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66-450: 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

132-457: A dreamcatcher made of yarn like a web is placed above a bed or sleeping area to protect sleeping children from nightmares. Items and symbols such as crosses, crucifixes , silver bullets , wild roses and garlic were believed to ward off or destroy vampires . Peisistratus hung the figure of a kind of grasshopper before the Acropolis of Athens as apotropaic magic. In Roman art , envy

198-485: 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

264-404: 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

330-510: 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

396-450: 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,

462-497: A pair of eyes were painted on Greek drinking vessels called kylikes ( eye-cups ) from the 6th century BCE up until the end of the end of the classical period . The exaggerated eyes may have been intended to prevent evil spirits from entering the mouth while drinking. Fishing boats in some parts of the Mediterranean region still have stylised eyes painted on the bows. The defunct Turkish budget airline , Fly Air , adopted

528-599: 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

594-590: A voyage. Similarly, the burial of an old boot or shoe by the lintel of the back door of a house seems to have had a similar intention. In Ireland and Great Britain , magpies are traditionally thought to bring bad luck. Many people repeated various rhymes or salutations to placate them. Apotropaic marks such as the initials of the Virgin Mary were scratched near the openings of buildings in England to ward off witches. Ancient Greeks and Romans used to spit into

660-399: A wooden post was used to support a chimney opening, this was often an easier material for amateur carving. To discourage witchcraft, rowan wood may have been chosen for the post or mantel. Similarly the grotesque faces carved into pumpkin lanterns (and their earlier counterparts, made from turnips , swedes or beets ) at Halloween are meant to avert evil: this season was Samhain ,

726-415: Is a circle of space marked out by practitioners of some branches of ritual magic , which they generally believe will contain energy and form a sacred space, or will provide them a form of magical protection, or both. It may be marked physically, drawn in a material like salt, flour, or chalk, or merely visualised. Ashkenazi Jews ' apotropaic names were often given not at birth but during serious illness. In

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792-517: 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

858-605: 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

924-409: Is hung over doors and windows to protect the household from fire, lightning, illness and evil spirits. In southern Ireland, it was formerly the custom at Samhain to weave a cross of sticks and straw called a 'parshell' or 'parshall', which was fixed over the doorway to ward off bad luck, illness, and witchcraft . Eyes were often painted to ward off the evil eye . An exaggerated apotropaic eye or

990-514: The Celtic new year. As a "time between times", it was believed to be a period when souls of the dead and other dangerous spirits walked the earth. Many European peoples had such associations with the period following the harvest in the fall (for instance the Celtic calendar ). In Ancient Greece , phalloi were believed to have apotropaic qualities. Often stone reliefs would be placed above doorways, and three-dimensional versions were built across

1056-557: The Gorgon , the head of which now may be called the Gorgoneion , which features wild eyes, fangs, and protruding tongue. The full figure of the Gorgon holds the apex of the oldest remaining Greek temple where she is flanked by two lionesses. The Gorgon head was mounted on the aegis and shield of Athena . People believed that the doorways and windows of buildings were particularly vulnerable to

1122-424: 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

1188-610: 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

1254-657: 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

1320-517: The Tower of London , and many churches. A collection of over 100 marks – previously thought to be graffiti – was discovered in 2019 on the walls of a cave network at Creswell Crags in Nottinghamshire. Gainsborough Old Hall has 20, the most of any English Heritage property, concentrated in the servant's quarters alongside curses about the owner William Hickman . In some Native American cultures ,

1386-582: The evil eye . Apotropaic observances may also be practiced out of superstition or out of tradition, as in good luck charms (perhaps some token on a charm bracelet ), amulets , or gestures such as crossed fingers or knocking on wood . Many different objects and charms were used for protection throughout history. Apotropaic magical rituals were practiced throughout the ancient Near East and ancient Egypt . Fearsome deities were invoked via ritual in order to protect individuals by warding away evil spirits. In ancient Egypt, these household rituals (performed in

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1452-476: 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

1518-461: 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

1584-421: The " averting gods " ( ἀποτρόπαιοι θεοί , apotropaioi theoi ), chthonic deities and heroes who grant safety and deflect evil and for the protection of the infants they wore on them amulets with apotropaic powers and committed the child to the care of kourotrophic (child-nurturing) deities. Greeks placed talismans in their houses and wore amulets to protect them from the evil eye . Peisistratus hung

1650-522: 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

1716-648: The 19th century, the custom of driving cattle between two fires was still practiced across most of Ireland and parts of Scotland. Also in Ireland and Scotland, bonfires were lit for the festivals Beltane and Samhain , and 18th–19th century accounts suggest the fires, smoke and ashes were deemed to have protective powers. In some areas, torches of burning fir or turf from the bonfire were carried sunwise around homes and fields to protect them. In central and northern Europe, bonfires lit on Walpurgis Night and at Midsummer were also believed to ward off evil. A magic circle

1782-419: 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

1848-500: The Greek world. Most notable of these were the urban monuments found on the island of Delos . The phallus was also an apotropaic symbol for the ancient Romans. These are known as fascinum . A similar use of phallic representations to ward off the evil eye remains popular in modern Bhutan . It is associated with the 500-year-old Buddhist tradition of Drukpa Kunley . Mirrors and other shiny reflective objects were believed to deflect

1914-495: 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: Apotropaic charm Apotropaic magic (from Greek αποτρέπω , apotrépō  'to ward off') or protective magic is a type of magic intended to turn away harm or evil influences, as in deflecting misfortune or averting

1980-627: The bodies of dried cats , as well as shoes (see concealed shoes ). Apotropaic marks, also called 'witch marks' or 'anti-witch marks' in Europe, are symbols or patterns scratched on the walls, beams and thresholds of buildings to protect them from witchcraft or evil spirits. They have many forms; in Britain they are often flower-like patterns of overlapping circles . such as hexafoils . Taper burn marks on thresholds of early modern buildings are also thought to be apotropaic marks. Other types of mark include

2046-700: The case of a family who had already lost a child, the parents may name the next child Alter and Alte (both meaning "old" in Yiddish) in an effort to confuse the Angel of Death. Another example is Nekras ( Некрас , "not handsome" in Russian) which was given with the hope the child would be handsome. Among Serbian names are many apotropaic names ( zaštitna imena , "protective names"), such as Vuk ("wolf") (and its many derivatives) and Staniša ("stone"). Historical Chinese given names sometimes had apotropaic meanings, such as in

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2112-537: 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

2178-511: 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

2244-478: The early 20th century. In Western culture , a horseshoe was often nailed up over, or close by, doorways (see Oakham's horseshoes ). Model horseshoes (of card or plastic) are given as good-luck tokens, particularly at weddings , and small paper horseshoes feature in confetti . In early modern Europe, certain objects were buried in the walls of houses to protect the household from witchcraft . These included specially-prepared witch bottles , horse skulls and

2310-474: 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,

2376-565: The entry or passage of evil . In ancient Greece, grotesque, satyr -like bearded faces, sometimes with the pointed cap of the workman, were carved over the doors of ovens and kilns, to protect the work from fire and mishap. Later, on churches and castles , gargoyles or other grotesque faces and figures such as sheela na gigs and hunky punks were carved to frighten away witches and other malign influences. Figures may also have been carved at fireplaces or chimneys; in some cases, simple geometric or letter carvings were used for these. When

2442-584: The entryways of residences. An example of the use of shiny apotropaic objects in Judaism can be found in the so-called "Halsgezeige" or textile neckbands used in the birthing customs of the Franco-German border region. Shiny coins or colourful stones would be sewn onto the neckband or on a central amulet in order to distract the evil eye . These neckbands were worn by women in childbirth and by young boys during their Brit Milah ceremony. This custom continued until

2508-502: The evil eye. Traditional English "Plough Jags" (performers of a regional variant of the mummers play ) sometimes decorated their costumes (particularly their hats) with shiny items, to the extent of borrowing silver plate for the purpose. "Witch balls" are shiny blown glass ornaments, such as Christmas baubles , that were hung in windows. Similarly, the Chinese Bagua mirror is usually installed to ward off negative energy and protect

2574-403: The figure of a kind of grasshopper before the Acropolis of Athens for protection. Another way for protection from enchantment used by the ancient Greeks was by spitting into the folds of the clothes. Ancient Greeks also had an old custom of dressing boys as girls in order to avert the evil eye. In Ireland, it is customary on St Brigid's Day to weave a Brigid's cross from rushes, which

2640-401: 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

2706-536: 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

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2772-420: The folds of clothes as a way of protection from enchantment. Ancient Greeks also had an old custom of dressing boys as girls in order to avert the evil eye. Achilles is said to have been dressed in his youth as a girl at the court of Lycomedes, king of Scyros in order to avert the evil eye. Fire was used in rituals of protection in many parts of Europe up to the early modern era. The need-fire or force-fire

2838-630: 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

2904-582: The gods. One of the most commonly found magical objects, the ivory apotropaic wand ( birth tusk ), gained widespread popularity in the Middle Kingdom (c. 1550 – 1069 BCE). These wands were used to protect expectant mothers and children from malevolent forces, and were adorned with processions of apotropaic solar deities . Likewise, protective amulets bearing the likenesses of gods and goddesses such as Taweret were commonly worn. Water came to be used frequently in ritual as well, wherein libation vessels in

2970-413: The home, not in state-run temples ) were embodied by the deity who personified magic itself, Heka . The two gods most frequently invoked in these rituals were the hippopotamus -formed fertility goddess , Taweret , and the lion-deity, Bes (who developed from the early apotropaic dwarf god, Aha , literally "fighter"). Objects were often used in these rituals in order to facilitate communication with

3036-717: 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

3102-598: The intertwined letters V and M or a double V (for the protector, the Virgin Mary , alias Virgo Virginum ), and crisscrossing lines to confuse any spirits that might try to follow them. At the Bradford-on-Avon Tithe Barn , a flower-like pattern of overlapping circles is incised into a stone in the wall. Similar marks of overlapping circles have been found on a window sill dated about 1616 at Owlpen Manor in Gloucestershire, as well as taper burn marks on

3168-624: The jambs of a medieval door frame. The marks are most common near places where witches were thought to be able to enter, whether doors, windows or chimneys. For example, during works at Knole , near Sevenoaks in Kent, in 1609, oak beams beneath floors, particularly near fireplaces, were scorched and carved with scratched witch marks to prevent witches and demons from coming down the chimney. Marks have been found in buildings including Knole House , Shakespeare's Birthplace in Stratford-upon-Avon ,

3234-455: 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

3300-509: 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

3366-481: 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

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3432-563: The prow of sailing ships are considered to have been a replacement for the sacrifice of a thrall during the Age of Invasions by Saxon and Viking sailors, to avoid bad luck on the voyage. Dredging the Thames under London Bridge led to the discovery of a large number of bent and broken knives, daggers, swords and coins, from the modern period and dating back to Celtic times. This custom seems to have been to avoid bad luck, particularly when setting off on

3498-405: 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

3564-585: 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

3630-428: 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 :

3696-599: The shape of Taweret were used to pour healing water over an individual. In much later periods (when Egypt came under the Greek Ptolemies ), stele featuring the god Horus were used in similar rituals; water would be poured over the stele and—after ritually acquiring healing powers—was collected in a basin for an afflicted person to drink. The ancient Greeks had various protective symbols and objects, with various names, such as apotropaia, probaskania, periammata, periapta and profylaktika . The Greeks made offerings to

3762-616: 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

3828-423: 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

3894-407: The symbol nazar boncuğu ( nazar bonjuk ) on the vertical stabilizer (fin) of its aeroplanes. The apotropaic Yiddish expression, קיין עין הרע , kain ein horeh , 'no evil eye' (in modern Hebrew , בלי עין הרע , bli ein ha'ra ), is somewhat equivalent to the expression, " knock on wood ." Among the ancient Greeks, the most widely used image intended to avert evil was that of

3960-644: 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

4026-463: 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,

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4092-470: 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

4158-545: 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

4224-573: Was a special fire kindled to ward off plague and murrain (infectious diseases affecting livestock) in parts of western, northern and eastern Europe. It could only be kindled by friction between wood, by a group of certain people, after all other fires in the area were doused. The livestock would be driven around the need-fire or over its embers, and all other fires would be re-lit from it. Two early medieval Irish texts say that druids used to drive cattle between two bonfires "with great incantations", to protect them from disease. Almost 1,000 years later, in

4290-516: 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

4356-435: Was thought to bring bad luck to the person envied. To avoid envy, Romans sought to incite laughter in their guests by using humorous images. Images such as large phalluses (see fascinus ), deformities such as hunchbacks, or Pygmies and other non-Roman subjects were common. Romans saw deformity as comical and believed that such images could be used to deflect the evil eye. In Europe, apotropaic figureheads carved onto

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