Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 19

In mechanics, Archimedes defined the principle of the lever and is credited with inventing the compound pulley. During his stay in Egypt he invented the hydraulic screw for raising water from a lower to a higher level. He is best known for discovering the law of hydrostatics, often called Archimedes' principle, which states that a body immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. This discovery is said to have been made as Archimedes stepped into his bath and perceived the displaced water overflowing.

Archimedes spent the major part of his life in Sicily, in and around Syracuse. He did not hold any public office but devoted his entire lifetime to research and experiment. During the Roman conquest of Sicily, however, he placed his gifts at the disposal of the state, and several of his mechanical devices were employed in the defense of Syracuse. Among the war machines attributed to him are the catapult and—perhaps legendary—a mirror system for focusing the sun's rays on the invaders' boats and igniting them.

After the capture of Syracuse during the Second Punic War, Archimedes was killed by a Roman soldier who found him drawing a mathematical diagram in the sand. It is said that Archimedes was so absorbed in calculation that he offended the intruder merely by remarking, “Do not disturb my diagrams.” Several of his works on mathematics and mechanics survive, including Floating Bodies, The Sand Reckoner, Measurement of the Circle, Spirals, and Sphere and Cylinder. They all exhibit the rigor and imaginativeness of his mathematical thinking.

Archimedes's Principle

An object is subject to an upward force when it is immersed in liquid. The force is equal to the weight of the liquid displaced. The apparent weight of a block of aluminium (1) immersed in water is reduced by an amount equal to the weight of water displaced. If a block of wood (2) is completely immersed in water, the upward force is greater than the weight of the wood. (Wood is less dense than water, so the weight of the block of wood is less than that of the same volume of water.) So the block rises and partly emerges to displace less water until the upward force exactly equals the weight of the block.

Parallelogram

Parallelogram, in plane geometry, a four-sided polygon, or quadrilateral, in which each side is equal in length to its opposite side, and the opposite sides are parallel to each other. Three special kinds of parallelogram are the square, in which all four sides are equal in length and meet at right angles; the rectangle, in which only opposite sides are equal in length but all sides meet at right angles; and the rhombus, in which all sides are equal in length but do not necessarily meet at right angles. Area is the product of one side, taken as base, times the shortest distance to the opposite side. See also Vector.

Examples of Parallelograms

A parallelogram is a four-sided figure in which opposite sides are parallel. Each side of a parallelogram is also equal in length to its opposite side. The area of a parallelogram is equal to the length of the base times the shortest distance to the opposite side.

Symmetry

Symmetry, orderly, mutually corresponding arrangement of various parts of a body, producing a proportionate, balanced form. The principle of symmetry is of great importance in the fields of biology, mathematics, and mineralogy.

In biology, the regular distribution of various parts of an animal's body on two opposite sides of a linear axis, or a median plane, is known as bilateral symmetry. The proportional arrangement of similar parts of a body around a central axis, as in the case of jellyfish or starfish, is known as radial symmetry. The bodies of protozoans, such as those of the order Radiolaria, which have a round form about a central point or nucleus, are said to have a spherical symmetry.

In geometry, symmetry is a feature of certain plane and solid shapes. So-called symmetry operations are those mathematical transformations that produce a figure identical to the original or a mirror image of the original figure. Symmetry operations are defined with respect to a given point (center of symmetry), line (axis of symmetry), and plane (plane of symmetry).

In mineralogy, laws of symmetry apply to the angular structure of crystals. All classes of crystal are divided into six systems that are based on the length of their axes and other details of symmetry. See Crystal; Metallography.

In physics, a system is said to exhibit symmetry if it remains unchanged in the course of operations such as mirror reversal, reversal in the direction of time, and space-time translation. Many physical systems obey such symmetries, to which the conservation laws of physics are also related. This relationship has come to be of particular importance in particle physics, where certain symmetries called internal symmetries are observed. Such symmetries exist in the mathematical “space” of that realm and underlie the conservation of such quantities as charge, parity, baryon and lepton number, and total strangeness, even as certain particles are substituted for one another. In current theoretical physics, however, such symmetries are now known to be only approximate. Except for baryon and lepton number, that is, they are violated in their physical manifestations. When internal symmetries do not operate the same way but instead can be different at each point in space-time, they are called gauge symmetries. Theorists currently hope to reduce all such symmetries to gauge symmetries in their effort to develop a grand unification theory that can incorporate all of the fundamental interactions of matter (see Unified Field Theory). See also Elementary Particles.

Symmetry

The property of symmetry describes certain physical phenomena, geometric shapes, and mathematical equations that remain unchanged despite changes in orientation or other properties. This illustration shows three geometric shapes and indicates the manner in which they are symmetrical. The top two figures remain unchanged as they revolve around a central point, while the lower triangle can rotate 180 degrees about a central line without changing.

Cone (geometry)

Cone (geometry), in geometry, surface generated by a straight line that moves along a closed curve while always passing through a fixed point. The straight line is called the generatrix, the fixed point is called the vertex, and the closed curve is called the directrix. If the generatrix is of infinite length, it generates two conical surfaces on opposite sides of the vertex. If the directrix of the cone is a circle, the cone is usually referred to as a circular cone.

Mathematicians study oblique cones (in which the axis is not perpendicular to the plane of the base) and various noncircular cones (in which the base is in the form of an ellipse or some other closed curve). The surface that is commonly called a cone, however, is the right circular cone.

Definition of a Cone

A cone is defined as the surface swept out by a straight line, called the generatrix, as it travels all the way around a closed curve, called the directrix, while remaining in contact with a fixed point, called the vertex. An infinitely long generatrix forms two cones, point to point.