Geometry (from the Ancient Greek: γεωμετρία;
geo- "earth", -metron "measurement") is a
branch of mathematics concerned with questions
of shape, size, relative position of figures,
and the properties of space. A mathematician
who works in the field of geometry is called
a geometer.
Geometry arose independently in a number of
early cultures as a practical way for dealing
with lengths, areas, and volumes. Geometry
began to see elements of formal mathematical
science emerging in the West as early as the
6th century BC. By the 3rd century BC, geometry
was put into an axiomatic form by Euclid,
whose treatment, Euclid's Elements, set a
standard for many centuries to follow. Geometry
arose independently in India, with texts providing
rules for geometric constructions appearing
as early as the 3rd century BC. Islamic scientists
preserved Greek ideas and expanded on them
during the Middle Ages. By the early 17th
century, geometry had been put on a solid
analytic footing by mathematicians such as
René Descartes and Pierre de Fermat. Since
then, and into modern times, geometry has
expanded into non-Euclidean geometry and manifolds,
describing spaces that lie beyond the normal
range of human experience.While geometry has
evolved significantly throughout the years,
there are some general concepts that are more
or less fundamental to geometry. These include
the concepts of points, lines, planes, surfaces,
angles, and curves, as well as the more advanced
notions of manifolds and topology or metric.Geometry
has applications to many fields, including
art, architecture, physics, as well as to
other branches of mathematics.
== Overview ==
Contemporary geometry has many subfields:
Euclidean geometry is geometry in its classical
sense. The mandatory educational curriculum
of the majority of nations includes the study
of points, lines, planes, angles, triangles,
congruence, similarity, solid figures, circles,
and analytic geometry. Euclidean geometry
also has applications in computer science,
crystallography, and various branches of modern
mathematics.
Differential geometry uses techniques of calculus
and linear algebra to study problems in geometry.
It has applications in physics, including
in general relativity.
Topology is the field concerned with the properties
of geometric objects that are unchanged by
continuous mappings. In practice, this often
means dealing with large-scale properties
of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes
in the Euclidean space and its more abstract
analogues, often using techniques of real
analysis. It has close connections to convex
analysis, optimization and functional analysis
and important applications in number theory.
Algebraic geometry studies geometry through
the use of multivariate polynomials and other
algebraic techniques. It has applications
in many areas, including cryptography and
string theory.
Discrete geometry is concerned mainly with
questions of relative position of simple geometric
objects, such as points, lines and circles.
It shares many methods and principles with
combinatorics.
Computational geometry deals with algorithms
and their implementations for manipulating
geometrical objects. Although being a young
area of geometry, it has many applications
in computer vision, image processing, computer-aided
design, medical imaging, etc.
== History ==
The earliest recorded beginnings of geometry
can be traced to ancient Mesopotamia and Egypt
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, construction, astronomy,
and various crafts. The earliest known texts
on geometry are the Egyptian Rhind Papyrus
(2000–1800 BC) and Moscow Papyrus (c. 1890
BC), 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. 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.
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.
Indian 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's Aryabhatiya
(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 (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. 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 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 on quadrilaterals,
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 (1288–1344),
Alfonso, 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 and equations,
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.
The second geometric development of this period
was the systematic study of projective geometry
by Girard Desargues (1591–1661). Projective
geometry is a geometry without measurement
or parallel lines, just the study of how points
are related to each other.
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.
== Important concepts in geometry ==
The following are some of the most important
concepts in geometry.
=== Axioms ===
Euclid 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 or synthetic geometry.
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 ===
Points are considered fundamental objects
in Euclidean geometry. They have been defined
in a variety of ways, including Euclid's definition
as 'that which has no part' and through the
use of algebra or nested sets. In many areas
of geometry, such as analytic geometry, differential
geometry, and topology, all objects are considered
to be built up from points. However, there
has been some study of geometry without reference
to points.
=== 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. In differential
geometry, a geodesic is a generalization of
the notion of a line to curved spaces.
=== Planes ===
A plane is a flat, two-dimensional surface
that extends infinitely far. Planes are used
in every area 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.
=== 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 polygons and triangles, 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 and calculus, the angles
between plane curves or space curves or surfaces
can be calculated using the derivative.
=== Curves ===
A curve 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 of
dimension one.
=== Surfaces ===
A surface is a two-dimensional object, such
as a sphere or paraboloid. In differential
geometry and topology, surfaces are described
by two-dimensional 'patches' (or neighborhoods)
that are assembled by diffeomorphisms or homeomorphisms,
respectively. In algebraic geometry, surfaces
are described by polynomial equations.
=== 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 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
=== 
Topologies and metrics ===
A topology is a mathematical structure on
a set that tells how elements of the set relate
spatially to each other. The best-known examples
of topologies come from metrics, which are
ways of measuring distances between points.
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.
=== 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 allowed in geometric
constructions are the compass and straightedge.
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 parabolas and other curves,
as well as 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 have used higher dimensions
for nearly two centuries. The concept of dimension
has gone through stages of being any natural
number n, to being possibly infinite with
the introduction of Hilbert space, to being
any positive real number in fractal geometry.
Dimension theory is a technical area, initially
within general topology, that discusses definitions;
in common with most mathematical ideas, dimension
is now defined rather than an intuition. Connected
topological manifolds have a well-defined
dimension; this is a theorem (invariance of
domain) rather than anything a priori.
The issue of dimension still matters to geometry
as many classic questions still lack complete
answers. For instance, many open problems
in topology depend on the dimension of an
object for the result. In physics, dimensions
3 of space and 4 of space-time are special
cases in geometric topology, and dimensions
10 and 11 are key ideas in string theory.
Currently, the existence of the theoretical
dimensions is purely defined by technical
reasons; it is likely that further research
may result in a geometric reason for the significance
of 10 or 11 dimensions in the theory, lending
credibility or possibly disproving string
theory.
=== Symmetry ===
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 M.C. Escher. Nonetheless,
it was not until the second half of 19th century
that the unifying role of symmetry in foundations
of geometry was recognized. 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' proved most influential.
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 (see Duality
(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 you will get an equally true theorem.
A similar and closely related form of duality
exists between a vector space and its dual
space.
=== Non-Euclidean geometry ===
In the nearly two thousand years since Euclid,
while the range of geometrical questions asked
and answered inevitably expanded, the basic
understanding of space remained essentially
the same. 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. This dominant view was 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 Einstein's general relativity theory, and
Riemannian geometry, that considers very general
spaces in which the notion of length is defined,
is a mainstay of modern geometry.
== Contemporary geometry ==
=== Euclidean geometry ===
Euclidean geometry has become closely connected
with computational geometry, computer graphics,
convex geometry, incidence geometry, finite
geometry, discrete geometry, and some areas
of combinatorics. Attention was given to further
work on Euclidean geometry and the Euclidean
groups by crystallography and the work of
H. S. M. Coxeter, and can be seen in theories
of Coxeter groups and polytopes. Geometric
group theory is an expanding area of the theory
of more general discrete groups, drawing on
geometric models and algebraic techniques.
=== Differential geometry ===
Differential geometry has been of increasing
importance to mathematical physics due to
Einstein's general relativity postulation
that the universe is curved. Contemporary
differential geometry is 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, and not a priori
parts of some ambient flat Euclidean space.
=== Topology and geometry ===
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 dictum
'topology is rubber-sheet geometry'. Contemporary
geometric topology and differential topology,
and particular subfields such as Morse theory,
would be counted by most mathematicians as
part of geometry. Algebraic topology and general
topology have gone their own ways.
=== Algebraic geometry ===
The field of algebraic geometry is the modern
incarnation of the Cartesian geometry of co-ordinates.
From late 1950s through 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.
The study of low-dimensional algebraic varieties,
algebraic curves, algebraic surfaces and algebraic
varieties of dimension 3 ("algebraic threefolds"),
has been far advanced. Gröbner basis theory
and real algebraic geometry are among more
applied subfields of modern algebraic geometry.
Arithmetic geometry is an active field combining
algebraic geometry and number theory. Other
directions of research involve moduli spaces
and complex geometry. Algebro-geometric methods
are commonly applied in string and brane theory.
== 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.
=== Architecture ===
Mathematics and architecture are related,
since, as with other arts, architects use
mathematics for several reasons. Apart from
the mathematics needed when engineering buildings,
architects use geometry: to define the spatial
form of a building; from the Pythagoreans
of the sixth century BC onwards, to create
forms considered harmonious, and thus to lay
out buildings and their surroundings according
to mathematical, aesthetic and sometimes religious
principles; to decorate buildings with mathematical
objects such as tessellations; and to meet
environmental goals, such as to minimise wind
speeds around the bases of tall buildings.
=== 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.
Modern geometry has many ties to physics as
is exemplified by the links between pseudo-Riemannian
geometry and general relativity. One of the
youngest physical theories, string theory,
is also very geometric in flavour.
=== Other fields of mathematics ===
Geometry has also had a large effect on other
areas of mathematics. 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. The subject of geometry was
further enriched by the study of the intrinsic
structure of geometric objects that originated
with Euler and Gauss and led to the creation
of topology and differential geometry.
An important area of application is number
theory. In ancient Greece the Pythagoreans
considered the role of numbers in geometry.
However, the discovery of incommensurable
lengths, which contradicted their philosophical
views, made them abandon abstract numbers
in favor of concrete geometric quantities,
such as length and area of figures. 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.
While the visual nature of geometry makes
it initially more accessible than other mathematical
areas such as algebra or number theory, geometric
language is also used in contexts far removed
from its traditional, Euclidean provenance
(for example, in fractal geometry and algebraic
geometry).Analytic geometry applies methods
of algebra to geometric questions, typically
by relating geometric curves to algebraic
equations. These ideas played a key role in
the development of calculus in the 17th century
and led to the discovery of many new properties
of plane curves. Modern algebraic geometry
considers similar questions on a vastly more
abstract level.
Leonhard Euler, in studying problems like
the Seven Bridges of Königsberg, considered
the most fundamental properties of geometric
figures based solely on shape, independent
of their metric properties. Euler called this
new branch of geometry geometria situs (geometry
of place), but it is now known as topology.
Topology grew out of geometry, but turned
into a large independent discipline. It does
not differentiate between objects that can
be continuously deformed into each other.
The objects may nevertheless retain some geometry,
as in the case of hyperbolic knots.
== See also ==
=== Lists ===
List of geometers
Category:Algebraic geometers
Category:Differential geometers
Category:Geometers
Category:Topologists
List of formulas in elementary geometry
List of geometry topics
List of important publications in geometry
List of mathematics articles
=== 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
Interactive geometry software
=== Other fields ===
Molecular geometry
== Notes ==
== Sources ==
Boyer, C.B. (1991) [1989]. A History of Mathematics
(Second edition, revised by Uta C. Merzbach
ed.). New York: Wiley. ISBN 978-0-471-54397-8.
Cooke, Roger (2005), The History of Mathematics:,
New York: Wiley-Interscience, 632 pages, ISBN
978-0-471-44459-6
Hayashi, Takao (2003), "Indian Mathematics",
in Grattan-Guinness, Ivor, Companion Encyclopedia
of the History and Philosophy of the Mathematical
Sciences, 1, Baltimore, MD: The Johns Hopkins
University Press, 976 pages, pp. 118–130,
ISBN 978-0-8018-7396-6
Hayashi, Takao (2005), "Indian Mathematics",
in Flood, Gavin, The Blackwell Companion to
Hinduism, Oxford: Basil Blackwell, 616 pages,
pp. 360–375, ISBN 978-1-4051-3251-0
Nikolai I. Lobachevsky, Pangeometry, translator
and editor: A. Papadopoulos, Heritage of European
Mathematics Series, Vol. 4, European Mathematical
Society, 2010.
== Further reading ==
Jay Kappraff, A Participatory Approach to
Modern Geometry, 2014, World Scientific Publishing,
ISBN 978-981-4556-70-5.
Leonard Mlodinow, Euclid's Window – The
Story of Geometry from Parallel Lines to Hyperspace,
UK edn. Allen Lane, 1992.
== External links ==
A geometry course from Wikiversity
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
