1.
Florence (stad)
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Florence is the capital city of the Italian region of Tuscany and of the Metropolitan City of Florence. It is the most populous city in Tuscany, with 383,083 inhabitants, Florence was a centre of medieval European trade and finance and one of the wealthiest cities of the time. It is considered the birthplace of the Renaissance, and has called the Athens of the Middle Ages. A turbulent political history includes periods of rule by the powerful Medici family, from 1865 to 1871 the city was the capital of the recently established Kingdom of Italy. The Historic Centre of Florence attracts 13 million tourists each year and it was declared a World Heritage Site by UNESCO in 1982. The city is noted for its culture, Renaissance art and architecture, the city also contains numerous museums and art galleries, such as the Uffizi Gallery and the Palazzo Pitti, and still exerts an influence in the fields of art, culture and politics. Due to Florences artistic and architectural heritage, it has been ranked by Forbes as one of the most beautiful cities in the world, in 2008, the city had the 17th highest average income in Italy. Florence originated as a Roman city, and later, after a period as a flourishing trading and banking medieval commune. According to the Encyclopædia Britannica, it was politically, economically, and culturally one of the most important cities in Europe, the language spoken in the city during the 14th century was, and still is, accepted as the Italian language. Starting from the late Middle Ages, Florentine money—in the form of the gold florin—financed the development of all over Europe, from Britain to Bruges, to Lyon. Florentine bankers financed the English kings during the Hundred Years War and they similarly financed the papacy, including the construction of their provisional capital of Avignon and, after their return to Rome, the reconstruction and Renaissance embellishment of Rome. Florence was home to the Medici, one of European historys most important noble families, Lorenzo de Medici was considered a political and cultural mastermind of Italy in the late 15th century. Two members of the family were popes in the early 16th century, Leo X, catherine de Medici married king Henry II of France and, after his death in 1559, reigned as regent in France. Marie de Medici married Henry IV of France and gave birth to the future king Louis XIII, the Medici reigned as Grand Dukes of Tuscany, starting with Cosimo I de Medici in 1569 and ending with the death of Gian Gastone de Medici in 1737. The Etruscans initially formed in 200 BC the small settlement of Fiesole and it was built in the style of an army camp with the main streets, the cardo and the decumanus, intersecting at the present Piazza della Repubblica. Situated along the Via Cassia, the route between Rome and the north, and within the fertile valley of the Arno, the settlement quickly became an important commercial centre. Peace returned under Lombard rule in the 6th century, Florence was conquered by Charlemagne in 774 and became part of the Duchy of Tuscany, with Lucca as capital. The population began to again and commerce prospered

2.
Sint-Pietersbasiliek
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The Papal Basilica of St. Peter in the Vatican, or simply St. Peters Basilica, is an Italian Renaissance church in Vatican City, the papal enclave within the city of Rome. While it is neither the church of the Catholic Church nor the cathedral of the Diocese of Rome. It has been described as holding a position in the Christian world. Catholic tradition holds that the Basilica is the site of Saint Peter, one of Christs Apostles. Saint Peters tomb is supposedly directly below the altar of the Basilica. For this reason, many Popes have been interred at St. Peters since the Early Christian period, construction of the present basilica, which would replace Old St. Peters Basilica from the 4th century AD, began on 18 April 1506 and was completed on 18 November 1626. St. Peters is famous as a place of pilgrimage and for its liturgical functions. The Pope presides at a number of liturgies throughout the year, drawing audiences of 15,000 to over 80,000 people, either within the Basilica or the adjoining St. Peters Square. St. Peters has many associations, with the Early Christian Church, the Papacy. As a work of architecture, it is regarded as the greatest building of its age, St. Peters is one of the four churches in the world that hold the rank of Major Basilica, all four of which are in Rome. Contrary to popular misconception, it is not a cathedral because it is not the seat of a bishop, St. Peters is a church built in the Renaissance style located in the Vatican City west of the River Tiber and near the Janiculum Hill and Hadrians Mausoleum. Its central dome dominates the skyline of Rome, the basilica is approached via St. Peters Square, a forecourt in two sections, both surrounded by tall colonnades. The first space is oval and the second trapezoid, the basilica is cruciform in shape, with an elongated nave in the Latin cross form but the early designs were for a centrally planned structure and this is still in evidence in the architecture. The central space is dominated both externally and internally by one of the largest domes in the world, the entrance is through a narthex, or entrance hall, which stretches across the building. One of the bronze doors leading from the narthex is the Holy Door. The interior is of vast dimensions when compared with other churches and this in its turn overwhelms us. The nave which leads to the dome is in three bays, with piers supporting a barrel-vault, the highest of any church. The nave is framed by wide aisles which have a number of chapels off them, there are also chapels surrounding the dome

3.
Vaticaanstad
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Vatican City, officially Vatican City State or the State of Vatican City, is a walled enclave within the city of Rome. With an area of approximately 44 hectares, and a population of 842, however, formally it is not sovereign, with sovereignty being held by the Holy See, the only entity of public international law that has diplomatic relations with almost every country in the world. It is an ecclesiastical or sacerdotal-monarchical state ruled by the Bishop of Rome – the Pope, the highest state functionaries are all Catholic clergy of various national origins. Vatican City is distinct from the Holy See, which dates back to early Christianity and is the episcopal see of 1.2 billion Latin. According to the terms of the treaty, the Holy See has full ownership, exclusive dominion, within Vatican City are religious and cultural sites such as St. Peters Basilica, the Sistine Chapel and the Vatican Museums. They feature some of the worlds most famous paintings and sculptures, the unique economy of Vatican City is supported financially by the sale of postage stamps and tourist mementos, fees for admission to museums, and the sale of publications. The name Vatican City was first used in the Lateran Treaty, signed on 11 February 1929, the name is taken from Vatican Hill, the geographic location of the state. Vatican is derived from the name of an Etruscan settlement, Vatica or Vaticum meaning garden, located in the area the Romans called vaticanus ager. The official Italian name of the city is Città del Vaticano or, more formally, Stato della Città del Vaticano, although the Holy See and the Catholic Church use Ecclesiastical Latin in official documents, the Vatican City officially uses Italian. The Latin name is Status Civitatis Vaticanæ, this is used in documents by not just the Holy See. The name Vatican was already in use in the time of the Roman Republic for an area on the west bank of the Tiber across from the city of Rome. Under the Roman Empire, many villas were constructed there, after Agrippina the Elder drained the area and laid out her gardens in the early 1st century AD. In AD40, her son, Emperor Caligula built in her gardens a circus for charioteers that was completed by Nero, the Circus Gaii et Neronis, usually called, simply. Even before the arrival of Christianity, it is supposed that this originally uninhabited part of Rome had long considered sacred. A shrine dedicated to the Phrygian goddess Cybele and her consort Attis remained active long after the Constantinian Basilica of St. Peter was built nearby, the particularly low quality of Vatican water, even after the reclamation of the area, was commented on by the poet Martial. The Vatican Obelisk was originally taken by Caligula from Heliopolis in Egypt to decorate the spina of his circus and is thus its last visible remnant and this area became the site of martyrdom of many Christians after the Great Fire of Rome in AD64. Ancient tradition holds that it was in this circus that Saint Peter was crucified upside-down, opposite the circus was a cemetery separated by the Via Cornelia. Peters in the first half of the 4th century, the Constantinian basilica was built in 326 over what was believed to be the tomb of Saint Peter, buried in that cemetery

4.
Taj Mahal
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The Taj Mahal is an ivory-white marble mausoleum on the south bank of the Yamuna river in the Indian city of Agra. It was commissioned in 1632 by the Mughal emperor, Shah Jahan, to house the tomb of his favourite wife, Mumtaz Mahal. The tomb is the centrepiece of a 17-hectare complex, which includes a mosque and a guest house, construction of the mausoleum was essentially completed in 1643 but work continued on other phases of the project for another 10 years. The construction project employed some 20,000 artisans under the guidance of a board of architects led by the architect to the emperor. The Taj Mahal was designated as a UNESCO World Heritage Site in 1983 for being the jewel of Muslim art in India and one of the universally admired masterpieces of the worlds heritage. Described by Nobel laureate Rabindranath Tagore as the tear-drop on the cheek of time, it is regarded by many as the best example of Mughal architecture, the Taj Mahal attracts 7–8 million visitors a year. In 2007, it was declared a winner of the New7Wonders of the World initiative. The Taj Mahal was commissioned by Shah Jahan in 1631, to be built in the memory of his wife Mumtaz Mahal, construction of the Taj Mahal began in 1632. The imperial court documenting Shah Jahans grief after the death of Mumtaz Mahal illustrate the story held as the inspiration for Taj Mahal. The principal mausoleum was completed in 1643 and the surrounding buildings, the Taj Mahal incorporates and expands on design traditions of Persian and earlier Mughal architecture. Specific inspiration came from successful Timurid and Mughal buildings including the Gur-e Amir, Humayuns Tomb, Itmad-Ud-Daulahs Tomb, while earlier Mughal buildings were primarily constructed of red sandstone, Shah Jahan promoted the use of white marble inlaid with semi-precious stones. Buildings under his patronage reached new levels of refinement, the tomb is the central focus of the entire complex of the Taj Mahal. It is a large, white marble standing on a square plinth and consists of a symmetrical building with an iwan topped by a large dome. Like most Mughal tombs, the elements are Persian in origin. The base structure is a large multi-chambered cube with chamfered corners forming an unequal eight-sided structure that is approximately 55 metres on each of the four long sides. Each side of the iwan is framed with a huge pishtaq or vaulted archway with two similarly shaped arched balconies stacked on either side and this motif of stacked pishtaqs is replicated on the chamfered corner areas, making the design completely symmetrical on all sides of the building. Four minarets frame the tomb, one at each corner of the plinth facing the chamfered corners, the main chamber houses the false sarcophagi of Mumtaz Mahal and Shah Jahan, the actual graves are at a lower level. The most spectacular feature is the dome that surmounts the tomb

5.
Bol (lichaam)
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3

6.
Ellipsoïde
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An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by any of the two following properties, every planar cross section is either an ellipse, or is empty, or is reduced to a single point. It is bounded, which means that it may be enclosed in a large sphere. An ellipsoid has three perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, if the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, and the axes are uniquely defined. If two of the axes have the length, then the ellipsoid is an ellipsoid of revolution. In this case, the ellipsoid is invariant under a rotation around the third axis, if the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the length, the ellipsoid is a sphere. The points, and lie on the surface, the line segments from the origin to these points are called the semi-principal axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the axis and semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid, if a = b < c, one has a prolate spheroid, if a = b = c, one has a sphere. It is easy to check, The intersection of a plane, remark, The contour of an ellipsoid, seen from a point outside the ellipsoid or from infinity, is in any case a plane section, hence an ellipse. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin and these parameters may be interpreted as spherical coordinates. More precisely, π /2 − θ is the polar angle, and φ is the azimuth angle of the point of the ellipsoid. More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the x to the equation T A =1. The eigenvectors of A define the axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes

7.
Ui (bouwkunst)
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An onion dome is a dome whose shape resembles an onion. Such domes are often larger in diameter than the drum upon which they sit and these bulbous structures taper smoothly to a point. Other types of Eastern Orthodox cupolas include helmet domes, Ukrainian pear domes, art historians disagree on when and why onion domes became a typical feature of Russian architecture. Byzantine churches and the architecture of Kievan Rus were characterized by broader, flatter domes without a special framework erected above the drum. In contrast to this ancient form, each drum of a Russian church is surmounted by a structure of metal or timber. By the end of the century, most Russian churches from before the Petrine period had bulbous domes. The largest onion domes were erected in the century in the area around Yaroslavl. Quite a few had more complicated bud-shaped domes, whose form derived from Baroque models of the seventeenth century. Pear-shaped domes are usually associated with Ukrainian Baroque, while cone-shaped domes are typical for Orthodox churches of Transcaucasia, Russian icons painted before the Mongol invasion of Rus do not feature churches with onion domes. Two highly venerated pre-Mongol churches that have been rebuilt—the Assumption Cathedral, restoration work on several other ancient churches revealed some fragments of former helmet-like domes below newer onion cupolas. It has been posited that onion domes first appeared during the reign of Ivan the Terrible, the domes of Saint Basils Cathedral have not been altered since the reign of Ivans son Fyodor I, indicating the presence of onion domes in the sixteenth-century Russia. Eight of the nine domes featured on St. Basils Cathedral represent each attack on Kazan, the ninth dome was constructed 36 years after the siege of Kazan as a tomb for Basil. The ornate finishes of these domes are bright in color and bold in shape as they are adorned with pyramids and stripes, some believe that onion domes first appeared in Russian wooden architecture above tent-like churches. According to this theory, onion domes were strictly utilitarian, as they prevented snow from piling on the roof, one example of such restoration is the Dormition Cathedral in the Moscow Kremlin. These findings demonstrated that Russian onion domes could not be imported from the Orient, sergey Zagraevsky, a modern art historian, surveyed hundreds of Russian icons and miniatures, from the eleventh century onward. He concluded that most icons painted after the Mongol invasion of Rus display only onion domes, first onion domes displayed on some pictures of twelfth century. He found only one icon from the fifteenth century displaying a dome resembling the helmet instead of an onion. Zagraevsky also indicated that the oldest depictions of the two Vladimir cathedrals represent them as having onion domes, prior to their replacement by classicizing helmet domes

8.
Cirkel
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles

9.
Ellips (wiskunde)
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin

10.
Veelhoek
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular