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  • Geometric basis of patterns
  • The beginnings of these design studies
  • Old Babylonian mathematics and Plimpton 322: A new understanding of the OB tablet Plimpton 322
  • Old Babylonian mathematics and Plimpton 322: A new understanding of the OB tablet Plimpton 322
  • Secrets of the Giza Pyramids

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    Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia modern Iraq from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity.

    The beginnings of these design studies

    The first few hundred years of the second millennium BC Old Babylonian periodand the last few centuries of the first millennium BC Seleucid period.

    Later under the Arab EmpireMesopotamia, especially Baghdadonce again became an important center of study for Islamic mathematics. In contrast to the sparsity of sources in Egyptian mathematicsour knowledge of Babylonian mathematics is derived from more than clay tablets unearthed since the s.

    Some of these appear to be graded homework. They developed a complex system of metrology from BC. From around BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.

    It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and The problem includes a diagram indicating the dimensions of the truncated pyramid. Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic periodGreek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematicswhen Arabic became the written language of Egyptian scholars.

    The most extensive Egyptian mathematical text is the Rhind papyrus sometimes also called the Ahmes Papyrus after its authordated to c. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, [27] including composite and prime numbers ; arithmeticgeometric and harmonic means ; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory namely, that of the number 6.

    One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum truncated pyramid. Finally, the Berlin Papyrus c. Greek mathematics The Pythagorean theorem.

    The Pythagoreans are generally credited with the first proof of the theorem. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb.

    Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.

    Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.

    Old Babylonian mathematics and Plimpton 322: A new understanding of the OB tablet Plimpton 322

    He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.

    As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.

    The Pythagoreans are credited with the first proof of the Pythagorean theorem[38] though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers. The diagram accompanies Book II, Proposition 5. Though he made no specific technical mathematical discoveries, Aristotle —c. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.

    Euclid also wrote extensively on other subjects, such as conic sectionsopticsspherical geometryand mechanics, but only half of his writings survive.

    Arabic / Islamic geometry 02

    Around the same time, Eratosthenes of Cyrene c. AD 90—a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. His main work was the Arithmetica, a collection of algebraic problems dealing with exact solutions to determinate and indeterminate equations. He is known for his hexagon theorem and centroid theoremas well as the Pappus configuration and Pappus graph.

    His Collection is a major source of knowledge on Greek mathematics as most of it has survived. The first woman mathematician recorded by history was Hypatia of Alexandria AD — She succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics.

    Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletusthe architects of the Haghia Sophia.
    The sexagesimal method developed by the Babylonians has a far greater computational potential than what was actually needed for the older problem texts.

    With the development of mathematical astronomy in the Seleucid period, however, it became indispensable. Astronomers sought to predict future occurrences of… The scope of astronomy Since the late 19th century astronomy has expanded to include astrophysicsthe application of physical and chemical knowledge to an understanding of the nature of celestial objects and the physical processes that control their formation, evolution, and emission of radiation. In addition, the gases and dust particles around and between the stars have become the subjects of much research.

    Study of the nuclear reactions that provide the energy radiated by stars has shown how the diversity of atoms found in nature can be derived from a universe that, following the first few minutes of its existence, consisted only of hydrogenheliumand a trace of lithium.

    Concerned with phenomena on the largest scale is cosmologythe study of the evolution of the universe. Astrophysics has transformed cosmology from a purely speculative activity to a modern science capable of predictions that can be tested. Its great advances notwithstanding, astronomy is still subject to a major constraint: Almost all measurements must be performed at great distances from the objects of interest, with no control over such quantities as their temperaturepressureor chemical composition.

    There are a few exceptions to this limitation—namely, meteorites most of which are from the asteroid belt, though some are from the Moon or Marsrock and soil samples brought back from the Moonsamples of comet and asteroid dust returned by robotic spacecraftand interplanetary dust particles collected in or above the stratosphere. These can be examined with laboratory techniques to provide information that cannot be obtained in any other way. In the future, space missions may return surface materials from Mars, or other objects, but much of astronomy appears otherwise confined to Earth-based observations augmented by observations from orbiting satellites and long-range space probes and supplemented by theory.

    Without a knowledge of astronomical distances, the size of an observed object in space would remain nothing more than an angular diameter and the brightness of a star could not be converted into its true radiated power, or luminosity. Within the inner solar systemsome distances can now be better determined through the timing of radar reflections or, in the case of the Moonthrough laser ranging. For the outer planets, triangulation is still used. Stellar distances are commonly expressed by astronomers in parsecs pckiloparsecs, or megaparsecs.

    Distances can be measured out to around a kiloparsec by trigonometric parallax see star: The Gaia satellite is expected to measure stars as far away as 10 kiloparsecs to an accuracy of 20 percent. Less-direct measurements must be used for more-distant stars and for galaxies. Two general methods for determining galactic distances are described here. In the first, a clearly identifiable type of star is used as a reference standard because its luminosity has been well determined.

    This requires observation of such stars that are close enough to Earth that their distances and luminosities have been reliably measured. Once the luminosities of such nearer standard candles have been calibratedthe distance to a farther standard candle can be calculated from its calibrated luminosity and its actual measured intensity. A standard candle can be identified by means of its spectrum or the pattern of regular variations in brightness.

    Corrections may have to be made for the absorption of starlight by interstellar gas and dust over great distances. This method forms the basis of measurements of distances to the closest galaxies. A region of the spiral galaxy M bottomwith three frames top showing a Cepheid variable increasing in brightness. Freedman, Observatories of the Carnegie Institution of Washington, and NASA The second method for galactic distance measurements makes use of the observation that the distances to galaxies generally correlate with the speeds with which those galaxies are receding from Earth as determined from the Doppler shift in the wavelengths of their emitted light.

    This correlation is expressed in the Hubble law: H has been used to determine distances to remote galaxies in which standard candles have not been found.

    Old Babylonian mathematics and Plimpton 322: A new understanding of the OB tablet Plimpton 322

    For additional discussion of the recession of galaxies, the Hubble law, and galactic distance determination, see physical science: Study of the solar system The solar system took shape 4.

    Gravitational attraction holds the planets in their elliptical orbits around the Sun. In addition to Earthfive major planets MercuryVenusMarsJupiterand Saturn have been known from ancient times. Since then only two more have been discovered: Uranus by accident in and Neptune in after a deliberate search following a theoretical prediction based on observed irregularities in the orbit of Uranus. Plutodiscovered in after a search for a planet predicted to lie beyond Neptune, was considered a major planet untilwhen it was redesignated a dwarf planet by the International Astronomical Union.

    Jupiter, the fifth planet from the Sun and largest planet in the solar system. The Great Red Spot is visible in the lower left. This image is based on observations made by the Voyager 1 spacecraft in Most of the mass of the solar system is concentrated in the Sun, with its 1. Together, all of the planets amount to 2. The solar system also contains five known objects of intermediate size classified as dwarf planets and a very large number of much smaller objects collectively called small bodies.

    The small bodies, roughly in order of decreasing size, are the asteroidsor minor planets; cometsincluding Kuiper beltCentaurand Oort cloud objects; meteoroids ; and interplanetary dust particles. Because of their starlike appearance when discovered, the largest of these bodies were termed asteroids, and that name is widely used, but, now that the rocky nature of these bodies is understood, their more descriptive name is minor planets.

    The eight planets of the solar system and Pluto, in a montage of images scaled to show the approximate sizes of the bodies relative to one another. Outward from the Sun, which is represented to scale by the yellow segment at the extreme left, are the four rocky terrestrial planets Mercury, Venus, Earth, and Marsthe four hydrogen-rich giant planets Jupiter, Saturn, Uranus, and Neptuneand icy, comparatively tiny Pluto.

    The compositions of these two groups of planets must therefore be significantly different. This dissimilarity is thought to be attributable to conditions that prevailed during the early development of the solar system see below Theories of origin. These are average temperatures; large variations exist between dayside and nightside for planets closest to the Sun, except for Venus with its thick atmosphere. Saturn and its spectacular rings, in a natural-colour composite of images taken by the Cassini spacecraft on October 6, The view is directed toward Saturn's southern hemisphere, which is tipped toward the Sun.

    Shadows cast by the rings are visible against the bluish northern hemisphere, while the planet's shadow is projected on the rings to the left. On Earth, with its large quantities of water and an active atmospheremany of these cosmic footprints have erodedbut remnants of very large craters can be seen in aerial and spacecraft photographs of the terrestrial surface.

    On Mercury, Mars, and the Moonthe absence of water and any significant atmosphere has left the craters unchanged for billions of years, apart from disturbances produced by infrequent later impacts. Volcanic activity has been an important force in the shaping of the surfaces of the Moon and the terrestrial planets. Seismic activity on the Moon has been monitored by means of seismometers left on its surface by Apollo astronauts and by Lunokhod robotic rovers.

    MercuryMeteorite crater surrounded by rays of ejected material on Mercury, in a photograph taken by the Messenger probe, January 14, A chain of craters crosses the centre of the rayed crater. The four outer dwarf planets have moons; Pluto has at least five moons, including one, Charonfully half the size of Pluto itself. Over asteroids and 80 Kuiper belt objects also have moons. Four planets Jupiter, Saturn, Uranus, and Neptuneone dwarf planet Haumeaand one Centaur object Chariklo have rings, disklike systems of small rocks and particles that orbit their parent bodies.

    Apollo missions a total weight of The opportunity to employ a wide range of laboratory techniques on these lunar samples has revolutionized planetary science. The results of the analyses have enabled investigators to determine the composition and age of the lunar surface. Seismic observations have made it possible to probe the lunar interior. This experiment, which has provided data used in calculations of the dynamics of the Earth-Moon system, has shown that the separation of the two bodies is increasing by 4.

    For additional information on lunar studies, see Moon. Apollo 17 geologist-astronaut Harrison Schmitt at the foot of a huge split boulder, December 13,during the mission's third extravehicular exploration of the Taurus-Littrow Valley landing site. Below the upper clouds, Venus has a hostile atmosphere containing clouds of sulfuric acid droplets. The outer planets have atmospheres composed largely of light gasesmainly hydrogen and helium.

    Photo mosaic of Mercury, taken by the Mariner 10 spacecraft, The two exceptions are Venus, which rotates in the clockwise direction beneath its cloud cover, and Uranuswhich has its rotation axis very nearly in the plane of the ecliptic. Some of the planets have magnetic fields. Through processes not yet fully understood, particles from the solar wind and galactic cosmic rays high-speed particles from outside the solar system populate two doughnut-shaped regions called the Van Allen radiation belts.

    The Van Allen radiation belts contained within Earth's magnetosphere. Pressure from the solar wind is responsible for the asymmetrical shape of the magnetosphere and the belts. Bursts of increased radio emission are correlated with the position of Iothe innermost of the four Galilean moons of Jupiter. No field has been detected around Venus or Mars.

    Secrets of the Giza Pyramids

    Jupiter's magnetosphereJupiter's magnetosphere as observed by the Cassini spacecraft in The magnetosphere is the largest object in the solar system. Hundreds of thousands more have been seen, but their orbits have not been as well determined. It is estimated that several million asteroids exist, but most are small, and their combined mass is estimated to be less than a thousandth that of Earth. Most of the asteroids have orbits close to the ecliptic and move in the asteroid beltbetween 2.

    Because some asteroids travel in orbits that can bring them close to Earth, there is a possibility of a collision that could have devastating results see Earth impact hazard. GaspraAsteroid Gaspra, composite of two images taken by the Galileo spacecraft on October 29, Galileo observed some impact craters on Gaspra, from the large concavity visible on the lower right to craters as small as metres feet in diameter.

    The asteroid's irregular shape and fracture lines suggest that it was once part of a larger body. Many comets have been observed over the centuries. Most make only a single pass through the inner solar system, but some are deflected by Jupiter or Saturn into orbits that allow them to return at predictable times. Very few comet masses have been well determined, but most are probably less than grams, one-billionth the mass of Earth.

    Smith Since the s more than a thousand comet nuclei in the Kuiper belt have been observed with large telescopes; a few are about half the size of Pluto, and Pluto is the largest Kuiper belt object. Consequently, in astronomers at the general assembly of the International Astronomical Union elected to create the new category of dwarf planets for objects with such qualifications.

    Pluto, Erisand Ceresthe latter being the largest member of the asteroid belt, were given this distinction. Two other Kuiper belt objects, Makemake and Haumeawere also designated as dwarf planets. PlutoPluto as seen by the New Horizons spacecraft, July 14, Meteoroids vary from small rocks to boulders weighing a ton or more. Most meteorites that have been collected on Earth are probably from asteroids. A few have been identified as being from the MoonMarsor the asteroid Vesta.

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