Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together. Mathematical symmetry may be observed with respect to the passage of time; this article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion; this means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry if there is a line going through it which divides it into two pieces which are mirror images of each other.
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry. An object has helical symmetry if it can be translated and rotated in three-dimensional space along a line known as a screw axis. An object contracted. Fractals exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection rotoreflection symmetry. A dyadic relation R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary Mary is the same age as Paul. Symmetric binary logical connectives are and, or, nand and nor. Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object; the set of operations that preserve a given property of the object form a group.
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include and odd functions in calculus. In statistics, it appears as symmetric probability distributions, as skewness, asymmetry of distributions. Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations; this concept has become one of the most powerful tools of theoretical physics, as it has become evident that all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his read 1972 article More is Different that "it is only overstating the case to say that physics is the study of symmetry." See Noether's theorem. Important symmetries in physics include discrete symmetries of spacetime. In biology, the notion of symmetry is used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves.
Animals that move in one direction have upper and lower sides and tail ends, therefore a left and a right. The head becomes specialized with a mouth and sense organs, the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs remain asymmetric. Plants and sessile animals such as sea anemones have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, sea lilies. In biology, the notion of symmetry is used as in physics, to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. Symmetry is important to chemistry because it undergirds all specific interactions between molecules in nature; the control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer the
Ravioli are a type of dumpling comprising a filling enveloped in thin pasta dough. Served in broth or with a sauce, they originated as a traditional food in Italian cuisine. Ravioli are square, though other forms are used, including circular and semi-circular; the earliest known mention of ravioli appears in the personal letters of Francesco di Marco Datini, a merchant of Prato in the 14th century. In Venice, the mid-14th-century manuscript Libro per cuoco offers ravioli of green herbs blanched and minced, mixed with beaten egg and fresh cheese, simmered in broth and seasoned with "sweet and strong spices". In Rome, ravioli were well-known when Bartolomeo Scappi served them with boiled chicken to the papal conclave of 1549. Ravioli were known in 14th century England, appearing in the Anglo-Norman vellum manuscript Forme of Cury under the name of rauioles. Sicilian ravioli and Malta's ravjul may thus be older than North Italian ones. Maltese ravjul are stuffed with irkotta, the locally produced sheep's-milk ricotta, or with gbejna, the traditional fresh sheep's-milk cheese.
Traditionally, ravioli are made at home. The filling varies according to the area. In Rome and Latium the filling is made with ricotta cheese, spinach and black pepper. In Sardinia, ravioli are filled with grated lemon rind. Modern ravioli is mass-produced by machine. In Europe and the United States, fresh-packed ravioli have several weeks of shelf life. Canned ravioli was pioneered by the Italian Army in the First World War and was popularized by Heinz and Buitoni in the UK and Europe, Chef Boyardee in the United States. Canned ravioli may be filled with beef, processed cheese, chicken, or Italian sausage and served in a tomato, tomato-meat, or tomato-cheese sauce. Toasted ravioli was developed in St. Louis, is a popular appetizer and snack food. Ravioli are encountered in the cooking of Nice, the broader Côte d'Azur, the surrounding regions in the south of France; the contents of these vary but most idiosyncratic to the region is the use of leftover daube meat. Miniaturized cheese-filled ravioli, locally called "ravioles", are a specialty of the Drôme department in the Rhône-Alpes region the commune of Romans-sur-Isère, are served au gratin.
In Turkey, Mantı, similar to ravioli is a popular dish. It is served with paprika sauce and yoghurt. Similar dishes in China are the wonton. In India, a popular dish called. However, it is prepared sweet, with a filling of dry fruits, a mixture of sweet spices deep fried in vegetable oil. Different stuffings are used in different parts of India; the dish is a popular food prepared during festivals all over the country. Jewish cuisine has a similar dish called kreplach, a pocket of meat or other filling, with an egg pasta based covering, it is simmered in chicken soup. In that method of preparation it appears to be the direct descendent or inspiration of the original dish, simmered in "broth". Claudia Roden argues it originated in the Venetian Ghetto at about the same time ravioli was developed, in time became a mainstay of Jewish cuisine. A similar Middle Eastern dish called shishbarak contains pasta filled with minced beef meat and cooked in hot yogurt. Adamson, Melitta Weiss. Regional Cuisines of Medieval Europe: A Book of Essays.
Routledge. ISBN 0-415-92994-6. Davidson, Alan, ed.. The Oxford Companion to Food. Oxford University Press. ISBN 978-0-192-11579-9. Dickie, John. Delizia! The Epic History of the Italians and Their Food. Free Press. ISBN 978-0-7432-7799-0. McNulty, Mary F. "Pasta". How Products are Made. Madehow.com. Retrieved 1 September 2013. Smith, Andrew F. ed.. The Oxford Companion to American Food and Drink. Oxford University Press. ISBN 978-0-19-530796-2. Retrieved 5 September 2012. Wolfert, Paula. Mediterranean Clay Pot Cooking: Traditional and Modern Recipes to Savor and Share. Hoboken, N. J.: John Wiley & Sons. P. 176. ISBN 978-0-764-57633-1. How to make ravioli from scratch General catalog of double sheet ravioli shapes Machine-made ravioli, commercial demonstration of machine producing different kinds of pasta, including ravioli
A serpentine shape is any of certain curved shapes of an object or design, which are suggestive of the shape of a snake. Serpentine shapes occur in architecture, in furniture, in mathematics; the serpentine shape is observed in many architectural settings. It may provide strength, as in serpentine walls, it may allow the facade of a building to face in multiple directions, or it may be chosen for purely aesthetic reasons. At the University of Virginia, serpentine walls extend down the length of the main lawn at the University of Virginia and flank both sides of the rotunda, they are one of the many structures. The sinusoidal path of the wall provides strength against toppling over, allowing the wall to be only a single brick thick. At the Massachusetts Institute of Technology, the Baker House dormitory has a serpentine shape which allows most rooms a view of the Charles River, gives many of the rooms a wedge-shaped layout. At San Carlo alle Quattro Fontane, Italy, designed by Francesco Borromini, is a serpentine facade constructed towards the end of Borromini's life.
The concave-convex facade of the church undulates in a non-classic way. Tall corinthian columns support the main entablatures. Between the columns, smaller columns with their entablatures weave behind the main columns and in turn they frame many architectural features of the church; the London parks Hyde Park and Kensington Gardens contain'The Serpentine', a lake that spans both parks. It received the name from its curving shape. A central bridge divides the lake into two parts, defines the boundaries between Hyde Park and Kensington Gardens. Among Castle Howard's gardens is a large, formal path behind the building, where a serpentine path is situated on a ridge, it merges back into the park. When buildings and site elements are set into the landscape, a serpentine path connecting every location is placed in-between features; the path merges into the landscape due to the natural shape, which allows convenient garden-path integration. A serpentine street is a winding roadway sometimes used to slow traffic in residential neighbourhoods bordered by landscaping features.
In furniture, serpentine-front dressers and cabinets have a convex section between two concave ones. This design was common in the Rococo period. Examples include 18th-century English furniture. Furniture with a concave section between two convex ones is sometimes referred to as reverse serpentine or oxbow; the serpentine curve is a cubic curve as described by Isaac Newton, given by the cartesian equation y = abx. The origin is a point of inflection, the axis of x being an asymptote and the curve lies between the parallel lines 2y = ±b. BACH motif Serpent This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed.. "Serpentine". Encyclopædia Britannica. Cambridge University Press
Romanesque architecture is an architectural style of medieval Europe characterized by semi-circular arches. There is no consensus for the beginning date of the Romanesque style, with proposals ranging from the 6th to the 11th century, this date being the most held. In the 12th century it developed into the Gothic style, marked by pointed arches. Examples of Romanesque architecture can be found across the continent, making it the first pan-European architectural style since Imperial Roman architecture; the Romanesque style in England is traditionally referred to as Norman architecture. Combining features of ancient Roman and Byzantine buildings and other local traditions, Romanesque architecture is known by its massive quality, thick walls, round arches, sturdy pillars, barrel vaults, large towers and decorative arcading; each building has defined forms of regular, symmetrical plan. The style can be identified right across Europe, despite regional characteristics and different materials. Many castles were built during this period, but they are outnumbered by churches.
The most significant are the great abbey churches, many of which are still standing, more or less complete and in use. The enormous quantity of churches built in the Romanesque period was succeeded by the still busier period of Gothic architecture, which or rebuilt most Romanesque churches in prosperous areas like England and Portugal; the largest groups of Romanesque survivors are in areas that were less prosperous in subsequent periods, including parts of southern France, rural Spain and rural Italy. Survivals of unfortified Romanesque secular houses and palaces, the domestic quarters of monasteries are far rarer, but these used and adapted the features found in church buildings, on a domestic scale. According to the Oxford English Dictionary, the word "Romanesque" means "descended from Roman" and was first used in English to designate what are now called Romance languages; the French term "romane" was first used in the architectural sense by archaeologist Charles de Gerville in a letter of 18 December 1818 to Auguste Le Prévost to describe what Gerville sees as a debased Roman architecture.
In 1824 Gerville's friend Arcisse de Caumont adopted the label "roman" to describe the "degraded" European architecture from the 5th to the 13th centuries, in his Essai sur l'architecture religieuse du moyen-âge, particulièrement en Normandie, at a time when the actual dates of many of the buildings so described had not been ascertained: The name Roman we give to this architecture, which should be universal as it is the same everywhere with slight local differences has the merit of indicating its origin and is not new since it is used to describe the language of the same period. Romance language is degenerated Latin language. Romanesque architecture is debased Roman architecture; the first use in a published work is in William Gunn's An Inquiry into the Origin and Influence of Gothic Architecture. The word was used by Gunn to describe the style, identifiably Medieval and prefigured the Gothic, yet maintained the rounded Roman arch and thus appeared to be a continuation of the Roman tradition of building.
The term is now used for the more restricted period from the late 10th to 12th centuries. The term "Pre-romanesque" is sometimes applied to architecture in Germany of the Carolingian and Ottonian periods and Visigothic and Asturian constructions between the 8th and the 10th centuries in the Iberian Peninsula while "First Romanesque" is applied to buildings in north of Italy and Spain and parts of France that have Romanesque features but pre-date the influence of the Abbey of Cluny. Typical Romanesque architectural forms Buildings of every type were constructed in the Romanesque style, with evidence remaining of simple domestic buildings, elegant town houses, grand palaces, commercial premises, civic buildings, city walls, village churches, abbey churches, abbey complexes and large cathedrals. Of these types of buildings and commercial buildings are the most rare, with only a handful of survivors in the United Kingdom, several clusters in France, isolated buildings across Europe and by far the largest number unidentified and altered over the centuries, in Italy.
Many castles exist, the foundations of. Most have been altered, many are in ruins. By far the greatest number of surviving Romanesque buildings are churches; these range from tiny chapels to large cathedrals. Although many have been extended and altered in different styles, a large number remain either intact or sympathetically restored, demonstrating the form and decoration of Romanesque church architecture; the scope of Romanesque architecture Romanesque architecture was the first distinctive style to spread across Europe since the Roman Empire. With the decline of Rome, Roman building methods survived to an extent in Western Europe, where successive Merovingian and Ottonian architects continued to build large stone buildings such as monastery churches and palaces. In the more northern countries, Roman building styles and techniques had never been adopted except for official buildings, while in Scandinavia they were unknown. Although the round arch continued in use, the engineering skills required to vault large spaces and build large domes were lost.
There was a loss of stylistic continuity apparent in the decline of the formal vocabulary of the Classical Orders. In Rome several great Constantinian basilicas continued in use as an inspiration to builders; some traditions of Rom
A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave, it is possible to approximate a triangle wave with additive synthesis by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 and multiplying the amplitude of the harmonics by one over the square of their mode number, n. The above can be summarised mathematically as follows: x t r i a n g l e = ∑ i = 0 N i n − 2 where N is the number of harmonics to include in the approximation, t is the independent variable, i is the harmonic label, related to its mode number by n = 2 i + 1; this infinite Fourier series converges to the triangle wave as N tends to infinity, as shown in the animation. Another definition of the triangle wave, with range from -1 to 1 and period p, is: x = 4 p ⌊ 2 t p + 1 2 ⌋ where the symbol ⌊ n ⌋ is the floor function of n.
The triangle wave can be the absolute value of the sawtooth wave: x = 2 | t p − ⌊ t p + 1 2 ⌋ | or, for a range from -1 to +1: x = 2 | 2 | − 1. The triangle wave can be expressed as the integral of the square wave: x = ∫ 0 t sgn d u. Here is a simple equation with a period of 4 and initial value y = 1: y = | x mod 4 − 2 | − 1; as this only uses the modulo operation and absolute value, this can be used to implement a triangle wave on hardware electronics with less CPU power. The previous equation can be generalized for a period of p, amplitude a, initial value y = a / 2: y = 2 a p | − p 2 | − 2 a 4; the former function is a specialization of the latter for a=2 and p=4: y = 2 × 2 4 | − 4 2 | − 2 ×
A ruler, sometimes called a rule or line gauge, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure or draw straight lines. Rulers have long been made in multiple sizes; some are wooden. Plastics have been used since they were invented. Metal is used for more durable rulers for use in the workshop. 12 in or 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket. Longer rulers, e.g. 18 in are necessary in some cases. Rigid wooden or plastic yardsticks, 1 yard long, meter sticks, 1 meter long, are used. Classically, long measuring rods were used for larger projects, now superseded by tape measure, surveyor's wheel or laser rangefinders. Desk rulers are used for three main purposes: to measure, to aid in drawing straight lines and as a straight guide for cutting and scoring with a blade. Practical rulers have distance markings along their edges. A line gauge is a type of ruler used in the printing industry.
These may be made from a variety of materials metal or clear plastic. Units of measurement on a basic line gauge include inches, agate and points. More detailed line gauges may contain sample widths of lines, samples of common type in several point sizes, etc. Measuring instruments similar in function to rulers are made portable by folding or retracting into a coil when not in use; when extended for use, they are straight, like a ruler. The illustrations on this page show a 2 m carpenter's rule, which folds down to a length of 25 cm to fit in a pocket, a 5 m tape, which retracts into a small housing. A flexible length measuring instrument, not straight in use is the tailor's fabric tape measure, a length of tape calibrated in inches and centimeters, it is used to measure around a solid body, e.g. a person's waist measurement, as well as linear measurement, e.g. inside leg. It is rolled up. A contraction rule is made having larger divisions than standard measures to allow for shrinkage of a metal casting.
They may be known as a shrinkage or shrink rule. A ruler software program can be used to measure pixels on mobile phone; these programs are known as screen rulers. In geometry, a ruler without any marks on it may be used only for drawing straight lines between points. A straightedge is used to help draw accurate graphs and tables. A ruler and compass construction refers to constructions using a compass, it is possible to bisect an angle into two equal parts with ruler and compass. It can be proved, that it is impossible to divide an angle into three equal parts using only a compass and straightedge — the problem of angle trisection. However, should two marks be allowed on the ruler, the problem becomes solvable. In the history of measurement many distance units have been used which were based on human body parts such as the cubit and foot and these units varied in length by era and location. In the late 18th century the metric system came into use and has been adopted to varying degrees in all countries in the world.
The oldest preserved measuring rod is a copper-alloy bar that dates from c. 2650 BCE and was found by the German Assyriologist Eckhard Unger while excavating at Nippur. Rulers made of Ivory were in use by the Indus Valley Civilization period prior to 1500 BCE. Excavations at Lothal have yielded one such ruler calibrated to about 1⁄16 inch. Ian Whitelaw holds that the Mohenjo-Daro ruler is divided into units corresponding to 1.32 inches and these are marked out in decimal subdivisions with amazing accuracy, to within 0.005 inches. Ancient bricks found throughout the region have dimensions. Anton Ullrich invented the folding ruler in 1851. Frank Hunt made the flexible ruler in 1902; the equivalent of a ruler for drawing or reproducing a smooth curve, where it takes the form of a rigid template, is known as a French curve. A flexible device which can be bent to the desired shape is known as a flat spline, or a flexible curve. A flexible lead rule used by masons that could be bent to the curves of a molding was known as a lesbian rule.
Ludwig Wittgenstein famously used rulers as an example in his discussion of language games in the Philosophical Investigations. He pointed out that the standard meter bar in Paris was the criterion against which all other rulers were determined to be one meter long, but that there was no analytical way to demonstrate that the standard meter bar itself was one meter long, it could only be asserted as one meter as part of a language game. Cherry, Dan. "Collector's guide to rules", Cabinetmaking, no. 259, July 2017, ISSN 1365-4292, pp. 52–6 Rees and Mark. The Rule Book: Measuring for the Trades. Lakeville, MN: Astragal Press ISBN 978-1-931626-26-2 OCLC 907853704 Russell, David R.. Antique Woodworking Tools: Their Craftsmanship from the Earliest Times to the Twentieth Century, Cambridge: John Adamson ISBN 978-1-898565-05-5 OCLC 727125586, pp. 64–74 Whitelaw, Ian. A Measure of All Things: The Story of Man and Measurement. Macmillan ISBN 0-312-37026-1 OCLC 938084552
Byzantine architecture is the architecture of the Byzantine Empire, or Eastern Roman Empire. The Byzantine era is dated from 330 CE, when Constantine the Great moved the Roman capital to Byzantium, which became Constantinople, until the fall of the Byzantine Empire in 1453. However, there was no hard line between the Byzantine and Roman empires, early Byzantine architecture is stylistically and structurally indistinguishable from Roman architecture; this terminology was introduced by modern historians to designate the medieval Roman Empire as it evolved as a distinct artistic and cultural entity centered on the new capital of Constantinople rather than the city of Rome and its environs. Its architecture influenced the medieval architecture throughout Europe and the Near East, became the primary progenitor of the Renaissance and Ottoman architectural traditions that followed its collapse. Early Byzantine architecture drew upon earlier elements of Roman architecture. Stylistic drift, technological advancement, political and territorial changes meant that a distinct style resulted in the Greek cross plan in church architecture.
Buildings increased in geometric complexity and plaster were used in addition to stone in the decoration of important public structures, classical orders were used more mosaics replaced carved decoration, complex domes rested upon massive piers, windows filtered light through thin sheets of alabaster to illuminate interiors. Most of the surviving structures are sacred in nature, with secular buildings known only through contemporaneous descriptions. Prime examples of early Byzantine architecture date from the Emperor Justinian I's reign and survive in Ravenna and Istanbul, as well as in Sofia. One of the great breakthroughs in the history of Western architecture occurred when Justinian's architects invented a complex system providing for a smooth transition from a square plan of the church to a circular dome by means of pendentives. In Ravenna, the longitudinal basilica of Sant'Apollinare Nuovo, the octagonal, centralized structure of the church of San Vitale, commissioned by Emperor Justinian but never seen by him, was built.
Justinian's monuments in Istanbul include the domed churches of Hagia Sophia and Hagia Irene, but there is an earlier, smaller church of Saints Sergius and Bacchus, which might have served as a model for both in that it combined the elements of a longitudinal basilica with those of a centralized building. Secular structures include the ruins of the Great Palace of Constantinople, the innovative walls of Constantinople and Basilica Cistern. A frieze in the Ostrogothic palace in Ravenna depicts an early Byzantine palace. Hagios Demetrios in Thessaloniki, Saint Catherine's Monastery on Mount Sinai, Jvari Monastery in present-day Georgia, three Armenian churches of Echmiadzin all date from the 7th century and provide a glimpse on architectural developments in the Byzantine provinces following the age of Justinian. Remarkable engineering feats include the 430 m long Sangarius Bridge and the pointed arch of Karamagara Bridge; the period of the Macedonian dynasty, traditionally considered the epitome of Byzantine art, has not left a lasting legacy in architecture.
It is presumed that Basil I's votive church of the Theotokos of the Pharos and the Nea Ekklesia served as a model for most cross-in-square sanctuaries of the period, including the Cattolica di Stilo in southern Italy, the monastery church of Hosios Lukas in Greece, Nea Moni of Chios, the Daphni Monastery near Athens. The cross-in-square type became predominant in the Slavic countries which were progressively Christianized by missionaries during the Macedonian period; the Hagia Sophia church in Ochrid, the eponymous cathedral in Kiev testify to a vogue for multiple subsidiary domes set on drums, which would gain in height and narrowness with the progress of time. In Istanbul and Asia Minor the architecture of the Komnenian period is non-existent, with the notable exceptions of the Elmali Kilise and other rock sanctuaries of Cappadocia, of the Churches of the Pantokrator and of the Theotokos Kyriotissa in Istanbul. Most examples of this architectural style and many of the other older Byzantine styles only survive on the outskirts of the Byzantine world, as most of the most significant and ancient churches/ buildings were in Asia Minor, but in World War I all churches that ended up within Turkish borders were destroyed,converted into mosques, or abandoned in the Greek and Christian genocides spanning from 1915–1923.
Only national forms of architecture can be found in abundance due to this. Those styles can be found in many Transcaucasian countries; the Paleologan period is well represented in a dozen former churches in Istanbul, notably St Saviour at Chora and St Mary Pammakaristos. Unlike their Slavic counterparts, the Paleologan architects never accented the vertical thrust of structures; as a result, there is little grandeur in the late medieval architecture of Byzantium. The Church of the Holy Apostles is cited as an archetypal structure of the late period, when the exterior walls were intricately decorated with complex brickwork patterns or with glazed ceramics. Other churches from the years predati