plane tree, sycamore, or buttonwood, any species of the genus Platanus, deciduous trees of most temperate regions of the Northern Hemisphere, exclusive of Africa. The hairy, dry, seedlike fruits are densely packed into a hard brown ball, hence the name buttonwood. The large American sycamore, or plane (P. occidentalis), supplies a hardwood popular for carpentry, furniture, and butchers' blocks. The Oriental plane (P. orientalis) of the Mediterranean area and W Asia is used for wood pulp. The London plane (P. acerifolia), sometimes miscalled Oriental plane, is much planted as an ornamental shade tree in cities; it is probably a hybrid of the American and Oriental planes. Sycamore is sometimes spelled sycomore, but should not be confused with the unrelated sycomore fig (Ficus sycomorus) of the family Moraceae. Plane trees are classified in the division Magnoliophyta, class Magnoliopsida, order Hamamelidales, family Platanaceae.

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.

inclined plane, simple machine, consisting of a sloping surface, whose purpose is to reduce the force that must be applied to raise a load. To raise a body vertically a force must be applied that is equal to the weight of the body, i.e., the product of its mass and the acceleration of gravity. The amount of work done (i.e., energy expended) in raising the body is equal to its weight times the distance through which it is raised. By means of an inclined plane a force smaller than the weight of the body can be exerted over a distance greater than the direct vertical distance, doing work equal to the product of the force and the distance through which it acts. If friction is ignored, the work done using the inclined plane will be exactly equal to the work done in lifting the body directly. In any real system some work is done to overcome friction between the plane and the load. The actual mechanical advantage of an inclined plane is the ratio of the load lifted to the force applied; ideally it is equal to the ratio of the length of the sloping plane to its vertical rise. An inclined plane whose sloping length is 5 m and whose vertical rise is 1 m has a mechanical advantage of 5; a 300-newton load can be moved up such a plane by a 60-newton force. The inclined plane has been modified in many ways. The screw and wedge are applications of the principle of the inclined plane but do not require that the load be moved vertically for their successful operation. The chisel, carpenter's plane, auger bit, and ax are some of the many tools based on this principle. Switchbacks on mountain roads are inclined planes that reduce the effort of an automobile engine but increase the distance a car must travel to ascend the mountain.

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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