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. Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one A five-pointed star (☆ is a very common Ideogram throughout the world 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". Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly

Pentagrams were used symbolically in ancient Greece and Babylonia. The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital The pentagram has magical associations, and many people who practice Neopagan faiths wear jewelry incorporating the symbol. Magic, sometimes known as sorcery, is a Conceptual system that asserts human ability to control the natural world (including events objects people and Neopaganism or Neo-Paganism is an Umbrella term used to identify a wide variety of modern religious movements particularly those influenced by historical Christians once more commonly used the pentagram to represent the five wounds of Jesus,^[1][2] and it also has associations within Freemasonry. Christianity ( Greek Χριστιανισμός from the word Xριστός ( Christ)is a monotheistic Religion centered on the life and teachings Jesus of Nazareth (7–2 BC / BCE —26–36 AD / CE)

The pentagram has long been associated with the planet Venus, and the worship of the goddess Venus, or her equivalent. The VENUS ( V ictoria E xperimental N etwork U nder the S ea project is a cabled sea floor observatory operated by the University Venus was a major Roman Goddess principally associated with Love, Beauty and fertility, the equivalent of the Greek goddess 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. Roman mythology, or more appropriately Latin mythology, refers to the mythological beliefs of the Italic people inhabiting the region of Latium and its Lucifer is a name frequently given to Satan in Christian belief It is most likely to have originated from the observations of prehistoric astronomers. ^[3] When viewed from Earth, successive inferior conjunctions of Venus plot a nearly perfect pentagram shape around the zodiac every eight years. Conjunction is a term used in Positional astronomy and Astrology. Zodiac denotes an annual cycle of twelve stations along the Ecliptic, the apparent path of the sun across the heavens through the Constellations that divide the ecliptic ^[4]

The word "pentacle" is sometimes used synonymously with "pentagram", although their technical usages are different, and their etymologies may be unrelated. A pentacle (or pantacle in Thelema) is an Amulet used in magical Evocation, generally made of Parchment, paper or metal ^[5]

Contents 1 Early history 1.1 Sumer 1.2 Pythagoreans 1.3 European occultism 2 Religious symbolism 2.1 Christianity 2.2 Mormonism 2.3 Judaism 2.4 Satanism 2.5 Neopaganism 2.6 Bahá'í Faith 2.7 Thelema 2.8 Samael Aun Weor 3 Political symbolism 3.1 Flags 4 Other organizations 4.1 Order of the Eastern Star 5 In literature 6 Geometry 6.1 Construction 6.2 Golden ratio 6.3 Trigonometric values 6.4 Three dimensional figures 7 See also 8 Notes 9 References 10 External links

Early history

Sumer

The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC. Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding 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). Sumer ( Sumerian: sux-Latn [[Ki (earth ki]]-[[EN (cuneiform en]]-'''ĝir15''', Akkadian: Šumeru; possibly Biblical Shinar In René Labat's index system of Sumerian hieroglyphs/pictograms it is shown with two points up. Sumer ( Sumerian: sux-Latn [[Ki (earth ki]]-[[EN (cuneiform en]]-'''ĝir15''', Akkadian: Šumeru; possibly Biblical Shinar Hieroglyph ( Greek grc-Grek ἱερογλύφος " sacred carving " or hieroglyphics ( = grc-Grek τὰ ἱερογλυφικά A pictogram ( also spelled pictogramme) or pictograph is a Symbol representing a Concept, object, activity place or event ^[6] In the Babylonian context, the edges of the pentagram were probably orientations: forward, backward, left, right, and "above". Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital 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. Astrology (from Greek grc ἄστρον astron, "constellation star" and grc -λογία -logia) is a group of Systems The VENUS ( V ictoria E xperimental N etwork U nder the S ea project is a cabled sea floor observatory operated by the University Ishtar ( D IŠTAR 𒀭𒌋𒁯 is the Assyrian and Babylonian counterpart to the Sumerian Inanna and to

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). Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced In Greek mythology, Hygieia ( or Hygeia ( was a daughter of Asclepius.

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. Many ancient philosophies used a set of archetypal classical "elements" to explain patterns in Nature. Water is a common Chemical substance that is essential for the survival of all known forms of Life. Gaia (ˈgeɪə or /ˈgaɪə/ (" land " or " Earth " from the Ancient Greek Γαîα also Gæa or Gea EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 An idea is a form (such as a Thought) formed by Consciousness (including Mind) through the Process of ideation. In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature Fire is the heat and light energy released during a Chemical reaction, in particular a combustion reaction. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five 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. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. See also Pherecydes of Leros Pherecydes of Syros (in Greek: Φερεκύδης was a Greek thinker from the island of Syros ^[7] 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". In classic Greek mythology below Heaven, Earth, and Pontus is Tartarus, or Tartaros ( Greek Τάρταρος deep place

In very early Greek thought, Tartaros (or Chaos, according to Hesiod) was the primordial Darkness from which the cosmos is born. For the state of disarray see Chaos. In Greek mythology Chaos ( Xάος) or Khaos is the original state of existence from which Hesiod ( Greek: Hesiodos) was an early Greek Poet and Rhapsode, who presumably lived around 700 BCE 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". Homer ( Ancient Greek:, Homēros) is a legendary ancient Greek epic Poet, traditionally said to be the author of the epic poems the The Boundless Darkness held influence through Mychos or Krater. For the Landform crater see Crater. A krater (in Greek: κρατήρ kratēr, from the Verb κεράννυμι 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. For the Landform crater see Crater. A krater (in Greek: κρατήρ kratēr, from the Verb κεράννυμι In classic Greek mythology below Heaven, Earth, and Pontus is Tartarus, or Tartaros ( Greek Τάρταρος deep place Hades (from Greek, Hadēs, originally, Haidēs or, Aidēs, probably from Indo-European *n̥-wid- 'unseen' refers both to the ancient 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. Biography Early life Birth and family Plato was born in Athens Greece Cratylus ( Greek: Κράτυλος is the name of a dialogue by Plato. 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. Medea (Μήδεια Mēdeia) in Greek mythology was the daughter of King Aeëtes of Colchis, niece of Circe, granddaughter of Euripides ( Ancient Greek:) (ca 480 BC–406 BC was the last of the three great tragedians of classical Athens (the other two being Aeschylus Hecate ( Greek: Ἑκάτη, "far-shooting") Hekate ( Hekátê " 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 Greek mythology, Persephone ( Kore or Cora) was the embodiment of the Earth's fertility at the same time that she was the Queen 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. Heinrich Cornelius Agrippa von Nettesheim ( September 14, 1486 – February 18 1535) was a German magician, Occult 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. "^[8]

"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. "^[9]

Man inscribed in a pentagram, from Heinrich Cornelius Agrippa's Libri tres de occulta philosophia. Heinrich Cornelius Agrippa von Nettesheim ( September 14, 1486 – February 18 1535) was a German magician, Occult Libri tres de occulta philosophia ( Three Books about Occult Philosophy) is Heinrich Cornelius Agrippa 's study of occult philosophy acknowledged as The five signs at the pentagram's vertices are astrological. In Geometry, a vertex (plural "vertices" is a special kind of point. Astrology (from Greek grc ἄστρον astron, "constellation star" and grc -λογία -logia) is a group of Systems

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

The occultist and magician Eliphas Levi's pentagram, which he considered to be a symbol of the microcosm, or human. The word occult comes from the Latin word occultus (clandestine hidden secret referring to "knowledge of the hidden" A magician is a person skilled in the mysterious and hidden art of magic, the ability to attain objectives acquire knowledge or perform works of wonder using Supernatural Eliphas Lévi, born Alphonse Louis Constant, ( February 8, 1810 - May 31, 1875) was a French Occult author Macrocosm and microcosm is an ancient Greek schema of seeing the same patterns reproduced in all levels of the Cosmos, from the largest scale (macrocosm or universe-level

Religious symbolism

Christianity

The pentagram was used as a Christian symbol for the five senses,^[10] 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). Christ is the English term for the Greek ( Khristós) meaning "the anointed " Piero Valeriano Bolzani, born Giampietro Valeriano Bolzani (1477–1558 was an Italian Renaissance humanist, favored by the Medici. Senses are the physiological methods of Perception. The senses and their operation classification and theory are overlapping topics studied by a variety of fields Health is a state of complete physical mental and social well-being and not merely the absence of disease or infirmity Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.

Medieval Christians believed it to symbolise the five wounds of Christ. The pentagram was believed to protect against witches and demons. ^[11]

The pentagram figured in a heavily symbolic Arthurian romance:^[11] it appears on the shield of Sir Gawain in the 14th century poem Sir Gawain and the Green Knight. King Arthur is a legendary British leader who according to medieval histories and romances, led the defence of Britain against the Saxon invaders Gawain (ˈgɔːwɪn or /gəˈweɪn/ also called Gwalchmei Gawan Gauvain Walewein etc Sir Gawain and the Green Knight is a late 14th-century 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,^[12] 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. Christ is the English term for the Greek ( Khristós) meaning "the anointed " In Christianity the Annunciation ( grc Ευαγγελισμός της Θεοτόκου, Evangelismós tēs Theotókou in Greek) is the revelation For depictions in painting and sculpture see Nativity of Jesus in art. Within the body of Christian beliefs the resurrection of Jesus is a core event on which much of Christian doctrine and theology depend The general and most common understanding of the Christian Doctrine of Ascension holds that Jesus bodily ascended to Heaven in the presence This article is about the theological concept For the works of art with this title see Assumption of the Virgin Mary in Art and Roman Catholic Marian art. Knight is the English term for a social position originating in the Middle Ages.

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. Ceremonial magic is a broad term used to encompass a wide variety of long elaborate and complex Rituals it is named as such because the works included are characterized by ^[11]

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^[13] and the Salt Lake Temple. The Church of Jesus Christ of Latter-day Saints is the fourth largest Christian denomination in the United States and the largest and most well-known In the The Nauvoo Illinois Temple is the 113th dedicated temple of The Church of Jesus Christ of Latter-day Saints. The Salt Lake Temple is the largest (of more than 120 and best-known temple of The Church of Jesus Christ of Latter-day Saints. These symbols derived from traditional morning star pentagrams that are no longer commonly used in mainstream Christianity. ^[14]

Judaism

The pentagram was the official seal of the city of Jerusalem for a time. Jerusalem (יְרוּשָׁלַיִם, he-Latn Yerushaláyim; Arabic: ar القُدس, ar-Latn al-Quds) is the ^[11] Due to the similarity of the star shapes, it is occasionally confused with the Star of David by those unfamiliar with the symbols. The Star of David or Shield of David ( Magen David in Hebrew with nikkud or מגן דוד without academically transcribed Māḡēn Dāwīḏ by

Satanism

Satanists use a pentagram with two points up, often inscribed in a double circle, with the head of a goat inside the pentagram. For other uses of the word see Satanism (disambiguation. Satanism can refer to a number of belief systems depending on the user and context This is referred to as the Sigil of Baphomet. The Sigil of Baphomet is the official insignia for the Church of Satan, chosen by Anton Szandor LaVey. 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). In most Christian traditions a fallen angel is an Angel that has been Exiled or banished from Heaven. The Pythagorean Greek letters are most often replaced by the Hebrew letters לויתן forming the name Leviathan. The Hebrew alphabet (אָלֶף-בֵּית עִבְרִי alephbet ’ivri) consists of 22 letters used for writing the Hebrew language. Leviathan ( is a Biblical Sea monster referred to in the Old Testament ( Psalm 74:13-14 Job 41 Isaiah 271 Less esoteric LaVeyan Satanists use it as a sign of rebellion or religious identification, the three downward points symbolising rejection of the holy Trinity. For other uses of the word see Satanism and Satanism (disambiguation.

Neopaganism

Many Neopagans, especially Wiccans, use the pentagram as a symbol of faith similar to the Christian cross or the Jewish Star of David. Neopaganism or Neo-Paganism is an Umbrella term used to identify a wide variety of modern religious movements particularly those influenced by historical A Christian is a person who adheres to Christianity, a monotheistic Religion centered on the life and teachings of Jesus of Nazareth A cross is a geometrical figure consisting of two lines or bars perpendicular to each other dividing one or two of the lines in half PLEASE TAKE NOTE************ The Star of David or Shield of David ( Magen David in Hebrew with nikkud or מגן דוד without academically transcribed Māḡēn Dāwīḏ by It is not, however, a universal symbol for Neopaganism, and is rarely used by Reconstructionists. Polytheistic reconstructionism, or simply Reconstructionism, is an approach to Neopaganism first emerging in the late 1960s to early 1970s and gatherig momentum 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. Many ancient philosophies used a set of archetypal classical "elements" to explain patterns in Nature. 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. An elemental is a mythological being first appearing in the alchemical works of Paracelsus.

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. Initiation is a Rite of passage Ceremony marking entrance or acceptance into a group or society

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. The word occult comes from the Latin word occultus (clandestine hidden secret referring to "knowledge of the hidden" ^[15][16][17] In public schools, such actions by administrators have been determined to be in violation of students' First Amendment right to free exercise of religion. The First Amendment to the United States Constitution is part of the United States Bill of Rights that expressly prohibits the United States Congress The Free Exercise Clause is the accompanying clause with the Establishment Clause of the First Amendment to the United States Constitution. ^[18]

Bahá'í Faith

Main article: Bahá'í symbols

The pentagram is the official symbol of the Bahá'í Faith. The official symbol of the Bahá'í Faith is the Five-pointed star, as mentioned by Shoghi Effendi: "Strictly speaking the 5-pointed The Bahá'í Faith is a Religion founded by Bahá'u'lláh in nineteenth-century Persia, emphasizing the spiritual unity of all humankind ^[19] In the Bahá'í Faith, the pentagram is known as the Haykal (Arabic: "temple"), and it was initiated and established by the Báb. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language Siyyid `Alí Muḥammad ( ( October 20, 1819 – July 9, 1850) was the founder of Bábism, and one of three central figures of the Both Báb and Bahá'u'lláh wrote various works in the form of a pentagram. Bahá'u'lláh ( ba-haa-ol-laa "Glory of God" ( November 12, 1817 – May 29, 1892) born Mírzá Ḥusayn-`Alí Nuri

An unidentified work of the Báb. Siyyid `Alí Muḥammad ( ( October 20, 1819 – July 9, 1850) was the founder of Bábism, and one of three central figures of the

An unidentified work of the Báb. Siyyid `Alí Muḥammad ( ( October 20, 1819 – July 9, 1850) was the founder of Bábism, and one of three central figures of the

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. Aleister Crowley, born Edward Alexander Crowley (ˈkroʊli (12 October 1875 – 1 December 1947 was a British Occultist Writer, mountaineer Thelema is a philosophy of life based on the rule or law "Do what thou wilt Magick, in the broadest sense is any act designed to cause intentional change Evil, in many cultures is used to describe acts or thoughts which are contrary to some particular religion The Hermetic Order of the Golden Dawn (or more commonly the Golden Dawn) was a magical order of the late 19th and early 20th centuries practicing a form of

Samael Aun Weor

Samael Aun Weor used the Pentagram to represent man's Atman, or Internal Christ. Samael Aun Weor ( March 6, 1917 - December 24, 1977) was a spiritual teacher Occultist esotericist and author The Ātman ( IAST: Ātman Sanskrit: आत्मन्‍ is a philosophical term used within Hinduism and Vedanta to identify the Soul Christ is the English term for the Greek ( Khristós) meaning "the anointed " 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. A mantra ( Devanāgarī मन्त्र (or mantram is a religious or mystical syllable or poem typically from the Sanskrit language 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. grc-Latn Logos (ˈloʊːgɒs ( Greek, logos) is an important term in Philosophy, Analytical psychology, Rhetoric and Religion

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. The Witches' Sabbath or Sabbat is a supposed meeting of those who practice Witchcraft, Satanism, or other rites

Political symbolism

Flags

While a solid five-pointed star is found on many flags, the pentagram is relatively rare. A five-pointed star (☆ is a very common Ideogram throughout the world It appears on two national flags, those of Ethiopia and Morocco and in some coats of arms. NOTE This intro is the result of careful NPOV work Please do not make potentially controversial edits to it without first discussing on the talk page Morocco (المغرب "al-Maghrib" officially the Kingdom of Morocco (المملكة المغربية is a country located in North Africa

According to Ivan Sache, on the Moroccan flags, the pentagram represents the link between God and the nation. Morocco (المغرب "al-Maghrib" officially the Kingdom of Morocco (المملكة المغربية is a country located in North Africa God is the principal or sole Deity in Religions and other belief systems that worship one deity. ^[20] 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. King Solomon ( Ge'ez: ስለሞን Arabic: ar سليمان, Sulayman, all from the Triliteral root S-L-M, "peace" Sigillum Solomonis is also a name of the Polygonatum genus In Medieval Jewish, Christian and Islamic legends the Seal PLEASE TAKE NOTE************ A Christian is a person who adheres to Christianity, a monotheistic Religion centered on the life and teachings of Jesus of Nazareth A Muslim (مسلم pronounced Muslim, not Muzlim) is an adherent of the Religion

Morocco's flag

Ethiopia's flag

Other organizations

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. Morocco (المغرب "al-Maghrib" officially the Kingdom of Morocco (المملكة المغربية is a country located in North Africa NOTE This intro is the result of careful NPOV work Please do not make potentially controversial edits to it without first discussing on the talk page The Order of the Eastern Star is the largest Fraternal organization in the world that both men and women can join The Order of the Eastern Star is the largest Fraternal organization in the world that both men and women can join A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line 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). Sir Gawain and the Green Knight is a late 14th-century Gawain (ˈgɔːwɪn or /gəˈweɪn/ also called Gwalchmei Gawan Gauvain Walewein etc

In Goethe's Faust, the pentagram prevents Mephistopheles from leaving a room. ˈjoːhan ˈvɔlfgaŋ fɔn ˈgøːtə (in English generally ˈgɝːtə 28 August 1749 22 March 1832 was a German writer Faust or Faustus ( Latin for "auspicious" or "lucky" is the protagonist of a classic German Legend in which he makes Mephistopheles (also Mephistophilus Mephistophilis Mephostopheles Mephisto and variants is a name often given to one representation of The devil or Satan

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. Howard Phillips Lovecraft ( August 20, 1890 – March 15, 1937) was an American author of horror, fantasy The Cthulhu Mythos is a Shared universe created in the 1920s by American horror writer H August William Derleth ( February 24 1909 &ndash July 4 1971) was an American writer and anthologist 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. Dan Brown (born June 22 1964 is an American Author of Thriller fiction, best known for the 2003 bestselling novel The Da Vinci Code The Da Vinci Code is a controversial mystery / detective Novel by US author Dan Brown, published in 2003 by Doubleday Zodiac denotes an annual cycle of twelve stations along the Ecliptic, the apparent path of the sun across the heavens through the Constellations that divide the ecliptic

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. For a topic outline on this subject see List of basic Japan topics. is a traditional Japanese Esoteric cosmology, a mixture of Natural science and Occultism. (921?-1005? was an onmyōji, a leading specialist of Onmyōdō during the middle of the Heian Period in Japan. In traditional Chinese philosophy, natural phenomena can be classified into the Wu Xing ( or the Five Phases, usually translated as five 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

The pentagram is the simplest regular star polygon. In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an General properties These properties apply to both convex and star regular polygons Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one 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}. In Mathematics, the Schläfli symbol is a notation of the form {pqr 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. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

Construction

The pentagram can be constructed by connecting alternate vertices of a pentagon; see details of the construction. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to It can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect. Stellation is a process of constructing new Polygons (in two Dimensions, new Polyhedra in three dimensions or in general new Polytopes in

Golden ratio

The golden ratio, φ = (1+√5)/2 ≈ 1. In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the 618, satisfying

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:

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. A golden triangle is an isosceles Triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third smaller side The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. A golden triangle is an isosceles Triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third smaller side

Trigonometric values

See Exact trigonometric constants: Pentagon

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. Iteration means the act of repeating Mathematics Iteration in mathematics may refer to the process of iterating a function, or to the techniques used Exact constant expressions for Trigonometric expressions are sometimes useful mainly for simplifying solutions into radical forms which allow further simplification

Three dimensional figures

See Uniform polyhedron: Icosahedral symmetry

Several polyhedra incorporate pentagrams:

Pentagrammic prism	Pentagrammic antiprism	Pentagrammic crossed antiprism	Small stellated dodecahedron
Small ditrigonal icosidodecahedron	Small dodecahemicosahedron	Snub dodecadodecahedron	16th stellation of icosidodecahedron

See also

Notes

References

• Grünbaum, B. and G. Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. 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 . Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. . . etc. (Toronto 1993), ed T. Bisztriczky et al, Kluwer Academic (1994) pp. 43-70.

External links

• Eric W. Weisstein, Pentagram at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• The Pythagorean Pentacle from the Biblioteca Arcana.
• In-depth analysis of the Golden Ratio
• The pentagram and Freemasonry

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