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Regular pentagram
Regular star polygon 5-2.svg
A regular pentagram
Type Regular star polygon
Edges and vertices 5
Schläfli symbol {5/2}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.png
Symmetry group Dihedral (D5)
Internal angle (degrees)36°
Dual polygon self
Properties Star, cyclic, equilateral, isogonal, isotoxal

A pentagram (sometimes known as a pentalpha, pentangle or star pentagon) is the shape of a five-pointed star.


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 the Jews. The pentagram has magical associations. Many people who practice Neopagan faiths wear jewelry incorporating the symbol. Christians once commonly used the pentagram to represent the five wounds of Jesus. [1] [2] The pentagram has associations with Freemasonry and is also used as a symbol by other belief systems.

Ancient Greece Civilization belonging to an early period of Greek history

Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Roughly three centuries after the Late Bronze Age collapse of Mycenaean Greece, Greek urban poleis began to form in the 8th century BC, ushering in the Archaic period and colonization of the Mediterranean Basin. This was followed by the period of Classical Greece, an era that began with the Greco-Persian Wars, lasting from the 5th to 4th centuries BC. Due to the conquests by Alexander the Great of Macedon, Hellenistic civilization flourished from Central Asia to the western end of the Mediterranean Sea. The Hellenistic period came to an end with the conquests and annexations of the eastern Mediterranean world by the Roman Republic, which established the Roman province of Macedonia in Roman Greece, and later the province of Achaea during the Roman Empire.

Babylonia Ancient Akkadian region in Mesopotamia

Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia. A small Amorite-ruled state emerged in 1894 BC, which contained the minor administrative town of Babylon. It was merely a small provincial town during the Akkadian Empire but greatly expanded during the reign of Hammurabi in the first half of the 18th century BC and became a major capital city. During the reign of Hammurabi and afterwards, Babylonia was called "the country of Akkad", a deliberate archaism in reference to the previous glory of the Akkadian Empire.

Wicca modern pagan, witchcraft religion

Wicca, also termed Pagan Witchcraft, is a contemporary Pagan new religious movement. It was developed in England during the first half of the 20th century and was introduced to the public in 1954 by Gerald Gardner, a retired British civil servant. Wicca draws upon a diverse set of ancient pagan and 20th-century hermetic motifs for its theological structure and ritual practices.

The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), [3] from πέντε (pente), "five" + γραμμή (grammē), "line". [4] The word "pentacle" is sometimes used synonymously with "pentagram". [5] The word pentalpha is a learned modern (17th-century) revival of a post-classical Greek name of the shape. [6]

Greek language Language spoken in Greece, Cyprus and Southern Albania

Greek is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning more than 3000 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.

A pentacle is a talisman that is used in magical evocation, and is usually made of parchment, paper, cloth, or metal, upon which a magical design is drawn. Protective symbols may also be included, a common one being the six-point form of the Seal of Solomon.


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).[ citation needed ]

First Dynasty of Ur

The First Dynasty of Ur was a 26th century-25th century BCE dynasty of rulers of the city of Ur in ancient Sumer. It is part of the Early Dynastic period III of the History of Mesopotamia. It was preceded by the earlier First dynasty of Kish and First Dynasty of Uruk.

Cuneiform Old writing system used for many languages, including Akkadian and Hittite

Cuneiform or Sumerian cuneiform, one of the earliest systems of writing, was invented by the Sumerians. It is distinguished by its wedge-shaped marks on clay tablets, made by means of a blunt reed for a stylus. The name cuneiform itself simply means "wedge shaped".

Logogram Grapheme which represents a word or a morpheme

In written language, a logogram or logograph is a written character that represents a word or phrase. Chinese characters are logograms; some Egyptian hieroglyphs and some graphemes in cuneiform script are also logograms. The use of logograms in writing is called logography, and a writing system that is based on logograms is called a logographic system.

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] [ clarification needed ]

Cosmogony (from the Koine Greek Greek: κοσμογονία and the root of γί νομαι / γέγονα is any model concerning the origin of either the cosmos or the universe.

Pherecydes of Syros Greek philosopher

Pherecydes of Syros was a Greek thinker from the island of Syros. Pherecydes authored a cosmogony, derived from three divine principles, Zas (Zeus), Cthonie (Earth) and Chronos (Time), known as the "Pentemychos". It formed a bridge between the mythological thought of Hesiod and pre-Socratic philosophy. His work is lost, but it survived into the Hellenistic period and we are informed on part of its content indirectly. Pherecydes was said to have been the first writer to communicate philosophical musings in prose. According to William, Aristotle regarded him partly a mythological writer and Plutarch, as well as many other writers gave him the title of Theologus.

Chronos Ancient Greek mythological Titan and personification of time

Chronos is the personification of time in pre-Socratic philosophy and later literature.

A Pythagorean "Hugieia Pentagram" Hugieia-pentagram.svg
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" [11]

Neoplatonism A strand of Platonic philosophy that emerged in the third century AD

Neoplatonism is a term used to designate a strand of Platonic philosophy that emerged in the third century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as it encapsulates a chain of thinkers which began with Ammonius Saccas and his student Plotinus and which stretches to the sixth century AD. Even though neoplatonism primarily circumscribes the thinkers who are now labeled Neoplatonists and not their ideas, there are some ideas that are common to neoplatonic systems, for example, the monistic idea that all of reality can be derived from a single principle, "the One".

Hygieia Ancient Greek goddess of good health and cleanliness

In Greek as well as Roman mythology, Hygieia, was one of the Aeclepiadae; the sons and daughters of the god of medicine, Asclepius, and the goddess of healing, Epione. She was the goddess/personification of health, cleanliness and hygiene.

Western symbolism

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]

Sense Physiological capacity of organisms that provides data for perception

A sense is a physiological capacity of organisms that provides data for perception. The senses and their operation, classification, and theory are overlapping topics studied by a variety of fields, most notably neuroscience, cognitive psychology, and philosophy of perception. The nervous system has a specific sensory nervous system, and a sense organ, or sensor, dedicated to each sense.

<i>Sir Gawain and the Green Knight</i> late 14th-century Middle English chivalric romance

Sir Gawain and the Green Knight is a late 14th-century Middle English chivalric romance. It is one of the best known Arthurian stories, with its plot combining two types of folklore motifs, the beheading game and the exchange of winnings. Written in stanzas of alliterative verse, each of which ends in a rhyming bob and wheel, it draws on Welsh, Irish, and English stories, as well as the French chivalric tradition. It is an important example of a chivalric romance, which typically involves a hero who goes on a quest which tests his prowess. It remains popular to this day in modern English renderings from J. R. R. Tolkien, Simon Armitage, and others, as well as through film and stage adaptations.

Gawain King Arthurs nephew and a Knight of the Round Table in the Arthurian legend

Gawain, also known as Gawaine or Gauwaine, among various other forms and spellings, is King Arthur's nephew and a Knight of the Round Table in the Arthurian legend. Under the name Gwalchmei, he appears very early in the legend's development, being mentioned in some of the earliest Welsh Arthurian sources. As Gawain, he appears in Latin, French, English, Dutch, German and Italian literature, notably as the protagonist of the story of Sir Gawain and the Green Knight. Other tales of Gawain include Historia Regum Britanniae, Roman de Brut, De Ortu Waluuanii, Diu Crône, The Awntyrs off Arthure, L'âtre périlleux, Le Chevalier à l'épée, and The Weddynge of Syr Gawen and Dame Ragnell, as well as the works of Chrétien de Troyes and the prose cycle Lancelot-Grail.

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.

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):

I must confess, my stepping o'er
Thy threshold a slight hindrance doth impede;
The wizard-foot [Drudenfuss] doth me retain.
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?
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's five phases. Wuxing en.svg
Wu Xing's five phases.

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 ), Metal (金 jīn), Water (水 shuǐ), and Wood (木 ).

Uses in modern occultism

Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists. It's major use is a continuation of the ancient Babylonian use of the pentagram as an apotropiac charm to protect against evil forces [17] . Eliphas Levi 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." [18] 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. [19] 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


The five-pointed star is a symbol of the Bahá'í Faith. [20] [21] 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. [22]

Latter Day Saints

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. [23] 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." [24]


A typical Neopagan pentagram (circumscribed). Pentagram within circle.svg
A typical Neopagan pentagram (circumscribed).
USVA headstone emblem 37 USVA headstone emb-37.svg
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. [25] 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. [26]

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. [27] [28]

Other religious use

Serer religion

Five Pointed Star Lined.svg
Yoonir, symbol of the universe in Serer religion and creation mythology. [29] [30]
Serer cosmogony - representation of the universe.jpg
Representation of the three worlds in Serer cosmogony : the invisible world, the terrestrial world and the nocturnal world. [31]

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. [29] [30]


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

Other modern use


Fractal pentagram drawn with a vector iteration program PentagramFractal.PNG
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.

It can be seen as a net of a pentagonal pyramid although with isosceles triangles.


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}.

Regular star truncation 5-3 2.svg
shallow t{5/2}
Regular star truncation 5-3 1.svg
t{5/3} = {10/3}

Golden ratio

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

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:

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

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.

Spherical pentagram

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.

Three-dimensional figures

Several polyhedra incorporate pentagrams:

Higher dimensions

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

4-simplex t0.svg
The regular 5-cell (4-simplex) has five vertices and 10 edges.
4-simplex t1.svg
The rectified 5-cell has 10 vertices and 30 edges.
5-simplex t1 A4.svg
The rectified 5-simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams.
5-simplex t2 A4.svg
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.

Pentagram of Venus

The pentagram of Venus Venus geocentric orbit curve simplified Line (pentagram).svg
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 approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction. [33] 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.

See also

Related Research Articles

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  1. Becker, Udo (1994). "Pentagram". The Continuum Encyclopedia of Symbols. Translated by Garmer, Lance W. Continuum. p. 230. ISBN   978-0-8264-1221-8.
  2. Ferguson, George Wells (1966). Signs & Symbols in Christian Art. Oxford University Press. p. 59. ISBN   978-0-19-501432-7.
  3. πεντάγραμμον, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus; a noun form of adjectival πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "five-lined" or "five lines"
  4. πέντε, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus; γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  5. this usage is borne out by the Oxford English Dictionary, although that work specifies that a circumscription makes the sthe form of a five-pointed star and its etymon post-classical Latin pentaculum [...] A pentagram, esp. one enclosed in a circle; a talisman or magical symbol in the shape of or inscribed with a pentagram. Also, in extended use: any similar magical symbol (freq. applied to a hexagram formed by two intersecting or interlaced equilateral triangles)."
  6. πένταλϕα, "five Alphas", interpreting the shape as five Α shapes overlapping at 72-degree angles.
  7. πεντέμυχος, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  8. This is a lost book, but its contents are preserved in Damascius, De principiis, quoted in Kirk and Raven, (1983) [1956], p. 55.
  9. "the divine products of Chronos' seed, when disposed in five recesses, were called πεντέμυχος (Pentemychos)" Kirk, Geoffrey Stephen; Raven, John Earle; Schofield, Malcolm (1983) [1957]. The Presocratic Philosophers: A Critical History with a Selection of Texts (2nd, illustrated, revised, reprint ed.). Cambridge University Press. pp. 51–52, 55. ISBN   978-0-521-27455-5. the only other place in Homer where Ortygie [ sic ] is mentioned is Odyssey V, 123, where Orion, having been carried off by Eos [the dawn], is slain... by Artemis... since solstices would normally be observed at sunrise in summer, and so in the north-east-by-east direction, that is what the phrase might suggest... the dwelling-place of Eos... Aia..
  10. Allman, G. J., Greek Geometry From Thales to Euclid (1889), p.26.
  11. Allman, G. J., Greek Geometry From Thales to Euclid, part I (1877), in Hermathena 3.5, pp. 183, 197, citing Iamblichus and the Scholiast on Aristophanes. The pentagram was said to have been so called from Pythagoras himself having written the letters Υ, Γ, Ι, Θ (= /ei/), Α on its vertices.
  12. Christian Symbols Ancient and Modern, Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, ISBN   0-7135-1960-6.
  13. Morgan, Gerald (1979). "The Significance of the Pentangle Symbolism in "Sir Gawain and the Green Knight"". The Modern Language Review. 74 (4): 769–790. JSTOR   3728227.
  14. Lévi, Éliphas (1999) [1896 (translated), 1854 (first published)]. Transcendental Magic, its Doctrine and Ritual[Dogme et rituel de la haute magie]. Trans. by A. E. Waite. York Beach: Weiser. OCLC   263626874.
  15. Lévi, Éliphas (2002) [1939 (translated), 1859 (first published)]. The Key of the Mysteries[la Clef des grands mystères suivant Hénoch, Abraham, Hermès Trismégiste et Salomon]. Trans. by Aleister Crowley. Boston: Weiser. p. 69. OCLC   49053462.
  16. Hartmann, Franz (1895) [1886]. Magic, White and Black (5th ed.). New York: The Path. OCLC   476635673.
  17. Schouten, Jan (1968). The Pentagram as Medical Symbol: An Iconological Study. Hes & De Graaf. p. 18. ISBN   978-9060041666.
  18. Waite, Arthur Edwar (1886). The Mysteries of Magic: A Digest of the Writings of Eliphas Lévi. George Redway. p. 136.
  19. DuQuette, Lon Milo (2003). The Magick of Aleister Crowley: A Handbook of the Rituals of Thelema. Weiser Books. pp. 93, 247. ISBN   978-1-57863-299-2.
  20. Bahá'í Reference Library – Directives from the Guardian, Pages 51–52
  21. Nine-Pointed Star, The:History and Symbolism by Universal House of Justice 24 January 1999
  22. Haykal - Baha'i Five Pointed Star Symbol
  23. See the Nauvoo Temple website discussing its architecture, and particularly the page on Nauvoo Temple exterior symbolism. Retrieved 16 December 2006.
  24. Brown, Matthew B (2002). "Inverted Stars on LDS Temples" (PDF). Archived from the original (PDF) on 29 February 2008.
  25. "Religious Clothing in School", Robinson, B.A., Ontario Consultants on Religious Tolerance, 20 August 1999, updated 29 April 2005. Retrieved 10 February 2006. "ACLU Defends Honor Student Witch Pentacle" (Press release). American Civil Liberties Union of Michigan. 10 February 1999. Archived from the original on 8 November 2003. Retrieved 10 February 2006.CS1 maint: BOT: original-url status unknown (link) "Witches and wardrobes: Boy says he was suspended from school for wearing magical symbol" Rouvalis, Cristina; Pittsburgh Post-Gazette, 27 September 2000. Retrieved 10 February 2006.
  26. "Federal judge upholds Indiana students' right to wear Wiccan symbols". Associated Press. 1 May 2000. Archived from the original on 30 March 2014. Retrieved 21 September 2007.CS1 maint: BOT: original-url status unknown (link)
  27. Associated Press (23 April 2007). "Wiccan symbol OK for soldiers' graves". Archived from the original on 26 April 2007.
  28. "Burial and Memorials: Available Emblems of Belief for Placement on Government Headstones and Markers". United States Department of Veterans Affairs. 3 July 2013. Retrieved 13 January 2014.
  29. 1 2 Gravrand 1990, p. 20.
  30. 1 2 Madiya, Clémentine Faïk-Nzuji (1996). Tracing Memory: A Glossery of Graphic Signs and Symbols in African Art and Culture. Mercury series, no. 71. Hull, Québec: Canadian Museum of Civilization. pp. 27, 155. ISBN   0-660-15965-1.
  31. Gravrand 1990, p. 216.
  32. Ritual of the Order of the Eastern Star, 1976
  33. Baez, John (4 January 2014). "The Pentagram of Venus". Azimuth. Archived from the original on 14 December 2015. Retrieved 7 January 2016.