In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
English heraldry is the form of coats of arms and other heraldic bearings and insignia used in England. It lies within the so-called Gallo-British tradition. Coats of arms in England are regulated and granted to individuals by the English kings of arms of the College of Arms. An individual's arms may be borne ‘by courtesy' by members of the holder's nuclear family, subject to a system of cadency marks, to difference those displays from the arms of the original holder; the English heraldic style is exemplified in the arms of British royalty, is reflected in the civic arms of cities and towns, as well as the noble arms of individuals in England. Royal orders in England, such as the Order of the Garter maintain notable heraldic bearings. Like many countries' heraldry, there is a classical influence within English heraldry, such as designs on Greek and Roman pottery. Many coats of arms feature charges related to the bearer's name or profession, a practice known as "canting arms"; some canting arms make references to foreign languages French, such as the otter in the arms of the Luttrel family.
Representations in person of Saints or other figure are rare, although there are however a few uses originating from seals, where there have never been such limitations. Although many places have dropped such iconography, the Metropolitan Borough of St Marylebone, includes a rendering of the Virgin Mary, although this is never stated; this is the case in many other examples those depicting Christ, to remove religious complications. Unlike in mainland Europe where family crests make a large use of their eponymous Saints, these are few and far between in England; the lion is the most common charge in Royal heraldry. Heraldic roses are common in English heraldry, as in the War of the Roses where both houses and York, used them, in the ensuing Tudor dynasty; the heraldic eagle, while common on the European continent and in Germany, is rare in English heraldry and, in early English heraldry, was associated with alliances with German princes. The coat of arms of Richard Neville, 16th Earl of Warwick, pictured on the left, uses all typical forms of heraldry in England: The first quarter consists of his father-in-law, Richard Beauchamp, who bore with an escutcheon of De Clare quartering Despenser, now shown in Neville's fourth quarter.
The second quarter shows the arms of the Montacutes. The third quarter shows the arms of Neville differenced by a label for Lancaster; the first Royal Coat of Arms was created in 1154 under Henry II, the idea of heraldry becoming popular among the knights on the first and second crusades, along with the idea of chivalry. Under Henry III, it gained a system of classification and a technical language, confirming its place as a science. However, over the next two centuries, the system was abused, leading to the swamping of true coats-of-arms. For the rest of the medieval period, it was popular within the upper classes to have a distinctive family mark for competitions and tournaments, it was popular within the lower classes, it found particular use with knights, for practice and in the mêlée of battle, where heraldry was worn on embroidered fabric covering their armour. Indeed, their houses' signs became known as coats-of-arms in this way, they were worn on shields, where they were known as shields-of-arms.
As well as military uses, the main charge was used in the seals of households. These were used to prove the authenticity of documents carried by heralds and is the basis of the word heraldry in English. One example of this is the seal of John Mundegumri. Prior to the 16th century, there was no regulation on the use of arms in England. One of the first contemporary records of medieval heraldry is a roll of arms called Falkirk Rolls, written soon after the Battle of Falkirk in 1298, it designs. This demonstrates that English heraldry was developed at this time, although the language is not quite identical, much of the terminology is the same as is still used, it is an occasional roll of arms. Other rolls of arms covering England include Glover's Roll; the position of herald in England was well defined, so on January 5, 1420, William Bruges was appointed by King Henry V to be Garter King of Arms. No such position had been created in other countries. A succession of different titles was introduced over the next four centuries for principal governor of arms, including King of Arms.
Some were members of the College of Arms, some were not. Other holders of positions included the Falcon King of Arms, a position created under King Edward III. Other positions were created for important counties, such as the Lancastrian King of Arms, but the balance of power between them and those charged with larger regions remains unclear. During the Tudor period, grants of arms were made for significant contributions to the country by one of the Herald and Kings of Arms in a standard format, as in the case of Thomas Bertie, granted arms on 10 July 1550; this was given as a passage read out by the herald. Although many are written in English, it is possible they were read out in Latin; the introduction in his case read: To all noble and gentled the present letters reading hearing or seeing, Thomas Hawley alias Clarencieulx principal Herald and King of Arms of the south
Ambiguous images or reversible figures are optical illusion images which exploit graphical similarities and other properties of visual system interpretation between two or more distinct image forms. These are famous for inducing the phenomenon of multistable perception. Multistable perception is the occurrence of an image being able to provide multiple, although stable, perceptions. Classic examples of this are the Rubin vase. Ambiguous images are important to the field of psychology because they are research tools used in experiments. There is varying evidence on whether ambiguous images can be represented mentally, but a majority of research has theorized that they cannot be properly represented mentally; the rabbit-duck image seems to be one of the earliest of this type. Middle vision is the stage in visual processing that combines all the basic features in the scene into distinct, recognizable object groups; this stage of vision comes after early vision. When perceiving and recognizing images, mid-level vision comes into use when we need to classify the object we are seeing.
Higher-level vision is used when the object classified must now be recognized as a specific member of its group. For example, through mid-level vision we perceive a face through high-level vision we recognize a face of a familiar person. Mid-level vision and high-level vision are crucial for understanding a reality, filled with ambiguous perceptual inputs; when we see an image, the first thing we do is attempt to organize all the parts of the scene into different groups. To do this, one of the most basic methods used is finding the edges. Edges can include obvious perceptions such as the edge of a house, can include other perceptions that the brain needs to process deeper, such as the edges of a person's facial features; when finding edges, the brain's visual system detects a point on the image with a sharp contrast of lighting. Being able to detect the location of the edge of an object aids in recognizing the object. In ambiguous images, detecting edges still seems natural to the person perceiving the image.
However, the brain undergoes deeper processing to resolve the ambiguity. For example, consider an image that involves an opposite change in magnitude of luminance between the object and the background; the opposing gradients will come to a point where there is an equal degree of luminance of the object and the background. At this point, there is no edge to be perceived. To counter this, the visual system connects the image as a whole rather than a set of edges, allowing one to see an object rather than edges and non-edges. Although there is no complete image to be seen, the brain is able to accomplish this because of its understanding of the physical world and real incidents of ambiguous lighting. In ambiguous images, an illusion is produced from illusory contours. An illusory contour is a perceived contour without the presence of a physical gradient. In examples where a white shape appears to occlude black objects on a white background, the white shape appears to be brighter than the background, the edges of this shape produce the illusory contours.
These illusory contours are processed by the brain in a similar way as real contours. The visual system accomplishes this by making inferences beyond the information, presented in much the same way as the luminance gradient. In mid-level vision, the visual system utilizes a set of heuristic methods, called Gestalt grouping rules, to identify a basic perception of an object that helps to resolve an ambiguity; this allows perception to be fast and easy by observing patterns and familiar images rather than a slow process of identifying each part of a group. This aids in resolving ambiguous images because the visual system will accept small variations in the pattern and still perceive the pattern as a whole; the Gestalt grouping rules are the result of the experience of the visual system. Once a pattern is perceived it is stored in memory and can be perceived again without the requirement of examining the entire object again. For example, when looking at a chess board, we perceive a checker pattern and not a set of alternating black and white squares.
The principle of good continuation provides the visual system a basis for identifying continuing edges. This means that when a set of lines is perceived, there is a tendency for a line to continue in one direction; this allows the visual system to identify the edges of a complex image by identifying points where lines cross. For example, two lines crossed in an "X" shape will be perceived as two lines travelling diagonally rather than two lines changing direction to form "V" shapes opposite to each other. An example of an ambiguous image would be two curving lines intersecting at a point; this junction would be perceived the same way as the "X", where the intersection is seen as the lines crossing rather than turning away from each other. Illusions of good continuation are used by magicians to trick audiences; the rule of similarity states that images that are similar to each other can be grouped together as being the same type of object or part of the same object. Therefore, the more similar two images or objects are, the more it will be that they can be grouped together.
For example, two squares among many circles will be grouped together. They can vary in similarity of colour, size and other properties, but will be grouped together with varying degrees of membership; the grouping property of pro
A mosaic is a piece of art or image made from the assembling of small pieces of colored glass, stone, or other materials. It is used in decorative art or as interior decoration. Most mosaics are made of small, flat square, pieces of stone or glass of different colors, known as tesserae; some floor mosaics, are made of small rounded pieces of stone, called "pebble mosaics". Mosaics have a long history, starting in Mesopotamia in the 3rd millennium BC. Pebble mosaics were made in Tiryns in Mycenean Greece. Early Christian basilicas from the 4th century onwards were decorated with ceiling mosaics. Mosaic art flourished in the Byzantine Empire from the 6th to the 15th centuries. Mosaic fell out of fashion in the Renaissance, though artists like Raphael continued to practise the old technique. Roman and Byzantine influence led Jewish artists to decorate 5th and 6th century synagogues in the Middle East with floor mosaics. Mosaic was used on religious buildings and palaces in early Islamic art, including Islam's first great religious building, the Dome of the Rock in Jerusalem, the Umayyad Mosque in Damascus.
Mosaic went out of fashion in the Islamic world after the 8th century. Modern mosaics are made by professional artists, street artists, as a popular craft. Many materials other than traditional stone and ceramic tesserae may be employed, including shells and beads; the earliest known examples of mosaics made of different materials were found at a temple building in Abra and are dated to the second half of 3rd millennium BC. They consist of pieces of colored stones and ivory. Excavations at Susa and Chogha Zanbil show evidence of the first glazed tiles, dating from around 1500 BC. However, mosaic patterns were not used until the times of Roman influence. Bronze age pebble mosaics have been found at Tiryns. Mythological subjects, or scenes of hunting or other pursuits of the wealthy, were popular as the centrepieces of a larger geometric design, with emphasized borders. Pliny the Elder mentions the artist Sosus of Pergamon by name, describing his mosaics of the food left on a floor after a feast and of a group of doves drinking from a bowl.
Both of these themes were copied. Greek figural mosaics could have been copied or adapted paintings, a far more prestigious artform, the style was enthusiastically adopted by the Romans so that large floor mosaics enriched the floors of Hellenistic villas and Roman dwellings from Britain to Dura-Europos. Most recorded names of Roman mosaic workers are Greek, suggesting they dominated high quality work across the empire. Splendid mosaic floors are found in Roman villas across North Africa, in places such as Carthage, can still be seen in the extensive collection in Bardo Museum in Tunis, Tunisia. There were two main techniques in Greco-Roman mosaic: opus vermiculatum used tiny tesserae cubes of 4 millimeters or less, was produced in workshops in small panels which were transported to the site glued to some temporary support; the tiny tesserae allowed fine detail, an approach to the illusionism of painting. Small panels called emblemata were inserted into walls or as the highlights of larger floor-mosaics in coarser work.
The normal technique was opus tessellatum, using larger tesserae, laid on site. There was a distinct native Italian style using black on a white background, no doubt cheaper than coloured work. In Rome and his architects used mosaics to cover some surfaces of walls and ceilings in the Domus Aurea, built 64 AD, wall mosaics are found at Pompeii and neighbouring sites; however it seems that it was not until the Christian era that figural wall mosaics became a major form of artistic expression. The Roman church of Santa Costanza, which served as a mausoleum for one or more of the Imperial family, has both religious mosaic and decorative secular ceiling mosaics on a round vault, which represent the style of contemporary palace decoration; the mosaics of the Villa Romana del Casale near Piazza Armerina in Sicily are the largest collection of late Roman mosaics in situ in the world, are protected as a UNESCO World Heritage Site. The large villa rustica, owned by Emperor Maximian, was built in the early 4th century.
The mosaics were covered and protected for 700 years by a landslide that occurred in the 12th Century. The most important pieces are the Circus Scene, the 64m long Great Hunting Scene, the Little Hunt, the Labours of Hercules and the famous Bikini Girls, showing women undertaking a range of sporting activities in garments that resemble 20th Century bikinis; the peristyle, the imperial apartments and the thermae were decorated with ornamental and mythological mosaics. Other important examples of Roman mosaic art in Sicily were unearthed on the Piazza Vittoria in Palermo where two houses were discovered; the most important scenes there depicted are an Orpheus mosaic, Alexander the Great's Hunt and the Four Seasons. In 1913 the Zliten mosaic, a Roman mosaic famous for its many scenes from gladiatorial contests and everyday life, was discovered in the Libyan town of Zliten. In 2000 archaeologists working
The island of Delos, near Mykonos, near the centre of the Cyclades archipelago, is one of the most important mythological and archaeological sites in Greece. The excavations in the island are among the most extensive in the Mediterranean. Delos had a position as a holy sanctuary for a millennium before Olympian Greek mythology made it the birthplace of Apollo and Artemis. From its Sacred Harbour, the horizon shows the three conical mounds that have identified landscapes sacred to a goddess in other sites: one, retaining its Pre-Greek name Mount Kynthos, is crowned with a sanctuary of Zeus. Established as a cult center, Delos had an importance that its natural resources could never have offered. In this vein Leto, searching for a birthing-place for Artemis and Apollo, addressed the island: Delos, if you would be willing to be the abode of my son Phoebus Apollo and make him a rich temple –, but if you have the temple of far-shooting Apollo, all men will bring you hecatombs and gather here, incessant savour of rich sacrifice will always arise, you will feed those who dwell in you from the hand of strangers.
Investigation of ancient stone huts found on the island indicate that it has been inhabited since the 3rd millennium BCE. Thucydides identifies the original inhabitants as piratical Carians who were expelled by King Minos of Crete. By the time of the Odyssey the island was famous as the birthplace of the twin gods Apollo and Artemis. Indeed, between 900 BCE and 100 CE, sacred Delos was a major cult centre, where Dionysus is in evidence as well as the Titaness Leto, mother of the above-mentioned twin deities. Acquiring Panhellenic religious significance, Delos was a religious pilgrimage for the Ionians. A number of "purifications" were executed by the city-state of Athens in an attempt to render the island fit for the proper worship of the gods; the first took place in the 6th century BCE, directed by the tyrant Pisistratus who ordered that all graves within sight of the temple be dug up and the bodies moved to another nearby island. In the 5th century BCE, during the 6th year of the Peloponnesian war and under instruction from the Delphic Oracle, the entire island was purged of all dead bodies.
It was ordered that no one should be allowed to either die or give birth on the island due to its sacred importance and to preserve its neutrality in commerce, since no one could claim ownership through inheritance. After this purification, the first quinquennial festival of the Delian games were celebrated there. Four years all inhabitants of the island were removed to Atramyttium in Asia as a further purification. After the Persian Wars the island became the natural meeting-ground for the Delian League, founded in 478 BCE, the congresses being held in the temple; the League's common treasury was kept here as well until 454 BCE. The island had no productive capacity for fiber, or timber, with such being imported. Limited water was exploited with an extensive cistern and aqueduct system and sanitary drains. Various regions operated agoras. Strabo states that in 166 BCE the Romans converted Delos into a free port, motivated by seeking to damage the trade of Rhodes, at the time the target of Roman hostility.
In 167 or 166 BCE, after the Roman victory in the Third Macedonian War, the Roman Republic ceded the island of Delos to the Athenians, who expelled most of the original inhabitants. Roman traders came to purchase tens of thousands of slaves captured by the Cilician pirates or captured in the wars following the disintegration of the Seleucid Empire, it became the center of the slave trade, with the largest slave market in the larger region being maintained here. The Roman destruction of Corinth in 146 BCE allowed Delos to at least assume Corinth's role as the premier trading center of Greece. However, Delos' commercial prosperity, construction activity, population waned after the island was assaulted by the forces of Mithridates VI of Pontus in 88 and 69 BCE, during the Mithridatic Wars with Rome. Before the end of the 1st century BCE, trade routes had changed. Due to the inadequate natural sources of food and water, the above history, unlike other Greek islands, Delos did not have an indigenous, self-supporting community of its own.
As a result, in times it was uninhabited. Since 1872 the École française d'Athènes has been excavating the island, the complex of buildings of which compares with those of Delphi and Olympia. In 1990, UNESCO inscribed Delos on the World Heritage List, citing it as the "exceptionally extensive and rich" archaeological site which "conveys the image of a great cosmopolitan Mediterranean port"; the small Sacred Lake in its circular bowl, now intentionally left dry by the island's caretakers to suppress disease-spreading bacteria, is a topographical feature that determined the placement of features. The Minoan Fountain was a rectangular public we
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane. Angles are formed by the intersection of two planes in Euclidean and other spaces; these are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is used to designate the measure of an angle or of a rotation; this measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation; the word angle comes from the Latin word angulus, meaning "corner".
Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, do not lie straight with respect to each other. According to Proclus an angle must be a relationship; the first concept was used by Eudemus. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower case Roman letters are used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted ∠BAC or B A C ^. Sometimes, where there is no risk of confusion, the angle may be referred to by its vertex. An angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign.
However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise angle from B to C, ∠CAB to the anticlockwise angle from C to B. An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle are called acute angles. An angle equal to 1/4 turn is called a right angle. Two lines that form a right angle are said to be orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle. Angles larger than a straight angle but less than 1 turn are called reflex angles. An angle equal to 1 turn is called complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles; the names and measured units are shown in a table below: Angles that have the same measure are said to be equal or congruent.
An angle is not dependent upon the lengths of the sides of the angle. Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by subtracting or adding straight angle, to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, an angle of 150 degrees has a reference angle of 30 degrees. An angle of 750 degrees has a reference angle of 30 degrees; when two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other. A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles, they are abbreviated as vert. opp. ∠s. The equality of vertically opposite angles is called the vertical angle theorem.
Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note, w