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## Point, Line, Plane and Solid

❶The field of astronomy , especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.
## GEOMETRY Defined for English Language Learners

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Examples of geometry in a Sentence the geometry of Sydney's famed opera house is suggestive of some modernistic sailing ship. Recent Examples of geometry from the Web Moreover, geometry helped Lewis not only think, but also imagine. David Calvis Opinion ," 6 May The latest Metroid side-scroller and first for the Nintendo 3DS is built on exactly the kind of 3D- geometry , climb-and-clamber engine that speedrunners love to exploit. There was some futuristic geometry happening—a lot of shapes and a lot of insane lines.

Remember when your math teacher said geometry would come in handy one day? While those types of information are mutable -- even Social Security numbers can be changed -- biometric data for retinas, fingerprints, hands, face geometry and blood samples are unique identifiers.

They admired especially the works of the Greek mathematicians and physicians and the philosophy of Aristotle. By the late 9th century they were already able to add to the geometry of Euclid, Archimedes, and Apollonius. In the 10th century they went beyond Ptolemy. Stimulated by the problem of finding the effective orientation for prayer the qiblah , or direction from the place of worship to Mecca , Islamic geometers and astronomers developed the stereographic projection invented to project the celestial sphere onto a two-dimensional map or instrument as well as plane and spherical trigonometry.

Here they incorporated elements derived from India as well as from Greece. Their achievements in geometry and geometrical astronomy materialized in instruments for drawing conic sections and, above all, in the beautiful brass astrolabes with which they reduced to the turn of a dial the toil of calculating astronomical quantities.

There they presided over translations of the Greek classics. He translated Archimedes and Apollonius, some of whose books now are known only in his versions. In a notable addition to Euclid, he tried valiantly to prove the parallel postulate discussed later in Non-Euclidean geometries.

Among the pieces of Greek geometrical astronomy that the Arabs made their own was the planispheric astrolabe , which incorporated one of the methods of projecting the celestial sphere onto a two-dimensional surface invented in ancient Greece.

As Ptolemy showed in his Planisphaerium , the fact that the stereographic projection maps circles into circles or straight lines makes the astrolabe a very convenient instrument for reckoning time and representing the motions of celestial bodies.

The earliest known Arabic astrolabes and manuals for their construction date from the 9th century. The Islamic world improved the astrolabe as an aid for determining the time for prayers, for finding the direction to Mecca, and for astrological divination. Contacts among Christians, Jews, and Arabs in Catalonia brought knowledge of the astrolabe to the West before the year The Elements Venice, was one of the first technical books ever printed.

Archimedes also came West in the 12th century, in Latin translations from Greek and Arabic sources. Apollonius arrived only by bits and pieces. Not until the humanists of the Renaissance turned their classical learning to mathematics, however, did the Greeks come out in standard printed editions in both Latin and Greek.

These texts affected their Latin readers with the strength of revelation. Europeans discovered the notion of proof, the power of generalization, and the superhuman cleverness of the Greeks; they hurried to master techniques that would enable them to improve their calendars and horoscopes, fashion better instruments, and raise Christian mathematicians to the level of the infidels. It took more than two centuries for the Europeans to make their unexpected heritage their own.

By the 15th century, however, they were prepared to go beyond their sources. The most novel developments occurred where creativity was strongest, in the art of the Italian Renaissance. The theory of linear perspective, the brainchild of the Florentine architect-engineers Filippo Brunelleschi — and Leon Battista Alberti —72 and their followers, was to help remake geometry during the 17th century.

Imagine, as Alberti directed, that the painter studies a scene through a window, using only one eye and not moving his head; he cannot know whether he looks at an external scene or at a glass painted to present to his eye the same visual pyramid. Supposing this decorated window to be the canvas, Alberti interpreted the painting-to-be as the projection of the scene in life onto a vertical plane cutting the visual pyramid.

At the same time, cartographers tried various projections of the sphere to accommodate the record of geographical discoveries that began in the midth century with Portuguese exploration of the west coast of Africa. Cartographers therefore adopted the stereographic projection that had served astronomers. After cutting the cylinder along a vertical line and flattening the resulting rectangle, the result was the now-familiar Mercator map shown in the photograph.

The intense cultivation of methods of projection by artists, architects, and cartographers during the Renaissance eventually provoked mathematicians into considering the properties of linear perspective in general. The most profound of these generalists was a sometime architect named Girard Desargues — Desargues observed that neither size nor shape is generally preserved in projections, but collinearity is, and he provided an example, possibly useful to artists, in images of triangles seen from different points of view.

Despite his generality of approach, Apollonius needed to prove all his theorems for each type of conic separately. Desargues saw that he could prove them all at once and, moreover, by treating a cylinder as a cone with vertex at infinity, demonstrate useful analogies between cylinders and cones.

Following his lead, Pascal made his surprising discovery that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic lie on a straight line. What Descartes had in mind was the use of compasses with sliding members to generate curves. To classify and study such curves, Descartes took his lead from the relations Apollonius had used to classify conic sections, which contain the squares, but no higher powers, of the variables.

To describe the more complicated curves produced by his instruments or defined as the loci of points satisfying involved criteria , Descartes had to include cubes and higher powers of the variables. He thus overcame what he called the deceptive character of the terms square , rectangle , and cube as used by the ancients and came to identify geometric curves as depictions of relationships defined algebraically.

By reducing relations difficult to state and prove geometrically to algebraic relations between coordinates usually rectangular of points on curves, Descartes brought about the union of algebra and geometry that gave birth to the calculus.

The familiar use of infinity, which underlay much of perspective theory and projective geometry, also leavened the tedious Archimedean method of exhaustion. Not surprisingly, a practical man, the Flemish engineer Simon Stevin — , who wrote on perspective and cartography among many other topics of applied mathematics, gave the first effective impulse toward redefining the object of Archimedean analysis.

Instead of confining the circle between an inscribed and a circumscribed polygon, the new view regarded the circle as identical to the polygons, and the polygons to one another, when the number of their sides becomes infinitely great.

A second geometrical inspiration for the calculus derived from efforts to define tangents to curves more complicated than conics. Let the sides sought for the rectangle be denoted by a and b. Fermat observed what Kepler had perceived earlier in investigating the most useful shapes for wine casks, that near its maximum or minimum a quantity scarcely changes as the variables on which it depends alter slightly.

The figure with maximum area is a square. To obtain the tangent to a curve by this method, Fermat began with a secant through two points a short distance apart and let the distance vanish see figure. Part of the motivation for the close study of Apollonius during the 17th century was the application of conic sections to astronomy. His astronomy thus made pressing and practical the otherwise merely difficult problem of the quadrature of conics and the associated theory of indivisibles.

With the methods of Apollonius and a few infinitesimals, an inspired geometer showed that the laws regarding both area and ellipse can be derived from the suppositions that bodies free from all forces either rest or travel uniformly in straight lines and that each planet constantly falls toward the Sun with an acceleration that depends only on the distance between their centres.

Besides the problem of planetary motion, questions in optics pushed 17th-century natural philosophers and mathematicians to the study of conic sections. As Archimedes is supposed to have shown or shone in his destruction of a Roman fleet by reflected sunlight, a parabolic mirror brings all rays parallel to its axis to a common focus. The story of Archimedes provoked many later geometers, including Newton, to emulation.

Eventually they created instruments powerful enough to melt iron. Descartes emphasized the desirability of lenses with hyperbolic surfaces, which focus bundles of parallel rays to a point spherical lenses of wide apertures give a blurry image , and he invented a machine to cut them—which, however, proved more ingenious than useful. A final example of early modern applications of geometry to the physical world is the old problem of the size of the Earth. Measuring the Earth, Modernized.

On the hypothesis that the Earth cooled from a spinning liquid blob, Newton calculated that it is an oblate spheroid obtained by rotating an ellipse around its minor axis , not a sphere, and he gave the excess of its equatorial over its polar diameter. During the 18th century many geodesists tried to find the eccentricity of the terrestrial ellipse. At first it appeared that all the measurements might be compatible with a Newtonian Earth.

By the end of the century, however, geodesists had uncovered by geometry that the Earth does not, in fact, have a regular geometrical shape.

The dominance of analysis algebra and the calculus during the 18th century produced a reaction in favour of geometry early in the 19th century. Fundamental new branches of the subject resulted that deepened, generalized, and violated principles of ancient geometry. The cultivators of these new fields, such as Jean-Victor Poncelet — and his self-taught disciple Jakob Steiner — , vehemently urged the claims of geometry over analysis.

Poncelet relied on this information to keep himself alive. The result was projective geometry. Poncelet employed three basic tools. One he took from Desargues: The second tool, continuity , allows the geometer to claim certain things as true for one figure that are true of another equally general figure provided that the figures can be derived from one another by a certain process of continual change.

Poncelet and his defender Michel Chasles — extended the principle of continuity into the domain of the imagination by considering constructs such as the common chord in two circles that do not intersect. Similarly, parallelism had to go. Efforts were well under way by the middle of the 19th century, by Karl George Christian von Staudt — among others, to purge projective geometry of the last superfluous relics from its Euclidean past.

The first possibility gives Euclidean geometry. Saccheri devoted himself to proving that the obtuse and the acute alternatives both end in contradictions, which would thereby eliminate the need for an explicit parallel postulate. He then destroyed the obtuse hypothesis by an argument that depended upon allowing lines to increase in length indefinitely.

If this is disallowed, the hypothesis of the obtuse angle produces a system equivalent to standard spherical geometry, the geometry of figures drawn on the surface of a sphere. As for the acute angle, Saccheri could defeat it only by appealing to an arbitrary hypothesis about the behaviour of lines at infinity. One of his followers, the Swiss-German polymath Johann Heinrich Lambert —77 , observed that, based on the acute hypothesis, the area of a triangle is the negative of that of a spherical triangle.

Although both Saccheri and Lambert aimed to establish the hypothesis of the right angle, their arguments seemed rather to indicate the unimpeachability of the alternatives. The earliest published non-Euclidean geometric systems were the independent work of two young men from the East who had nothing to lose by their boldness. Both Lobachevsky and Bolyai had worked out their novel geometries by Lobachevsky and Bolyai reasoned about the hypothesis of the acute angle in the manner of Saccheri and Lambert and recovered their results about the areas of triangles.

They advanced beyond Saccheri and Lambert by deriving an imaginary trigonometry to go with their imaginary geometry. Another of the profound impulses Gauss gave geometry concerned the general description of surfaces.

Typically—with the notable exception of the geometry of the sphere—mathematicians had treated surfaces as structures in three-dimensional Euclidean space. However, as these surfaces occupy only two dimensions, only two variables are needed to describe them. For example, the shortest distance, or path, between two points on the surface of a sphere is the lesser arc of the great circle joining them, whereas, considered as points in three-dimensional space, the shortest distance between them is an ordinary straight line.

The shortest path between two points on a surface lying wholly within that surface is called a geodesic, which reflects the origin of the concept in geodesy, in which Gauss took an active interest. Riemann began with an abstract space of n dimensions. That was in the s, when mathematicians and mathematical physicists were beginning to use n -dimensional Euclidean space to describe the motions of systems of particles in the then-new kinetic theory of gases.

When this very general differential geometry came down to two-dimensional surfaces of constant curvature, it revealed excellent models for non-Euclidean geometries. Since the hypothesis of the obtuse angle correctly characterizes Euclidean geometry applied to the surface of a sphere, the non-Euclidean geometry based on it must be exactly as consistent as Euclidean geometry.

The case of the acute angle treated by Lobachevsky and Bolyai required a sharper tool. Beltrami found it in a projection into a disc in the Euclidean plane of the points of a non-Euclidean space, in which each geodesic from the non-Euclidean space corresponds to a chord of the disc. Geometry built on the hypothesis of the acute angle has the same consistency as Euclidean geometry. The key role of Euclidean geometry in proofs of the consistency of non-Euclidean geometries exposed the Elements to ever-deeper scrutiny.

The choice of undefined concepts and axioms is free, apart from the constraint of consistency. Bolyai apparently could not free himself from the persuasion that Euclidean geometry represented reality.

By his calculation, based on stellar parallaxes then just detected, his geometry could be physically meaningful only in gargantuan triangles spanning interstellar space. In fact, non-Euclidean geometries apply to the cosmos more locally than Lobachevsky imagined.

This was an extravagant piece of geometrizing—the replacement of gravitational force by the curvature of a surface. But it was not all. In relativity theory time is considered to be a dimension along with the three dimensions of space. On the closed four-dimensional world thus formed, the history of the universe stands revealed as describable by motion within a vast congeries of geodesics in a non-Euclidean universe. We welcome suggested improvements to any of our articles. You can make it easier for us to review and, hopefully, publish your contribution by keeping a few points in mind.

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Indeed, until the second half of the 19th…. Here a very effective approximating rule is being used that the square root of the sum…. The geometric problems in the papyri seek measurements of figures, like rectangles and triangles of given base and height, by means of suitable arithmetic operations. In Felix Klein suggested in his inaugural lecture at the University of Erlangen, Germany, that group theoretical ideas might be fruitfully put to use in the context of geometry.

Since the beginning of the 19th century, the study of projective geometry had attained…. Syntax and semantics fallacies and paradoxes In number game: Geometric fallacies and paradoxes foundations In mathematics: The foundations of geometry structure of coordination compounds In coordination compound: Geometry applications Earth measurement In geoid incommensurables In Incommensurables physics In relativity: Articles from Britannica Encyclopedias for elementary and high school students.

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Geometry (from the Ancient Greek: γεωμετρία; geo-"earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.. Geometry arose independently in a number of early cultures as .

Geometry. Geometry is all about shapes and their properties.. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles shapes that can be drawn on a piece of paper.

Learn high school geometry for free—transformations, congruence, similarity, trigonometry, analytic geometry, and more. Full curriculum of exercises and videos. Set students up for success in Geometry and beyond! Explore the entire Geometry curriculum: angles, geometric constructions, and more. Try it free!

the geometry of Sydney's famed opera house is suggestive of some modernistic sailing ship. Geometry: Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in.