Geometry is the study of figures in a space of a given number of dimensions and of a given type. The most common types of geometry are plane geometry (dealing with objects like the point, line, circle, triangle, and polygon), solid geometry (dealing with objects like the line, sphere, and polyhedron), and spherical geometry (dealing with objects like the spherical triangle and spherical polygon). Geometry was part of the quadrivium taught in medieval universities.
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- Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space.
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A mathematical pun notes that without geometry, life is pointless. An old children's joke asks, 'What does an acorn say when it grows up?' and answers, 'Geometry' ('gee, I'm a tree').
Historically, the study of geometry proceeds from a small number of accepted truths (axioms or postulates), then builds up true statements using a systematic and rigorous step-by-step proof. However, there is much more to geometry than this relatively dry textbook approach, as evidenced by some of the beautiful and unexpected results of projective geometry (not to mention Schubert's powerful but questionable enumerative geometry).
The late mathematician E. T. Bell has described geometry as follows (Coxeter and Greitzer 1967, p. 1): 'With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics.' While the literature of algebra, arithmetic, and analysis has grown extensively since Bell's day, the remainder of his commentary holds even more so today.
Formally, a geometry is defined as a complete locally homogeneous Riemannian manifold. In , the possible geometries are Euclidean planar, hyperbolic planar, and elliptic planar. In , the possible geometries include Euclidean, hyperbolic, and elliptic, but also include five other types.
SEE ALSO:Absolute Geometry, Affine Geometry, Analytic Geometry, Cartesian Coordinates, Combinatorial Geometry, Computational Geometry, Differential Geometry, Discrete Geometry, Enumerative Geometry, Finsler Geometry, Inversive Geometry, Kawaguchi Geometry, Nil Geometry, Non-Euclidean Geometry, Ordered Geometry, Plane Geometry, Projective Geometry, Sol Geometry, Solid Geometry, Spherical Geometry, Stochastic Geometry, Thurston's Geometrization ConjectureREFERENCES:Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952.
Bogomolny, A. 'Geometry.' https://www.cut-the-knot.org/geometry.shtml.
Bold, B. FamousProblems of Geometry and How to Solve Them. New York: Dover, 1964.
Cinderella, Inc. 'Cinderella: The Interactive Geometry Software.' https://www.cinderella.de/.
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Coxeter, H. S. M. Introductionto Geometry, 2nd ed. New York: Wiley, 1969.
Coxeter, H. S. M. TheBeauty of Geometry: Twelve Essays. New York: Dover, 1999.
Coxeter, H. S. M. and Greitzer, S. L. GeometryRevisited. Washington, DC: Math. Assoc. Amer., 1967.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. UnsolvedProblems in Geometry. New York: Springer-Verlag, 1994.
Davis, C.; Grünbaum, B.; and Scherk, F. A. TheGeometric Vein: The Coxeter Festschrift. New York: Springer, 1981.
Eppstein, D. 'Geometry Junkyard.' https://www.ics.uci.edu/~eppstein/junkyard/.
Eppstein, D. 'Many-Dimensional Geometry.' https://www.ics.uci.edu/~eppstein/junkyard/highdim.html.
Eppstein, D. 'Planar Geometry.' https://www.ics.uci.edu/~eppstein/junkyard/2d.html.
Eppstein, D. 'Three-Dimensional Geometry.' https://www.ics.uci.edu/~eppstein/junkyard/3d.html.
Eves, H. W. ASurvey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, 1972.
Geometry Center. https://www.geom.umn.edu.
Ghyka, M. C. TheGeometry of Art and Life, 2nd ed. New York: Dover, 1977.
Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: The Open Court Publishing Co., 1921.
Ivins, W. M. Artand Geometry. New York: Dover, 1964.
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.
King, J. and Schattschneider, D. (Eds.). Geometry Turned On: Dynamic Software in Learning, Teaching and Research. Washington, DC: Math. Assoc. Amer., 1997.
Klee, V. 'Some Unsolved Problems in Plane Geometry.' Math. Mag.52,131-145, 1979.
Klein, F. FamousProblems of Elementary Geometry and Other Monographs. New York: Dover, 1956.
MathPages. 'Geometry.' https://www.mathpages.com/home/igeometr.htm.
Melzak, Z. A. Invitationto Geometry. New York: Wiley, 1983.
Meschkowski, H. Unsolvedand Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966.
Moise, E. E. Elementary Geometry from an Advanced Standpoint, 3rd ed. Reading, MA: Addison-Wesley, 1990.
Ogilvy, C. S. 'Some Unsolved Problems of Modern Geometry.' Ch. 11 in Excursions in Geometry. New York: Dover, pp. 143-153, 1990.
Playfair, J. Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Circle and the Geometry of Solids to which are added Elements of Plane and Spherical Trigonometry. New York: W. E. Dean, 1861.
Simon, M. Über die Entwicklung der Elementargeometrie im XIX. Jahrhundert.Leipzig: Teubner, pp. 97-105, 1906.
Townsend, R. Chapters on the Modern Geometry of the Point, Line, and Circle, Being the Substance of Lectures Delivered in the University of Dublin to the Candidates for Honors of the First Year in Arts, 2 vols. Dublin: Hodges, Smith and Co., 1863.
Uspenskii, V. A. SomeApplications of Mechanics to Mathematics. New York: Blaisdell, 1961.
Weisstein, E. W. 'Books about Geometry.' https://www.ericweisstein.com/encyclopedias/books/Geometry.html.
Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, 1961.
Referenced on Wolfram|Alpha: GeometryCITE THIS AS:Weisstein, Eric W. 'Geometry.' From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Geometry.html
- Major branches of geometry
- History of geometry
- Ancient geometry: practical and empirical
- Ancient geometry: abstract and applied
- The three classical problems
- Astronomy and trigonometry
- Ancient geometry: cosmological and metaphysical
- The post-classical period
- Transformation
- French circles
- Relaxation and rigour
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Join Britannica's Publishing Partner Program and our community of experts to gain a global audience for your work!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 surveying, and its name is derived from Greek words meaning “Earth measurement.” Eventually it was realized that geometry need not be limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) but that even the most abstract thoughts and images might be represented and developed in geometric terms.
This article begins with a brief guidepost to the major branches of geometry and then proceeds to an extensive historical treatment. For information on specific branches of geometry, seeEuclidean geometry, analytic geometry, projective geometry, differential geometry, non-Euclidean geometries, and topology.
Major branches of geometry
Euclidean geometry
In several ancient cultures there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic. The Elements epitomized the axiomatic-deductive method for many centuries.
Analytic geometry
Analytic geometry was initiated by the French mathematician René Descartes (1596–1650), who introduced rectangular coordinates to locate points and to enable lines and curves to be represented with algebraic equations. Algebraic geometry is a modern extension of the subject to multidimensional and non-Euclidean spaces.
Projective geometry
Projective geometry originated with the French mathematician Girard Desargues (1591–1661) to deal with those properties of geometric figures that are not altered by projecting their image, or “shadow,” onto another surface.
Differential geometry
Geometry
The German mathematician Carl Friedrich Gauss (1777–1855), in connection with practical problems of surveying and geodesy, initiated the field of differential geometry. Using differential calculus, he characterized the intrinsic properties of curves and surfaces. For instance, he showed that the intrinsic curvature of a cylinder is the same as that of a plane, as can be seen by cutting a cylinder along its axis and flattening, but not the same as that of a sphere, which cannot be flattened without distortion.
Non-Euclidean geometries
Beginning in the 19th century, various mathematicians substituted alternatives to Euclid’s parallel postulate, which, in its modern form, reads, “given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.” They hoped to show that the alternatives were logically impossible. Instead, they discovered that consistent non-Euclidean geometries exist.
Topology
Topology, the youngest and most sophisticated branch of geometry, focuses on the properties of geometric objects that remain unchanged upon continuous deformation—shrinking, stretching, and folding, but not tearing. The continuous development of topology dates from 1911, when the Dutch mathematician L.E.J. Brouwer (1881–1966) introduced methods generally applicable to the topic.
History of geometry
The earliest known unambiguous examples of written records—dating from Egypt and Mesopotamia about 3100 bce—demonstrate that ancient peoples had already begun to devise mathematical rules and techniques useful for surveying land areas, constructing buildings, and measuring storage containers. Beginning about the 6th century bce, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, from the combination of the Greek words geo (“Earth”) and metron (“measure”) for the measurement of the Earth.
In addition to describing some of the achievements of the ancient Greeks, notably Euclid’s logical development of geometry in the Elements, this article examines some applications of geometry to astronomy, cartography, and painting from classical Greece through medieval Islam and Renaissance Europe. It concludes with a brief discussion of extensions to non-Euclidean and multidimensional geometries in the modern age.
Ancient geometry: practical and empirical
The origin of geometry lies in the concerns of everyday life. The traditional account, preserved in Herodotus’s History (5th century bce), credits the Egyptians with inventing surveying in order to reestablish property values after the annual flood of the Nile. Similarly, eagerness to know the volumes of solid figures derived from the need to evaluate tribute, store oil and grain, and build dams and pyramids. Even the three abstruse geometrical problems of ancient times—to double a cube, trisect an angle, and square a circle, all of which will be discussed later—probably arose from practical matters, from religious ritual, timekeeping, and construction, respectively, in pre-Greek societies of the Mediterranean. And the main subject of later Greek geometry, the theory of conic sections, owed its general importance, and perhaps also its origin, to its application to optics and astronomy.
While many ancient individuals, known and unknown, contributed to the subject, none equaled the impact of Euclid and his Elements of geometry, a book now 2,300 years old and the object of as much painful and painstaking study as the Bible. Much less is known about Euclid, however, than about Moses. In fact, the only thing known with a fair degree of confidence is that Euclid taught at the Library of Alexandria during the reign of Ptolemy I (323–285/283 bce). Euclid wrote not only on geometry but also on astronomy and optics and perhaps also on mechanics and music. Only the Elements, which was extensively copied and translated, has survived intact.
Euclid’s Elements was so complete and clearly written that it literally obliterated the work of his predecessors. What is known about Greek geometry before him comes primarily from bits quoted by Plato and Aristotle and by later mathematicians and commentators. Among other precious items they preserved are some results and the general approach of Pythagoras (c. 580–c. 500 bce) and his followers. The Pythagoreans convinced themselves that all things are, or owe their relationships to, numbers. The doctrine gave mathematics supreme importance in the investigation and understanding of the world. Plato developed a similar view, and philosophers influenced by Pythagoras or Plato often wrote ecstatically about geometry as the key to the interpretation of the universe. Thus ancient geometry gained an association with the sublime to complement its earthy origins and its reputation as the exemplar of precise reasoning.
Finding the right angle
Ancient builders and surveyors needed to be able to construct right angles in the field on demand. The method employed by the Egyptians earned them the name “rope pullers” in Greece, apparently because they employed a rope for laying out their construction guidelines. One way that they could have employed a rope to construct right triangles was to mark a looped rope with knots so that, when held at the knots and pulled tight, the rope must form a right triangle. The simplest way to perform the trick is to take a rope that is 12 units long, make a knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop, as shown in the animation. However, the Egyptian scribes have not left us instructions about these procedures, much less any hint that they knew how to generalize them to obtain the Pythagorean theorem: the square on the line opposite the right angle equals the sum of the squares on the other two sides. Similarly, the Vedic scriptures of ancient India contain sections called sulvasutras, or “rules of the rope,” for the exact positioning of sacrificial altars. The required right angles were made by ropes marked to give the triads (3, 4, 5) and (5, 12, 13).
In Babylonian clay tablets (c. 1700–1500 bce) modern historians have discovered problems whose solutions indicate that the Pythagorean theorem and some special triads were known more than a thousand years before Euclid. A right triangle made at random, however, is very unlikely to have all its sides measurable by the same unit—that is, every side a whole-number multiple of some common unit of measurement. This fact, which came as a shock when discovered by the Pythagoreans, gave rise to the concept and theory of incommensurability.
Locating the inaccessible
By ancient tradition, Thales of Miletus, who lived before Pythagoras in the 6th century bce, invented a way to measure inaccessible heights, such as the Egyptian pyramids. Although none of his writings survives, Thales may well have known about a Babylonian observation that for similar triangles (triangles having the same shape but not necessarily the same size) the length of each corresponding side is increased (or decreased) by the same multiple. A determination of the height of a tower using similar triangles is demonstrated in the figure. The ancient Chinese arrived at measures of inaccessible heights and distances by another route, using “complementary” rectangles, as seen in the next figure, which can be shown to give results equivalent to those of the Greek method involving triangles.
Estimating the wealth
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A Babylonian cuneiform tablet written some 3,500 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise on circular enclosures with an implied value of π = 3. The contractor for King Solomon’s swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23), used the same value. However, the Hebrews should have taken their π from the Egyptians before crossing the Red Sea, for the Rhind papyrus (c. 2000 bce; our principal source for ancient Egyptian mathematics) implies π = 3.1605.
Knowledge of the area of a circle was of practical value to the officials who kept track of the pharaoh’s tribute as well as to the builders of altars and swimming pools. Ahmes, the scribe who copied and annotated the Rhind papyrus (c. 1650 bce), has much to say about cylindrical granaries and pyramids, whole and truncated. He could calculate their volumes, and, as appears from his taking the Egyptian seked, the horizontal distance associated with a vertical rise of one cubit, as the defining quantity for the pyramid’s slope, he knew something about similar triangles.
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