Geometry

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Geometry (from

plane, distance, angle, surface, and curve, as fundamental concepts.[3]

Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics.[4] 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.

During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is

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 the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—

continuity, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure
on which some geometry is defined.

History

Main article: History of geometry
Arab
practicing geometry in the 15th century

The earliest recorded beginnings of geometry can be traced to ancient

motion within time-velocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries.[8] South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.[9][10]

In the 7th century BC, the

spiral bearing his name and obtained formulas for the volumes of surfaces of revolution
.

Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).

Diophantine equations.[21]
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."[22] Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes.
rational triangles (i.e. triangles with rational sides and rational areas).[23]

In the

mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.[24][25] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[26] Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[27] Omar Khayyam (1048–1131) found geometric solutions to cubic equations.[28] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were part of a line of research on the parallel postulate continued by later European geometers, including Vitello (c. 1230 – c. 1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri, that by the 19th century led to the discovery of hyperbolic geometry.[29]

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 equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).[30] This was a necessary precursor to the development of calculus and a precise quantitative science of physics.[31] The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661).[32] Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.[33]

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

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 dynamical systems. 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.[35]

Main concepts

The following are some of the most important concepts in geometry.[3][36]

Axioms

An illustration of Euclid's parallel postulate
See also: Euclidean geometry and Axiom

Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others[41] 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.[42]

Objects

Points

Main article: Point (geometry)

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",

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.[44][45] One of the oldest such geometries is Whitehead's point-free geometry, formulated by Alfred North Whitehead in 1919–1920.

Lines

Main article: Line (geometry)

Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself".[43] 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,[46] 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.[47] In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.[48]

Planes

Main article: Euclidean plane

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely;[43] 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;[49] it can be studied as an affine space, where collinearity and ratios can be studied but not distances;[50] it can be studied as the complex plane using techniques of complex analysis;[51] and so on.

Angles

Main article: Angle

rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[52]

Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.

In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.[43] The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.[53]

In

derivative.[54][55]

Curves

Main article:
Curve (geometry)

A

space curves.[56]

In topology, a curve is defined by a function from an interval of the real numbers to another space.

algebraic varieties of dimension one.[58]

Surfaces

Main article: Surface (mathematics)
A sphere is a surface that can be defined parametrically (by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly (by x2 + y2 + z2 − r2 = 0).

A

polynomial equations.[58]

Solids

Main article: Solid geometry
In Euclidean space, a ball is the volume bounded by a sphere.

A

solid is a three-dimensional object bounded by a closed surface; for example, a ball
is the volume bounded by a sphere.

Manifolds

Main article: Manifold

A

neighborhood that is homeomorphic to Euclidean space.[49] In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.[57]

Manifolds are used extensively in physics, including in general relativity and string theory.[60]

Measures: length, area, and volume

Main articles: Length, Area, and Volume
See also: Area § List of formulas, and Volume § Volume formulas

Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.[61]

In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.[62]

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.

Lebesgue integral.[64]

Other geometrical measures include the

angular measure, curvature, compactness measures
.

Metrics and measures

Main articles:
Metric (mathematics) and Measure (mathematics)
Euclidean metric
.

The concept of length or distance can be generalized, leading to the idea of

Riemannian metrics of general relativity.[66]

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.[67]

Congruence and similarity

Main articles: Congruence (geometry) and Similarity (geometry)

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

Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.[70]

Compass and straightedge constructions

Main article:
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 straightedge.[c] 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.

Rotation and orientation

Main articles:
Rotation (geometry) and Orientation (geometry)

The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space.

Dimension

For broader coverage of this topic, see
Dimension (mathematics)
.
topological dimension
=1

Where the traditional geometry allowed dimensions 1 (a

higher dimensions for nearly two centuries.[71] 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.[72]

In

fractal geometry).[73] 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.[74]

Symmetry

Main article: Symmetry
tiling of the hyperbolic plane

The theme of symmetry in geometry is nearly as old as the science of geometry itself.[75] Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers[76] and were investigated in detail before the time of Euclid.[39] 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.[77] 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.[78] Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines.[79] 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.[80] Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,[81][82] the latter in Lie theory and Riemannian geometry.[83][84]

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.[85] A similar and closely related form of duality exists between a vector space and its dual space.[86]

Contemporary geometry

Euclidean geometry

Main article: Euclidean geometry

solid figures, circles, and analytic geometry.[94]

Euclidean vectors

Main article: Euclidean vector

Euclidean vectors are used for a myriad of applications in physics and engineering, such as position, displacement, deformation, velocity, acceleration, force, etc.

Differential geometry

Differential geometry uses tools from calculus to study problems involving curvature.
Main article: Differential geometry

Differential geometry uses techniques of calculus and linear algebra to study problems in geometry.[95] It has applications in physics,[96] econometrics,[97] and bioinformatics,[98] among others.

In particular, differential geometry is of importance to

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).[100]

Non-Euclidean geometry

This section is an excerpt from Non-Euclidean geometry.[edit]
Behavior of lines with a common perpendicular in each of the three types of geometry
In
metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries
that have also been called non-Euclidean geometry.

Topology

Main article: Topology
A thickening of the trefoil knot

Topology is the field concerned with the properties of

compactness.[49]

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 homeomorphisms.[103] 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.[104]

Algebraic geometry

Main article: Algebraic geometry
Quintic Calabi–Yau threefold

Algebraic geometry is fundamentally the study by means of

Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.[107]

Algebraic geometry has applications in many areas, including cryptography[108] and string theory.[109]

Complex geometry

Main article: Complex geometry

several complex variables, and has found applications to string theory and mirror symmetry.[113]

Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces.[114][115][116] 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.[117][118] The primary objects of study in complex geometry are

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, worldsheets 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

Main article: Discrete geometry
Discrete geometry includes the study of various sphere packings.

Discrete geometry is a subject that has close connections with convex geometry.[119][120][121] It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc.[122][123] It shares many methods and principles with combinatorics.

Computational geometry

Main article: Computational geometry

implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal, and linear programming.[124]

Although being a young area of geometry, it has many applications in

image processing, computer-aided design, medical imaging, etc.[125]

Geometric group theory

Main article: Geometric group theory
The Cayley graph of the free group on two generators a and b

Geometric group theory uses large-scale geometric techniques to study finitely generated groups.[126] 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.[127]

Geometric group theory often revolves around the

right angled Artin groups.[126][128]

Convex geometry

Main article: Convex geometry

optimization and functional analysis and important applications in number theory
.

Convex geometry dates back to antiquity.

tilings and lattices
.

Applications

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

Art

Main article: Mathematics and art
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations

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.[130]

Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure.[131] These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.[132]

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.[133]

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.[134] 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.[135][136]

Architecture

Main articles: Mathematics and architecture and Architectural geometry

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

tessellations,[90] and the use of symmetry.[90]

Physics

Main article: Mathematical physics

The 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.[140]

quantum information theory.[143]

Other fields of mathematics

The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

infinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.[144][145]

Another important area of application is

scheme theory, which is used in Wiles's proof of Fermat's Last Theorem.[148]

See also
Main category: Geometry
Lists
Related topics
Other applications

Notes

  1. Desargues in the 17th century, all the way back to the implicit use of spherical geometry to understand the Earth geodesy
    and to navigate the oceans since antiquity.
  2. ^ Pythagorean triples are triples of integers with the property: . Thus, , , etc.
  3. ^ The ancient Greeks had some constructions using other instruments.

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Sources

Further reading

External links
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Geometry
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