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 p r ² h, 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 2p rh,, and the total surface area is 2p 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.

After Einstein’s work of 1917, several scientists, including the abbé Georges Lemaître in Belgium, Willem de Sitter in Holland, and Alexander Friedmann in Russia, succeeded in finding solutions to Einstein’s field equations. The universes described by the different solutions varied. De Sitter’s model had no matter in it. This model is actually not a bad approximation since the average density of the universe is extremely low. Lemaître’s universe expanded from a “primeval atom.” Friedmann’s universe also expanded from a very dense clump of matter, but did not involve the cosmological constant. These models explained how the universe behaved shortly after its creation, but there was still no satisfactory explanation for the beginning of the universe.

In the 1940s George Gamow was joined by his students Ralph Alpher and Robert Herman in working out details of Friedmann’s solutions to Einstein’s theory. They expanded on Gamow’s idea that the universe expanded from a primordial state of matter called ylem consisting of protons, neutrons, and electrons in a sea of radiation. They theorized the universe was very hot at the time of the big bang (the point at which the universe explosively expanded from its primordial state), since elements heavier than hydrogen can be formed only at a high temperature. Alpher and Hermann predicted that radiation from the big bang should still exist. Cosmic background radiation roughly corresponding to the temperature predicted by Gamow’s team was detected in the 1960s, further supporting the big bang theory, though the work of Alpher, Herman, and Gamow had been forgotten.

The big bang theory seeks to explain what happened at or soon after the beginning of the universe. Scientists can now model the universe back to 10^-43 seconds after the big bang. For the time before that moment, the classical theory of gravity is no longer adequate. Scientists are searching for a theory that merges quantum mechanics and gravity, but have not found one yet. Many scientists have hope that string theory will tie together gravity and quantum mechanics and help scientists explore further back in time (see Physics: Unified Field Theory).

Because scientists cannot look back in time beyond that early epoch, the actual big bang is hidden from them. There is no way at present to detect the origin of the universe. Further, the big bang theory does not explain what existed before the big bang. It may be that time itself began at the big bang, so that it makes no sense to discuss what happened “before” the big bang.

According to the big bang theory, the universe expanded rapidly in its first microseconds. A single force existed at the beginning of the universe, and as the universe expanded and cooled, this force separated into those we know today: gravity, electromagnetism, the strong nuclear force, and the weak nuclear force. A theory called the electroweak theory now provides a unified explanation of electromagnetism and the weak nuclear force theory (see Unified Field Theory). Physicists are now searching for a grand unification theory to also incorporate the strong nuclear force. String theory seeks to incorporate the force of gravity with the other three forces.

One widely accepted version of big bang theory includes the idea of inflation. In this model, the universe expanded much more rapidly at first, to about 10⁵⁰ times its original size in the first 10^-32 second, then slowed its expansion. The theory was advanced in the 1980s by American cosmologist Alan Guth and elaborated upon by American astronomer Paul Steinhardt, Russian American scientist Andrei Linde, and British astronomer Andreas Albrecht. The inflationary universe theory (see Inflationary Theory) solves a number of problems of cosmology. For example, it shows that the universe now appears close to the type of flat space described by the laws of Euclid’s geometry: We see only a tiny region of the original universe, similar to the way we do not notice the curvature of the earth because we see only a small part of it. The inflationary universe also shows why the universe appears so homogeneous. If the universe we observe was inflated from some small, original region, it is not surprising that it appears uniform.

Once the expansion of the initial inflationary era ended, the universe continued to expand more slowly. The inflationary model predicts that the universe is on the boundary between being open and closed. If the universe is open, it will keep expanding forever, even though the rate of expansion will gradually slow. If the universe is closed, the expansion of the universe will eventually stop and the universe will begin contracting until it collapses. Whether the universe is open or closed depends on the density, or concentration of mass, in the universe. If the universe is dense enough, it is closed.

The universe cooled as it expanded. After about one second, protons formed. In the following few minutes—often referred to as the “first three minutes,” combinations of protons and neutrons formed the isotope of hydrogen known as deuterium as well as some of the other light elements, principally helium, as well as some lithium, beryllium, and boron. The study of the distribution of deuterium, helium, and the other light elements is now a major field of research. The uniformity of the helium abundance around the universe supports the big bang theory and the abundance of deuterium can be used to estimate the density of matter in the universe.

From about 300,000 to about 1 million years after the big bang, the universe cooled to about 3000° C (about 5000° F) and protons and electrons combined to make hydrogen atoms. Hydrogen atoms can only absorb and emit specific colors, or wavelengths, of light. The formation of atoms allowed many other wavelengths of light, wavelengths that had been interfering with the free electrons, to travel much farther than before. This change set free radiation that we can detect today. After billions of years of cooling, this cosmic background radiation is at about 3° K (-270° C/-454° F).The cosmic background radiation was first detected and identified in 1965 by American astrophysicists Arno Penzias and Robert Wilson.

The National Aeronautics and Space Administration’s Cosmic Background Explorer (COBE) spacecraft mapped the cosmic background radiation between 1989 and 1993. It verified that the distribution of intensity of the background radiation precisely matched that of matter that emits radiation because of its temperature, as predicted for the big bang theory. It also showed that the cosmic background radiation is not uniform, that it varies slightly. These variations are thought to be the seeds from which galaxies and other structures in the universe grew.

Evidence indicates that the matter that scientists detect in the universe is only a small fraction of all the matter that exists. For example, observations of the speeds with which individual galaxies move within clusters of galaxies show that there must be a great deal of unseen matter exerting gravitational forces to keep the clusters from flying apart.

Cosmologists now think that much of the universe—perhaps 99 percent— is dark matter, or matter that has gravity but that we cannot see or otherwise detect. Theorized kinds of dark matter include cold dark matter, with slowly moving (cold) massive particles. No such particles have yet been detected, though astronomers have given them names like Weakly Interacting Massive Particles (WIMPs). Other cold dark matter could be nonradiating stars or planets, which are known as MACHOs (Massive Compact Halo Objects). An alternative model includes hot dark matter, where hot implies that the particles are moving very fast. The fundamental particles known as neutrinos are the prime example of hot dark matter. If the inflationary version of big bang theory is correct, then the amount of dark matter that exists is just enough to bring the universe to the boundary between open and closed.

Scientists develop theoretical models to show how the universe’s structures, such as clusters of galaxies, have formed. Their models invoke hot dark matter, cold dark matter, or a mixture of the two. This unseen matter would have provided the gravitational force needed to hold large structures such as clusters of galaxies together. The theories continue to match the observations, though there is no consensus on the type or types of dark matter that must be included. Supercomputers are important for making such models.

Astronomers are making new observations that are interpreted within the framework of the big bang theory. Scientists have not found any major problems with the big bang theory, but the theory is being constantly adjusted to match the observed universe.

Trapezoid

Trapezoid, in plane geometry, quadrilateral (four-sided) figure with two parallel sides, or bases, of unequal length. The perpendicular distance between the bases is known as the altitude. The sides that are not parallel are called legs, and a line from the midpoint of one leg to the midpoint of the other is called the median; the multiplication of the altitude and the median yields the area of the trapezoid. When the legs of a trapezoid are of equal length, the figure is called an isosceles trapezoid.

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