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61-450: A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon , formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon . Drawing a circle around the five points creates a similar symbol referred to as the pentacle , which is used widely by Wiccans and in paganism , or as a sign of life and connections. The word pentagram comes from

122-489: A numeral prefix , such as penta- , with the Greek suffix -gram (in this case generating the word pentagram ). The prefix is normally a Greek cardinal , but synonyms using other prefixes exist. For example, a nine-pointed polygon or enneagram is also known as a nonagram , using the ordinal nona from Latin . The -gram suffix derives from γραμμή ( grammḗ ), meaning a line. The name star polygon reflects

183-404: 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. The inverted pentagram

244-427: A "Vittra-way", or moved from houses that are deemed a "Vittra-place" (Vittra ställe) because of bad luck – although this is rather uncommon. In tales told in the north of Sweden, Vittra often take the place that trolls, tomte and vättar hold in the same stories told in other parts of the country. Vittra are believed to sometimes "borrow" cattle that later would be returned to the owner with the ability to give more milk as

305-434: A Mara. Nøkken , näcken , or strömkarlen , is a dangerous fresh water -dwelling creature. The nøkk plays a fiddle to lure his victims out onto thin ice on foot or onto water in leaky boats, then draws them down to the bottom of the water where he is waiting for them. The nøkk is also a shapeshifter , who usually changes into a horse or a man in order to lure victims to him. Storsjöodjuret , often referred to as

366-447: A Swedish story in which a farmer and his wife entered their barn early in the morning and found a little, old, grey man sweeping the floor. They saw his clothing, which was nothing more than torn rags, and the wife decided to make him some new ones; when the nisse found them in the barn, however, he considered himself too elegant to perform any more farm labour and thus disappeared from the farm. Nisser are also associated with Christmas and

427-460: A daily basis.) There was some discrepancy as to how they came into being. Some stories say that the Maras are restless children, whose souls leave their body at night to haunt the living. Another tale explains that if a pregnant woman pulled a horse placenta over her head before giving birth, the child would be delivered safely; however, if it were a son, he would become a werewolf, and if a daughter,

488-510: A degenerate polygon will result with coinciding vertices and edges. For example, {6/2} will appear as a triangle, but can be labeled with two sets of vertices: 1-3 and 4-6. This should be seen not as two overlapping triangles, but as a double-winding single unicursal hexagon. Alternatively, a regular star polygon can also be obtained as a sequence of stellations of a convex regular core polygon. Constructions based on stellation also allow regular polygonal compounds to be obtained in cases where

549-450: A difference when retrograde polygons are incorporated in higher-dimensional polytopes. For example, an antiprism formed from a prograde pentagram {5/2} results in a pentagrammic antiprism ; the analogous construction from a retrograde "crossed pentagram" {5/3} results in a pentagrammic crossed-antiprism . Another example is the tetrahemihexahedron , which can be seen as a "crossed triangle" {3/2} cuploid . If p and q are not coprime,

610-586: A fox, which she ties under her skirt in order to hide it from men. If she can manage to get married in a church, her tail falls off and she becomes human. The Huldufólk are a race of fairies or elves who are said to live in the mountains, hills, and rocks of the Faroes. They are said to be similar in appearance to humans, but they are much smaller and have pale skin and long, dark hair. The huldufólk are generally benevolent creatures, but they can be mischievous if they are angered. In Scandinavia, there has been

671-687: A mix thereof. They are closely linked to the mist and it is often said in Sweden that, "the Elves are dancing in the mist". The female form of Elves may have originated from the female deities called Dís (singular) and Díser (plural) found in pre-Christian Scandinavian religion. They were very powerful spirits closely linked to the seid magic. Even today the word "dis" is a synonym for mist or very light rain in Swedish, Norwegian and Danish. Particularly in Denmark,

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732-599: A tessellation pattern. In his 1619 work Harmonices Mundi , among periodic tilings, Johannes Kepler includes nonperiodic tilings, like that with three regular pentagons and one regular star pentagon fitting around certain vertices, 5.5.5.5/2, and related to modern Penrose tilings . The interior of a star polygon may be treated in different ways. Three such treatments are illustrated for a pentagram. Branko Grünbaum and Geoffrey Shephard consider two of them, as regular star n -gons and as isotoxal concave simple 2 n -gons. [REDACTED] These three treatments are: When

793-828: A troll king, lives inside the Dovre Mountains with his court, as described in detail in Ibsen 's Peer Gynt . Elves (in Swedish, Älva if female and Alv if male, Alv in Norwegian, and Elver in Danish) are in some parts mostly described as female (in contrast to the light and dark elves in the Edda), otherworldly, beautiful and seductive residents of forests, meadows and mires. They are skilled in magic and illusions. Sometimes they are described as small fairies, sometimes as full-sized women and sometimes as half transparent spirits, or

854-452: A troll. Sometimes saints' legends involve a holy man tricking an enormous troll to build a church. Trolls come in many different shapes and forms, and are generally not fair to behold, as they can have as many as nine heads. Trolls live throughout the land. They dwell in mountains, under bridges, and at the bottom of lakes. Trolls who live in the mountains may be rich and, hoarding mounds of gold and silver in their cliff dwellings. Dovregubben ,

915-515: A widespread belief in the Mara . The Mara (or, in English, " nightmare ") appears in many different forms, but would terrorize the sleeping by "riding" on their chest, thus giving them nightmares. (This appears to describe "apparitions" commonly seen and/or felt during episodes of sleep paralysis .) The Mara traditionally could ride on cattle, which would be left drained of energy and with tangled fur at

976-536: Is a legendary creature from Faroese folklore. It is said to be a type of undead being that inhabits the mountains and hills of the Faroe Islands. The dreygur is typically described as a large, strong creature with pale skin and long, dark hair. It is often depicted as being cannibalistic . The Nisse (in southern Sweden, Norway and Denmark) or tomte (in Sweden) is a benevolent wight who takes care of

1037-530: Is a continuation of the ancient Babylonian use of the pentagram as an apotropaic charm to protect against evil forces. Éliphas Lévi claimed that "The Pentagram expresses the mind's domination over the elements and it is by this sign that we bind the demons of the air, the spirits of fire, the spectres of water, and the ghosts of earth." In this spirit, the Hermetic Order of the Golden Dawn developed

1098-544: Is a designation for several types of human-like supernatural beings in Scandinavian folklore. They are mentioned in the Edda (1220) as a monster with many heads. Later, trolls became characters in fairy tales, legends and ballads. They play a main part in many of the fairy tales from Asbjørnsen and Moes collections of Norwegian tales (1844). Trolls may be compared to many supernatural beings in other cultures, for instance

1159-688: Is a sea creature that looks like a man with three toes on each foot. Its skin is black and red. It has a long left foot and a small right foot which drags behind, making it lean left when walking. The Kraken is a legendary sea-monster, resembling a giant octopus or squid said to appear off the coasts of Norway . The Selkie is a mythical creature that is part-human and part-seal. According to legend, Selkies can shed their seal skins and transform into humans. There are many stories in Faroese folklore about Selkies falling in love with humans and leaving their sea life behind to live on land. The dreygur

1220-621: Is broadly used in Satanism , sometimes depicted with the goat 's head of Baphomet , as popularized by the Church of Satan since 1968. LaVeyan Satanists pair the goat head with Hebrew letters at the five points of the pentagram to form the Sigil of Baphomet . The Baphomet sigil was adapted for the Joy of Satan Ministries logo, using cuneiform characters at the five points of the pentagram, reflecting

1281-531: The Cyclopes of Homer 's Odyssey . In Swedish, such beings are often termed 'jätte' (giant), a word related to the Norse 'jotun'. The origins of the word troll is uncertain. Trolls are described in many ways in Scandinavian folk literature, but they are often portrayed as stupid, and slow to act. In fairy tales and legends about trolls, the plot is often that a human with courage and presence of mind can outwit

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1342-424: The Greek word πεντάγραμμον ( pentagrammon ), from πέντε ( pente ), "five" + γραμμή ( grammē ), "line". The word pentagram refers to just the star and the word pentacle refers to the star within a circle, although there is some overlap in usage. The word pentalpha is a 17th-century revival of a post-classical Greek name of the shape. Early pentagrams have been found on Sumerian pottery from Ur c. 3500 BCE , and

1403-617: The Hermetic Order of the Golden Dawn , who, following Levi, considered this orientation of the symbol evil and associated it with the triumph of matter over spirit. The five-pointed star is a symbol of the Baháʼí Faith . In the Baháʼí Faith, the star is known as the Haykal ( Arabic : "temple" ), and it was initiated and established by the Báb . The Báb and Bahá'u'lláh wrote various works in

1464-578: The Low Countries , the Baltic countries , Finland and Sápmi . Folklore is a concept encompassing expressive traditions of a particular culture or group. The peoples of Scandinavia are heterogenous, as are the oral genres and material culture that has been common in their lands. However, there are some commonalities across Scandinavian folkloric traditions, among them a common ground in elements from Norse mythology as well as Christian conceptions of

1525-660: The Sun approximately 13 times for every eight 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, and each group of five intersections equidistant from the figure's center have the same geometric relationship. The pentagram has these Unicode code points that enable them to be included in documents: Star polygon Branko Grünbaum identified two primary usages of this terminology by Johannes Kepler , one corresponding to

1586-482: The five joys that Mary had of Jesus, and exemplifies the five virtues of knighthood , which are generosity, friendship, chastity, chivalry, and piety. The North rose of Amiens Cathedral (built in the 13th century) exhibits a pentagram-based motif. Some sources interpret the unusual downward-pointing star as symbolizing the Holy Spirit descending on people. Heinrich Cornelius Agrippa and others perpetuated

1647-461: The regular star polygons with intersecting edges that do not generate new vertices, and the other one to the isotoxal concave simple polygons . Polygrams include polygons like the pentagram , but also compound figures like the hexagram . One definition of a star polygon , used in turtle graphics , is a polygon having q ≥ 2 turns ( q is called the turning number or density ), like in spirolaterals . Star polygon names combine

1708-678: The "Great Lake Monster," is steeped in the folklore of Sweden, specifically with Lake Storsjön . Notably the legendary creature was briefly granted a protected status by the Swedish Environmental Protection Agency , but this was later removed by the Swedish Parliament . Selma is a legendary sea serpent said to live in the 13-kilometre-long (8-mile) Lake Seljord ( Seljordsvatnet ) in Seljord , Vestfold og Telemark , Norway . The circhos

1769-534: 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

1830-419: The 5th vertex, from the 5th to the 2nd vertex, from the 2nd to the 4th vertex, and from the 4th to the 1st vertex. If q ≥ p /2, then the construction of { p / q } will result in the same polygon as { p /( p − q )}; connecting every third vertex of the pentagon will yield an identical result to that of connecting every second vertex. However, the vertices will be reached in the opposite direction, which makes

1891-567: The Mara's touch. Trees would curl up and wilt at the Mara's touch as well. In some tales, like the Banshee , they served as an omen of death. If one were to leave a dirty doll in a family living room, one of the members would soon fall ill and die of tuberculosis . ("Lung soot", another name for tuberculosis, referred to the effect of proper chimneys in 18th through 19th century homes. Inhabitants would therefore contract diseases due to inhaling smoke on

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1952-605: The area of the polygon is calculated, each of these approaches yields a different result. Star polygons feature prominently in art and culture. Such polygons may or may not be regular , but they are always highly symmetrical . Examples include: Icelandic folklore Nordic folklore is the folklore of Denmark , Norway , Sweden , Iceland and the Faroe Islands . It has common roots with, and has been under mutual influence with, folklore in England , Germany ,

2013-483: The context of Norway's evolving cultural landscape, these churches endure as emblematic symbols of enduring identity and heritage, encapsulating the nuanced interplay between religious, mythological, and societal dimensions. A large number of different mythological creatures from Scandinavian folklore have become well known in other parts of the world, mainly through popular culture and fantasy genres. Some of these are: Troll (Norwegian and Swedish), trolde (Danish)

2074-542: The density q and amount p of vertices are not coprime. When constructing star polygons from stellation, however, if q > p /2, the lines will instead diverge infinitely, and if q = p /2, the lines will be parallel, with both resulting in no further intersection in Euclidean space. However, it may be possible to construct some such polygons in spherical space, similarly to the monogon and digon ; such polygons do not yet appear to have been studied in detail. When

2135-521: The diagram happened to be imperfectly drawn): Mephistopheles: Faust: Mephistopheles: Also protective is the use in Icelandic folklore of a gestured or carved rather than painted pentagram (called smèrhnút in Icelandic), according to 19th century folklorist Jón Árnason : Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists. Its major use

2196-474: 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,

2257-444: The elves. In more modern tales, it isn't uncommon for a rather ugly male Tomte, Troll, Vätte or a Dwarf to fall in love with a beautiful Elven female, as the beginning of a story of impossible or forbidden love. The Huldra , Hylda, Skogsrå or Skogfru (Forest wife/woman) is a dangerous seductress who lives in the forest. The Huldra is said to lure men with her charm. She has a long cow's tail, or according to some traditions, that of

2318-560: The female elves have merged with the dangerous and seductive huldra, skogsfrun or "keeper of the forest", often called hylde. In some parts of Sweden the elves also share features with the Skogsfrun, "Huldra", or "Hylda", and can seduce and bewitch careless men and suck the life out of them or make them go down in the mire and drown. But at the same time the Skogsrå exists as its own being, with other distinct features clearly separating it from

2379-401: The first one, and continuing the process until the original vertex is reached again. Alternatively, for integers p and q , it can be considered as being constructed by connecting every q th point out of p points regularly spaced in a circular placement. For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the 1st to the 3rd vertex, from the 3rd to

2440-541: The five-pointed star was at various times the symbol of Ishtar or Marduk . Pentagram symbols from about 5,000 years ago were found in the Liangzhu culture of China . The pentagram was known to the ancient Greeks , with a depiction on a vase possibly dating back to the 7th century BCE. Pythagoreanism originated in the 6th century BCE and used the pentagram as a symbol of mutual recognition, of wellbeing, and to recognize good deeds and charity. From around 300–150 BCE

2501-660: The form of a pentagram. The Church of Jesus Christ of Latter-day Saints is theorized to have begun 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

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2562-427: The house and barn when the farmer is asleep, but only if the farmer reciprocates by setting out food for the nisse and he himself also takes care of his family, farm and animals. If the nisse is ignored or maltreated or the farm is not cared for, he is likely to sabotage the work instead to teach the farmer a lesson. Although the nisse should be treated with respect, some tales warn against treating him too kindly. There's

2623-722: The intersecting line segments are removed from a regular star n -gon, the resulting figure is no longer regular, but can be seen as an isotoxal concave simple 2 n -gon, alternating vertices at two different radii. Branko Grünbaum , in Tilings and patterns , represents such a star that matches the outline of a regular polygram { n / d } as | n / d |, or more generally with { n 𝛼 }, which denotes an isotoxal concave or convex simple 2 n -gon with outer internal angle 𝛼. These polygons are often seen in tiling patterns. The parametric angle 𝛼 (in degrees or radians) can be chosen to match internal angles of neighboring polygons in

2684-455: The obtuse triangle. A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It

2745-514: The pentagram stood as the symbol of Jerusalem, marked by the 5 Hebrew letters ירשלם spelling its name. 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" The pentagram was used in ancient times as a Christian symbol for the five senses , or of the five wounds of Christ . The pentagram plays an important symbolic role in

2806-481: 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 but controversial writer Éliphas Lévi , known for believing that magic

2867-405: 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 . 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

2928-668: The region. An exploration of these dances unveils insights into social dynamics, community cohesion, and the perpetuation of mythological themes across generations. Stave churches in Norway represent a unique synthesis of Christian and Norse cultural influences, evident in their architectural and ornamental features. These wooden structures, characterized by intricate carvings, serve as tangible artifacts linking contemporary communities to historical narratives. Beyond mere historical relics, Stave churches function as active centers for cultural preservation, hosting various ceremonies and events. In

2989-585: 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. The golden ratio , φ = (1 + √ 5 ) / 2 ≈ 1.618, satisfying: plays an important role in regular pentagons and pentagrams. Each intersection of edges sections

3050-428: The resemblance of these shapes to the diffraction spikes of real stars. A regular star polygon is a self-intersecting, equilateral, and equiangular polygon . A regular star polygon is denoted by its Schläfli symbol { p / q }, where p (the number of vertices) and q (the density ) are relatively prime (they share no factors) and where q ≥ 2. The density of a polygon can also be called its turning number :

3111-638: The shape's earliest use in Sumeria . The inverted pentagram also appears in The Satanic Temple logo, with an alternative depiction of Baphomet's head. Other depictions of the Satanic goat's head resemble the inverted pentagram without its explicit outline. 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

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3172-423: The sum of the turn angles of all the vertices, divided by 360°. The symmetry group of { p / q } is the dihedral group D p , of order 2 p , independent of q . Regular star polygons were first studied systematically by Thomas Bradwardine , and later Johannes Kepler . Regular star polygons can be created by connecting one vertex of a regular p -sided simple polygon to another vertex, non-adjacent to

3233-650: 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." 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. 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 . The encircled pentagram (referred to as

3294-409: The toilet so they can move out of the way) or building your home too close to or, even worse, on top of their home, disturbing their cattle or blocking their roads. They can make your life very very miserable or even dangerous – they do whatever it takes to drive you away, even arrange accidents that will harm or even kill you. Even in modern days, people have rebuilt or moved houses in order not to block

3355-468: The universe in the Serer creation myth , and also represents the star Sirius . 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,

3416-475: 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 the system of magick used in Thelema : 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

3477-673: The wight Tomten originates from some sort of general house god or deity prior to Old Norse religion . A Nisse/Tomte is said to be able to change his size between that of a 5-year-old child and a thumb, and also to have the ability to make himself invisible. A type of wight from Northern Sweden called Vittra lives underground, is invisible most of the time and has its own cattle. Most of the time Vittra are rather distant and do not meddle in human affairs, but are fearsome when enraged. This can be achieved by not respecting them properly, for example by neglecting to perform certain rituals (such as saying "look out" when putting out hot water or going to

3538-476: The world. Among the many tales common in Scandinavian oral traditions, some have become known beyond Scandinavian borders – examples include The Three Billy Goats Gruff and The Giant Who Had No Heart in His Body . Nordic folklore's traditional dances, intricately linked to celebrations, rituals, and communal assemblies, exhibit specific movements, patterns, and music deeply ingrained in the cultural fabric of

3599-490: The yule time. Farmers customarily place bowls of rice porridge on their doorsteps to please the nisser, comparative to the cookies and milk left out for Santa Claus in other cultures. Some believe that the nisse brings them presents as well. In Swedish, the word "tomten" (definite form of "tomte") is very closely linked to the word for the plot of land where a house or cottage is built, which is called "tomten" as well (definite form of "tomt"). Therefore, some scholars believe that

3660-557: Was a real science, had called it evil whenever the symbol appeared the other way up: The apotropaic (protective) use in German folklore of the pentagram symbol (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

3721-516: Was studied later by Carl Friedrich Gauss . Several polyhedra incorporate pentagrams: Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures: All ten 4-dimensional Schläfli–Hess 4-polytopes have either pentagrammic faces or vertex figure elements. 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

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