- Geometry (from Ancient Greek γεωμετρία (geōmetría) ‘land measurement’; from γῆ (gê) ‘earth, land’, and μέτρον (métron) ‘a measure’) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with 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 Gauss‘ Theorema Egregium (“remarkable theorem”) that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space.
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 sciences, 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.This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.
A European and an Arab practicing geometry in the 15th century
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia
in the 2nd millennium BC.
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
, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus
(2000–1800 BC) and Moscow Papyrus
(c. 1890 BC), 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 ancient Nubians
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’ theorem
Pythagoras established the Pythagorean School
, which is credited with the first proof of the Pythagorean theorem
though the statement of the theorem has a long history. Eudoxus
(408–c. 355 BC) 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.
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. Archimedes
(c. 287–212 BC) of Syracuse
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 volumes
of surfaces of revolution
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid’s Elements, (c. 1310).
mathematicians also made many important contributions in geometry. The Satapatha Brahmana
(3rd century BC) contains rules for ritual geometric constructions that are similar to the Sulba Sutras
According to (Hayashi 2005
, p. 363), 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
which are particular cases of Diophantine equations
In the Bakhshali manuscript
, there is 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
(499) includes the computation of areas and volumes. Brahmagupta
wrote his astronomical work Brāhma Sphuṭa Siddhānta
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
triangles with rational sides and rational areas).
In the Middle Ages
, mathematics in medieval Islam
contributed to the development of geometry, especially algebraic geometry
(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 ratios
of geometrical quantities, and contributed to the development of analytic geometry
. Omar Khayyám
(1048–1131) found geometric solutions to cubic equations
The theorems of Ibn al-Haytham
(Alhazen), Omar Khayyam and Nasir al-Din al-Tusi
, 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 Witelo
(c. 1230–c. 1314), Gersonides
, John Wallis
, and Giovanni Girolamo Saccheri
In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates
, by René Descartes
(1596–1650) and Pierre de Fermat
This was a necessary precursor to the development of calculus
and a precise quantitative science of physics
The second geometric development of this period was the systematic study of projective geometry
by Girard Desargues
Projective geometry studies properties of shapes which are unchanged under projections
, especially as they relate to artistic perspective
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 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
The following are some of the most important concepts in geometry.
took an abstract approach to geometry in his Elements
one of the most influential books ever written.
Euclid introduced certain axioms
, or postulates
, expressing primary or self-evident properties of points, lines, and planes.
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
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
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.
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”,
or in synthetic geometry
. In modern mathematics, they are generally defined as elements
of a set
, which is itself axiomatically
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 has 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
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.
In differential geometry, a geodesic
is a generalization of the notion of a line to curved spaces
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;
it can be studied as an affine space
, where collinearity and ratios can be studied but not distances;
it can be studied as the complex plane
using techniques of complex analysis
and so on.
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.
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
, 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
In differential geometry
, the angles between plane curves
or space curves
can be calculated using the derivative
is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves
and those in 3-dimensional space are called space curves
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 algebraic curves
, which are defined as algebraic varieties
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.)
is a two-dimensional object, such as a sphere or paraboloid.
In differential geometry
surfaces are described by two-dimensional ‘patches’ (or neighborhoods
) that are assembled by diffeomorphisms
, respectively. In algebraic geometry, surfaces are described by polynomial equations
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 homeomorphic
to Euclidean space.
In differential geometry
, a differentiable manifold
is a space where each neighborhood is diffeomorphic
to Euclidean space.
Manifolds are used extensively in physics, including in general relativity
and string theory
Length, area, and volume
, and volume
describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.
In Euclidean geometry
and analytic geometry
, the length of a line segment can often be calculated by the Pythagorean theorem
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 integrals
, such as the Riemann integral
or the Lebesgue integral
Metrics and measures
The concept of length or distance can be generalized, leading to the idea of metrics
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
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
, where the measures follow rules similar to those of classical area and volume.
Congruence and similarity
are concepts that describe when two shapes have similar characteristics.
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. 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.
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
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.
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 higher dimensions
for nearly two centuries.
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 natural numbers
, to infinite dimension (Hilbert spaces
, for example) and positive real numbers
(in fractal geometry
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.
The theme of symmetry
in geometry is nearly as old as the science of geometry itself.
Symmetric shapes such as the circle
, regular polygons
and platonic solids
held deep significance for many ancient philosophers
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.
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
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.
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
the latter in Lie theory
and Riemannian geometry
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
, lies in
, 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
is geometry in its classical sense.
As it models the space of the physical world, it is used in many scientific areas, such as mechanics
and many technical fields, such as engineering
The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points
, solid figures
, and analytic geometry
uses techniques of calculus
and linear algebra
to study problems in geometry.
It has applications in physics
In particular, differential geometry is of importance to mathematical physics
due to Albert Einstein
‘s general relativity
postulation that the universe
Differential geometry can either be intrinsic
(meaning that the spaces it considers are smooth manifolds
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).
Euclidean geometry was not the only historical form of geometry studied. Spherical geometry
has long been used by astronomers, astrologers, and navigators.
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
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.
is the field concerned with the properties of continuous mappings
and can be considered a generalization of Euclidean geometry.
In practice, topology often means dealing with large-scale properties of spaces, such as connectedness
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
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
The field of algebraic geometry
developed from the Cartesian geometry
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.
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. 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
It has applications in many areas, including cryptography
and string theory
studies the nature of geometric structures modelled on, or arising out of, the complex plane
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
Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann
in his study of Riemann surfaces
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.
The primary objects of study in complex geometry are complex manifolds
, 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, 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.
is a subject that has close connections with convex geometry
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
, the Kneser-Poulsen conjecture, etc.
It shares many methods and principles with combinatorics
deals with algorithms
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
Although being a young area of geometry, it has many applications in computer vision
, image processing
, computer-aided design
, medical imaging
Geometric group theory
Geometric group theory
The Cayley graph of the free group
on two generators a
uses large-scale geometric techniques to study finitely generated groups
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
Geometric group theory often revolves around the Cayley graph
, which is a geometric representation of a group. Other important topics include quasi-isometries
, Gromov-hyperbolic groups
, and right angled Artin groups
shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis
and discrete mathematics
It has close connections to convex analysis
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
, and later Kepler
all studied convex polytopes
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
Geometry has found applications in many fields, some of which are described below.
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
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.
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.
, 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
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.
Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.
Applications of geometry to architecture include the use of projective geometry
to create forced perspective
the use of conic sections
in constructing domes and similar objects,
the use of tessellations
and the use of symmetry.
The field of astronomy
, especially as it relates to mapping the positions of stars
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 are used in general relativity
. String theory
makes use of several variants of geometry,
as does quantum information theory
Other fields of mathematics
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
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 plane curves
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.
Another important area of application is number theory
In ancient Greece
considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.
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