plane

plane, in mathematics, flat surface of infinite extent but no thickness. An example of a plane, or more exactly of a bounded portion of a plane, is the surface forming one face, or side, of a cube. A plane is determined, or defined, by any of the following: (1) three points not in a straight line; (2) a straight line and a point not on the line; (3) two intersecting lines; or (4) two parallel lines. Two straight lines in space do not usually lie in the same plane. For a given plane in space, a line can either lie outside and parallel to it, intersect the plane in a single point, or lie entirely in the plane; if more than one point of a straight line lies in the plane, then the entire line must lie in the plane.

Fixed-wing aircraft that is heavier than air, propelled by a screw propeller or a high-velocity jet, and supported by the dynamic reaction of the air against its wings. An airplane's essential components are the body or fuselage, a flight-sustaining wing system, stabilizing tail surfaces, altitude-control devices such as rudders, a thrust-providing power source, and a landing support system. Beginning in the 1840s, several British and French inventors produced designs for engine-powered aircraft, but the first powered, sustained, and controlled flight was only achieved by Wilbur and Orville Wright in 1903. Later airplane design was affected by the development of the jet engine; most airplanes today have a long nose section, swept-back wings with jet engines placed behind the plane's midsection, and a tail stabilizing section. Most airplanes are designed to operate from land; seaplanes are adapted to touch down on water, and carrier-based planes are modified for high-speed short takeoff and landing. Seealso airfoil; aviation; glider; helicopter.

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Any of ten species of large trees that make up the genus Platanus, sole genus of the family Platanaceae, native to North America, eastern Europe, and Asia. Plane trees are planted widely in cities for their resistance to diseases and to air pollution and because they grow rapidly and furnish quick shade. They are characterized by scaling bark; large, deciduous, usually lobed leaves; and globular heads of flower and seed. Ball-shaped smooth or bristly seed clusters, which dangle singly and often persist after leaf fall, are key identifiers. Winter bark is patchy and picturesque; as the outer bark flakes off, inner bark shows shades of white, gray, green, and yellow.

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Any theory of the nature of geometric space differing from the traditional view held since Euclid's time. These geometries arose in the 19th century when several mathematicians working independently explored the possibility of rejecting Euclid's parallel postulate. Different assumptions about how many lines through a point not on a given line could be parallel to that line resulted in hyperbolic geometry and elliptic geometry. Mathematicians were forced to abandon the idea of a single correct geometry; it became their task not to discover mathematical systems but to create them by selecting consistent axioms and studying the theorems that could be derived from them. The development of these alternative geometries had a profound impact on the notion of space and paved the way for the theory of relativity. Seealso Nikolay Lobachevsky, Bernhard Riemann.

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Study of points, lines, angles, surfaces, and solids based on Euclid's axioms. Its importance lies less in its results than in the systematic method Euclid used to develop and present them. This axiomatic method has been the model for many systems of rational thought, even outside mathematics, for over 2,000 years. From 10 axioms and postulates, Euclid deduced 465 theorems, or propositions, concerning aspects of plane and solid geometric figures. This work was long held to constitute an accurate description of the physical world and to provide a sufficient basis for understanding it. During the 19th century, rejection of some of Euclid's postulates resulted in two non-Euclidean geometries that proved just as valid and consistent.

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Music composed to accompany a play. The practice dates back to ritualistic Greek drama, and it is thus connected to the use of music in other kinds of ritual. Sometimes limited to the role of introduction or interlude (setting a mood or a historical period, for example), it may also accompany spoken dialogue (see melodrama). Film and television music is sometimes considered incidental music.

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In mathematics, the upper half-plane H is the set of complex numbers

$mathbb\{H\}\; =\; \{x\; +\; iy\; ;|\; y\; >\; 0;\; x,\; y\; in\; mathbb\{R\}\; \}$

with positive imaginary part y.

The term is associated with a common visualization of complex numbers with points in the plane endowed with Cartesian coordinates, with the Y-axis pointing upwards: the "upper half-plane" corresponds to the half-plane above the X-axis.

When endowed with a particular metric, the upper half-plane may be called the hyperbolic plane, Poincaré half-plane, or Lobachevsky plane, particularly in texts by Russian authors. Some authors prefer the symbol $mathfrak\{h\}.$

It is the domain of many functions of interest in complex analysis, especially elliptic modular forms. The lower half-plane, defined by y < 0, is equally good, but less used by convention. The open unit disk D (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping (see "Poincaré metric"), meaning that it is usually possible to pass between H and D.

It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.

The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature.

Generalizations

One natural generalization in differential geometry is hyperbolic n-space Hⁿ, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is H² since it has real dimension 2.

In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product Hⁿ of n copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space H_n, which is the domain of Siegel modular forms.

Let

$mathbb\{H\}\_n=\{Fin\; M\_\{n\}(mathbb\{C\})\; ;\; |\; F=F^T\; ;textrm\{and\};\; Im\; (F)\; >0\; \}$

be the set of symmetric square matrices whose imaginary part is positive definite; that is the set of square matrices whose imaginary parts have positive eigenvalues. The set H_n is called the Siegel upper half-space of genus n.

See also

• Cusp neighborhood
• Fuchsian group
• Fundamental domain
• Hyperbolic geometry
• Kleinian group
• Modular group
• Riemann surface
• Schwarz-Ahlfors-Pick theorem

External links

• Visual presentation of the upper half-plane