1.
Maclaurin spheroid
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A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes, in other words, an ellipsoid with two equal semi-diameters. If the ellipse is rotated about its axis, the result is a prolate spheroid. If the ellipse is rotated about its axis, the result is an oblate spheroid. If the generating ellipse is a circle, the result is a sphere, because of the combined effects of gravity and rotation, the Earths shape is not quite a sphere but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere, the current World Geodetic System model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles. The semi-major axis a is the radius of the spheroid. There are two cases, c < a, oblate spheroid c > a, prolate spheroid The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area S o b l a t e =2 π a 2 where e 2 =1 − c 2 a 2. The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. A prolate spheroid with c > a has surface area S p r o l a t e =2 π a 2 where e 2 =1 − a 2 c 2. The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a and these formulas are identical in the sense that the formula for Soblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity, both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid is 4π/3a2c ≈4. 19a2c, if A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is π/6A2C ≈0. 523A2C. Both of these curvatures are always positive, so every point on a spheroid is elliptic. These are just two of different parameters used to define an ellipse and its solid body counterparts. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal, deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects. An extreme example of a planet in science fiction is Mesklin, in Hal Clements novel Mission of Gravity. The prolate spheroid is the shape of the ball in several sports, several moons of the Solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids
2.
Quadric
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In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is an hypersurface in a space, and is defined as the zero set of an irreducible polynomial of degree two in D +1 variables. When the defining polynomial is not absolutely irreducible, the set is generally not considered as a quadric. The values Q, P and R are often taken to be real numbers or complex numbers. A quadric is an algebraic variety, or, if it is reducible. Quadrics may also be defined in spaces, see Quadric. Quadrics in the Euclidean plane are those of dimension D =1, in this case, one talks of conic sections, or conics. In three-dimensional Euclidean space, quadrics have dimension D =2 and they are classified and named by their orbits under affine transformations. More precisely, if an affine transformation maps a quadric onto another one, they belong to the same class, each of these 17 normal forms correspond to a single orbit under affine transformations. In three cases there are no points, ε1 = ε2 =1, ε1 =0, ε2 =1. In one case, the cone, there is a single point. If ε4 =1, one has a line, for ε4 =0, one has a double plane. For ε4 =1, one has two intersecting planes and it remains nine true quadrics, a cone and three cylinders and five non-degenerated quadrics, which are detailed in the following table. In a three-dimensional Euclidean space there are 17 such normal forms, of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all, the quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the coordinates on RD+1 are one introduces new coordinates on RD+2 related to the original coordinates by x i = X i / X0. In the new variables, every quadric is defined by an equation of the form Q = ∑ i j a i j X i X j =0 where the coefficients aij are symmetric in i and j. Regarding Q =0 as an equation in projective space exhibits the quadric as an algebraic variety
3.
Surface (mathematics)
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In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line, there are several more precise definitions, depending on the context and the mathematical tools that are used for the study. Often, a surface is defined by equations that are satisfied by the coordinates of its points and this is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface, if the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the sphere is an algebraic surface, as it may be defined by the implicit equation x 2 + y 2 + z 2 −1 =0. A surface may also be defined as the image, in space of dimension at least 3. In this case, one says that one has a parametric surface, for example, the unit sphere may be parametrized by the Euler angles, also called longitude u and latitude v by x = cos cos y = sin cos z = sin . Parametric equations of surfaces are often irregular at some points, for example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles. For the remaining two points, one has cos v =0, and the longitude u may take any values, also, there are surfaces for which there cannot exits a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations and this allows defining surfaces in spaces of dimension higher than three, and even abstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that have singularities, in classical geometry, a surface is generally defined as a locus of a point or a line. A ruled surface is the locus of a moving line satisfying some constraints, in modern terminology, a surface is a surface. In this article, several kinds of surfaces are considered and compared, a non-ambiguous terminology is thus necessary for distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two and we call differential surfaces the surfaces that are differentiable manifolds. Every differential surface is a surface, but the converse is false. For simplicity, unless stated, surface will mean a surface in the Euclidean space of dimension 3 or in R3. A surface, that is not supposed to be included in another space, is called an abstract surface, the graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differential surface, a plane is together an algebraic surface and a differentiable surface
4.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
5.
Ellipsoid
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An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by any of the two following properties, every planar cross section is either an ellipse, or is empty, or is reduced to a single point. It is bounded, which means that it may be enclosed in a large sphere. An ellipsoid has three perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, if the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, and the axes are uniquely defined. If two of the axes have the length, then the ellipsoid is an ellipsoid of revolution. In this case, the ellipsoid is invariant under a rotation around the third axis, if the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the length, the ellipsoid is a sphere. The points, and lie on the surface, the line segments from the origin to these points are called the semi-principal axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the axis and semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid, if a = b < c, one has a prolate spheroid, if a = b = c, one has a sphere. It is easy to check, The intersection of a plane, remark, The contour of an ellipsoid, seen from a point outside the ellipsoid or from infinity, is in any case a plane section, hence an ellipse. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin and these parameters may be interpreted as spherical coordinates. More precisely, π /2 − θ is the polar angle, and φ is the azimuth angle of the point of the ellipsoid. More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the x to the equation T A =1. The eigenvectors of A define the axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes
6.
American football
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The offense must advance at least ten yards in four downs, or plays, or else they turn over the football to the opposing team, if they succeed, they are given a new set of four downs. Points are primarily scored by advancing the ball into the teams end zone for a touchdown or kicking the ball through the opponents goalposts for a field goal. The team with the most points at the end of a game wins, American football evolved in the United States, originating from the sports of association football and rugby football. The first game of American football was played on November 6,1869, during the latter half of the 1870s, colleges playing association football switched to the Rugby Union code, which allowed carrying the ball. American football as a whole is the most popular sport in the United States, Professional football and college football are the most popular forms of the game, with the other major levels being high school and youth football. As of 2012, nearly 1.1 million high school athletes and 70,000 college athletes play the sport in the United States annually, almost all of them men, in the United States, American football is referred to as football. The term football was established in the rulebook for the 1876 college football season. The terms gridiron or American football are favored in English-speaking countries where other codes of football are popular, such as the United Kingdom, Ireland, New Zealand, American football evolved from the sports of association football and rugby football. What is considered to be the first American football game was played on November 6,1869 between Rutgers and Princeton, two college teams, the game was played between two teams of 25 players each and used a round ball that could not be picked up or carried. It could, however, be kicked or batted with the feet, hands, head or sides, Rutgers won the game 6 goals to 4. Collegiate play continued for years in which matches were played using the rules of the host school. Representatives of Yale, Columbia, Princeton and Rutgers met on October 19,1873 to create a set of rules for all schools to adhere to. Teams were set at 20 players each, and fields of 400 by 250 feet were specified, Harvard abstained from the conference, as they favored a rugby-style game that allowed running with the ball. An 1875 Harvard-Yale game played under rugby-style rules was observed by two impressed Princeton athletes and these players introduced the sport to Princeton, a feat the Professional Football Researchers Association compared to selling refrigerators to Eskimos. Princeton, Harvard, Yale and Columbia then agreed to play using a form of rugby union rules with a modified scoring system. These schools formed the Intercollegiate Football Association, although Yale did not join until 1879, the introduction of the snap resulted in unexpected consequences. Prior to the snap, the strategy had been to punt if a scrum resulted in bad field position, however, a group of Princeton players realized that, as the snap was uncontested, they now could hold the ball indefinitely to prevent their opponent from scoring. In 1881, both teams in a game between Yale-Princeton used this strategy to maintain their undefeated records, each team held the ball, gaining no ground, for an entire half, resulting in a 0-0 tie
7.
Rugby football
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Rugby is a type of football developed at Rugby School in Rugby, Warwickshire, one of many versions of football played at English public schools in the 19th century. The two main types of rugby are rugby league and rugby union, although rugby league initially used rugby union rules, they are now wholly separate sports. Following the 1895 split in rugby football, the two rugby league and rugby union differed in administration only. Soon the rules of rugby league were modified, resulting in two different forms of rugby. After 100 years, in 1995 rugby union joined rugby league, the Greeks and Romans are known to have played many ball games, some of which involved the use of the feet. These games appear to have resembled rugby football, the Roman politician Cicero describes the case of a man who was killed whilst having a shave when a ball was kicked into a barbers shop. Roman ball games already knew the air-filled ball, the follis, episkyros is recognised as an early form of football by FIFA. In 1871, English clubs met to form the Rugby Football Union, in 1892, after charges of professionalism were made against some clubs for paying players for missing work, the Northern Rugby Football Union, usually called the Northern Union, was formed. The existing rugby union authorities responded by issuing sanctions against the clubs, players, after the schism, the separate clubs were named rugby league and rugby union. Rugby union is both a professional and amateur game, and is dominated by the first tier unions, Argentina, Australia, England, France, Ireland, Italy, New Zealand, Scotland, South Africa and Wales. Rugby Union is administered by World Rugby, whose headquarters are located in Dublin and it is the national sport in New Zealand, Wales, Fiji, Samoa, Tonga and Madagascar, and is the most popular form of rugby globally. The Olympic Games have admitted the seven-a-side version of the game, known as Rugby sevens, there was a possibility sevens would be a demonstration sport at the 2012 London Olympics but many sports including sevens were dropped. In Canada and the United States, rugby union evolved into gridiron football, during the late 1800s, the two forms of the game were very similar, but numerous rule changes have differentiated the gridiron-based game from its rugby counterpart. Rugby league is also both a professional and amateur game, administered on a level by the Rugby League International Federation. International Rugby League is dominated by Australia, England and New Zealand, in Papua New Guinea it is the national sport. Other nations from the South Pacific and Europe also play in the Pacific Cup, distinctive features common to both rugby codes include the oval ball and throwing the ball forward is not allowed, so that players can gain ground only by running with the ball or by kicking it. As the sport of rugby league moved further away from its counterpart, rule changes were implemented with the aim of making a faster-paced. League players may not contest possession after making a tackle, play is continued with a play-the-ball, in league, if the team in possession fails to score before a set of six tackles, it surrenders possession
8.
Lentil
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The lentil is an edible pulse. It is an annual plant of the legume family, known for its lens-shaped seeds. It is about 40 cm tall, and the seeds grow in pods, in South Asian cuisine, split lentils are known as lentils. Usually eaten with rice or rotis, the lentil is a staple throughout regions of India, Pakistan, Bangladesh. As a food crop, the majority of production comes from Canada. Lentils have been part of the diet since aceramic Neolithic times. Archeological evidence shows they were eaten 9,500 to 13,000 years ago, Lentil colors range from yellow to red-orange to green, brown and black. Lentils also vary in size, and are sold in forms, with or without the skins. Raw lentils are 8% water, 63% carbohydrates including 11% dietary fiber, 25% protein, lentils are a rich source of numerous essential nutrients, including folate, thiamin, pantothenic acid, vitamin B6, phosphorus, iron and zinc, among others. When lentils are cooked by boiling, protein content declines to 9% of total composition, lentils have the second-highest ratio of protein per calorie of any legume, after soybeans. The low levels of readily digestible starch, and high levels of slowly digested starch, the remaining 65% of the starch is a resistant starch classified as RS1. A minimum of 10% in starch from lentils escapes digestion and absorption in the small intestine, lentils also have anti-nutrient factors, such as trypsin inhibitors and a relatively high phytate content. Trypsin is an involved in digestion, and phytates reduce the bioavailability of dietary minerals. The phytates can be reduced by prolonged soaking and fermentation or sprouting, lentils are relatively tolerant to drought, and are grown throughout the world. FAOSTAT reported that the production of lentils for calendar year 2013 was 4,975,621 metric tons, primarily coming from Canada. About a quarter of the production of lentils is from India. Canada is the largest export producer of lentils in the world, Statistics Canada estimates that Canadian lentil production for the 2009/10 year was a record 1.5 million metric tons. The most commonly grown type is the Laird lentil, the Palouse region of eastern Washington and the Idaho panhandle, with its commercial center at Pullman, Washington, constitute the most important lentil-producing region in the United States
9.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
10.
Gravity
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Gravity, or gravitation, is a natural phenomenon by which all things with mass are brought toward one another, including planets, stars and galaxies. Since energy and mass are equivalent, all forms of energy, including light, on Earth, gravity gives weight to physical objects and causes the ocean tides. Gravity has a range, although its effects become increasingly weaker on farther objects. The most extreme example of this curvature of spacetime is a hole, from which nothing can escape once past its event horizon. More gravity results in time dilation, where time lapses more slowly at a lower gravitational potential. Gravity is the weakest of the four fundamental interactions of nature, the gravitational attraction is approximately 1038 times weaker than the strong force,1036 times weaker than the electromagnetic force and 1029 times weaker than the weak force. As a consequence, gravity has an influence on the behavior of subatomic particles. On the other hand, gravity is the dominant interaction at the macroscopic scale, for this reason, in part, pursuit of a theory of everything, the merging of the general theory of relativity and quantum mechanics into quantum gravity, has become an area of research. While the modern European thinkers are credited with development of gravitational theory, some of the earliest descriptions came from early mathematician-astronomers, such as Aryabhata, who had identified the force of gravity to explain why objects do not fall out when the Earth rotates. Later, the works of Brahmagupta referred to the presence of force, described it as an attractive force. Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and this was a major departure from Aristotles belief that heavier objects have a higher gravitational acceleration. Galileo postulated air resistance as the reason that objects with less mass may fall slower in an atmosphere, galileos work set the stage for the formulation of Newtons theory of gravity. In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. Newtons theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the position of the planet. A discrepancy in Mercurys orbit pointed out flaws in Newtons theory, the issue was resolved in 1915 by Albert Einsteins new theory of general relativity, which accounted for the small discrepancy in Mercurys orbit. The simplest way to test the equivalence principle is to drop two objects of different masses or compositions in a vacuum and see whether they hit the ground at the same time. Such experiments demonstrate that all objects fall at the rate when other forces are negligible
11.
Earth's rotation
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Earths rotation is the rotation of the planet Earth around its own axis. The Earth rotates from the west towards east, as viewed from North Star or polestar Polaris, the Earth turns counter-clockwise. The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earths axis of rotation meets its surface and this point is distinct from the Earths North Magnetic Pole. The South Pole is the point where the Earths axis of rotation intersects its surface. The Earth rotates once in about 24 hours with respect to the sun, Earths rotation is slowing slightly with time, thus, a day was shorter in the past. This is due to the effects the Moon has on Earths rotation. Atomic clocks show that a modern-day is longer by about 1.7 milliseconds than a century ago, analysis of historical astronomical records shows a slowing trend of 2.3 milliseconds per century since the 8th century BCE. Among the ancient Greeks, several of the Pythagorean school believed in the rotation of the rather than the apparent diurnal rotation of the heavens. Perhaps the first was Philolaus, though his system was complicated, in the third century BCE, Aristarchus of Samos suggested the suns central place. However, Aristotle in the fourth century criticized the ideas of Philolaus as being based on rather than observation. He established the idea of a sphere of fixed stars that rotated about the earth and this was accepted by most of those who came after, in particular Claudius Ptolemy, who thought the earth would be devastated by gales if it rotated. In the 10th century, some Muslim astronomers accepted that the Earth rotates around its axis, treatises were written to discuss its possibility, either as refutations or expressing doubts about Ptolemys arguments against it. At the Maragha and Samarkand observatories, the Earths rotation was discussed by Tusi and Qushji, in medieval Europe, Thomas Aquinas accepted Aristotles view and so, reluctantly, did John Buridan and Nicole Oresme in the fourteenth century. Not until Nicolaus Copernicus in 1543 adopted a heliocentric world system did the earths rotation begin to be established, Copernicus pointed out that if the movement of the earth is violent, then the movement of the stars must be very much more so. He acknowledged the contribution of the Pythagoreans and pointed to examples of relative motion, for Copernicus this was the first step in establishing the simpler pattern of planets circling a central sun. Tycho Brahe, who produced accurate observations on which Kepler based his laws, in 1600, William Gilbert strongly supported the earths rotation in his treatise on the earths magnetism and thereby influenced many of his contemporaries. Those like Gilbert who did not openly support or reject the motion of the earth about the sun are often called semi-Copernicans, however, the contributions of Kepler, Galileo and Newton gathered support for the theory of the rotation of the Earth. The earths rotation implies that the bulges and the poles are flattened
12.
Figure of the Earth
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The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and this is, in fact, the surface on which actual Earth measurements are made. The topographic surface is generally the concern of topographers and hydrographers, the Pythagorean concept of a spherical Earth offers a simple surface that is mathematically easy to deal with. Many astronomical and navigational computations use it as a representing the Earth. The idea of a planar or flat surface for Earth, however, is sufficient for surveys of small areas, as the local topography is far more significant than the curvature. Plane-table surveys are made for small areas, and no account is taken of the curvature of the Earth. A survey of a city would likely be computed as though the Earth were a surface the size of the city. For such small areas, exact positions can be determined relative to each other without considering the size, in the mid- to late 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the figure of the Earth. The primary utility of this improved accuracy was to provide geographical and gravitational data for the guidance systems of ballistic missiles. This funding also drove the expansion of geoscientific disciplines, fostering the creation, the models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth. The simplest model for the shape of the entire Earth is a sphere, the Earths radius is the distance from Earths center to its surface, about 6,371 kilometers. The concept of a spherical Earth dates back to around the 6th century BC, the first scientific estimation of the radius of the earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes’s measurement ranging from 2% to 15%. The Earth is only approximately spherical, so no single value serves as its natural radius, distances from points on the surface to the center range from 6,353 km to 6,384 km. Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 kilometers, regardless of the model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km. The difference 21 kilometers correspond to the polar radius being approximately 0. 3% shorter than the equator radius, since the Earth is flattened at the poles and bulges at the equator, geodesy represents the shape of the earth with an oblate spheroid. The oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis and it is the regular geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other body is called a reference ellipsoid. The reference ellipsoid for Earth is called an Earth ellipsoid, an ellipsoid of revolution is uniquely defined by two numbers, two dimensions, or one dimension and a number representing the difference between the two dimensions
13.
Planet
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The term planet is ancient, with ties to history, astrology, science, mythology, and religion. Several planets in the Solar System can be seen with the naked eye and these were regarded by many early cultures as divine, or as emissaries of deities. As scientific knowledge advanced, human perception of the planets changed, in 2006, the International Astronomical Union officially adopted a resolution defining planets within the Solar System. This definition is controversial because it excludes many objects of mass based on where or what they orbit. The planets were thought by Ptolemy to orbit Earth in deferent, at about the same time, by careful analysis of pre-telescopic observation data collected by Tycho Brahe, Johannes Kepler found the planets orbits were not circular but elliptical. As observational tools improved, astronomers saw that, like Earth, the planets rotated around tilted axes, and some shared such features as ice caps and seasons. Since the dawn of the Space Age, close observation by space probes has found that Earth and the planets share characteristics such as volcanism, hurricanes, tectonics. Planets are generally divided into two types, large low-density giant planets, and smaller rocky terrestrials. Under IAU definitions, there are eight planets in the Solar System, in order of increasing distance from the Sun, they are the four terrestrials, Mercury, Venus, Earth, and Mars, then the four giant planets, Jupiter, Saturn, Uranus, and Neptune. Six of the planets are orbited by one or more natural satellites, several thousands of planets around other stars have been discovered in the Milky Way. e. in the habitable zone. On December 20,2011, the Kepler Space Telescope team reported the discovery of the first Earth-sized extrasolar planets, Kepler-20e and Kepler-20f, orbiting a Sun-like star, Kepler-20. A2012 study, analyzing gravitational microlensing data, estimates an average of at least 1.6 bound planets for every star in the Milky Way, around one in five Sun-like stars is thought to have an Earth-sized planet in its habitable zone. The idea of planets has evolved over its history, from the lights of antiquity to the earthly objects of the scientific age. The concept has expanded to include not only in the Solar System. The ambiguities inherent in defining planets have led to much scientific controversy, the five classical planets, being visible to the naked eye, have been known since ancient times and have had a significant impact on mythology, religious cosmology, and ancient astronomy. In ancient times, astronomers noted how certain lights moved across the sky, as opposed to the fixed stars, ancient Greeks called these lights πλάνητες ἀστέρες or simply πλανῆται, from which todays word planet was derived. In ancient Greece, China, Babylon, and indeed all pre-modern civilizations, it was almost universally believed that Earth was the center of the Universe and that all the planets circled Earth. The first civilization known to have a theory of the planets were the Babylonians
14.
Cartography
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Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively, the fundamental problems of traditional cartography are to, Set the maps agenda and select traits of the object to be mapped. This is the concern of map editing, traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. Represent the terrain of the object on flat media. This is the concern of map projections, eliminate characteristics of the mapped object that are not relevant to the maps purpose. This is the concern of generalization, reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization, orchestrate the elements of the map to best convey its message to its audience. This is the concern of map design, modern cartography constitutes many theoretical and practical foundations of geographic information systems. The earliest known map is a matter of debate, both because the term map isnt well-defined and because some artifacts that might be maps might actually be something else. A wall painting that might depict the ancient Anatolian city of Çatalhöyük has been dated to the late 7th millennium BCE, the oldest surviving world maps are from 9th century BCE Babylonia. One shows Babylon on the Euphrates, surrounded by Assyria, Urartu and several cities, all, in turn, another depicts Babylon as being north of the world center. The ancient Greeks and Romans created maps since Anaximander in the 6th century BCE, in the 2nd century AD, Ptolemy wrote his treatise on cartography, Geographia. This contained Ptolemys world map – the world known to Western society. As early as the 8th century, Arab scholars were translating the works of the Greek geographers into Arabic, in ancient China, geographical literature dates to the 5th century BCE. The oldest extant Chinese maps come from the State of Qin, dated back to the 4th century BCE, in the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song, a star map on the equidistant cylindrical projection. Early forms of cartography of India included depictions of the pole star and these charts may have been used for navigation. Mappa mundi are the Medieval European maps of the world, approximately 1,100 mappae mundi are known to have survived from the Middle Ages. Of these, some 900 are found illustrating manuscripts and the remainder exist as stand-alone documents, the Arab geographer Muhammad al-Idrisi produced his medieval atlas Tabula Rogeriana in 1154
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World Geodetic System
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The World Geodetic System is a standard for use in cartography, geodesy, and navigation including GPS. It comprises a standard system for the Earth, a standard spheroidal reference surface for raw altitude data. The latest revision is WGS84, established in 1984 and last revised in 2004, earlier schemes included WGS72, WGS66, and WGS60. WGS84 is the coordinate system used by the Global Positioning System. The coordinate origin of WGS84 is meant to be located at the Earths center of mass, the error is believed to be less than 2 cm. The WGS84 meridian of longitude is the IERS Reference Meridian,5.31 arc seconds or 102.5 metres east of the Greenwich meridian at the latitude of the Royal Observatory. The WGS84 datum surface is a spheroid with major radius a =6378137 m at the equator. The polar semi-minor axis b then equals a times, or 6356752.3142 m, currently, WGS84 uses the EGM96 geoid, revised in 2004. This geoid defines the sea level surface by means of a spherical harmonics series of degree 360. The deviations of the EGM96 geoid from the WGS84 reference ellipsoid range from about −105 m to about +85 m, EGM96 differs from the original WGS84 geoid, referred to as EGM84. Efforts to supplement the national surveying systems began in the 19th century with F. R. Helmerts famous book Mathematische und Physikalische Theorien der Physikalischen Geodäsie. Austria and Germany founded the Zentralbüro für die Internationale Erdmessung, a unified geodetic system for the whole world became essential in the 1950s for several reasons, International space science and the beginning of astronautics. The lack of inter-continental geodetic information, efforts of the U. S. Army, Navy and Air Force were combined leading to the DoD World Geodetic System 1960. Heritage surveying methods found elevation differences from a local horizontal determined by the level, plumb line. As a result, the elevations in the data are referenced to the geoid, the latter observational method is more suitable for global mapping. The sole contribution of data to the development of WGS60 was a value for the ellipsoid flattening which was obtained from the nodal motion of a satellite. Prior to WGS60, the U. S. Army, the Army performed an adjustment to minimize the difference between astro-geodetic and gravimetric geoids. By matching the relative astro-geodetic geoids of the selected datums with an earth-centered gravimetric geoid, since the Army and Air Force systems agreed remarkably well for the NAD, ED and TD areas, they were consolidated and became WGS60
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Equator
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The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north
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Geodesy
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Geodesists also study geodynamical phenomena such as crustal motion, tides, and polar motion. For this they design global and national networks, using space and terrestrial techniques while relying on datums. Geodesy — from the Ancient Greek word γεωδαισία geodaisia — is primarily concerned with positioning within the temporally varying gravity field, such geodetic operations are also applied to other astronomical bodies in the solar system. It is also the science of measuring and understanding the earths geometric shape, orientation in space and this applies to the solid surface, the liquid surface and the Earths atmosphere. For this reason, the study of the Earths gravity field is called physical geodesy by some, the geoid is essentially the figure of the Earth abstracted from its topographical features. It is an idealized surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. The geoid, unlike the ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between the geoid and the ellipsoid is called the geoidal undulation. It varies globally between ±110 m, when referred to the GRS80 ellipsoid, a reference ellipsoid, customarily chosen to be the same size as the geoid, is described by its semi-major axis a and flattening f. The quantity f = a − b/a, where b is the axis, is a purely geometrical one. The mechanical ellipticity of the Earth can be determined to high precision by observation of satellite orbit perturbations and its relationship with the geometrical flattening is indirect. The relationship depends on the density distribution, or, in simplest terms. The 1980 Geodetic Reference System posited a 6,378,137 m semi-major axis and this system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics. It is essentially the basis for geodetic positioning by the Global Positioning System and is also in widespread use outside the geodetic community. The locations of points in space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such systems are typically geocentric. The X-axis lies within the Greenwich observatorys meridian plane, the coordinate transformation between these two systems is described to good approximation by sidereal time, which takes into account variations in the Earths axial rotation. A more accurate description also takes polar motion into account, a closely monitored by geodesists
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Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
19.
Surface area
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The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
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Eccentricity (mathematics)
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In mathematics, the eccentricity, denoted e or ε, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular, in particular, The eccentricity of a circle is zero. The eccentricity of an ellipse which is not a circle is greater than zero, the eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1, the eccentricity of a line is infinite. Furthermore, two sections are similar if and only if they have the same eccentricity. Any conic section can be defined as the locus of points whose distances to a point and that ratio is called eccentricity, commonly denoted as e. The eccentricity can also be defined in terms of the intersection of a plane, for β =0 the plane section is a circle, for β = α a parabola. The linear eccentricity of an ellipse or hyperbola, denoted c, is the distance between its center and either of its two foci, the eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a, that is, e = c a. The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity, the eccentricity is also sometimes called numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation, three notational conventions are in common use, e for the eccentricity and c for the linear eccentricity. ε for the eccentricity and e for the linear eccentricity, E or ϵ for the eccentricity and f for the linear eccentricity. This article uses the first notation, where, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity of an ellipse is strictly less than 1, for any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis. The eccentricity is also the ratio of the axis a to the distance d from the center to the directrix. The eccentricity can be expressed in terms of the g, e = g. Define the maximum and minimum radii r max and r min as the maximum and minimum distances from either focus to the ellipse, the eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a hyperbola is 2. The eccentricity of a quadric is the eccentricity of a designated section of it
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Imaginary number
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An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2, for example, 5i is an imaginary number, and its square is −25. Zero is considered to be real and imaginary. Originally coined in the 17th century as a term and regarded as fictitious or useless. Some authors use the term pure imaginary number to denote what is called here an imaginary number, the concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as numbers, were poorly understood and regarded by some as fictitious or useless. The use of numbers was not widely accepted until the work of Leonhard Euler. The geometric significance of numbers as points in a plane was first described by Caspar Wessel. This idea first surfaced with the articles by James Cockle beginning in 1848, geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a number line, positively increasing in magnitude to the right. This vertical axis is called the imaginary axis and is denoted iℝ, I. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin, note that a 90-degree rotation in the negative direction also satisfies this interpretation. This reflects the fact that −i also solves the equation x2 = −1, in general, multiplying by a complex number is the same as rotating around the origin by the complex numbers argument, followed by a scaling by its magnitude. Care must be used when working with numbers expressed as the principal values of the square roots of negative numbers. For example,6 =36 = ≠ −4 −9 = =6 i 2 = −6, Imaginary unit de Moivres formula NaN Octonion Quaternion Nahin, Paul. An Imaginary Tale, the Story of the Square Root of −1, explains many applications of imaginary expressions. How can one show that imaginary numbers really do exist, – an article that discusses the existence of imaginary numbers. In our time, Imaginary numbers Discussion of imaginary numbers on BBC Radio 4, 5Numbers programme 4 BBC Radio 4 programme Why Use Imaginary Numbers
22.
Earth radius
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Earth radius is the distance from the Earths center to its surface, about 6,371 km. This length is used as a unit of distance, especially in astronomy and geology. This article deals primarily with spherical and ellipsoidal models of the Earth, see Figure of the Earth for a more complete discussion of the models. The Earth is only approximately spherical, so no single value serves as its natural radius, distances from points on the surface to the center range from 6,353 km to 6,384 km. Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 km. It can also mean some kind of average of such distances, Aristotle, writing in On the Heavens around 350 BC, reports that the mathematicians guess the circumference of the Earth to be 400,000 stadia. Due to uncertainty about which stadion variant Aristotle meant, scholars have interpreted Aristotles figure to be anywhere from highly accurate to almost double the true value, the first known scientific measurement and calculation of the radius of the Earth was performed by Eratosthenes about 240 BC. Estimates of the accuracy of Eratosthenes’s measurement range from within 0. 5% to within 17%, as with Aristotles report, uncertainty in the accuracy of his measurement is due to modern uncertainty over which stadion definition he used. Earths rotation, internal density variations, and external tidal forces cause its shape to deviate systematically from a perfect sphere, local topography increases the variance, resulting in a surface of profound complexity. Our descriptions of the Earths surface must be simpler than reality in order to be tractable, hence, we create models to approximate characteristics of the Earths surface, generally relying on the simplest model that suits the need. Each of the models in use involve some notion of the geometric radius. Strictly speaking, spheres are the solids to have radii. In the case of the geoid and ellipsoids, the distance from any point on the model to the specified center is called a radius of the Earth or the radius of the Earth at that point. It is also common to refer to any mean radius of a model as the radius of the earth. When considering the Earths real surface, on the hand, it is uncommon to refer to a radius. Rather, elevation above or below sea level is useful, regardless of the model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km. Hence, the Earth deviates from a sphere by only a third of a percent. While specific values differ, the concepts in this article generalize to any major planet
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Latitude
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In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earths surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles, lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the location of features on the surface of the Earth. Without qualification the term latitude should be taken to be the latitude as defined in the following sections. Also defined are six auxiliary latitudes which are used in special applications, there is a separate article on the History of latitude measurements. Two levels of abstraction are employed in the definition of latitude and longitude, in the first step the physical surface is modelled by the geoid, a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface, the simplest choice for the reference surface is a sphere, but the geoid is more accurately modelled by an ellipsoid. The definitions of latitude and longitude on such surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface, latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO19111 standard. This is of importance in accurate applications, such as a Global Positioning System, but in common usage, where high accuracy is not required. In English texts the latitude angle, defined below, is denoted by the Greek lower-case letter phi. It is measured in degrees, minutes and seconds or decimal degrees, the precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its field is the science of geodesy. These topics are not discussed in this article and this article relates to coordinate systems for the Earth, it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface, the plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. Planes parallel to the plane intersect the surface in circles of constant latitude. The Equator has a latitude of 0°, the North Pole has a latitude of 90° North, the latitude of an arbitrary point is the angle between the equatorial plane and the radius to that point. The latitude, as defined in this way for the sphere, is termed the spherical latitude
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Gaussian curvature
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In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point, K = κ1 κ2. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane, the Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is a measure of curvature, depending only on distances that are measured on the surface. This is the content of the Theorema egregium, Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827. At any point on a surface, we can find a vector that is at right angles to the surface. The intersection of a plane and the surface will form a curve called a normal section. For most points on most surfaces, different normal sections will have different curvatures, the Gaussian curvature is the product of the two principal curvatures Κ = κ1 κ2. The sign of the Gaussian curvature can be used to characterise the surface, if both principal curvatures are of the same sign, κ1κ2 >0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane, all sectional curvatures will have the same sign. If the principal curvatures have different signs, κ1κ2 <0, then the Gaussian curvature is negative, at such points, the surface will be saddle shaped. If one of the principal curvatures is zero, κ1κ2 =0, the Gaussian curvature is zero, most surfaces will contain regions of positive Gaussian curvature and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line. When a surface has a constant zero Gaussian curvature, then it is a developable surface, when a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry. When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface, in differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point. We represent the surface by the implicit function theorem as the graph of a function, f, then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f. This definition allows one immediately to grasp the distinction between cup/cap versus saddle point and it is also given by K = ⟨ e 1, e 2 ⟩ det g, where ∇ i = ∇ e i is the covariant derivative and g is the metric tensor. At a point p on a surface in R3, the Gaussian curvature is also given by K = det. A useful formula for the Gaussian curvature is Liouvilles equation in terms of the Laplacian in isothermal coordinates, the surface integral of the Gaussian curvature over some region of a surface is called the total curvature
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Mean curvature
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The concept was introduced by Sophie Germain in her work on elasticity theory. Let p be a point on the surface S, each plane through p containing the normal line to S cuts S in a curve. Fixing a choice of unit normal gives a signed curvature to that curve, as the plane is rotated by an angle θ that curvature can vary. The maximal curvature κ1 and minimal curvature κ2 are known as the principal curvatures of S, the mean curvature at p ∈ S is then the average of the signed curvature over all angles θ, H =12 π ∫02 π κ d θ. By applying Eulers theorem, this is equal to the average of the principal curvatures, more generally, for a hypersurface T the mean curvature is given as H =1 n ∑ i =1 n κ i. More abstractly, the curvature is the trace of the second fundamental form divided by n. A surface is a surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the curvature of the surface S, is said to obey a heat-type equation called the mean curvature flow equation. The sphere is the embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the embedded surface is weakened to immersed surface. For a surface defined in 3D space, the curvature is related to a unit normal of the surface,2 H = − ∇ ⋅ n ^ where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal, the formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean Curvature may also be calculated 2 H = Trace where I and II denote first and second quadratic form matrices, in particular at a point where ∇ S =0, the mean curvature is half the trace of the Hessian matrix of S. The mean curvature of a surface specified by an implicit equation F =0 can be calculated by using the gradient ∇ F = and the Hessian matrix Hess =. The mean curvature is given by, H = ∇ F Hess ∇ F T − | ∇ F |2 Trace 2 | ∇ F |3 Another form is as the divergence of the unit normal. A unit normal is given by ∇ F | ∇ F |, an alternate definition is occasionally used in fluid mechanics to avoid factors of two, H f =. A minimal surface is a surface which has zero mean curvature at all points, classic examples include the catenoid, helicoid and Enneper surface. Recent discoveries include Costas minimal surface and the Gyroid, an extension of the idea of a minimal surface are surfaces of constant mean curvature
26.
Aspect ratio
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The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. For example, the ratio of a rectangle is the ratio of its longer side to its shorter side – the ratio of width to height. The aspect ratio is expressed as two separated by a colon. The values x and y do not represent actual widths and heights but, rather, as an example,8,5,16,10 and 1.6,1 are three ways of representing the same aspect ratio. In objects of more than two dimensions, such as hyperrectangles, the ratio can still be defined as the ratio of the longest side to the shortest side. The term is most commonly used reference to, Graphic / image Image aspect ratio Display aspect ratio. A square has the smallest possible ratio of 1,1. An ellipse with a ratio of 1,1 is a circle. A circle has the minimal DWAR which is 1, a square has a DWAR of sqrt. The Cube-Volume Aspect Ratio of a set is the d-th root of the ratio of the d-volume of the smallest enclosing axes-parallel d-cube. A square has the minimal CVAR which is 1, a circle has a CVAR of sqrt. An axis-parallel rectangle of width W and height H, where W>H, has a CVAR of sqrt = sqrt, if the dimension d is fixed, then all reasonable definitions of aspect ratio are equivalent to within constant factors. Aspect ratios are mathematically expressed as x, y, in digital images there is a subtle distinction between the Display Aspect Ratio and the Storage Aspect Ratio, see Distinctions. Ratio Equidimensional ratios in 3D List of film formats Squeeze mapping Vertical orientation
27.
Flattening
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Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution respectively. Other terms used are ellipticity, or oblateness, the usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is f l a t t e n i n g = f = a − b a. The compression factor is b/a in each case, for the ellipse, this factor is also the aspect ratio of the ellipse. There are two variants of flattening and when it is necessary to avoid confusion the above flattening is called the first flattening. The following definitions may be found in texts and online web texts In the following. All flattenings are zero for a circle, the flattenings are related to other parameters of the ellipse. For example, b = a = a, e 2 =2 f − f 2 =4 n 2, other values in the Solar System are Jupiter, f=1/16, Saturn, f= 1/10, the Moon f= 1/900. The flattening of the Sun is about 9×10−6, in 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid of revolution. The amount of flattening depends on the density and the balance of gravitational force, astronomy Earth ellipsoid Earths rotation Eccentricity Equatorial bulge Gravitational field Gravity formula Ovality Planetology Sphericity Roundness
28.
Atomic nucleus
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After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg. Almost all of the mass of an atom is located in the nucleus, protons and neutrons are bound together to form a nucleus by the nuclear force. The diameter of the nucleus is in the range of 6985175000000000000♠1.75 fm for hydrogen to about 6986150000000000000♠15 fm for the heaviest atoms and these dimensions are much smaller than the diameter of the atom itself, by a factor of about 23,000 to about 145,000. The branch of physics concerned with the study and understanding of the nucleus, including its composition. The nucleus was discovered in 1911, as a result of Ernest Rutherfords efforts to test Thomsons plum pudding model of the atom, the electron had already been discovered earlier by J. J. Knowing that atoms are electrically neutral, Thomson postulated that there must be a charge as well. In his plum pudding model, Thomson suggested that an atom consisted of negative electrons randomly scattered within a sphere of positive charge, to his surprise, many of the particles were deflected at very large angles. This justified the idea of an atom with a dense center of positive charge. The term nucleus is from the Latin word nucleus, a diminutive of nux, in 1844, Michael Faraday used the term to refer to the central point of an atom. The modern atomic meaning was proposed by Ernest Rutherford in 1912, the adoption of the term nucleus to atomic theory, however, was not immediate. In 1916, for example, Gilbert N, the nuclear strong force extends far enough from each baryon so as to bind the neutrons and protons together against the repulsive electrical force between the positively charged protons. The nuclear strong force has a short range, and essentially drops to zero just beyond the edge of the nucleus. The collective action of the charged nucleus is to hold the electrically negative charged electrons in their orbits about the nucleus. The collection of negatively charged electrons orbiting the nucleus display an affinity for certain configurations, which chemical element an atom represents is determined by the number of protons in the nucleus, the neutral atom will have an equal number of electrons orbiting that nucleus. Individual chemical elements can create more stable electron configurations by combining to share their electrons and it is that sharing of electrons to create stable electronic orbits about the nucleus that appears to us as the chemistry of our macro world. Protons define the entire charge of a nucleus, and hence its chemical identity, neutrons are electrically neutral, but contribute to the mass of a nucleus to nearly the same extent as the protons. Neutrons explain the phenomenon of isotopes – varieties of the chemical element which differ only in their atomic mass. They are sometimes viewed as two different quantum states of the particle, the nucleon
29.
Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
30.
Electromagnetic force
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Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as fields, magnetic fields. The other three fundamental interactions are the interaction, the weak interaction, and gravitation. The word electromagnetism is a form of two Greek terms, ἤλεκτρον, ēlektron, amber, and μαγνῆτις λίθος magnētis lithos, which means magnesian stone. The electromagnetic force plays a role in determining the internal properties of most objects encountered in daily life. Ordinary matter takes its form as a result of forces between individual atoms and molecules in matter, and is a manifestation of the electromagnetic force. Electrons are bound by the force to atomic nuclei, and their orbital shapes. The electromagnetic force governs the processes involved in chemistry, which arise from interactions between the electrons of neighboring atoms, there are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential, although electromagnetism is considered one of the four fundamental forces, at high energy the weak force and electromagnetic force are unified as a single electroweak force. In the history of the universe, during the epoch the unified force broke into the two separate forces as the universe cooled. Originally, electricity and magnetism were considered to be two separate forces, Magnetic poles attract or repel one another in a manner similar to positive and negative charges and always exist as pairs, every north pole is yoked to a south pole. An electric current inside a wire creates a corresponding magnetic field outside the wire. Its direction depends on the direction of the current in the wire. A current is induced in a loop of wire when it is moved toward or away from a field, or a magnet is moved towards or away from it. While preparing for a lecture on 21 April 1820, Hans Christian Ørsted made a surprising observation. As he was setting up his materials, he noticed a compass needle deflected away from north when the electric current from the battery he was using was switched on. At the time of discovery, Ørsted did not suggest any explanation of the phenomenon. However, three later he began more intensive investigations
31.
Surface tension
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Surface tension is the elastic tendency of a fluid surface which makes it acquire the least surface area possible. Surface tension allows insects, usually denser than water, to float, at liquid-air interfaces, surface tension results from the greater attraction of liquid molecules to each other than to the molecules in the air. The net effect is a force at its surface that causes the liquid to behave as if its surface were covered with a stretched elastic membrane. Thus, the surface becomes under tension from the imbalanced forces, because of the relatively high attraction of water molecules for each other through a web of hydrogen bonds, water has a higher surface tension compared to that of most other liquids. Surface tension is an important factor in the phenomenon of capillarity, Surface tension has the dimension of force per unit length, or of energy per unit area. The two are equivalent, but when referring to energy per unit of area, it is common to use the surface energy. In materials science, surface tension is used for either surface stress or surface free energy, the cohesive forces among liquid molecules are responsible for the phenomenon of surface tension. In the bulk of the liquid, each molecule is pulled equally in every direction by neighboring liquid molecules, the molecules at the surface do not have the same molecules on all sides of them and therefore are pulled inwards. This creates some internal pressure and forces liquid surfaces to contract to the minimal area, Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of water tend to be pulled into a shape by the imbalance in cohesive forces of the surface layer. In the absence of forces, including gravity, drops of virtually all liquids would be approximately spherical. The spherical shape minimizes the necessary wall tension of the surface according to Laplaces law. Another way to view surface tension is in terms of energy, a molecule in contact with a neighbor is in a lower state of energy than if it were alone. The interior molecules have as many neighbors as they can possibly have, for the liquid to minimize its energy state, the number of higher energy boundary molecules must be minimized. The minimized quantity of boundary molecules results in a surface area. As a result of surface area minimization, a surface will assume the smoothest shape it can, since any curvature in the surface shape results in greater area, a higher energy will also result. Consequently, the surface will push back against any curvature in much the way as a ball pushed uphill will push back to minimize its gravitational potential energy. Bubbles in pure water are unstable, the addition of surfactants, however, can have a stabilizing effect on the bubbles
32.
Quantum mechanics
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Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms, in one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. This phase is known as old quantum theory, according to Planck, each energy element is proportional to its frequency, E = h ν, where h is Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, lower energy/frequency means increased time and vice versa, photons of differing frequencies all deliver the same amount of action, but do so in varying time intervals. High frequency waves are damaging to human tissue because they deliver their action packets concentrated in time, the Copenhagen interpretation of Niels Bohr became widely accepted. In the mid-1920s, developments in mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons
33.
Nuclear shell model
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The first shell model was proposed by Dmitry Ivanenko in 1932. The shell model is analogous to the atomic shell model which describes the arrangement of electrons in an atom. When adding nucleons to a nucleus, there are points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are certain magic numbers of nucleons,2,8,20,28,50,82,126 which are tightly bound than the next higher number, is the origin of the shell model. The shells for protons and for neutrons are independent of each other, therefore, one can have magic nuclei where one nucleon type or the other is at a magic number, and doubly magic nuclei, where both are. Some semimagic numbers have been found, notably Z=40 giving nuclear shell filling for the elements,16 may also be a magic number. In order to get these numbers, the shell model starts from an average potential with a shape something between the square well and the harmonic oscillator. To this potential a spin orbit term is added, nevertheless, the magic numbers of nucleons, as well as other properties, can be arrived at by approximating the model with a three-dimensional harmonic oscillator plus a spin-orbit interaction. A more realistic but also complicated potential is known as Woods Saxon potential and this would give, for example, in the first two levels We can imagine ourselves building a nucleus by adding protons and neutrons. These will always fill the lowest available level, thus the first two protons fill level zero, the next six protons fill level one, and so on. Therefore nuclei which have a full outer shell will have a higher binding energy than other nuclei with a similar total number of protons. All this is true for neutrons as well and this means that the magic numbers are expected to be those in which all occupied shells are full. We see that for the first two numbers we get 2 and 8, in accord with experiment, however the full set of magic numbers does not turn out correctly. These can be computed as follows, In a three-dimensional harmonic oscillator the total degeneracy at level n is 2, due to the spin, the degeneracy is doubled and is. Thus the magic numbers would be ∑ n =0 k =3 for all integer k and this gives the following magic numbers,2,8,20,40,70,112. Which agree with experiment only in the first three entries and these numbers are twice the tetrahedral numbers from the Pascal Triangle. In particular, the first six shells are, level 0,2 states =2, level 2,2 states +10 states =12. Level 3,6 states +14 states =20, level 4,2 states +10 states +18 states =30
34.
Jupiter
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Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a giant planet with a mass one-thousandth that of the Sun, Jupiter and Saturn are gas giants, the other two giant planets, Uranus and Neptune are ice giants. Jupiter has been known to astronomers since antiquity, the Romans named it after their god Jupiter. Jupiter is primarily composed of hydrogen with a quarter of its mass being helium and it may also have a rocky core of heavier elements, but like the other giant planets, Jupiter lacks a well-defined solid surface. Because of its rotation, the planets shape is that of an oblate spheroid. The outer atmosphere is visibly segregated into several bands at different latitudes, resulting in turbulence, a prominent result is the Great Red Spot, a giant storm that is known to have existed since at least the 17th century when it was first seen by telescope. Surrounding Jupiter is a faint planetary ring system and a powerful magnetosphere, Jupiter has at least 67 moons, including the four large Galilean moons discovered by Galileo Galilei in 1610. Ganymede, the largest of these, has a greater than that of the planet Mercury. Jupiter has been explored on several occasions by robotic spacecraft, most notably during the early Pioneer and Voyager flyby missions and later by the Galileo orbiter. In late February 2007, Jupiter was visited by the New Horizons probe, the latest probe to visit the planet is Juno, which entered into orbit around Jupiter on July 4,2016. Future targets for exploration in the Jupiter system include the probable ice-covered liquid ocean of its moon Europa, Earth and its neighbor planets may have formed from fragments of planets after collisions with Jupiter destroyed those super-Earths near the Sun. Astronomers have discovered nearly 500 planetary systems with multiple planets, Jupiter moving out of the inner Solar System would have allowed the formation of inner planets, including Earth. Jupiter is composed primarily of gaseous and liquid matter and it is the largest of the four giant planets in the Solar System and hence its largest planet. It has a diameter of 142,984 km at its equator, the average density of Jupiter,1.326 g/cm3, is the second highest of the giant planets, but lower than those of the four terrestrial planets. Jupiters upper atmosphere is about 88–92% hydrogen and 8–12% helium by percent volume of gas molecules, a helium atom has about four times as much mass as a hydrogen atom, so the composition changes when described as the proportion of mass contributed by different atoms. Thus, Jupiters atmosphere is approximately 75% hydrogen and 24% helium by mass, the atmosphere contains trace amounts of methane, water vapor, ammonia, and silicon-based compounds. There are also traces of carbon, ethane, hydrogen sulfide, neon, oxygen, phosphine, the outermost layer of the atmosphere contains crystals of frozen ammonia. The interior contains denser materials - by mass it is roughly 71% hydrogen, 24% helium, through infrared and ultraviolet measurements, trace amounts of benzene and other hydrocarbons have also been found
35.
Astronomical object
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An astronomical object or celestial object is a naturally occurring physical entity, association, or structure that current astronomy has demonstrated to exist in the observable universe. In astronomy, the object and body are often used interchangeably. Examples for astronomical objects include planetary systems, star clusters, nebulae and galaxies, while asteroids, moons, planets, and stars are astronomical bodies. A comet may be identified as both body and object, It is a body when referring to the nucleus of ice and dust. The universe can be viewed as having a hierarchical structure, at the largest scales, the fundamental component of assembly is the galaxy. Galaxies are organized groups and clusters, often within larger superclusters. Disc galaxies encompass lenticular and spiral galaxies with features, such as spiral arms, at the core, most galaxies have a supermassive black hole, which may result in an active galactic nucleus. Galaxies can also have satellites in the form of dwarf galaxies, the constituents of a galaxy are formed out of gaseous matter that assembles through gravitational self-attraction in a hierarchical manner. At this level, the fundamental components are the stars. The great variety of forms are determined almost entirely by the mass, composition. Stars may be found in systems that orbit about each other in a hierarchical organization. A planetary system and various objects such as asteroids, comets and debris. The various distinctive types of stars are shown by the Hertzsprung–Russell diagram —a plot of stellar luminosity versus surface temperature. Each star follows a track across this diagram. If this track takes the star through a region containing a variable type. An example of this is the instability strip, a region of the H-R diagram that includes Delta Scuti, RR Lyrae, the table below lists the general categories of bodies and objects by their location or structure. International Astronomical Naming Commission List of light sources List of Solar System objects Lists of astronomical objects SkyChart, Sky & Telescope Monthly skymaps for every location on Earth
36.
Saturn
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Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with a radius about nine times that of Earth. Although it has only one-eighth the average density of Earth, with its larger volume Saturn is just over 95 times more massive, Saturn is named after the Roman god of agriculture, its astronomical symbol represents the gods sickle. Saturns interior is composed of a core of iron–nickel and rock. This core is surrounded by a layer of metallic hydrogen, an intermediate layer of liquid hydrogen and liquid helium. Saturn has a yellow hue due to ammonia crystals in its upper atmosphere. Saturns magnetic field strength is around one-twentieth of Jupiters, the outer atmosphere is generally bland and lacking in contrast, although long-lived features can appear. Wind speeds on Saturn can reach 1,800 km/h, higher than on Jupiter, sixty-two moons are known to orbit Saturn, of which fifty-three are officially named. This does not include the hundreds of moonlets comprising the rings, Saturn is a gas giant because it is predominantly composed of hydrogen and helium. It lacks a definite surface, though it may have a solid core, Saturns rotation causes it to have the shape of an oblate spheroid, that is, it is flattened at the poles and bulges at its equator. Its equatorial and polar radii differ by almost 10%,60,268 km versus 54,364 km, Jupiter, Uranus, and Neptune, the other giant planets in the Solar System, are also oblate but to a lesser extent. Saturn is the planet of the Solar System that is less dense than water—about 30% less. Although Saturns core is considerably denser than water, the specific density of the planet is 0.69 g/cm3 due to the atmosphere. Jupiter has 318 times the Earths mass, while Saturn is 95 times the mass of the Earth, together, Jupiter and Saturn hold 92% of the total planetary mass in the Solar System. On 8 January 2015, NASA reported determining the center of the planet Saturn, the temperature, pressure, and density inside Saturn all rise steadily toward the core, which causes hydrogen to transition into a metal in the deeper layers. Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a rocky core surrounded by hydrogen. This core is similar in composition to the Earth, but more dense, in 2004, they estimated that the core must be 9–22 times the mass of the Earth, which corresponds to a diameter of about 25,000 km. This is surrounded by a liquid metallic hydrogen layer, followed by a liquid layer of helium-saturated molecular hydrogen that gradually transitions to a gas with increasing altitude
37.
Altair
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Altair, also designated Alpha Aquilae, is the brightest star in the constellation of Aquila and the twelfth brightest star in the night sky. It is currently in the G-cloud—a nearby accumulation of gas and dust known as an interstellar cloud, Altair is an A-type main sequence star with an apparent visual magnitude of 0.77 and is one of the vertices of the asterism known as the Summer Triangle. It is 16.7 light-years from the Sun and is one of the closest stars visible to the naked eye, Altair rotates rapidly, with a velocity at the equator of approximately 286 km/s. This is a significant fraction of the estimated breakup speed of 400 km/s. A study with the Palomar Testbed Interferometer revealed that Altair is not spherical, other interferometric studies with multiple telescopes, operating in the infrared, have imaged and confirmed this phenomenon. α Aquilae is the stars Bayer designation, the traditional name Altair has been used since medieval times. It is an abbreviation of the Arabic phrase النسر الطائر, al-nesr al-ṭā’ir, in 2016, the International Astronomical Union organized a Working Group on Star Names to catalog and standardize proper names for stars. The WGSNs first bulletin of July 2016 included a table of the first two batches of names approved by the WGSN, which included Altair for this star and it is now so entered in the IAU Catalog of Star Names. Along with Beta Aquilae and Gamma Aquilae, Altair forms the line of stars sometimes referred to as the Family of Aquila or Shaft of Aquila. Altair is a main sequence star with approximately 1.8 times the mass of the Sun and 11 times its luminosity. Altair possesses an extremely rapid rate of rotation, it has a period of approximately 9 hours. For comparison, the equator of the Sun requires a more than 25 days for a complete rotation. This rapid rotation forces Altair to be oblate, its diameter is over 20 percent greater than its polar diameter. As a result, it was identified in 2005 as a Delta Scuti variable star and its light curve can be approximated by adding together a number of sine waves, with periods that range between 0.8 and 1.5 hours. It is a source of coronal X-ray emission, with the most active sources of emission being located near the stars equator. This activity may be due to convection cells forming at the cooler equator, the angular diameter of Altair was measured interferometrically by R. Hanbury Brown and his co-workers at Narrabri Observatory in the 1960s. They found a diameter of 3 milliarcseconds, although Hanbury Brown et al. realized that Altair would be rotationally flattened, they had insufficient data to experimentally observe its oblateness. Altair was later observed to be flattened by infrared interferometric measurements made by the Palomar Testbed Interferometer in 1999 and 2000 and this work was published by G. T. van Belle, David R. Ciardi and their co-authors in 2001
38.
Christiaan Huygens
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Christiaan Huygens, FRS was a prominent Dutch mathematician and scientist. He is known particularly as an astronomer, physicist, probabilist and horologist, Huygens was a leading scientist of his time. His work included early telescopic studies of the rings of Saturn and the discovery of its moon Titan and he published major studies of mechanics and optics, and pioneered work on games of chance. Christiaan Huygens was born on 14 April 1629 in The Hague, into a rich and influential Dutch family, Christiaan was named after his paternal grandfather. His mother was Suzanna van Baerle and she died in 1637, shortly after the birth of Huygens sister. The couple had five children, Constantijn, Christiaan, Lodewijk, Philips, Constantijn Huygens was a diplomat and advisor to the House of Orange, and also a poet and musician. His friends included Galileo Galilei, Marin Mersenne and René Descartes, Huygens was educated at home until turning sixteen years old. He liked to play with miniatures of mills and other machines and his father gave him a liberal education, he studied languages and music, history and geography, mathematics, logic and rhetoric, but also dancing, fencing and horse riding. In 1644 Huygens had as his mathematical tutor Jan Jansz de Jonge Stampioen, Descartes was impressed by his skills in geometry. His father sent Huygens to study law and mathematics at the University of Leiden, Frans van Schooten was an academic at Leiden from 1646, and also a private tutor to Huygens and his elder brother, replacing Stampioen on the advice of Descartes. Van Schooten brought his mathematical education up to date, in introducing him to the work of Fermat on differential geometry. Constantijn Huygens was closely involved in the new College, which lasted only to 1669, Christiaan Huygens lived at the home of the jurist Johann Henryk Dauber, and had mathematics classes with the English lecturer John Pell. He completed his studies in August 1649 and he then had a stint as a diplomat on a mission with Henry, Duke of Nassau. It took him to Bentheim, then Flensburg and he took off for Denmark, visited Copenhagen and Helsingør, and hoped to cross the Øresund to visit Descartes in Stockholm. While his father Constantijn had wished his son Christiaan to be a diplomat, in political terms, the First Stadtholderless Period that began in 1650 meant that the House of Orange was not in power, removing Constantijns influence. Further, he realised that his son had no interest in such a career, Huygens generally wrote in French or Latin. While still a student at Leiden he began a correspondence with the intelligencer Mersenne. Mersenne wrote to Constantijn on his sons talent for mathematics, the letters show the early interests of Huygens in mathematics
39.
Earth
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Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
40.
Centrifugal force
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In Newtonian mechanics, the centrifugal force is an inertial force directed away from the axis of rotation that appears to act on all objects when viewed in a rotating reference frame. When they are analyzed in a coordinate system. The term has also been used for the force that is a reaction to a centripetal force. The centrifugal force is an outward force apparent in a reference frame. All measurements of position and velocity must be relative to some frame of reference. An inertial frame of reference is one that is not accelerating, the use of an inertial frame of reference, which will be the case for all elementary calculations, is often not explicitly stated but may generally be assumed unless stated otherwise. In terms of a frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newtons laws of motion, in its current usage the term centrifugal force has no meaning in an inertial frame. In an inertial frame, an object that has no acting on it travels in a straight line. When measurements are made with respect to a reference frame, however. If it is desired to apply Newtons laws in the frame, it is necessary to introduce new, fictitious. Consider a stone being whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is the tension in the string. There are no forces acting on the stone so there is a net force on the stone in the horizontal plane. In an inertial frame of reference, were it not for this net force acting on the stone, in order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed the stone moves in a straight line, in this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newtons laws of motion. In a frame of reference rotating with the stone around the axis as the stone. However, the tension in the string is still acting on the stone, if Newtons laws were applied in their usual form, the stone would accelerate in the direction of the net applied force, towards the axis of rotation, which it does not do. With this new the net force on the stone is zero, with the addition of this extra inertial or fictitious force Newtons laws can be applied in the rotating frame as if it were an inertial frame
41.
Reference ellipsoid
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In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Current practice uses the word alone in preference to the full term oblate ellipsoid of revolution or the older term oblate spheroid. In the rare instances where a more general shape is required as a model the term used is triaxial ellipsoid. A great many ellipsoids have been used with various sizes and centres, the shape of an ellipsoid is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated about its minor axis. The semi-major axis of the ellipse, a, is identified as the radius of the ellipsoid. For the Earth, f is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21 km, some precise values are given in the table below and also in Figure of the Earth. A great many other parameters are used in geodesy but they can all be related to one or two of the set a, b and f, a primary use of reference ellipsoids is to serve as a basis for a coordinate system of latitude, longitude, and elevation. For this purpose it is necessary to identify a zero meridian, for other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater Airy-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid, the longitude measures the rotational angle between the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed in degrees ranging from −180° to +180° For other bodies a range of 0° to 360° is used. The latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, the common or geodetic latitude is the angle between the equatorial plane and a line that is normal to the reference ellipsoid. Depending on the flattening, it may be different from the geocentric latitude. For non-Earth bodies the terms planetographic and planetocentric are used instead, see geodetic system for more detail. If these coordinates, i. e. N is the radius of curvature in the prime vertical, in contrast, extracting φ, λ and h from the rectangular coordinates usually requires iteration. A straightforward method is given in an OSGB publication and also in web notes, more sophisticated methods are outlined in geodetic system. Currently the most common reference used, and that used in the context of the Global Positioning System, is the one defined by WGS84. Traditional reference ellipsoids or geodetic datums are defined regionally and therefore non-geocentric, modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e. g. WGS84. Reference ellipsoids are also useful for mapping of other planetary bodies including planets, their satellites, asteroids
42.
Science fiction
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Science fiction often explores the potential consequences of scientific and other innovations, and has been called a literature of ideas. Science fiction is difficult to define, as it includes a range of subgenres and themes. Author and editor Damon Knight summed up the difficulty, saying science fiction is what we point to when we say it, a definition echoed by author Mark C. Glassy, who argues that the definition of science fiction is like the definition of pornography, you do not know what it is, in 1970 or 1971William Atheling Jr. According to science fiction writer Robert A, rod Serlings definition is fantasy is the impossible made probable. Science fiction is the improbable made possible, Science fiction is largely based on writing rationally about alternative possible worlds or futures. Science fiction elements include, A time setting in the future, in alternative timelines, a spatial setting or scenes in outer space, on other worlds, or on subterranean earth. Characters that include aliens, mutants, androids, or humanoid robots, futuristic or plausible technology such as ray guns, teleportation machines, and humanoid computers. Scientific principles that are new or that contradict accepted physical laws, for time travel, wormholes. New and different political or social systems, e. g. utopian, dystopian, post-scarcity, paranormal abilities such as mind control, telepathy, telekinesis Other universes or dimensions and travel between them. A product of the budding Age of Reason and the development of science itself. Isaac Asimov and Carl Sagan considered Keplers work the first science fiction story and it depicts a journey to the Moon and how the Earths motion is seen from there. Later, Edgar Allan Poe wrote a story about a flight to the moon, more examples appeared throughout the 19th century. Wells The War of the Worlds describes an invasion of late Victorian England by Martians using tripod fighting machines equipped with advanced weaponry and it is a seminal depiction of an alien invasion of Earth. In the late 19th century, the scientific romance was used in Britain to describe much of this fiction. This produced additional offshoots, such as the 1884 novella Flatland, the term would continue to be used into the early 20th century for writers such as Olaf Stapledon. In the early 20th century, pulp magazines helped develop a new generation of mainly American SF writers, influenced by Hugo Gernsback, the founder of Amazing Stories magazine. In 1912 Edgar Rice Burroughs published A Princess of Mars, the first of his series of Barsoom novels, situated on Mars
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Hal Clement
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Harry Clement Stubbs, better known by the pen name Hal Clement, was an American science fiction writer and a leader of the hard science fiction subgenre. He also painted astronomically oriented artworks under the name George Richard, in 1998 Clement was inducted by the Science Fiction and Fantasy Hall of Fame and named the 17th SFWA Grand Master by the Science Fiction and Fantasy Writers of America. Stubbs was born in Somerville, Massachusetts and died in Milton and he went to Harvard, graduating with a B. S. in astronomy in 1943. While there he wrote his first published story, Proof, which appeared in the June 1942 issue of Astounding Science Fiction, edited by John W. Campbell and his further educational background includes an M. Ed. and M. S. in chemistry. During World War II Clement was a pilot and copilot of a B-24 Liberator and flew 35 combat missions over Europe with the 68th Bomb Squadron, 44th Bomb Group, based in England with 8th Air Force. After the war, he served in the United States Air Force Reserve and he taught chemistry and astronomy for many years at Milton Academy in Milton, Massachusetts. The latter novel features a land and sea expedition across the superjovian planet Mesklin to recover a stranded scientific probe, the natives of Mesklin are centipede-like intelligent beings about 50 centimeters long. Clements article Whirligig World describes his approach to writing a fiction story, Writing a science fiction story is fun. He rules must be quite simple, for the author, the rule is to make as few such slips as he possibly can. Certain exceptions are made, but fair play demands that all such matters be mentioned as early as possible in the story. Clement was a frequent guest at science fiction conventions, especially in the eastern United States, Clement died in Milton Hospital at the age of 81. He died in his sleep, most likely due to complications of diabetes, for the 1945 short story Uncommon Sense he received a 50-year Retro Hugo Award at the 1996 World Science Fiction Convention. The Hal Clement Award for Young Adults for Excellence in Childrens Science Fiction Literature is presented in his memory at Worldcon each year, wayne Barlowe illustrated two of Clements fictional species, the Abyormenites and the Mesklinites, in his Barlowes Guide to Extraterrestrials. Compared with contemporary science fiction authors like Isaac Asimov or Poul Anderson and those that he created as settings include a number of unusual worlds. They include, Abyormen – A planet circling a dwarf star and this produces a hot and a cold season, each of 65 years duration. The native intelligent life forms undergo a seasonal mass death, dhrawn – A high-gravity world settled by Mesklinites in Star Light. Habranha - A planet that is locked with its sun, such that the dark side is a mix of solid CO2, solid methane, and ice. Hekla – An ice-age planet in Cold Front, kaihapa – An uninhabited ocean planet, twin of Kainui, in Noise
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Mission of Gravity
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Mission of Gravity is a science fiction novel by Hal Clement. The novel was serialized in Astounding Science Fiction magazine in April–July 1953 and its first hardcover book publication was in 1954, and it was first published as a paperback book in 1958. Along with the novel itself, many editions of the also include Whirligig World. Clement published three sequels to Mission of Gravity, a 1970 novel called Star Light, a 1973 short story called Lecture Demonstration, Mission of Gravity was nominated for a Retro Hugo Award for the year 1954. The story is set on an oblate planet named Mesklin. The story is told from the points of view of one of the intelligent life forms. The locals are centipede-like, in order to withstand the enormous gravity, the native protagonist, Barlennan, captain of the Bree, is on a trading expedition to the equator, where the gravity is a tiny fraction of what his culture is used to. Communication is achieved through an audio-visual radio built to function in a high-gravity environment and they are captured by various lifeforms similar to themselves, but who live in the lower-gravity areas and have developed projectile weapons and gliders. Gradually, with help, they gain an understanding of these. Mission was the runner-up for the 1955 International Fantasy Award for fiction, boucher and McComas found Mission compact and unified, with a good deal of adventurous excitement and characterized it as a splendid specimen of science fiction in the grandest of grand manners. Wayne Barlowe illustrated the Mesklinites in his Barlowes Guide to Extraterrestrials, a significant theme of the novel is the universality of physical law, regardless of the exotic nature of the location, the underlying rules of the universe are constant. A secondary theme deals with the method, and the necessity of proceeding one step at a time. It is most often praised for the thoroughness and care with which Clement designed and described Mesklin — even today, it is considered one of the definitive examples of worldbuilding