Geometry

Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures.

A mathematician who works in the field of geometry is called a geometer.

Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' italic=no ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of and Riemannian geometry.

Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.

Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others.

Originally developed to model the physical world, geometry has applications in almost all , and also in art, architecture, and other activities that are related to graphics.

Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

---

History The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Ancient Egypt in the 2nd millennium BC.

Otto Neugebauer (1969). 9780486223322, Dover Publications. ISBN 9780486223322

. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus (), and the Babylonian clay tablets, such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the Nubia established a system of geometry including early versions of sun clocks.

In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. Pythagoras established the Pythagoreans, which is credited with the first proof of the Pythagorean theorem,Eves, Howard, , Saunders, 1990, . though the statement of the theorem has a long history.

Eudoxus (408–) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.Howard Eves, , Saunders, 1990, p. 141: "No work, except The Bible, has been more widely used...." Archimedes () of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of pi. He also studied the spiral bearing his name and obtained formulas for the of surfaces of revolution.

Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to the Shulba Sutras. According to , the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples,Pythagorean triples are triples of integers $(a,b,c)$ with the property: $a^2+b^2=c^2$. Thus, $3^2+4^2=5^2$, $8^2+15^2=17^2$, $12^2+35^2=37^2$ etc. which are particular cases of Diophantine equations.: "The arithmetic content of the Śulva Sūtras consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others." In the Bakhshali manuscript, there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas).

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.

"Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."". Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyam (1048–1131) found geometric solutions to . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on , including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Vitello (), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, pp. 447–494 470, Routledge, London and New York:

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

C.H. Edwards Jr. (2023). 9781461262305, Springer Science & Business Media. . ISBN 9781461262305

The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661).

(2023). 9781461386926, Springer Science & Business Media. . ISBN 9781461386926

Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.

C. R. Wylie (2023). 9780486141701, Courier Corporation. . ISBN 9780486141701

Two developments in geometry in the 19th century changed the way it had been studied previously.

These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of . As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.

Eduardo Bayro-Corrochano (2023). 9783319748306, Springer. . ISBN 9783319748306

---

Main concepts The following are some of the most important concepts in geometry.

Morris Kline (1990). 9780195061376, Oxford University Press. . ISBN 9780195061376

---

Axioms

Euclid took an abstract approach to geometry in his Elements,

Victor J. Katz (2023). 9780883851630, Cambridge University Press. . ISBN 9780883851630

one of the most influential books ever written.

David Berlinski (2023). 9780465038633, Basic Books. . ISBN 9780465038633

Euclid introduced certain , or , expressing primary or self-evident properties of points, lines, and planes.

Robin Hartshorne (2023). 9780387226767, Springer Science & Business Media. . ISBN 9780387226767

He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.

(2023). 9781351973533, Taylor & Francis. . ISBN 9781351973533

At the start of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others

I.M. Yaglom (2023). 9781461261353, Springer Science & Business Media. . ISBN 9781461261353

led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.

Audun Holme (2023). 9783642144417, Springer Science & Business Media. . ISBN 9783642144417

---

Objects

---

Points

Points are generally considered fundamental objects for building geometry. They may be defined by the properties that they must have, as in Euclid's definition as "that which has no part", Euclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion Press . or in synthetic geometry. In modern mathematics, they are generally defined as elements of a set called space, which is itself axiomatically defined.

With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes.

However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry, formulated by Alfred North Whitehead in 1919–1920.

---

Lines

Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.

(1995). 9780444883551, Elsevier. . ISBN 9780444883551

In differential geometry, a geodesic is a generalization of the notion of a line to manifold.

---

Planes

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles;

James R. Munkres (2023). 9780131816299, Prentice Hall, Inc. . ISBN 9780131816299

it can be studied as an affine space, where collinearity and ratios can be studied but not distances;

Wanda Szmielew (1983). 9789027712431 . ISBN 9789027712431

it can be studied as the complex plane using techniques of complex analysis;

Lars V. Ahlfors (1979). 9780070006577, McGraw-Hill. . ISBN 9780070006577

and so on.

---

Angles

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

In Euclidean geometry, angles are used to study and , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.

I. M. Gelʹfand (2023). 9780817639143, Birkhäuser. . ISBN 9780817639143

In differential geometry and calculus, the angles between or or surfaces can be calculated using the derivative.Stewart, James (2012). Calculus: Early Transcendentals, 7th ed., Brooks Cole Cengage Learning.

Jürgen Jost (2023). 9783540426271, Springer-Verlag. ISBN 9783540426271

.

---

Curves

A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called and those in 3-dimensional space are called .Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.

In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable Algebraic geometry studies , which are defined as algebraic varieties of dimension one.

---

Surfaces

A surface is a two-dimensional object, such as a sphere or paraboloid.Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." . In differential geometry

Manfredo Perdigão do Carmo (1976). 9780132125895, Prentice-Hall. . ISBN 9780132125895

and topology, surfaces are described by two-dimensional 'patches' (or neighborhoods) that are assembled by or , respectively. In algebraic geometry, surfaces are described by polynomial equations.

David Mumford (1999). 9783540632931, Springer-Verlag. ISBN 9783540632931

---

Manifolds

A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighborhood that is homeomorphism to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphism to Euclidean space.

Manifolds are used extensively in physics, including in general relativity and string theory.Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. .

---

Length, area, and volume

Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.

Steven A. Treese (2023). 9783319775777, Springer International Publishing. . ISBN 9783319775777

In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.

James W. Cannon (2023). 9781470437145, American Mathematical Soc.. . ISBN 9781470437145

Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and volume can be defined in terms of , such as the Riemann integral

Gilbert Strang (1991). 9780961408824, SIAM. . ISBN 9780961408824

or the Lebesgue integral.

H. S. Bear (2023). 9780120839711, Academic Press. . ISBN 9780120839711

---

Metrics and measures

The concept of length or distance can be generalized, leading to the idea of metric space.Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, . For instance, the Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.

Robert M. Wald (1984). 9780226870335, University of Chicago Press. ISBN 9780226870335

In a different direction, the concepts of length, area and volume are extended by measure theory, which studies methods of assigning a size or measure to sets, where the measures follow rules similar to those of classical area and volume.

Terence Tao (2023). 9780821869192, American Mathematical Soc.. . ISBN 9780821869192

---

Congruence and similarity

Congruence and similarity are concepts that describe when two shapes have similar characteristics.

Shlomo Libeskind (2023). 9780763743666, Jones & Bartlett Learning. . ISBN 9780763743666

In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape.

Mark A. Freitag (2023). 9780618610082, Cengage Learning. . ISBN 9780618610082

Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by .

Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.

George E. Martin (2023). 9781461256809, Springer Science & Business Media. . ISBN 9781461256809

---

Compass and straightedge constructions

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the compass and ruler. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis, parabolas and other curves, or mechanical devices, were found.

---

Dimension

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians and physicists have used for nearly two centuries.

Mark Blacklock (2023). 9780198755487, Oxford University Press. . ISBN 9780198755487

One example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the system's degrees of freedom. For instance, the configuration of a screw can be described by five coordinates.

In general topology, the concept of dimension has been extended from , to infinite dimension (, for example) and positive (in fractal geometry).

Roger Temam (2023). 9781461206453, Springer Science & Business Media. . ISBN 9781461206453

In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.

(1994). 9780821851548, American Mathematical Soc.. . ISBN 9780821851548

---

Symmetry

The theme of symmetry in geometry is nearly as old as the science of geometry itself.

Ian Stewart (2023). 9780465082377, Basic Books. . ISBN 9780465082377

Symmetric shapes such as the circle, and held deep significance for many ancient philosophers

Stakhov Alexey (2023). 9789814472579, World Scientific. . ISBN 9789814472579

and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of Leonardo da Vinci, M. C. Escher, and others.

Werner Hahn (1998). 9789810223632, World Scientific. . ISBN 9789810223632

In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is.

Brian J. Cantwell (2023). 9781139431712, Cambridge University Press. . ISBN 9781139431712

Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by , geometric transformations that take straight lines into straight lines.

(2023). 9781475753257, Springer Science & Business Media. . ISBN 9781475753257

However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,

Michio Kaku (2023). 9781468403978, Springer Science & Business Media. . ISBN 9781468403978

(2023). 9781470412272, American Mathematical Soc.. . ISBN 9781470412272

the latter in Lie theory and Riemannian geometry.

W-H. Steeb (1996). 9789813105034, World Scientific Publishing Company. . ISBN 9789813105034

Charles W. Misner (2023). 9780521021395, Cambridge University Press. . ISBN 9780521021395

A different type of symmetry is the principle of duality in projective geometry, among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and the result is an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.

G. Gierz (2023). 9783540394372, Springer. . ISBN 9783540394372

---

Contemporary geometry

---

Euclidean geometry

Euclidean geometry is geometry in its classical sense.

(2023). 9789400909595, Springer Science & Business Media. . ISBN 9789400909595

As it models the space of the physical world, it is used in many scientific areas, such as mechanics, astronomy, crystallography, and many technical fields, such as engineering,

W. Abbot (2023). 9789401727426, Springer Science & Business Media. . ISBN 9789401727426

architecture,

(2023). 9780226327839, University of Chicago Press. . ISBN 9780226327839

geodesy,

(2023). 9781483290799, Elsevier. . ISBN 9781483290799

aerodynamics,

(2023). 9781107053748, Cambridge University Press. . ISBN 9781107053748

and navigation.

Roy Williams (1998). 9781898563464, Horwood Pub.. . ISBN 9781898563464

The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points, lines, planes, , , congruence, similarity, , , and analytic geometry.

---

Differential geometry

Differential geometry uses techniques of calculus and linear algebra to study problems in geometry.

Gerard Walschap (2023). 9783110369540, De Gruyter. . ISBN 9783110369540

It has applications in physics,

Harley Flanders (2023). 9780486139616, Courier Corporation. . ISBN 9780486139616

econometrics,

(2023). 9780521651165, Cambridge University Press. . ISBN 9780521651165

and bioinformatics,

(2023). 9781118099520, John Wiley & Sons. . ISBN 9781118099520

among others.

In particular, differential geometry is of importance to mathematical physics due to Albert Einstein's general relativity postulation that the universe is curvature.

P.A.M. Dirac (2023). 9781400884193, Princeton University Press. . ISBN 9781400884193

Differential geometry can either be intrinsic (meaning that the spaces it considers are whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point) or extrinsic (where the object under study is a part of some ambient flat Euclidean space).

(2023). 9783319564784, Springer. . ISBN 9783319564784

---

Non-Euclidean geometry

Euclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators.

Boris A. Rosenfeld (2023). 9781441986801, Springer Science & Business Media. . ISBN 9781441986801

Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosoph, Dover, 1965, p. 164. This view was at first somewhat challenged by thinkers such as Saccheri, then finally overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory). They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen ( On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Albert Einstein's general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.

Peter Pesic (2023). 9780486453507, Courier Corporation. . ISBN 9780486453507

---

Topology Topology is the field concerned with the properties of continuous mappings,

Martin D. Crossley (2023). 9781852337827, Springer Science & Business Media. . ISBN 9781852337827

and can be considered a generalization of Euclidean geometry.

(1988). 9780080570853, Elsevier. . ISBN 9780080570853

In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness.

The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are .

George E. Martin (1996). 9780387906362, Springer Science & Business Media. . ISBN 9780387906362

This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.

J. P. May (1999). 9780226511832, University of Chicago Press. . ISBN 9780226511832

---

Algebraic geometry The field of algebraic geometry developed from the Cartesian geometry of co-ordinates. It underwent periodic periods of growth, accompanied by the creation and study of projective geometry, birational geometry, algebraic varieties, and commutative algebra, among other topics.

Suzanne C. Dieudonne (1985). 9780412993718, CRC Press. . ISBN 9780412993718

From the late 1950s through the mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.

(2023). 9780821836798, American Mathematical Soc.. . ISBN 9780821836798

Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a long-standing problem of number theory.

In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials.

Robin Hartshorne (2023). 9781475738490, Springer Science & Business Media. . ISBN 9781475738490

It has applications in many areas, including cryptography

(2023). 9783319639314, Springer. . ISBN 9783319639314

and string theory.

(2023). 9783540798149, Springer. . ISBN 9783540798149

---

Complex geometry Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane.

Daniel Huybrechts (2023). 9783540266877, Springer. . ISBN 9783540266877

Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.

R. O. Jr. Wells (2023). 9780387738918, Springer-Verlag. . ISBN 9780387738918

Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables, and has found applications to string theory and mirror symmetry. Hori, K., Thomas, R., Katz, S., Vafa, C., Pandharipande, R., Klemm, A., ... & Zaslow, E. (2003). Mirror symmetry (Vol. 1). American Mathematical Soc.

Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of .Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media. Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.

Work in the spirit of Riemann was carried out by the Italian school of algebraic geometry in the early 1900s. Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre, who introduced the concept of sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry.Serre, J. P. (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.Serre, J. P. (1956). Géométrie algébrique et géométrie analytique. In Annales de l'Institut Fourier (vol. 6, pp. 1–42). The primary objects of study in complex geometry are , complex algebraic varieties, and complex analytic varieties, and holomorphic vector bundles and coherent sheaves over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi–Yau manifolds, and these spaces find uses in string theory. In particular, of strings are modelled by Riemann surfaces, and superstring theory predicts that the extra 6 dimensions of 10 dimensional spacetime may be modelled by Calabi–Yau manifolds.

---

Discrete geometry Discrete geometry is a subject that has close connections with convex geometry.

Jiří Matoušek (2023). 9781461300397, Springer Science & Business Media. . ISBN 9781461300397

Chuanming Zong (2023). 9780521855358, Cambridge University Press. . ISBN 9780521855358

Peter M. Gruber (2023). 9783540711339, Springer Science & Business Media. . ISBN 9783540711339

It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of , triangulations, the Kneser-Poulsen conjecture, etc.

(2023). 9781400838981, Princeton University Press. . ISBN 9781400838981

Károly Bezdek (2023). 9781441906007, Springer Science & Business Media. . ISBN 9781441906007

It shares many methods and principles with combinatorics.

---

Computational geometry Computational geometry deals with and their implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal, and linear programming.

(2023). 9781461210986, Springer Science & Business Media. . ISBN 9781461210986

Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc.

---

Geometric group theory Geometric group theory uses large-scale geometric techniques to study finitely generated groups.

Clara Löh (2023). 9783319722542, Springer. . ISBN 9783319722542

It is closely connected to low-dimensional topology, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem.

(2023). 9780821852019, American Mathematical Soc.. . ISBN 9780821852019

Geometric group theory often revolves around the Cayley graph, which is a geometric representation of a group. Other important topics include quasi-isometry, Gromov-hyperbolic groups, and right angled Artin groups.

---

Convex geometry Convex geometry investigates convex set shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics.

Gerard Meurant (2023). 9780080934396, Elsevier Science. . ISBN 9780080934396

It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.

Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity. The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.

---

Applications Geometry has found applications in many fields, some of which are described below.

---

Art Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.

Jürgen Richter-Gebert (2023). 9783642172861, Springer Science & Business Media. . ISBN 9783642172861

Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure.

These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.

Brad J. Guigar (2023). 9781440523052, Adams Media. . ISBN 9781440523052

The golden ratio is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.

Tilings, or tessellations, have been used in art throughout history. Islamic art makes frequent use of tessellations, as did the art of M. C. Escher.

Escher's work also made use of hyperbolic geometry.

Cézanne advanced the theory that all images can be built up from the sphere, the cone, and the cylinder. This is still used in art theory today, although the exact list of shapes varies from author to author.

Phyllis Gelineau (2023). 9781111301262, Cengage Learning. . ISBN 9781111301262

---

Architecture Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.

(2023). 9783990433713, Birkhäuser. . ISBN 9783990433713

Helmut Pottmann (2023). 9781934493045, Bentley Institute Press. . ISBN 9781934493045

Applications of geometry to architecture include the use of projective geometry to create forced perspective,

(2023). 9781856693714, Laurence King Publishing. . ISBN 9781856693714

the use of in constructing domes and similar objects, the use of tessellations, and the use of symmetry.

---

Physics

The field of astronomy, especially as it relates to mapping the positions of and on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.

(1985). 9780521317795, Cambridge University Press. . ISBN 9780521317795

Riemannian geometry and pseudo-Riemannian geometry are used in general relativity.

Dmitriĭ Vladimirovich Alekseevskiĭ (2023). 9783037190517, European Mathematical Society. . ISBN 9783037190517

String theory makes use of several variants of geometry,

(2023). 9780465022663, Basic Books. . ISBN 9780465022663

as does quantum information theory.

(2023). 9781107026254, Cambridge University Press. ISBN 9781107026254

---

Other fields of mathematics

Calculus was strongly influenced by geometry. For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.

(2023). 9781483262406, Elsevier Science. . ISBN 9781483262406

(2023). 9781464174995, W. H. Freeman. . ISBN 9781464174995

Another important area of application is number theory.

Álvaro Lozano-Robledo (2023). 9781470450168, American Mathematical Soc.. . ISBN 9781470450168

In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.

Arturo Sangalli (2023). 9780691049557, Princeton University Press. . ISBN 9780691049557

Since the 19th century, geometry has been used for solving problems in number theory, for example through the geometry of numbers or, more recently, scheme theory, which is used in Wiles's proof of Fermat's Last Theorem.

(2023). 9781461219743, Springer Science & Business Media. . ISBN 9781461219743

---

See also

---

Lists

• List of geometers
• Algebraic geometers
• Differential geometers
• Geometers
• Topologists
• List of formulas in elementary geometry
• List of geometry topics
• List of important publications in geometry
• Lists of mathematics topics

---

Related topics

• Descriptive geometry
• Finite geometry
• Flatland, a book written by Edwin Abbott Abbott about two- and three-dimensional space, to understand the concept of four dimensions
• List of interactive geometry software

---

Other fields

• Molecular geometry

---

Notes

---

Sources

• C.B. Boyer (1991). 9780471543978, Wiley. . ISBN 9780471543978
• Roger Cooke (2023). 9780471444596, Wiley-Interscience. ISBN 9780471444596
• Takao Hayashi (2023). 9780801873966, The Johns Hopkins University Press. ISBN 9780801873966
• Takao Hayashi (2023). 9781405132510, Basil Blackwell. ISBN 9781405132510

---

Further reading

• Jay Kappraff (2023). 9789814556705, World Scientific Publishing. . ISBN 9789814556705
• Leonard Mlodinow (2023). 9780713996340, Allen Lane. ISBN 9780713996340

---

External links

• A course from
• Unusual Geometry Problems
• The Math Forum – Geometry
• The Math Forum – K–12 Geometry
• The Math Forum – College Geometry
• The Math Forum – Advanced Geometry
• Nature Precedings – Pegs and Ropes Geometry at Stonehenge
• The Mathematical Atlas – Geometric Areas of Mathematics
• "4000 Years of Geometry", lecture by Robin Wilson given at Gresham College, 3 October 2007 (available for MP3 and MP4 download as well as a text file)
• Finitism in Geometry at the Stanford Encyclopedia of Philosophy
• The Geometry Junkyard
• Interactive geometry reference with hundreds of applets
• Dynamic Geometry Sketches (with some Student Explorations)
• Geometry classes at Khan Academy

Post Comment