7]

### Judaism

The pentagram was the official seal of the city of Jerusalem during the period of 300-150 B.C.E.[1] Due to the similarity of the star shapes, it is occasionally confused with the Star of David by those unfamiliar with the symbols. In the Jewish kabbalistic tradition, the pentagram represents justice, mercy, wisdom, understanding, and transcendent splendor.[3]

### 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 (Greek, baphe and metis, meaning "absorption of knowledge").[2] 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 symbolizing rejection of the holy Trinity.

Some pinpoint this symbol's first appearance to the brutal interrogations of the Knights Templar during the Christian Inquisition. However, there was no consensus as to the symbol's description.[2]

### Neopaganism

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.

A woman wears a pentacle as an amulet.

Its religious symbolism is commonly explained by reference to the neo-Pythagorean understanding that the five points of the pentagram represent the four elements (earth, fire, air, water) 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.[8] In public schools, such actions by administrators have been determined to be in violation of students' First Amendment right to free exercise of religion.[9]

### Bahá'í Faith

The pentagram is the official symbol of the Bahá'í Faith.[10] 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.

### 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 humanity's Atman, or Internal Christ. When a human'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. Similar to many other uses of the symbol, when the pentagram's inferior rays point upwards, it represents Satan.

### Asia

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, earth, air, water, wood, metal. As a predominantly non-Christian country, with a different set of associations attached to the symbol, there is no social stigma associated with it.

## 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.[11] It is also possible that both flags use the pentagram as a symbol of King Solomon, the archetypal wise king of Jewish, Christian, and Muslim lore.

## Other organizations

Order of the Eastern Star symbol.

### 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 Goethe's Faust, the pentagram prevents Mephistopholes 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 American gothic fiction writer 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 the planet Venus, based on the successive inferior conjunctions of Venus against the Zodiac.

## Notes

1. 1.0 1.1 1.2 1.3 Symbols.com, Pentagram, Symbols.com. Retrieved August 9, 2014.
2. 2.0 2.1 2.2 Symbols of Wicca, other Neopagan traditions, Satanism, etc., Religious tolerance.org. Ontario Consultants on Religious Tolerance. Retrieved August 9, 2014.
3. 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 The Pentagram in Depth, SymbolDictionary.net. Retrieved August 9, 2014.
4. G. S. Kirk and J. E. Raven, The Presocratic philosophers; a critical history with a selection of texts. (Cambridge, UK: University Press, 1957, ISBN 9780521274555), 55
5. Éliphas Lévi. Transcendental magic: its doctrine and ritual. (London: Bracken, 1995, ISBN 9781858913797).
6. Marshall University, Nauvoo Temple. Retrieved July 20, 2008.
7. Matthew B. Brown, Inverted Stars on LDS Temples, The Foundation for Apologetic Information and Research, Retrieved July 20, 2008.
8. B. A. Robinson, Religious Clothing & Jewelry in School, Religious tolerance.org. Ontario Consultants on Religious Tolerance. Retrieved July 20, 2008.
9. Associated Press, May 1, 2000, Federal judge upholds Indiana students' right to wear Wiccan symbols, First Amendment Center, Retrieved July 20, 2008.
10. Bahá'í Reference Library, 141: NINE (Number), Directives from the Guardian, (Bahá'í International Community), 51-52, Retrieved July 20, 2008.
11. Flags of the World, Moroccan flag, Flagspot. Retrieved July 20, 2008.

## References

• Grünbaum, B. "Polyhedra with Hollow Faces," Proc of NATO-ASI Conference on Polytopes, Edited by T. Bisztriczky et al. Kluwer Academic, 1994, 43-70. OCLC 197470331
• Grünbaum, Branko, and G. C. Shephard. Tilings and patterns. A Series of books in the mathematical sciences. New York: W.H. Freeman, 1987. ISBN 978-0716711933.
• Kirk, G. S., and J. E. Raven, The Presocratic Philosophers: A critical history with a selection of texts. Cambridge, UK: University Press, 1957, ISBN 978-0521274555.
• Lévi, Éliphas. Transcendental magic: its doctrine and ritual. London: Bracken, 1995, ISBN 978-1858913797.