Matrix Theory and Linear Algebra (562420), страница 17

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Mandelbrot Set

Polish-born French mathematician Benoit Mandelbrot coined the term “fractal” to describe complex geometric shapes that, when magnified, continue to resemble the shape’s larger structure. This property, in which the pattern of the whole repeats itself on smaller and smaller scales, is called self similarity. The fractal shown here, called the Mandelbrot set, is the graphical representation of a mathematical function.

Prism

Prism, in geometry, three-dimensional solid, of which the bases are two parallel planes. The faces of the prism in these planes are congruent polygons. The lateral faces of the prism are parallelograms (see Fig. 1). The intersections of the lateral faces, called the lateral edges, are parallel to each other. A prism is called a right prism if the lateral edges are perpendicular to the bases; if they are not, it is called an oblique prism. A prism is triangular, square, and so on, according as its bases are triangles, squares, or some other geometrical figure. A parallelopiped is a prism that has parallelograms as the bases; a rectangular parallelopiped, or box, is one in which all six faces (four lateral faces and two bases) are rectangles (see Fig. 2); in a cube the six faces are squares (see Fig. 3).

The altitude of a prism is the perpendicular distance between the planes of the bases. A truncated prism is that portion of a prism between a base and a section formed by a plane not parallel to the base but cutting all lateral edges (see Fig. 4). The volume, V, of a prism is given by the area, B, of a base multiplied by the altitude, h; in symbols, V = Bh. If a,b,c are the lengths of three edges of a rectangular parallelopiped that meet at one vertex (the length, width, and depth of a box), the volume is given by V = abc. In particular, if a is the length of one of the twelve equal edges of a cube, the volume of the cube is V = a³ and the total surface area, S, of the cube is S = 6a².

Prisms: Figures 1-4

Prisms are commonly used to bend the path of light in devices such as binoculars and periscopes. White light passed through a glass prism will divide into the colors of the spectrum. In geometry, prisms are three-dimensional solids in which the bases are two parallel planes, and the faces in these planes are congruent polygons. As shown in figure 1, the lateral faces of the prism are parallelograms. Figure 2 depicts a prism in which all six bases are rectangular parallelograms, a special case of a prism called a rectangular parallelopiped. Figure 3 is also a parallelopiped, but each of the faces is a square, forming a cubic solid. A truncated prism, shown in figure 4, is a portion of a prism formed by cutting all lateral edges with a plane not parallel to the base.

Separation of White Light into Colored Light

Light that contains many colors, such as sunlight, appears white. When white light passes through a prism-shaped transparent block, the prism separates the light into a spectrum of different colors. The prism separates the light by refracting, or bending, light of different colors at different angles. Rays of red light bend the least and rays of violet light bend the most.

Circle

Circle, in geometry, plane curve such that each point on the curve is the same distance from a fixed point. This point is called the center of the circle. The circle belongs to the class of curves known as conic sections because a circle can be described as the intersection of a right circular cone with a plane that is perpendicular to the axis of the cone (see Geometry: Conic Sections).

Any line segment that passes through the center and is terminated by the circle is called a diameter of the circle. A radius is a line segment from the center of the circle to the perimeter of the circle. A chord is any straight-line segment that is intercepted by the circle. An arc of a circle is a portion lying between two points on the circle. A central angle is an angle with the vertex at the center of the circle and with sides forming radii of the circle. A central angle is subtended by the arc that lies between the points at which the central angle's sides intersect the circle.

Of all plane figures having the same perimeter, the circle has the greatest area. The ratio of the circumference to the diameter of a circle is a constant designated by the symbol , or pi. Pi is one of the most important mathematical constants, and plays a role in many calculations and proofs in mathematics, physics, engineering, and other sciences. Pi is approximately 3.141592, although 3.1416 and even 3 are sufficiently accurate for ordinary purposes. The Greek mathematician Archimedes described the value of  as lying between 3 and 3.

The center of a circle is a point of symmetry, and any diameter of a circle is an axis of symmetry. Concentric circles—that is, circles having different perimeters but the same center—never intersect. The area of a circle is equal to pi multiplied by the square of the circle's radius. An arc of a circle is proportional to the angle subtended at the center, and conversely; this property forms the basis of angular measure. There are 360° in a circle.

Parts of a Circle

Parts of a circle include its diameter, circumference, radius, chord, arc, and sector. To explore the properties of a circle, wrap a string around a soda can and measure the length of the string (the circumference of the circle formed by the string). Then lay the string directly across the top of the soda can so that it divides the top in half. Measure the string’s length (the diameter of the circle formed by the top of the can). Divide the circumference (C) by the diameter (D). Do this for several circles. You should find that the ratio of C/D is about 3:1 for all of your circles, both large and small. This ratio is often represented by the symbol . Another experiment compares running around the circumference of a circular track to heading directly across along its diameter. For every 3 steps around the circular track, a runner only needs to take 1 step directly across.

Parabola

Parabola, in mathematics, plane curve formed by the intersection of a cone with a plane parallel to a straight line on the slanting surface of the cone (see Geometry). Each point of the curve is equidistant from a fixed point, called the focus, and a fixed straight line, known as the directrix. The parabola is symmetrical about a line passing through the focus and perpendicular to the directrix. For a parabola symmetric about the x-axis and with its vertex at the origin, the mathematical equation is y² = 2 px, in which p is the distance between the focus and the directrix.

A parabola is the curve that describes the trajectory of a projectile, such as a bullet or a ball, in the absence of air resistance. Because of air resistance, however, the curves in which projectiles travel only approximate true parabolas. Parabolic mirrors are reflectors that have the shape of a parabola rotated about the parabola's axis of symmetry. Parabolic mirrors reflect rays of light in parallel lines from a light source at the mirror's focus. Such reflectors are used in automobile headlights and all other forms of searchlights. Parabolic mirrors also bring parallel rays of light to a focus without spherical aberration (see Optics). This type of reflector is therefore valuable in astronomical telescopes (see Telescope). Parabolic reflectors are used also as antennas in radio astronomy and radar.

Parabola

A parabola is a curve that describes all of the points equidistant from a line, the directrix, and a fixed point, the focus. As illustrated, the distance between the focus and a point on the curve (d₁) always equals the distance between the directrix and the same point on the curve (d₂).

Cylinder

Cylinder, three-dimensional geometric figure. A circular cylinder consists of two circular bases of equal area that are in parallel planes, and are connected by a lateral surface that intersects the boundaries of the bases. The volume of a circular cylinder is r²h, where r is the radius of the bases, and h is the perpendicular distance between the planes that contain the bases. In a right circular cylinder, the lateral surface is perpendicular to the bases. The lateral surface area of a right circular cylinder is 2rh,, and the total surface area is 2r(r+h).

More generally, a cylinder need not have circular bases, nor must a cylinder form a closed surface. If MNPQ is a curve in a plane (Fig. 1), and APB is a line that is not in the plane and that intersects the curve at a point P, then all lines parallel to AB and intersecting MNQ when taken together form a cylindrical surface. If the curve MNPQ is closed, the volume enclosed is a cylindrical solid. The term cylinder may refer to either the solid or the surface. The line APB, or any other line of the surface that is parallel to APB, is called a generatrix or element of the cylinder, and the curve MNPQ is called a directrix or base. In a closed cylinder, all the elements taken together form the lateral surface. A closed cylinder is circular, elliptical, triangular, and so on, according to whether its directrix is a circle, ellipse, or triangle. In a right cylinder, all elements are perpendicular to the directrix; in an oblique cylinder, the elements are not perpendicular to the directrix. In general, the volume of a closed cylinder between the base and a plane parallel to it is given by Bh, in which B is the area of the base and h is the perpendicular distance between the two parallel planes; see Fig. 2. See Solid Geometry.

Pythagoras

Pythagoras (582?-500?bc), Greek philosopher and mathematician, whose doctrines strongly influenced Plato.

Born on the island of Sámos, Pythagoras was instructed in the teachings of the early Ionian philosophers Thales, Anaximander, and Anaximenes. Pythagoras is said to have been driven from Sámos by his disgust for the tyranny of Polycrates. About 530 bc Pythagoras settled in Crotona, a Greek colony in southern Italy, where he founded a movement with religious, political, and philosophical aims, known as Pythagoreanism. The philosophy of Pythagoras is known only through the work of his disciples.

The Pythagoreans adhered to certain mysteries, similar in many respects to the Orphic mysteries (see Mysteries; Orphism). Obedience and silence, abstinence from food, simplicity in dress and possessions, and the habit of frequent self-examination were prescribed. The Pythagoreans believed in immortality and in the transmigration of souls. Pythagoras himself was said to have claimed that he had been Euphorbus, a warrior in the Trojan War, and that he had been permitted to bring into his earthly life the memory of all his previous existences.

Among the extensive mathematical investigations carried on by the Pythagoreans were their studies of odd and even numbers and of prime and square numbers (see Number Theory). From this arithmetical standpoint they cultivated the concept of number, which became for them the ultimate principle of all proportion, order, and harmony in the universe. Through such studies they established a scientific foundation for mathematics. In geometry the great discovery of the school was the hypotenuse theorem, or Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

The astronomy of the Pythagoreans marked an important advance in ancient scientific thought, for they were the first to consider the earth as a globe revolving with the other planets around a central fire. They explained the harmonious arrangement of things as that of bodies in a single, all-inclusive sphere of reality, moving according to a numerical scheme. Because the Pythagoreans thought that the heavenly bodies are separated from one another by intervals corresponding to the harmonic lengths of strings, they held that the movement of the spheres gives rise to a musical sound—the “harmony of the spheres.”

Big Bang Theory

Big Bang Theory, currently accepted explanation of the beginning of the universe. The big bang theory proposes that the universe was once extremely compact, dense, and hot. Some original event, a cosmic explosion called the big bang, occurred about 10 billion to 20 billion years ago, and the universe has since been expanding and cooling.

The theory is based on the mathematical equations, known as the field equations, of the general theory of relativity set forth in 1915 by Albert Einstein. In 1922 Russian physicist Alexander Friedmann provided a set of solutions to the field equations. These solutions have served as the framework for much of the current theoretical work on the big bang theory. American astronomer Edwin Hubble provided some of the greatest supporting evidence for the theory with his 1929 discovery that the light of distant galaxies was universally shifted toward the red end of the spectrum (see Redshift). This proved that the galaxies were moving away from each other. He found that galaxies farther away were moving away faster, showing that the universe is expanding uniformly. However, the universe’s initial state was still unknown.

In the 1940s Russian American physicist George Gamow worked out a theory that fit with Friedmann’s solutions in which the universe expanded from a hot, dense state. In 1950 British astronomer Fred Hoyle, in support of his own opposing steady-state theory, referred to Gamow’s theory as a mere “big bang,” but the name stuck. Indeed, a contest in the 1990s by Sky & Telescope magazine to find a better (perhaps more dignified) name did not produce one.

The overall framework of the big bang theory came out of solutions to Einstein’s general relativity field equations and remains unchanged, but various details of the theory are still being modified today. Einstein himself initially believed that the universe was static. When his equations seemed to imply that the universe was expanding or contracting, Einstein added a constant term to cancel out the expansion or contraction of the universe. When the expansion of the universe was later discovered, Einstein stated that introducing this “cosmological constant” had been a mistake.

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