title: Jacobian matrix and determinant

text: formula_8. Specializing further, when , that is when is a scalar-valued function of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function .


title: Jacobian matrix and determinant

text: The Jacobian of a vector-valued function in several variables generalizes the gradient of a scalar-valued function in several variables, which in turn generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian matrix of a scalar-valued function in several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a single variable is its derivative.


title: Jacobian matrix and determinant

text: At each point where a function is differentiable, its Jacobian matrix can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that the function imposes locally near that point. For example, if is used to smoothly transform an image, the Jacobian matrix , describes how the image in the neighborhood of is transformed.


title: Jacobian matrix and determinant

text: If a function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However a function does not need to be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist.


title: Jacobian matrix and determinant

text: If is differentiable at a point in , then its differential is represented by . In this case, the linear transformation represented by is the best linear approximation of near the point , in the sense that


title: Jacobian matrix and determinant

text: where is a quantity that approaches zero much faster than the distance between and does as approaches . This approximation specializes to the approximation of a scalar function of a single variable by its Taylor polynomial of degree one, namely


title: Jacobian matrix and determinant

text: In this sense, the Jacobian may be regarded as a kind of "first-order derivative" of a vector-valued function of several variables. In particular, this means that the gradient of a scalar-valued function of several variables may too be regarded as its "first-order derivative".


title: Jacobian matrix and determinant

text: The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian matrix, which in a sense is the "second derivative" of the function in question.


title: Jacobian matrix and determinant

text: If , then is a function from to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jacobian".


title: Jacobian matrix and determinant

text: The Jacobian determinant at a given point gives important information about the behavior of near that point. For instance, the continuously differentiable function is invertible near a point if the Jacobian determinant at is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at is positive, then preserves orientation near ; if it is negative, reverses orientation. The absolute value of the Jacobian determinant at gives us the factor by which the function expands or shrinks volumes near ; this is why it occurs in the general substitution rule.


title: Jacobian matrix and determinant

text: The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. This is because the -dimensional element is in general a parallelepiped in the new coordinate system, and the -volume of a parallelepiped is the determinant of its edge vectors.


title: Jacobian matrix and determinant

text: The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point. Its applications include determining the stability of the disease-free equilibrium in disease modelling.


title: Jacobian matrix and determinant

text: According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the "inverse" function. That is, if the Jacobian of the function is continuous and nonsingular at the point in , then is invertible when restricted to some neighborhood of and


title: Jacobian matrix and determinant

text: In other words, if the Jacobian determinant is not zero at a point, then the function is "locally invertible" near this point, that is, there is a neighbourhood of this point in which the function is invertible.


title: Jacobian matrix and determinant

text: The (unproved) Jacobian conjecture is related to global invertibility in the case of a polynomial function, that is a function defined by "n" polynomials in "n" variables. It asserts that, if the Jacobian determinant is a non-zero constant (or, equivalently, that it does not have any complex zero), then the function is invertible and its inverse is a polynomial function.


title: Jacobian matrix and determinant

text: If is a differentiable function, a "critical point" of is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point. In other words, let be the maximal dimension of the open balls contained in the image of ; then a point is critical if all minors of rank of are zero.


title: Jacobian matrix and determinant

text: The transformation from polar coordinates to Cartesian coordinates ("x", "y"), is given by the function with components:


title: Jacobian matrix and determinant

text: The Jacobian determinant is equal to . This can be used to transform integrals between the two coordinate systems:


title: Jacobian matrix and determinant

text: The transformation from spherical coordinates to Cartesian coordinates ("x", "y", "z"), is given by the function with components:


title: Jacobian matrix and determinant

text: The determinant is . Since is the volume for a rectangular differential volume element (because the volume of a rectangular prism is the product of its sides), we can interpret as the volume of the spherical differential volume element. Unlike rectangular differential volume element's volume, this differential volume element's volume is not a constant, and varies with coordinates ( and ). It can be used to transform integrals between the two coordinate systems:


title: Jacobian matrix and determinant

text: From this we see that reverses orientation near those points where and have the same sign; the function is locally invertible everywhere except near points where or . Intuitively, if one starts with a tiny object around the point and apply to that object, one will get a resulting object with approximately times the volume of the original one, with orientation reversed.


title: Jacobian matrix and determinant

text: The Jacobian serves as a linearized design matrix in statistical regression and curve fitting; see non-linear least squares.


title: Jacobian matrix and determinant

text: Consider a dynamical system of the form formula_28, where formula_29 is the (component-wise) derivative of formula_30 with respect to the evolution parameter formula_31 (time), and formula_32 is differentiable. If formula_33, then formula_34 is a stationary point (also called a steady state). By the Hartman–Grobman theorem, the behavior of the system near a stationary point is related to the eigenvalues of formula_35, the Jacobian of formula_36 at the stationary point. Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point, if any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.


title: Jacobian matrix and determinant

text: A square system of coupled nonlinear equations can be solved iteratively by Newton's method. This method uses the Jacobian matrix of the system of equations.


title: Siddhant Chaturvedi

text: Siddhant Chaturvedi (born 29 April 1993) is an Indian actor who appears in Hindi films. He starred as a teenage cricketer in the Amazon Prime Video web series "Inside Edge" from 2017 to 2019 and ventured into films with the supporting role of a street rapper in the musical drama "Gully Boy" (2019), which won him the Filmfare Award for Best Supporting Actor. He has since starred in the romantic drama "Gehraiyaan" (2022).


title: Siddhant Chaturvedi

text: Chaturvedi was born on 29 April 1993 in Ballia, Uttar Pradesh and moved to Mumbai when he was five years old. His father is a chartered accountant and his mother is a homemaker. He attended Mithibai college in Mumbai and initially aspired to be a chartered accountant (CA), while acting on stage as a hobby. While pursuing his CA articleship, he became disillusioned and took some time off, during which he participated in and won "The Times of India" Fresh Face Contest in 2013.


title: Siddhant Chaturvedi

text: In 2016, Chaturvedi appeared in the web television sitcom "Life Sahi Hai", about four male roommates. Next year, he began starring as Prashant Kanaujia, a teenage cricketer, in the Amazon Prime Video web series "Inside Edge", which was inspired by the Indian Premier League. Reviewing the first season for Scroll.in, Devarsi Ghosh considered Chaturvedi and his co-star Tanuj Virwani to be the show's "two big acting discoveries". He later reprised his role for the second season in 2019.


title: Siddhant Chaturvedi

text: At a success party of "Inside Edge", filmmaker Zoya Akhtar, whose brother Farhan Akhtar produced the show, spotted and asked Chaturvedi to audition for her next directorial "Gully Boy" (2019), which marked his film debut and proved to be his breakout role. Starring Ranveer Singh and Alia Bhatt, the film saw Chaturvedi play the supporting role of a street rapper named MC Sher, who fuels the ambitions of Ranveer Singh's titular character. Jay Weissberg of "Variety" found Chaturvedi to be "impressive" in his part, and Rajeev Masand commended his "charming presence". At the annual Filmfare Awards ceremony, he was awarded with the Filmfare Award for Best Supporting Actor in addition to a nomination for Best Male Debut.


title: Siddhant Chaturvedi

text: Chaturvedi next appeared in Yash Raj Films's "Bunty Aur Babli 2" (2021), co-starring Rani Mukerji, Saif Ali Khan and Sharvari Wagh, a sequel to the crime comedy "Bunty Aur Babli" (2005). It emerged as a critical and commercial failure. He had a lead role in Shakun Batra's "Gehraiyaan" (2022), a drama about infidelity, co-starring Deepika Padukone and Ananya Panday. It was released on Amazon Prime Video. Sukanya Verma of Rediff.com found his portrayal of a "tricky part" to be "compelling". He then featured in the comedy horror film "Phone Bhoot" with Katrina Kaif and Ishaan Khatter. Reviewers were generally unimpressed with the picture, but Saibal Chatterjee of NDTV opined that Chaturvedi had played his "brash and quick-on-the-draw" character well.


title: Siddhant Chaturvedi

text: Chaturvedi will next star in the action film "Yudhra" with Malavika Mohanan, and with Ananya Panday and Adarsh Gourav in "Kho Gaye Hum Kahan".


title: M. C. Escher

text: Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints.


title: M. C. Escher

text: Despite wide popular interest, Escher was for most of his life neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held. In the late twentieth century, he became more widely appreciated, and in the twenty-first century he has been celebrated in exhibitions around the world.


title: M. C. Escher

text: His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographer Friedrich Haag, and conducted his own research into tessellation.


title: M. C. Escher

text: Early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. He traveled in Italy and Spain, sketching buildings, townscapes, architecture and the tilings of the Alhambra and the Mezquita of Cordoba, and became steadily more interested in their mathematical structure.


title: M. C. Escher

text: Escher's art became well known among scientists and mathematicians, and in popular culture, especially after it was featured by Martin Gardner in his April 1966 Mathematical Games column in "Scientific American". Apart from being used in a variety of technical papers, his work has appeared on the covers of many books and albums. He was one of the major inspirations of Douglas Hofstadter's Pulitzer Prize-winning 1979 book "Gödel, Escher, Bach".


title: M. C. Escher

text: Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, Friesland, the Netherlands, in a house that forms part of the Princessehof Ceramics Museum today. He was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the family moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as "Mauk", he was a sickly child and was placed in a special school at the age of seven; he failed the second grade. Although he excelled at drawing, his grades were generally poor. He took carpentry and piano lessons until he was thirteen years old.


title: M. C. Escher

text: In 1918, he went to the Technical College of Delft. From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and the art of making woodcuts. He briefly studied architecture, but he failed a number of subjects (due partly to a persistent skin infection) and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita.


title: M. C. Escher

text: In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, and Ravello. In the same year, he traveled through Spain, visiting Madrid, Toledo, and Granada. He was impressed by the Italian countryside and, in Granada, by the Moorish architecture of the fourteenth-century Alhambra. The intricate decorative designs of the Alhambra, based on geometrical symmetries featuring interlocking repetitive patterns in the coloured tiles or sculpted into the walls and ceilings, triggered his interest in the mathematics of tessellation and became a powerful influence on his work.


title: M. C. Escher

text: Escher returned to Italy and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, like himself attracted to Italy – whom he married in 1924. The couple settled in Rome where their first son, Giorgio (George) Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two more sons – Arthur and Jan.


title: M. C. Escher

text: He travelled frequently, visiting (among other places) Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the Amalfi coast in 1931 and 1934, and Gargano and Sicily in 1932 and 1935. The townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra and spending days at a time making detailed drawings of its mosaic patterns. It was here that he became fascinated, to the point of obsession, with tessellation, explaining:


title: M. C. Escher

text: The sketches he made in the Alhambra formed a major source for his work from that time on. He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba. This turned out to be the last of his long study journeys; after 1937, his artworks were created in his studio rather than in the field. His art correspondingly changed sharply from being mainly observational, with a strong emphasis on the realistic details of things seen in nature and architecture, to being the product of his geometric analysis and his visual imagination. All the same, even his early work already shows his interest in the nature of space, the unusual, perspective, and multiple points of view.


title: M. C. Escher

text: In 1935, the political climate in Italy under Mussolini became unacceptable to Escher. He had no interest in politics, finding it impossible to involve himself with any ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, George, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to Château-d'Œx, Switzerland, where they remained for two years.


title: M. C. Escher

text: The Netherlands post office had Escher design a semi-postal stamp for the "Air Fund" (Dutch: "Het Nationaal Luchtvaartfonds") in 1935, and again in 1949 he designed Dutch stamps. These were for the 75th anniversary of the Universal Postal Union; a different design was used by Suriname and the Netherlands Antilles for the same commemoration.


title: M. C. Escher

text: Escher, who had been very fond of and inspired by the landscapes in Italy, was decidedly unhappy in Switzerland. In 1937 the family moved again, to Uccle (Ukkel), a suburb of Brussels, Belgium. World War II forced them to move in January 1941, this time to Baarn, Netherlands, where Escher lived until 1970. Most of Escher's best-known works date from this period. The sometimes cloudy, cold, and wet weather of the Netherlands allowed him to focus intently on his work. After 1953, Escher lectured widely. A planned series of lectures in North America in 1962 was cancelled after an illness, and he stopped creating artworks for a time, but the illustrations and text for the lectures were later published as part of the book "Escher on Escher". He was awarded the Knighthood of the Order of Orange-Nassau in 1955; he was later made an Officer in 1967.


title: M. C. Escher

text: In July 1969 he finished his last work, a large woodcut with threefold rotational symmetry called "Snakes", in which snakes wind through a pattern of linked rings. These shrink to infinity toward both the center and the edge of a circle. It was exceptionally elaborate, being printed using three blocks, each rotated three times about the center of the image and precisely aligned to avoid gaps and overlaps, for a total of nine print operations for each finished print. The image encapsulates Escher's love of symmetry; of interlocking patterns; and, at the end of his life, of his approach to infinity. The care that Escher took in creating and printing this woodcut can be seen in a video recording.


title: M. C. Escher

text: Escher moved to the Rosa Spier Huis in Laren in 1970, an artists' retirement home in which he had his own studio. He died in a hospital in Hilversum on 27 March 1972, aged 73. He is buried at the New Cemetery in Baarn.


title: M. C. Escher

text: Escher's work is inescapably mathematical. This has caused a disconnect between his full-on popular fame and the lack of esteem with which he has been viewed in the art world. His originality and mastery of graphic techniques are respected, but his works have been thought too intellectual and insufficiently lyrical. Movements such as conceptual art have, to a degree, reversed the art world's attitude to intellectuality and lyricism, but this did not rehabilitate Escher, because traditional critics still disliked his narrative themes and his use of perspective. However, these same qualities made his work highly attractive to the public.


title: M. C. Escher

text: Escher is not the first artist to explore mathematical themes: Parmigianino (1503–1540) had explored spherical geometry and reflection in his 1524 "Self-portrait in a Convex Mirror", depicting his own image in a curved mirror, while William Hogarth's 1754 "Satire on False Perspective" foreshadows Escher's playful exploration of errors in perspective. Another early artistic forerunner is Giovanni Battista Piranesi (1720–1778), whose dark "fantastical" prints such as "The Drawbridge" in his "Carceri" ("Prisons") sequence depict perspectives of complex architecture with many stairs and ramps, peopled by walking figures. Escher greatly admired Piranesi and had several of Piranesi's prints hanging in his studio.


title: M. C. Escher

text: Only with 20th century movements such as Cubism, De Stijl, Dadaism, and Surrealism did mainstream art start to explore Escher-like ways of looking at the world with multiple simultaneous viewpoints. However, although Escher had much in common with, for example, Magritte's surrealism and Op art, he did not make contact with any of these movements.


title: M. C. Escher

text: In his early years Escher sketched landscapes and nature. He also sketched insects such as ants, bees, grasshoppers, and mantises, which appeared frequently in his later work. His early love of Roman and Italian landscapes and of nature created an interest in tessellation, which he called "Regular Division of the Plane"; this became the title of his 1958 book, complete with reproductions of a series of woodcuts based on tessellations of the plane, in which he described the systematic buildup of mathematical designs in his artworks. He wrote, "Mathematicians have opened the gate leading to an extensive domain".


title: M. C. Escher

text: After his 1936 journey to the Alhambra and to La Mezquita, Cordoba, where he sketched the Moorish architecture and the tessellated mosaic decorations, Escher began to explore the properties and possibilities of tessellation using geometric grids as the basis for his sketches. He then extended these to form complex interlocking designs, for example with animals such as birds, fish, and reptiles. One of his first attempts at a tessellation was his pencil, India ink, and watercolour "Study of Regular Division of the Plane with Reptiles" (1939), constructed on a hexagonal grid. The heads of the red, green, and white reptiles meet at a vertex; the tails, legs, and sides of the animals interlock exactly. It was used as the basis for his 1943 lithograph "Reptiles".


title: M. C. Escher

text: His first study of mathematics began with papers by George Pólya and by the crystallographer Friedrich Haag on plane symmetry groups, sent to him by his brother Berend, a geologist. He carefully studied the 17 canonical wallpaper groups and created periodic tilings with 43 drawings of different types of symmetry. From this point on, he developed a mathematical approach to expressions of symmetry in his artworks using his own notation. Starting in 1937, he created woodcuts based on the 17 groups. His "Metamorphosis I" (1937) began a series of designs that told a story through the use of pictures. In "Metamorphosis I", he transformed convex polygons into regular patterns in a plane to form a human motif. He extended the approach in his piece "Metamorphosis III", which is almost seven metres long.


title: M. C. Escher

text: In 1941 and 1942 Escher summarised his findings for his own artistic use in a sketchbook, which he labeled (following Haag) "Regelmatige vlakverdeling in asymmetrische congruente veelhoeken" ("Regular division of the plane with asymmetric congruent polygons"). The mathematician Doris Schattschneider unequivocally described this notebook as recording "a methodical investigation that can only be termed mathematical research." She defined the research questions he was following as


title: M. C. Escher

text: Although Escher did not have mathematical training – his understanding of mathematics was largely visual and intuitive – his art had a strong mathematical component, and several of the worlds that he drew were built around impossible objects. After 1924 Escher turned to sketching landscapes in Italy and Corsica with irregular perspectives that are impossible in natural form. His first print of an impossible reality was "Still Life and Street" (1937); impossible stairs and multiple visual and gravitational perspectives feature in popular works such as "Relativity" (1953). "House of Stairs" (1951) attracted the interest of the mathematician Roger Penrose and his father, the biologist Lionel Penrose. In 1956, they published a paper, "Impossible Objects: A Special Type of Visual Illusion" and later sent Escher a copy. Escher replied, admiring the Penroses' continuously rising flights of steps, and enclosed a print of "Ascending and Descending" (1960). The paper also contained the tribar or Penrose triangle, which Escher used repeatedly in his lithograph of a building that appears to function as a perpetual motion machine, "Waterfall" (1961).


title: M. C. Escher

text: Escher was interested enough in Hieronymus Bosch's 1500 triptych "The Garden of Earthly Delights" to re-create part of its right-hand panel, "Hell", as a lithograph in 1935. He reused the figure of a Mediaeval woman in a two-pointed headdress and a long gown in his lithograph "Belvedere" in 1958; the image is, like many of his other "extraordinary invented places", peopled with "jesters, knaves, and contemplators". Thus, Escher not only was interested in possible or impossible geometry but was, in his own words, a "reality enthusiast"; he combined "formal astonishment with a vivid and idiosyncratic vision".


title: M. C. Escher

text: Escher worked primarily in the media of lithographs and woodcuts, although the few mezzotints he made are considered to be masterpieces of the technique. In his graphic art, he portrayed mathematical relationships among shapes, figures, and space. Integrated into his prints were mirror images of cones, spheres, cubes, rings, and spirals.


title: M. C. Escher

text: Escher was also fascinated by mathematical objects such as the Möbius strip, which has only one surface. His wood engraving "Möbius Strip II" (1963) depicts a chain of ants marching forever over what, at any one place, are the two opposite faces of the object—which are seen on inspection to be parts of the strip's single surface. In Escher's own words:


title: M. C. Escher

text: The mathematical influence in his work became prominent after 1936, when, having boldly asked the Adria Shipping Company if he could sail with them as travelling artist in return for making drawings of their ships, they surprisingly agreed, and he sailed the Mediterranean, becoming interested in order and symmetry. Escher described this journey, including his repeat visit to the Alhambra, as "the richest source of inspiration I have ever tapped".


title: M. C. Escher

text: Escher's interest in curvilinear perspective was encouraged by his friend and "kindred spirit", the art historian and artist Albert Flocon, in another example of constructive mutual influence. Flocon identified Escher as a "thinking artist" alongside Piero della Francesca, Leonardo da Vinci, Albrecht Dürer, Wenzel Jamnitzer, Abraham Bosse, Girard Desargues, and Père Nicon. Flocon was delighted by Escher's "Grafiek en tekeningen" ("Graphics in Drawing"), which he read in 1959. This stimulated Flocon and André Barre to correspond with Escher and to write the book "La Perspective curviligne" ("Curvilinear perspective").


title: M. C. Escher

text: Escher often incorporated three-dimensional objects such as the Platonic solids such as spheres, tetrahedrons, and cubes into his works, as well as mathematical objects such as cylinders and stellated polyhedra. In the print "Reptiles", he combined two- and three-dimensional images. In one of his papers, Escher emphasized the importance of dimensionality:


title: M. C. Escher

text: Escher's artwork is especially well-liked by mathematicians such as Doris Schattschneider and scientists such as Roger Penrose, who enjoy his use of polyhedra and geometric distortions. For example, in "Gravitation", animals climb around a stellated dodecahedron.


title: M. C. Escher

text: The two towers of "Waterfall" impossible building are topped with compound polyhedra, one a compound of three cubes, the other a stellated rhombic dodecahedron now known as Escher's solid. Escher had used this solid in his 1948 woodcut "Stars", which also contains all five of the Platonic solids and various stellated solids, representing stars; the central solid is animated by chameleons climbing through the frame as it whirls in space. Escher possessed a 6 cm refracting telescope and was a keen-enough amateur astronomer to have recorded observations of binary stars.


title: M. C. Escher

text: Escher's artistic expression was created from images in his mind, rather than directly from observations and travels to other countries. His interest in the multiple levels of reality in art is seen in works such as "Drawing Hands" (1948), where two hands are shown, each drawing the other. The critic Steven Poole commented that


title: M. C. Escher

text: In 1954 the International Congress of Mathematicians met in Amsterdam, and N. G. de Bruin organised a display of Escher's work at the Stedelijk Museum for the participants. Both Roger Penrose and H. S. M. Coxeter were deeply impressed with Escher's intuitive grasp of mathematics. Inspired by "Relativity", Penrose devised his tribar, and his father, Lionel Penrose, devised an endless staircase. Roger Penrose sent sketches of both objects to Escher, and the cycle of invention was closed when Escher then created the perpetual motion machine of "Waterfall" and the endless march of the monk-figures of "Ascending and Descending".


title: M. C. Escher

text: In 1957 Coxeter obtained Escher's permission to use two of his drawings in his paper "Crystal symmetry and its generalizations". He sent Escher a copy of the paper; Escher recorded that Coxeter's figure of a hyperbolic tessellation "gave me quite a shock": the infinite regular repetition of the tiles in the hyperbolic plane, growing rapidly smaller towards the edge of the circle, was precisely what he wanted to allow him to represent infinity on a two-dimensional plane.


title: M. C. Escher

text: Escher carefully studied Coxeter's figure, marking it up to analyse the successively smaller circles with which (he deduced) it had been constructed. He then constructed a diagram, which he sent to Coxeter, showing his analysis; Coxeter confirmed it was correct, but disappointed Escher with his highly technical reply. All the same, Escher persisted with hyperbolic tiling, which he called "Coxetering". Among the results were the series of wood engravings "Circle Limit I–IV". In 1959, Coxeter published his finding that these works were extraordinarily accurate: "Escher got it absolutely right to the millimeter".


title: M. C. Escher

text: Escher's special way of thinking and rich graphics have had a continuous influence in mathematics and art, as well as in popular culture.


title: M. C. Escher

text: The Escher intellectual property is controlled by the M.C. Escher Company, while exhibitions of his artworks are managed separately by the M.C. Escher Foundation.


title: M. C. Escher

text: The primary institutional collections of original works by M.C. Escher are the Escher Museum in The Hague; the National Gallery of Art (Washington, DC); the National Gallery of Canada (Ottawa); the Israel Museum (Jerusalem); and the Huis ten Bosch (Nagasaki, Japan).


title: M. C. Escher

text: Despite wide popular interest, Escher was for a long time somewhat neglected in the art world; even in his native Netherlands, he was 70 before a retrospective exhibition was held. In the twenty-first century, major exhibitions have been held in cities around the world. An exhibition of his work in Rio de Janeiro attracted more than 573,000 visitors in 2011; its daily visitor count of 9,677 made it the most visited museum exhibition of the year, anywhere in the world. No major exhibition of Escher's work was held in Britain until 2015, when the Scottish National Gallery of Modern Art ran one in Edinburgh from June to September 2015, moving in October 2015 to the Dulwich Picture Gallery, London. The exhibition poster is based on "Hand with Reflecting Sphere", 1935, which shows Escher in his house reflected in a handheld sphere, thus illustrating the artist, his interest in levels of reality in art (e.g., is the hand in the foreground more real than the reflected one?), perspective, and spherical geometry. The exhibition moved to Italy in 2015–2016, attracting over 500,000 visitors in Rome and Bologna, and then Milan.


title: M. C. Escher

text: Doris Schattschneider identifies eleven strands of mathematical and scientific research anticipated or directly inspired by Escher. These are the classification of regular tilings using the edge relationships of tiles: two-color and two-motif tilings (counterchange symmetry or antisymmetry); color symmetry (in crystallography); metamorphosis or topological change; covering surfaces with symmetric patterns; Escher's algorithm (for generating patterns using decorated squares); creating tile shapes; local versus global definitions of regularity; symmetry of a tiling induced by the symmetry of a tile; orderliness not induced by symmetry groups; the filling of the central void in Escher's lithograph "Print Gallery" by H. Lenstra and B. de Smit.


title: M. C. Escher

text: The Pulitzer Prize-winning 1979 book "Gödel, Escher, Bach" by Douglas Hofstadter discusses the ideas of self-reference and strange loops expressed in Escher's art. The asteroid 4444 Escher was named in Escher's honor in 1985.


title: M. C. Escher

text: Escher's fame in popular culture grew when his work was featured by Martin Gardner in his April 1966 "Mathematical Games" column in "Scientific American". Escher's works have appeared on many album covers including The Scaffold's 1969 "L the P" with "Ascending and Descending"; Mott the Hoople's eponymous 1969 record with "Reptiles", Beaver & Krause's 1970 "In A Wild Sanctuary" with "Three Worlds"; and Mandrake Memorial's 1970 "Puzzle" with "House of Stairs" and (inside) "Curl Up". His works have similarly been used on many book covers, including some editions of Edwin Abbott's "Flatland", which used "Three Spheres"; E. H. Gombrich's "Meditations on a Hobby Horse" with "Horseman"; Pamela Hall's "Heads You Lose" with "Plane Filling 1"; Patrick A. Horton's "Mastering the Power of Story" with "Drawing Hands"; Erich Gamma et al.'s "Design Patterns: Elements of Reusable Object-oriented software" with "Swans"; and Arthur Markman's "Knowledge Representation" with "Reptiles". The "World of Escher" markets posters, neckties, T-shirts, and jigsaw puzzles of Escher's artworks. Both Austria and the Netherlands have issued postage stamps commemorating the artist and his works.


title: Logan Lucky

text: Logan Lucky is a 2017 American heist comedy film directed by Steven Soderbergh, based on a screenplay credited to Rebecca Blunt. Soderbergh came out of retirement to direct the film and distributed it independently through his own company, Fingerprint Releasing. The film features an ensemble cast consisting of Channing Tatum, Adam Driver, Riley Keough, Daniel Craig, Seth MacFarlane, Katie Holmes, Hilary Swank, Katherine Waterston, and Sebastian Stan. The film follows the Logan family and their attempt to rob the Charlotte Motor Speedway whilst avoiding security officers and the FBI.


title: Logan Lucky

text: "Logan Lucky" premiered in Knoxville, Tennessee on August 9, 2017, and was theatrically released in the United States on August 18, 2017 by Bleecker Street. The film received positive reviews, with many critics praising the cast's performances and Soderbergh's direction, and grossed $48 million worldwide.


title: Logan Lucky

text: West Virginian Jimmy Logan is laid off from his construction job in the tunnels underneath the Charlotte Motor Speedway. He is close to his daughter, Sadie, who lives with her mom, Bobbie Jo, and her new, rich husband, Moody. Sadie performs in beauty pageants, and Jimmy's hair stylist sister, Mellie, does her hair. Bobbie Jo and Moody plan on moving to Lynchburg as Moody plans to open up a new dealership.


title: Logan Lucky

text: Jimmy goes to a bar tended by his brother Clyde, who lost his lower arm fighting in Iraq. A pretentious NASCAR-team owner, Max Chilblain, makes fun of Clyde's disability, leading to a fight with Jimmy, while Clyde sets fire to Max's car. The next day, Jimmy tells Clyde his plan to rob the Speedway, exploiting his knowledge of its underground pneumatic tube system for moving its vast amount of money.


title: Logan Lucky

text: Clyde and Jimmy recruit their sister Mellie, incarcerated safe-cracker Joe Bang, and Joe's dimwitted brothers, Sam and Fish. They will break Joe out of prison and return him later that day. Clyde intentionally gets himself sent to prison on a minor charge. With the unknowing help of a woman working at the vault, Mellie, Sam, and Fish infest the Speedway's tube system with painted cockroaches, determining which tubes go to the vault. While gathering supplies, Jimmy meets former schoolmate Sylvia, who gives him a tetanus shot inside her mobile clinic, which is in need of donations. Jimmy learns that construction at the speedway is being completed ahead of schedule, forcing them to mount the heist a week earlier, during the much busier Coca-Cola 600 race on Memorial Day weekend.


title: Logan Lucky

text: Joe arranges for fellow inmates to stage a riot, resulting in a lockdown that hides his and Clyde's escape. Mellie meets them in a Ford Shelby GT350 stolen from Moody. At the speedway, Sam and Fish blow up a cell tower to disable the credit card machines, forcing vendors to accept cash only. Clyde, Jimmy and Joe enter the tube room, and Joe uses an explosive improvised from bleach, gummy bears and a dietary salt substitute to blow open the gate from the tube to the vault. They connect the tube to a giant vacuum pump, and suck up all the cash into garbage bags. Security guards investigate the smoke from the explosion, but are diverted by Earl, one of Clyde's bar patrons. Jimmy accidentally reverses the direction of the vacuum, sucking Clyde's prosthetic arm into the machine. Sam and Fish get the money out and into Jimmy's truck. On his way out, Clyde is recognized by Chilblain and hits him, then Clyde and Joe sneak back into the prison disguised as firefighters. Jimmy, having retrieved Clyde’s arm from the machine, makes it to his daughter's pageant just in time to see her perform his favorite song, "Take Me Home, Country Roads." Jimmy then abandons the money and anonymously informs the police of its location.


title: Logan Lucky

text: FBI agent Sarah Grayson investigates the heist, dubbed “Ocean's Seven-Eleven" in the press. While she suspects Jimmy, Clyde and Joe, she has no evidence. Due to the unwillingness of the prison warden to report the riot, the refuting of Chilblain's eyewitness account by his disgruntled and bitter race driver, Dayton White, and the speedway president's satisfaction with the money's recovery and insurance settlement, the case is closed after six months. When Joe returns home after prison he finds a garbage bag of money buried in his yard. Sylvia receives an anonymous donation for her clinic, the inmate who orchestrated the prison riot and the woman working at the vault each receive an envelope stuffed with cash.


title: Logan Lucky

text: Unknown to Joe and his brothers, Jimmy had filled extra trash bags during the heist, which he hid at the dump to retrieve later. Now working in a Lowe's store and having bought a house near his daughter and ex-wife's home, Jimmy happily reunites with Clyde, Mellie, Joe, and Sylvia at the bar. The group is watched by Grayson, who tells Clyde that she is new to the area but thinks she will stay for a while.


title: Logan Lucky

text: Six other NASCAR drivers make cameo appearances in the film. Carl Edwards and Kyle Busch are West Virginia state troopers, Brad Keselowski and Joey Logano are security guards, Kyle Larson is a limousine driver, and Ryan Blaney is a delivery boy. LeAnn Rimes and Jesco White also make cameo appearances.


title: Logan Lucky

text: Soderbergh's "Behind the Candelabra" (2013) was intended to be his final film as director. Soderbergh has said that he was initially given the "Logan Lucky" script in hopes that he could recommend a suitable director for the project, but he enjoyed reading it and decided to take it on for himself, specifically noting that it was an "anti-glam version of an "Ocean's" movie". He told "Entertainment Weekly", "Nobody dresses nice. Nobody has nice stuff. They have no money. They have no technology. It's all rubber band technology." At the time, Soderbergh was also theorizing a new distribution model and felt that the script gave him the perfect opportunity to do so.


title: Logan Lucky

text: The film was announced in February 2016, along with Channing Tatum's casting. "Variety" initially reported that Matt Damon was also to star, although "Deadline Hollywood" reported the same day that Damon was not involved, but rather Michael Shannon. Shannon later left the project due to scheduling conflicts. Much of the rest of the cast was announced between that May and August, among them Seth MacFarlane, Adam Driver and Daniel Craig in May, Hilary Swank in June, and Jim O'Heir in August.


title: Logan Lucky

text: There has been media speculation regarding the identity of the film's screenwriter, Rebecca Blunt. The film's production notes state that she is a native of Logan, West Virginia who now lives in New York City, and that she is a first-time screenwriter. No one other than Soderbergh and Adam Driver has personally vouched for her existence. Some people involved with the film have exchanged emails with a person they believed to be her, and think that she lives in the United Kingdom. Sources speculated that "Rebecca Blunt" was actually a pseudonym for Soderbergh's wife Jules Asner, for comedian John Henson, or for Soderbergh himself. According to "The Playlist", Asner was the sole screenwriter and chose to use a pseudonym because she did not want the perception of the film to be that "Soderbergh was directing his wife's script." Soderbergh denied the claim that Rebecca Blunt was not a real person.


title: Logan Lucky

text: "Logan Lucky" began filming on August 24, 2016. Production lasted 36 days, with much of the photography taking place at the Charlotte Motor Speedway and Atlanta Motor Speedway during the NASCAR race meetings at both circuits. The Atlanta outside barriers were repainted yellow, in an attempt to resemble Charlotte; the Charlotte track had yellow walls owing to then NASCAR's premiership title sponsor Sprint (they have since been replaced with white walls); Atlanta has white walls with red and blue pattern for Folds of Honor, the charity that is involved with (now spring) race sponsor QuikTrip. The scenes with the Fox commentators inside the commentary box were shot in Atlanta, because in Charlotte, the commentators are typically positioned on top of the grandstand roof for opening broadcast scenes and also since the road course can be seen from the Charlotte commentary box.


title: Logan Lucky

text: David Holmes composed the score for "Logan Lucky", having previously scored other Soderbergh films, including "Out of Sight", the "Ocean's Trilogy", and "Haywire". The soundtrack was released by Milan Records; it includes the cue "Original Score Medley" by David Holmes, and music by various artists. The film features "Flashing Lights" by Screaming Lord Sutch and Heavy Friends.


title: Logan Lucky

text: Fingerprint Releasing and Bleecker Street released the film on August 18, 2017. "Logan Lucky" is the first film distributed by Fingerprint Releasing, which Soderbergh created to distribute films independently, instead of through big studios. The distributors spent $20 million on prints and advertising.


title: Logan Lucky

text: "Logan Lucky" grossed $27.8 million in the United States and Canada, and $19.8 million in other territories, for a worldwide total of $47.5 million, against a production budget of $29 million.


title: Logan Lucky

text: In North America, the film was released alongside "The Hitman's Bodyguard" as well as the wide expansion of "Wind River", and was projected to gross $7–9 million from 3,008 theaters in its opening weekend. Having already covered the cost of the production budget through international advance sales, and the costs of prints and advertising through a deal with Amazon, Soderbergh said a debut of $15 million would be needed to be considered a success. The film grossed $2.8 million on its first day (including $525,000 from Thursday night previews). It went on to open to $7.6 million, finishing third at the box office, behind "The Hitman's Bodyguard" and "". In its second weekend, the film took in $4.2 million, dropping 44.2% and finishing 5th, in what was the lowest combined grossing weekend (for the top 10 films) since September 2001. During the following weekend, which preceded Labor Day, the film made $4.4 million over three days (an increase of 4%), and an estimated $5.5 million over the four-day weekend. It was again part of a historically low weekend, as it was the worst combined holiday weekend since 1998.


title: Logan Lucky

text: On review aggregation website Rotten Tomatoes, the film holds an approval rating of 92% based on reviews from 290 critics, with an average rating of 7.5/10. The website's critical consensus reads, "High-octane fun that's smartly assembled without putting on airs, "Logan Lucky" marks a welcome end to Steven Soderbergh's retirement — and proves he hasn't lost his ability to entertain." On Metacritic, which assigns a weighted average rating to reviews, the film has a score of 78 out of 100, based on reviews from 51 critics, indicating "generally favorable reviews". Audiences polled by CinemaScore gave the film an average grade of "B" on an A+ to F scale, while PostTrak reported filmgoers gave it an 80% overall positive score.


title: Logan Lucky

text: Owen Gleiberman of "Variety" gave the film a positive review, writing, "Steven Soderbergh's "Logan Lucky" is a high-spirited, low-down blast." Todd McCarthy of "The Hollywood Reporter" gave the film a positive review, saying, "This is a good-times film that doesn't put on airs, dress to impress or pretend to be something it isn't. It just aims to please and does a pretty good job of it." Matt Zoller Seitz of RogerEbert.com gave the film three-and-a-half out of four stars, praising the smooth direction and efficiency of storytelling Soderbergh brought to the film, and stating that other than needing additional scenes to give its "oddball characters" more depth, ["Logan Lucky" is] a precision-tooled entertainment made by experts, and sometimes more than that. Watching it is like finding money in the pocket of a coat that you haven't worn in years." Peter Travers of Rolling Stone called it "a terrific, twisty, funny-as-hell crime flick about so-called hicks who decide that making America great again starts right at home." Richard Roeper of the Chicago Sun-Times wrote: ""Logan Lucky" is great fun and one of the most purely entertaining movies of the year."


title: Logan Lucky

text: Rex Reed of "The New York Observer" was critical of the film, and described Soderbergh as an overrated director. Reed complained that the film takes a slower pace than "Ocean's Eleven", and although he praised the camera work, music and ham acting, he concluded, "It doesn't work. "Logan Lucky" is as charming and welcome as toenail fungus."


title: Logan Lucky

text: "Logan Lucky" was released on digital download on November 14, 2017, and on DVD, Blu-ray, and Ultra HD Blu-ray on November 28, 2017.


title: Eel

text: Eels are ray-finned fish belonging to the order Anguilliformes (), which consists of eight suborders, 19 families, 111 genera, and about 800 species. Eels undergo considerable development from the early larval stage to the eventual adult stage and are usually predators.


title: Eel

text: The term "eel" is also used for some other eel-shaped fish, such as electric eels (genus "Electrophorus"), spiny eels (family Mastacembelidae), swamp eels (family Synbranchidae), and deep-sea spiny eels (family Notacanthidae). However, these other clades evolved their eel-like shapes independently from the true eels. Eels live both in salt and fresh water, and some species are catadromous.


title: Eel

text: Eels are elongated fish, ranging in length from in the one-jawed eel ("Monognathus ahlstromi") to in the slender giant moray. Adults range in weight from to well over . They possess no pelvic fins, and many species also lack pectoral fins. The dorsal and anal fins are fused with the caudal fin, forming a single ribbon running along much of the length of the animal. Eels swim by generating waves that travel the length of their bodies. They can swim backward by reversing the direction of the wave.


title: Eel

text: Most eels live in the shallow waters of the ocean and burrow into sand, mud, or amongst rocks. Most eel species are nocturnal, and thus are rarely seen. Sometimes, they are seen living together in holes or "eel pits." Some eels also live in deeper water on the continental shelves and over the slopes deep as . Only members of the "Anguilla" regularly inhabit fresh water, but they, too, return to the sea to breed.


title: Eel

text: The heaviest true eel is the European conger. The maximum size of this species has been reported as reaching a length of and a weight of . Other eels are longer, but do not weigh as much, such as the slender giant moray, which reaches .


title: Eel

text: Eels begin life as flat and transparent larvae, called leptocephali. Eel larvae drift in the sea's surface waters, feeding on marine snow, small particles that float in the water. Eel larvae then metamorphose into glass eels and become elvers before finally seeking out their juvenile and adult habitats. Many eels remain in the sea throughout their lives, but freshwater elvers of eels in the family Anguillidae travel upstream and are forced to climb up obstructions, such as weirs, dam walls, and natural waterfalls.


title: Eel

text: Lady Colin Campbell found that the eel fisheries at Ballisodare were greatly improved by the hanging of loosely plaited grass ladders over barriers, enabling elvers to ascend more easily.


title: Eel

text: Several classifications of eels exist; some, such as FishBase, divide eels into 20 families, whereas other classifications such as ITIS and Systema Naturae 2000, include additional eel families, which are noted below the family with which they are synonymized in the FishBase system.


title: Eel

text: Identifying the origin of the freshwater species is considered to be problematic; however, genomic studies indicate they are a monophyletic group that originated among the deep-sea eels.


title: Eel

text: In some classifications, the family Cyematidae of bobtail snipe eels is included in the Anguilliformes, but in the FishBase system that family is included in the order Saccopharyngiformes.


title: Eel

text: The electric eel of South America is not a true eel but is a South American knifefish more closely related to the carps and catfishes.


title: Eel

text: Freshwater eels ("unagi") and marine eels (conger eel, "anago") are commonly used in Japanese cuisine; foods such as "unadon" and "unajū" are popular, but expensive. Eels are also very popular in Chinese cuisine, and are prepared in many different ways. Hong Kong eel prices have often reached 1000 HKD (128.86 US Dollars) per kg, and once exceeded 5000 HKD per kg. In India, eels are popularly eaten in the Northeast. Freshwater eels, known as "Kusia" in Assamese, are eaten with curry, often with herbs. The European eel and other freshwater eels are mostly eaten in Europe, the United States, and other places. Today, the European eel is considered critically endangered. A traditional east London food is jellied eels, although the demand has significantly declined since World War II. The Spanish cuisine delicacy "angulas" consists of elver (young eels) sautéed in olive oil with garlic; elvers usually reach prices of up to 1000 euro per kg. New Zealand longfin eel is a traditional Māori food in New Zealand. In Italian cuisine, eels from the Valli di Comacchio, a swampy zone along the Adriatic coast, are especially prized, along with freshwater eels of Bolsena Lake and pond eels from Cabras, Sardinia. In northern Germany, the Netherlands, the Czech Republic, Poland, Denmark, and Sweden, smoked eel is considered a delicacy.


title: Eel

text: Elvers, often fried, were formerly a cheap dish in the United Kingdom. During the 1990s, their numbers collapsed across Europe. They are now a delicacy, the UK's most expensive species.


title: Eel

text: Eel blood is toxic to humans and other mammals, but both cooking and the digestive process destroy the toxic protein. The toxin derived from eel blood serum was used by Charles Robert Richet in his Nobel Prize-winning research which discovered anaphylaxis (by injecting it into dogs and observing the effect). The poison used by Richet was actually obtained from sea anemones.


title: Eel

text: Eelskin leather is highly prized. It is very smooth and exceptionally strong. It does not come from eels. It comes from the Pacific hagfish, a jawless fish which is also known as the slime eel.


title: Eel

text: The large lake of Almere, which existed in the early Medieval Netherlands, got its name from the eels which lived in its water (the Dutch word for eel is aal or ael, so: "ael mere" = "eel lake"). The name is preserved in the new city of Almere in Flevoland, given in 1984 in memory of this body of water on whose site the town is located.


title: Eel

text: The daylight passage in the spring of elvers upstream along the Thames was at one time called "eel fare". The word 'elver' is thought to be a corruption of "eel fare."


title: Eel

text: A famous attraction on the French Polynesian island of Huahine (part of the Society Islands) is the bridge across a stream hosting three- to six-foot-long eels, deemed sacred by local culture.


title: Eel

text: Eel fishing in Nazi-era Danzig plays an important role in Günter Grass' novel "The Tin Drum". The cruelty of humans to eels is used as a metaphor for Nazi atrocities, and the sight of eels being killed by a fisherman triggers the madness of the protagonist's mother.


title: Eel

text: Sinister implications of eels fishing are also referenced in Jo Nesbø's "Cockroaches", the second book of the Harry Hole detective series. The book's background includes a Norwegian village where eels in the nearby sea are rumored to feed on the corpses of drowned humans, making the eating of these eels verge on cannibalism.


title: Eel

text: In 2010, Greenpeace International added the European eel, Japanese eel, and American eel to its seafood red list. Japan consumes more than 70% of the global eel catch.


title: Eel

text: The English name "eel" descends from Old English "ǣl", Common Germanic "*ēlaz". Also from the common Germanic are West Frisian "iel", Dutch "aal", German "Aal", and Icelandic "áll". Katz (1998) identifies a number of Indo-European cognates, among them the second part of the Latin word for eels, "anguilla", attested in its simplex form "illa" (in a glossary only), and the Greek word for "eel," "egkhelys" (the second part of which is attested in Hesychius as "elyes"). The first compound member, "anguis" ("snake"), is cognate to other Indo-European words for "snake" (compare Old Irish "escung" "eel," Old High German "unc" "snake," Lithuanian "angìs", Greek "ophis, okhis", Vedic Sanskrit "áhi", Avestan "aži", Armenian "auj, iž", Old Church Slavonic "*ǫžь", all from Proto-Indo-European "*h₁ogʷʰis"). The word also appears in the Old English word for "hedgehog," which is "igil" (meaning "snake eater"), and perhaps in the "egi-" of Old High German "egidehsa" "wall lizard."


title: Eel

text: According to this theory, the name Bellerophon (Βελλεροφόντης, attested in a variant Ἐλλεροφόντης in Eustathius of Thessalonica), is also related, translating to "the slayer of the serpent" ("ahihán"). In this theory, the ελλερο- is an adjective form of an older word, ελλυ, meaning "snake," which is directly comparable to Hittite "ellu-essar-" "snake pit." This myth likely came to Greece via Anatolia. In the Hittite version of the myth, the dragon is called "Illuyanka": the "illuy-" part is cognate to the word "illa", and the "-anka" part is cognate to "angu", a word for "snake." Since the words for "snake" (and similarly shaped animals) are often subject to taboo in many Indo-European (and non-Indo-European) languages, no unambiguous Proto-Indo-European form of the word for eel can be reconstructed. It may have been "*ēl(l)-u-", "*ēl(l)-o-", or something similar.


title: Dance

text: Dance is a performing art form consisting of sequences of movement, either improvised or purposefully selected. This movement has aesthetic and often symbolic value. Dance can be categorized and described by its choreography, by its repertoire of movements, or by its historical period or place of origin.


title: Dance

text: An important distinction is to be drawn between the contexts of theatrical and participatory dance, although these two categories are not always completely separate; both may have special functions, whether social, ceremonial, competitive, erotic, martial, or sacred/liturgical. Other forms of human movement are sometimes said to have a dance-like quality, including martial arts, gymnastics, cheerleading, figure skating, synchronized swimming, marching bands, and many other forms of athletics. There are many professional athletes like, professional football players and soccer players, who take dance classes to help with their skills. To be more specific professional athletes take many ballet classes because ballet is a foundation of many things we do in life. Ballet helps with balance, core strength, and leg strength.


title: Dance

text: Dance requires an equal amount of cognitive focus as well as physical strength. The demanding yet evolving art-form allows individuals to express themselves creatively through movement, while enabling them to adapt movement that possesses a rhythmical pattern and fluid motions that allure to an audience either onstage or on film. Dance is considered to be a very aesthetically pleasing art-form.


title: Dance

text: Theatrical dance, also called performance or concert dance, is intended primarily as a spectacle, usually a performance upon a stage by virtuoso dancers. It often tells a story, perhaps using mime, costume and scenery, or else it may interpret the musical accompaniment, which is often specially composed and performed in a theatre setting but it is not a requirement. Examples are western ballet and modern dance, Classical Indian dance such as Bharatanatyam and Chinese and Japanese song and dance dramas such as dragon dance. Most classical forms are centred upon dance alone, but performance dance may also appear in opera and other forms of musical theatre.


title: Dance

text: Participatory dance, on the other hand, whether it be a folk dance, a social dance, a group dance such as a line, circle, chain or square dance, or a partner dance such as is common in Western ballroom dancing, is undertaken primarily for a common purpose, such as social interaction or exercise, or building flexibility of participants rather than to serve any benefit to onlookers. Such dance seldom has any narrative. A group dance and a "corps de ballet", a social partner dance and a "pas de deux", differ profoundly. Even a solo dance may be undertaken solely for the satisfaction of the dancer. Participatory dancers often all employ the same movements and steps but, for example, in the rave culture of electronic dance music, vast crowds may engage in free dance, uncoordinated with those around them. On the other hand, some cultures lay down strict rules as to the particular dances in which, for example, men, women, and children may or must participate.


title: Dance

text: Archaeological evidence for early dance includes 10,000-year-old paintings in Madhya Pradesh, India at the Rock Shelters of Bhimbetka, and Egyptian tomb paintings depicting dancing figures, dated c. 3300 BC. It has been proposed that before the invention of written languages, dance was an important part of the oral and performance methods of passing stories down from one generation to the next. The use of dance in ecstatic trance states and healing rituals (as observed today in many contemporary "primitive" cultures, from the Brazilian rainforest to the Kalahari Desert) is thought to have been another early factor in the social development of dance.


title: Dance

text: References to dance can be found in very early recorded history; Greek dance ("horos") is referred to by Plato, Aristotle, Plutarch and Lucian. The Bible and Talmud refer to many events related to dance, and contain over 30 different dance terms. In Chinese pottery as early as the Neolithic period, groups of people are depicted dancing in a line holding hands, and the earliest Chinese word for "dance" is found written in the oracle bones. Dance is further described in the "Lüshi Chunqiu". Primitive dance in ancient China was associated with sorcery and shamanic rituals.


title: Dance

text: During the first millennium BCE in India, many texts were composed which attempted to codify aspects of daily life. Bharata Muni's "Natyashastra" (literally ""the text of dramaturgy"") is one of the earlier texts. It mainly deals with drama, in which dance plays an important part in Indian culture. It categorizes dance into four types--secular, ritual, abstract, and, interpretive--and into four regional varieties. The text elaborates various hand-gestures ("mudras") and classifies movements of the various limbs, steps and so on. A strong continuous tradition of dance has since continued in India, through to modern times, where it continues to play a role in culture, ritual, and, notably, the Bollywood entertainment industry. Many other contemporary dance forms can likewise be traced back to historical, traditional, ceremonial, and ethnic dance.


title: Dance

text: Dance is generally, however not exclusively, performed with the accompaniment of music and may or may not be performed "in time" to such music. Some dance (such as tap dance) may provide its own audible accompaniment in place of (or in addition to) music. Many early forms of music and dance were created for each other and are frequently performed together. Notable examples of traditional dance/music couplings include the jig, waltz, tango, disco, and salsa. Some musical genres have a parallel dance form such as baroque music and baroque dance; other varieties of dance and music may share nomenclature but developed separately, such as classical music and classical ballet. The choreography and music go hand in hand, as they complement each other to express a story told by the choreographer and or dancers.


title: Dance

text: Rhythm and dance are deeply linked in history and practice. The American dancer Ted Shawn wrote; "The conception of rhythm which underlies all studies of the dance is something about which we could talk forever, and still not finish." A musical rhythm requires two main elements; first, a regularly-repeating pulse (also called the "beat" or "tactus") that establishes the tempo and, second, a pattern of accents and rests that establishes the character of the metre or basic rhythmic pattern. The basic pulse is roughly equal in duration to a simple step or gesture.


title: Dance

text: Dances generally have a characteristic tempo and rhythmic pattern. The tango, for example, is usually danced in time at approximately 66 beats per minute. The basic slow step, called a "slow", lasts for one beat, so that a full "right–left" step is equal to one measure. The basic forward and backward walk of the dance is so counted – "slow-slow" – while many additional figures are counted "slow – quick-quick.


title: Dance

text: Just as musical rhythms (e.g., drum beats) are defined by a pattern of strong and weak beats, so repetitive body movements often depend on alternating "strong" and "weak" muscular movements. Given this alternation of left-right, of forward-backward and rise-fall, along with the bilateral symmetry of the human body, it is natural that many dances and much music are in duple and quadruple meter. Since some such movements require more time in one phase than the other – such as the longer time required to lift a hammer than to strike – some dance rhythms fall equally naturally into triple metre. Occasionally, as in the folk dances of the Balkans, dance traditions depend heavily on more complex rhythms. Further, complex dances composed of a fixed sequence of steps always require phrases and melodies of a certain fixed length to accompany that sequence.


title: Dance

text: The very act of dancing, the steps themselves, generate an "initial skeleton of rhythmic beats" that must have preceded any separate musical accompaniment, while dance itself, as much as music, requires time-keeping just as utilitarian repetitive movements such as walking, hauling and digging take on, as they become refined, something of the quality of dance.


title: Dance

text: Musical accompaniment, therefore, arose in the earliest dance, so that ancient Egyptians attributed the origin of the dance to the divine Athotus, who was said to have observed that music accompanying religious rituals caused participants to move rhythmically and to have brought these movements into proportional measure. The same idea, that dance arises from musical rhythm, is still found in renaissance Europe in the works of the dancing master Guglielmo Ebreo da Pesaro who speaks of dance as a physical movement that arises from and expresses inward, spiritual motion agreeing with the "measures and perfect concords of harmony" that fall upon the human ear, while, earlier, Mechthild of Magdeburg, seizing upon dance as a symbol of the holy life foreshadowed in Jesus' saying "I have piped and ye have not danced", writes;


title: Dance

text: Thoinot Arbeau's celebrated 16th-century dance-treatise "Orchésographie", indeed, begins with definitions of over eighty distinct drum-rhythms.


title: Dance

text: As has been shown above, dance has been represented through the ages as having emerged as a response to music yet, as Lincoln Kirstein implied, it is at least as likely that primitive music arose from dance. Shawn concurs, stating that dance "was the first art of the human race, and the matrix out of which all other arts grew" and that even the "metre in our poetry today is a result of the accents necessitated by body movement, as the dancing and reciting was performed simultaneously" – an assertion somewhat supported by the common use of the term "foot" to describe the fundamental rhythmic units of poetry.


title: Dance

text: Scholes, not a dancer but a musician, offers support for this view, stating that the steady measures of music, of two, three or four beats to the bar, its equal and balanced phrases, regular cadences, contrasts and repetitions, may all be attributed to the "incalculable" influence of dance upon music.


title: Dance

text: Émile Jaques-Dalcroze, primarily a musician and teacher, relates how a study of the physical movements of pianists led him "to the discovery that musical sensations of a rhythmic nature call for the muscular and nervous response of the whole organism", to develop "a special training designed to regulate nervous reactions and effect a co-ordination of muscles and nerves" and ultimately to seek the connections between "the art of music and the art of dance", which he formulated into his system of eurhythmics. He concluded that "musical rhythm is only the transposition into sound of movements and dynamisms spontaneously and involuntarily expressing emotion".


title: Dance

text: Hence, though doubtless, as Shawn asserts, "it is quite possible to develop the dance without music and... music is perfectly capable of standing on its own feet without any assistance from the dance", nevertheless the "two arts will always be related and the relationship can be profitable both to the dance and to music", the precedence of one art over the other being a moot point. The common ballad measures of hymns and folk-songs takes their name from dance, as does the carol, originally a circle dance. Many purely musical pieces have been named "waltz" or "minuet", for example, while many concert dances have been produced that are based upon abstract musical pieces, such as "2 and 3 Part Inventions, Adams Violin Concerto" and "Andantino". Similarly, poems are often structured and named after dances or musical works, while dance and music have both drawn their conception of "measure" or "metre" from poetry.


title: Dance

text: Shawn quotes with approval the statement of Dalcroze that, while the art of musical rhythm consists in differentiating and combining time durations, pauses and accents "according to physiological law", that of "plastic rhythm" (i.e. dance) "is to designate movement in space, to interpret long time-values by slow movements and short ones by quick movements, regulate pauses by their divers successions and express sound accentuations in their multiple nuances by additions of bodily weight, by means of muscular innervations".


title: Dance

text: Shawn nevertheless points out that the system of musical time is a "man-made, artificial thing... a manufactured tool, whereas rhythm is something that has always existed and depends on man not at all", being "the continuous flowing time which our human minds cut up into convenient units", suggesting that music might be revivified by a return to the values and the time-perception of dancing.


title: Dance

text: The early-20th-century American dancer Helen Moller stated that "it is rhythm and form more than harmony and color which, from the beginning, has bound music, poetry and dancing together in a union that is indissoluble."


title: Dance

text: Concert dance, like opera, generally depends for its large-scale form upon a narrative dramatic structure. The movements and gestures of the choreography are primarily intended to mime the personality and aims of the characters and their part in the plot. Such theatrical requirements tend towards longer, freer movements than those usual in non-narrative dance styles. On the other hand, the "ballet blanc", developed in the 19th century, allows interludes of rhythmic dance that developed into entirely "plotless" ballets in the 20th century and that allowed fast, rhythmic dance-steps such as those of the "petit allegro". A well-known example is "The Cygnets' Dance" in act two of "Swan Lake".


title: Dance

text: The ballet developed out of courtly dramatic productions of 16th- and 17th-century France and Italy and for some time dancers performed dances developed from those familiar from the musical suite, all of which were defined by definite rhythms closely identified with each dance. These appeared as character dances in the era of romantic nationalism.


title: Dance

text: Ballet reached widespread vogue in the romantic era, accompanied by a larger orchestra and grander musical conceptions that did not lend themselves easily to rhythmic clarity and by dance that emphasised dramatic mime. A broader concept of rhythm was needed, that which Rudolf Laban terms the "rhythm and shape" of movement that communicates character, emotion and intention, while only certain scenes required the exact synchronisation of step and music essential to other dance styles, so that, to Laban, modern Europeans seemed totally unable to grasp the meaning of "primitive rhythmic movements", a situation that began to change in the 20th century with such productions as Igor Stravinsky's "The Rite of Spring" with its new rhythmic language evoking primal feelings of a primitive past.


title: Dance

text: Indian classical dance styles, like ballet, are often in dramatic form, so that there is a similar complementarity between narrative expression and "pure" dance. In this case, the two are separately defined, though not always separately performed. The rhythmic elements, which are abstract and technical, are known as "nritta". Both this and expressive dance "(nritya)", though, are closely tied to the rhythmic system ("tala"). Teachers have adapted the spoken rhythmic mnemonic system called "bol" to the needs of dancers.


title: Dance

text: Japanese classical dance-theatre styles such as Kabuki and Noh, like Indian dance-drama, distinguish between narrative and abstract dance productions. The three main categories of kabuki are "jidaimono" (historical), "sewamono" (domestic) and "shosagoto" (dance pieces). Somewhat similarly, Noh distinguishes between "Geki Noh", based around the advancement of plot and the narration of action, and "Furyū Noh", dance pieces involving acrobatics, stage properties, multiple characters and elaborate stage action.


title: Dance

text: Social dances, those intended for participation rather than for an audience, may include various forms of mime and narrative, but are typically set much more closely to the rhythmic pattern of music, so that terms like waltz and polka refer as much to musical pieces as to the dance itself. The rhythm of the dancers' feet may even form an essential part of the music, as in tap dance. African dance, for example, is rooted in fixed basic steps, but may also allow a high degree of rhythmic interpretation: the feet or the trunk mark the basic pulse while cross-rhythms are picked up by shoulders, knees, or head, with the best dancers simultaneously giving plastic expression to all the elements of the polyrhythmic pattern.


title: Dance

text: Dance in Africa is deeply integrated into society and major events in a community are frequently reflected in dances: dances are performed for births and funerals, weddings and wars. Traditional dances impart cultural morals, including religious traditions and sexual standards; give vent to repressed emotions, such as grief; motivate community members to cooperate, whether fighting wars or grinding grain; enact spiritual rituals; and contribute to social cohesiveness.


title: Dance

text: Thousands of dances are performed around the continent. These may be divided into traditional, neotraditional, and classical styles: folkloric dances of a particular society, dances created more recently in imitation of traditional styles, and dances transmitted more formally in schools or private lessons. African dance has been altered by many forces, such as European missionaries and colonialist governments, who often suppressed local dance traditions as licentious or distracting. Dance in contemporary African cultures still serves its traditional functions in new contexts; dance may celebrate the inauguration of a hospital, build community for rural migrants in unfamiliar cities, and be incorporated into Christian church ceremonies.


title: Dance

text: All Indian classical dances are to varying degrees rooted in the "Natyashastra" and therefore share common features: for example, the "mudra"s (hand positions), some body positions, leg movement and the inclusion of dramatic or expressive acting or abhinaya. Indian classical music provides accompaniment and dancers of nearly all the styles wear bells around their ankles to counterpoint and complement the percussion.


title: Dance

text: There are now many regional varieties of Indian classical dance. Dances like ""Odra Magadhi"", which after decades-long debate, has been traced to present day Mithila, Odisha region's dance form of Odissi (Orissi), indicate influence of dances in cultural interactions between different regions.


title: Dance

text: The Punjab area overlapping India and Pakistan is the place of origin of Bhangra. It is widely known both as a style of music and a dance. It is mostly related to ancient harvest celebrations, love, patriotism or social issues. Its music is coordinated by a musical instrument called the 'Dhol'. Bhangra is not just music but a dance, a celebration of the harvest where people beat the dhol (drum), sing Boliyaan (lyrics) and dance. It developed further with the Vaisakhi festival of the Sikhs.


title: Dance

text: The dances of Sri Lanka include the devil dances ("yakun natima"), a carefully crafted ritual reaching far back into Sri Lanka's pre-Buddhist past that combines ancient "Ayurvedic" concepts of disease causation with psychological manipulation and combines many aspects including Sinhalese cosmology. Their influence can be seen on the classical dances of Sri Lanka.


title: Dance

text: Indonesian dances reflect the richness and diversity of Indonesian ethnic groups and cultures. There are more than 1,300 ethnic groups in Indonesia, it can be seen from the cultural roots of the Austronesian and Melanesian peoples, and various cultural influences from Asia and the west. Dances in Indonesia originate from ritual movements and religious ceremonies, this kind of dance usually begins with rituals, such as war dances, shaman dances to cure or ward off disease, dances to call rain and other types of dances. With the acceptance of dharma religion in the 1st century in Indonesia, Hinduism and Buddhist rituals were celebrated in various artistic performances. Hindu epics such as the Ramayana, Mahabharata and also the Panji became the inspiration to be shown in a dance-drama called "Sendratari" resembling "ballet" in the western tradition. An elaborate and highly stylized dance method was invented and has survived to this day, especially on the islands of Java and Bali. The Javanese Wayang wong dance takes footage from the Ramayana or Mahabharata episodes, but this dance is very different from the Indian version, indonesian dances do not pay as much attention to the "mudras" as Indian dances: even more to show local forms. The sacred Javanese ritual dance Bedhaya is believed to date back to the Majapahit period in the 14th century or even earlier, this dance originated from ritual dances performed by virgin girls to worship Hindu Gods such as Shiva, Brahma, and Vishnu. In Bali, dance has become an integral part of the sacred Hindu Dharma rituals. Some experts believe that Balinese dance comes from an older dance tradition from Java. Reliefs from temples in East Java from the 14th century feature crowns and headdresses similar to the headdresses used in Balinese dance today. Islam began to spread to the Indonesian archipelago when indigenous dances and dharma dances were still popular. Artists and dancers still use styles from the previous era, replacing stories with more Islamic interpretations and clothing that is more closed according to Islamic teachings.


title: Dance

text: The dances of the Middle East are usually the traditional forms of circle dancing which are modernized to an extent. They would include dabke, tamzara, Assyrian folk dance, Kurdish dance, Armenian dance and Turkish dance, among others. All these forms of dances would usually involve participants engaging each other by holding hands or arms (depending on the style of the dance). They would make rhythmic moves with their legs and shoulders as they curve around the dance floor. The head of the dance would generally hold a cane or handkerchief.


title: Dance

text: Folk dances vary across Europe and may date back hundreds or thousands of years, but many have features in common such as group participation led by a caller, hand-holding or arm-linking between participants, and fixed musical forms known as caroles. Some, such as the maypole dance are common to many nations, while others such as the céilidh and the polka are deeply-rooted in a single culture. Some European folk dances such as the square dance were brought to the New World and subsequently became part of American culture.


title: Dance

text: Ballet developed first in Italy and then in France from lavish court spectacles that combined music, drama, poetry, song, costumes and dance. Members of the court nobility took part as performers. During the reign of Louis XIV, himself a dancer, dance became more codified. Professional dancers began to take the place of court amateurs, and ballet masters were licensed by the French government. The first ballet dance academy was the Académie Royale de Danse (Royal Dance Academy), opened in Paris in 1661. Shortly thereafter, the first institutionalized ballet troupe, associated with the Academy, was formed; this troupe began as an all-male ensemble but by 1681 opened to include women as well.


title: Dance

text: 20th century concert dance brought an explosion of innovation in dance style characterized by an exploration of freer technique. Early pioneers of what became known as modern dance include Loie Fuller, Isadora Duncan, Mary Wigman and Ruth St. Denis. The relationship of music to dance serves as the basis for Eurhythmics, devised by Emile Jaques-Dalcroze, which was influential to the development of Modern dance and modern ballet through artists such as Marie Rambert. Eurythmy, developed by Rudolf Steiner and Marie Steiner-von Sivers, combines formal elements reminiscent of traditional dance with the new freer style, and introduced a complex new vocabulary to dance. In the 1920s, important founders of the new style such as Martha Graham and Doris Humphrey began their work. Since this time, a wide variety of dance styles have been developed; see Modern dance.


title: Dance

text: African American dance developed in everyday spaces, rather than in dance studios, schools or companies. Tap dance, disco, jazz dance, swing dance, hip hop dance, the lindy hop with its relationship to rock and roll music and rock and roll dance have had a global influence. Dance styles fusing classical ballet technique with African-American dance have also appeared in the 21st century, including Hiplet.


title: Dance

text: Dance is central to Latin American social life and culture. Brazilian Samba, Argentinian tango, and Cuban salsa are internationally popular partner dances, and other national dances—merengue, cueca, plena, jarabe, joropo, marinera, cumbia, bachata and others—are important components of their respective countries' cultures. Traditional Carnival festivals incorporate these and other dances in enormous celebrations.


title: Dance

text: Dance has played an important role in forging a collective identity among the many cultural and ethnic groups of Latin America. Dance served to unite the many African, European, and indigenous peoples of the region. Certain dance genres, such as capoeira, and body movements, especially the characteristic "quebradas" or pelvis swings, have been variously banned and celebrated throughout Latin American history.


title: Dance

text: Dance studies are offered through the arts and humanities programs of many higher education institutions. Some universities offer Bachelor of Arts and higher academic degrees in Dance. A dance study curriculum may encompass a diverse range of courses and topics, including dance practice and performance, choreography, ethnochoreology, kinesiology, dance notation, and dance therapy. Most recently, dance and movement therapy has been integrated in some schools into math lessons for students with learning disabilities, emotional/behavioral disabilities and/or attention deficit hyperactivity disorder (ADHD).


title: Dance

text: Professional dancers are usually employed on contract or for particular performances or productions. The professional life of a dancer is generally one of constantly changing work situations, strong competitive pressure and low pay. Consequently, professional dancers often must supplement their incomes to achieve financial stability. In the U.S. many professional dancers belong to unions (such as the American Guild of Musical Artists, Screen Actors Guild and Actors' Equity Association) that establish working conditions and minimum salaries for their members. Professional dancers must possess large amounts of athleticism. To lead a successful career, it is advantageous to be versatile in many styles of dance, have a strong technical background and to use other forms of physical training to remain fit and healthy.


title: Dance

text: Dance teachers typically focus on teaching dance performance, or coaching competitive dancers, or both. They typically have performance experience in the types of dance they teach or coach. For example, dancesport teachers and coaches are often tournament dancers or former dancesport performers. Dance teachers may be self-employed, or employed by dance schools or general education institutions with dance programs. Some work for university programs or other schools that are associated with professional classical dance (e.g., ballet) or modern dance companies. Others are employed by smaller, privately owned dance schools that offer dance training and performance coaching for various types of dance.


title: Dance

text: Choreographers are the ones that design the dancing movements within a dance, they are often university trained and are typically employed for particular projects or, more rarely may work on contract as the resident choreographer for a specific dance company.


title: Dance

text: A dance competition is an organized event in which contestants perform dances before a judge or judges for awards, and in some cases, monetary prizes. There are several major types of dance competitions, distinguished primarily by the style or styles of dances performed. Dance competitions are an excellent setting to build connections with industry leading faculty members, adjudicators, choreographers and other dancers from competing studios. A typical dance competition for younger pre-professional dancers can last anywhere between two to four days, depending whether it is a regional or national competition.


title: Dance

text: The purpose of dance competitions is to provide a fun and educative place for dancers and give them the opportunity to perform their choreographed routines from their current dance season onstage. Often times, competitions will take place in a professional setting or may vary to non-performance spaces, such as a high school theatre. The results of the dancers are then dictated by a credible panel of judges and are evaluated on their performance than given a score. As far as competitive categories go, most competitions base their categories according to the dance style, age, experience level and the number of dancers competing in the routine. Major types of dance competitions include:


title: Dance

text: There are numerous dance competition shows presented on television and other mass media outlets including, NBC's World Of Dance, NBC's Dancing With Myself, Dancing With The Stars, etc.


title: Dance

text: In most forms of dance the foot is the source of movement, and in some cases require specific shoes to aid in the health, safety ability of the dancer, depending on the type of dance, the intensity of the movements, and the surface that will be danced on.


title: Dance

text: Dance footwear can be potentially both supportive and or restrictive to the movement of the dancer. The effectiveness of the shoe is related to its ability to help the foot do something it is not intended to do, or to make easier a difficult movement. Such effects relate to health and safety because of the function of the equipment as unnatural to the bodies usual mobility.


title: Dance

text: Ballet is notable for the risks of injury due to the biomechanics of the ankle and the toes as the main support for the rest of the movements. With the pointe shoe, the design specifically brings all of the toes together to allow the toes to be stood on for longer periods of time.


title: Dance

text: There are accessories associated with pointe shoes that help to mitigate injury and soothe pain while dancing, including things such as toe pads, toe tape, and cushions.


title: Dance

text: Dancers are publicly thought to be very preoccupied with their body image to fit a certain mold in the industry. Research indicates that dancers do have greater difficulty controlling their eating habits as a large quantity strive for the art-form's ideal body mass. Some dancers often resort to abusive tactics to maintain a certain image. Common scenarios include dancers abusing laxatives for weight control and end up falling into unhealthy eating disorders. Studies show that a large quantity of dancers use at least one method of weight control including over exercising and food restriction. The pressure for dancers to maintain a below average weight affects their eating and weight controlling behaviours and their life-style. Due to its artistic nature, dancers tend to have many hostile self-critical tendencies. Commonly seen in performers, it is likely that a variety of individuals may be resistant to concepts of self-compassion.


title: Dance

text: Eating disorders in dancers are generally very publicly common. Through data analysis and studies published, sufficient data regarding the percentage and accuracy dancers have of realistically falling into unhealthy disordered eating habits or the development of an eating disorder were extracted. Dancers, in general, have a higher risk of suffering from eating disorders than the general public, primarily falling into anorexia nervosa and EDNOS. Research has yet to distinguish a direct correlation regarding dancers having a higher risk of suffering from bulimia nervosa. Studies concluded that dancers overall have a three times higher risk of suffering from eating disorders, more specifically anorexia nervosa and EDNOS.


title: Dance

text: Dance has become a fundamental aspect of the popular app and a primary category influencing the youth's culture today. Dance challenges have become a popular form of content across many social media platforms including TikTok. During 2020, TikTok dances offered an escape for isolated individuals to play and connect with one another through virtual format. With TikTok's easy accessibility to a variety of different filters and special effects, the app made filming yourself dancing to music a fun and easy past time. Since its debut back in 2017, the app attracted a small but growing audience of professional dancers in their early 20s to 30s. While the majority of this demographic is more accustomed to performing onstage, this app introduced a new era of dancing onscreen.


title: Paulo Coelho

text: Paulo Coelho de Souza (, ; born 24 August 1947) is a Brazilian lyricist and novelist and a member of the Brazilian Academy of Letters since 2002. His novel "The Alchemist" became an international best-seller and he has published 28 more books since then.


title: Paulo Coelho

text: Paulo Coelho was born in Rio de Janeiro, Brazil, and attended a Jesuit school. At 17, Coelho's parents committed him to a mental institution from which he escaped three times before being released at the age of 20. Coelho later remarked that "It wasn't that they wanted to hurt me, but they didn't know what to do... They did not do that to destroy me, they did that to save me."


title: Paulo Coelho

text: At his parents' wishes, Coelho enrolled in law school and abandoned his dream of becoming a writer. One year later, he dropped out and lived life as a hippie, traveling through South America, North Africa, Mexico, and Europe and started using drugs in the 1960s.


title: Paulo Coelho

text: Upon his return to Brazil, Coelho worked as a songwriter, composing lyrics for Elis Regina, Rita Lee, and Brazilian icon Raul Seixas. Composing with Raul led to Coelho being associated with magic and occultism, due to the content of some songs. He is often accused that these songs were rip-offs of foreign songs not well known in Brazil at the time. In 1974, by his account, he was arrested for "subversive" activities and tortured by the ruling military government, who had taken power ten years earlier and viewed his lyrics as left-wing and dangerous. Coelho also worked as an actor, journalist and theatre director before pursuing his writing career.


title: Paulo Coelho

text: Coelho married artist Christina Oiticica in 1980. Together they had previously spent half the year in Rio de Janeiro and the other half in a country house in the Pyrenees Mountains of France, but now the couple reside permanently in Geneva, Switzerland.


title: Paulo Coelho

text: In 1986 Coelho walked the 500-plus mile Road of Santiago de Compostela in northwestern Spain. On the path, he had a spiritual awakening, which he described autobiographically in "The Pilgrimage". In an interview, Coelho stated "[In 1986], I was very happy in the things I was doing. I was doing something that gave me food and water – to use the metaphor in "The Alchemist", I was working, I had a person whom I loved, I had money, but I was not fulfilling my dream. My dream was, and still is, to be a writer." Coelho would leave his lucrative career as a songwriter and pursue writing full-time.


title: Paulo Coelho

text: "The Pilgrim – Story of Paulo Coelho" is the international title for the biographical film "Não Pare na Pista", a co-production between Brazil’s Drama Films and the Spanish Babel Films, in which the younger and older Coelho are played by two different actors. One of the producers, Iôna de Macêdo, told "Screen International": "The film tells the story of a man who has a dream. It's a little like "Alice in Wonderland" – he's someone who is too big for his house." The film, shot in Portuguese, had its premiere in Brazilian theaters in 2014 and was internationally distributed in 2015.


title: Paulo Coelho

text: In 1982, Coelho published his first book, "Hell Archives", which failed to make a substantial impact. In 1986 he contributed to the "Practical Manual of Vampirism", although he later tried to take it off the shelves since he considered it "of bad quality." After making the pilgrimage to Santiago de Compostela in 1986, Coelho wrote "The Pilgrimage", published in 1987.


title: Paulo Coelho

text: While trying to overcome his procrastination about launching his writing career, Coelho decided, "If I see a white feather today, that is a sign that God is giving me that I have to write a new book." Seeing one in the window of a shop, he began writing that day. The following year, Coelho wrote "The Alchemist" and published it through a small Brazilian publishing house that made an initial print run of 900 copies and decided not to reprint it. He subsequently found a bigger publishing house, and with the publication of his next book "Brida", "The Alchemist" took off. HarperCollins decided to publish the book in 1994. Later it became an international bestseller. In a 2009 interview with the Syrian "Forward Magazine", Coelho stated that the Sufi tradition had been an influence on him, particularly when writing "The Alchemist" and later "The Zahir".


title: Paulo Coelho

text: Since the publication of "The Alchemist", Coelho has generally written at least one novel every two years. Four of them – "The Pilgrimage", "Hippie", "The Valkyries" and" Aleph" – are autobiographical, while the majority of the rest are broadly fictional. Other books, like "Maktub", "The Manual of the Warrior of Light" and "Like the Flowing River", are collections of essays, newspaper columns, or selected teachings. His work has been published in more than 170 countries and translated into eighty-three languages. Together, his books have sold 320 million copies.


title: Paulo Coelho

text: On 22 December 2016, Coelho was listed by UK-based company Richtopia at number 2 in the list of 200 most influential contemporary authors.


title: Paulo Coelho

text: However, reactions to his writing have not been without controversy. Though he was raised in a Catholic family, and describes himself as of that faith even now, his stance has been described as incompatible with the Catholic faith, because of its New Age, pantheist and relativist contents. And whatever his sales, reviews of Coelho's later work consistently note its superficiality.


title: Paulo Coelho

text: In 2016, he was contacted by basketball player Kobe Bryant, who wanted to discuss a children's book project with him. Some months before Bryant's death in a helicopter crash in January 2020, they started to write the book together, but upon hearing about his passing, Coelho decided to delete the whole draft, saying in an interview that "it didn't make any sense to publish without him." He did not say how many pages had been written or whether the book had a title.


title: Paulo Coelho

text: In 2018, it was announced that Coelho had signed for a TV series based on the characters of his novels "The Devil and Miss Prym", "Brida" and "The Witch of Portobello".


title: There's Something About Mary

text: There's Something About Mary is a 1998 American romantic comedy film directed by Peter Farrelly and Bobby Farrelly. It stars Cameron Diaz as the title character with Ben Stiller, Matt Dillon, Lee Evans, and Chris Elliott all playing men who are in love with Mary, and vying for her affection.


title: There's Something About Mary

text: "There's Something About Mary" was released theatrically on July 15, 1998, by 20th Century Fox. It received generally positive reviews from critics who praised its humor and Diaz's performance. The film became a major box office success, grossing over $369 million worldwide against its $23 million budget, becoming the fourth-highest-grossing film of the year. It is placed 27th in the American Film Institute's "100 Years, 100 Laughs: America's Funniest Movies", a list of the 100 funniest movies of the 20th century. In 2000, readers of "Total Film" magazine voted "There's Something About Mary" the fourth-greatest comedy film of all time.


title: There's Something About Mary

text: Diaz won a New York Film Critics Circle Award for Best Actress, an MTV Movie Award for Best Performance, an American Comedy Award for Best Actress, a Blockbuster Entertainment Award for Best Actress. Her performance additionally was nominated for the Golden Globe Award for Best Actress – Motion Picture Musical or Comedy. The film was also nominated for a Golden Globe Award for Best Motion Picture – Musical or Comedy. It won four out of eight MTV Movie Awards, including Best Movie.