GEOMETRY (from the
Ancient Greek
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
Geometry arose independently in a number of early cultures as a
practical way for dealing with lengths , areas , and volumes .
Geometry
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
Euclid , whose treatment,
Euclid\'s Elements , set a standard for many centuries to follow.
Geometry
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
Middle Ages . By the early 17th century, geometry had been put on
a solid analytic footing by mathematicians such as
René Descartes
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.
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
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
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 .
Geometry
Geometry has applications to many fields, including art,
architecture, physics, as well as to other branches of mathematics.
CONTENTS
* 1 History
* 2 Important concepts in geometry
* 2.1 Axioms
* 2.2 Points
* 2.3 Lines
* 2.4 Planes
* 2.5 Angles
* 2.6 Curves
* 2.7 Surfaces
* 2.8 Manifolds
* 2.9 Topologies and metrics
* 2.10
Compass and straightedge constructions
* 2.11
Dimension
Dimension
* 2.12
Symmetry
* 2.13 Non-
Euclidean geometry
* 3 Contemporary geometry
* 3.1
Euclidean geometry
* 3.2
Differential geometry
* 3.3
Topology
Topology and geometry
* 3.4
Algebraic geometry
Algebraic geometry
* 4 Applications
* 4.1 Art
* 4.2
Architecture
Architecture
* 4.3
Physics
Physics
* 4.4 Other fields of mathematics
* 5 See also
* 5.1 Lists
* 5.2 Related topics
* 5.3 Other fields
* 6 Notes
* 7 Sources
* 8 Further reading
* 9 External links
HISTORY
Main article:
History of geometry A European and an Arab
practicing geometry in the 15th century.
The earliest recorded beginnings of geometry can be traced to ancient
Mesopotamia
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
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
Pythagoras established the
Pythagorean School , which is credited with the first proof of the
Pythagorean theorem
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
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
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)
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
Aryabhata 's Aryabhatiya
(499) includes the computation of areas and volumes.
Brahmagupta
Brahmagupta wrote
his astronomical work Brāhma Sphuṭa Siddhānta in 628. Chapter 12,
containing 66
Sanskrit
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
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
Thābit ibn Qurra (known
as Thebit in
Latin
Latin ) (836–901) dealt with arithmetic operations
applied to ratios of geometrical quantities, and contributed to the
development of analytic geometry .
Omar Khayyám
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
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
Alfonso ,
John Wallis
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
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
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
Felix Klein (which
generalized the Euclidean and non-Euclidean geometries). Two of the
master geometers of the time were
Bernhard Riemann
Bernhard Riemann (1826–1866),
working primarily with tools from mathematical analysis , and
introducing the
Riemann surface
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
An illustration of Euclid's parallel postulate See also:
Euclidean geometry
Euclid
Euclid took an abstract approach to geometry in his Elements , one of
the most influential books ever written.
Euclid
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
Main article:
Point (geometry)
Point (geometry)
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
Main article:
Line (geometry)
Euclid
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
Main article:
Plane (geometry)
Plane (geometry)
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
Main article:
Angle
Angle
Euclid
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. 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.
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
Main article:
Curve
Curve (geometry)
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
Algebraic geometry studies algebraic curves , which
are defined as algebraic varieties of dimension one.
SURFACES
Main article:
Surface (mathematics)
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 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
Main article:
Manifold
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
Main article:
Topology
Topology Visual checking of the Pythagorean
theorem for the (3, 4, 5) triangle as in the
Zhoubi Suanjing 500–200
BC. The
Pythagorean theorem
Pythagorean theorem is a consequence of the
Euclidean metric .
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
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 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
Main article:
Dimension
Dimension The
Koch snowflake
Koch snowflake , with fractal
dimension =log4/log3 and topological dimension =1
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.
Dimension
Dimension has gone through stages of being any natural
number n, possibly infinite with the introduction of
Hilbert space
Hilbert space ,
and any positive real number in fractal geometry .
Dimension
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, in the absence of
complete answers to classic questions. Dimensions 3 of space and 4 of
space-time are special cases in geometric topology .
Dimension
Dimension 10 or
11 is a key number in string theory . Research may bring a
satisfactory geometric reason for the significance of 10 and 11
dimensions.
SYMMETRY
Main article:
Symmetry A tiling of the hyperbolic plane
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
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
Differential geometry uses tools from calculus to study problems
involving curvature.
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
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
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
Geometry
Geometry lessons in the 20th century
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
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
A thickening of the trefoil knot
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
Quintic Calabi–Yau threefold
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
Jean-Pierre Serre and
Alexander Grothendieck
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
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
Geometry has found applications in many fields, some of which are
described below.
ART
Main article:
Mathematics
Mathematics and art
Mathematics
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
Main articles:
Mathematics
Mathematics and architecture and Architectural
geometry
Mathematics
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
Main article:
Mathematical physics
Mathematical physics The 421polytope ,
orthogonally projected into the E8
Lie group
Lie group
Coxeter plane . Lie
groups have several applications in 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
Geometry has also had a large effect on other areas of mathematics.
For instance, the introduction of coordinates by
René Descartes
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
Euler and Gauss and led to the creation of topology and
differential geometry . The
Pythagoreans discovered that the
sides of a triangle could have incommensurable lengths.
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
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
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
Euler called this new branch of geometry geometria situs (geometry of
place), but it is now known as topology .
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
*
Geometry
Geometry portal
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
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
* ^ A B (Boyer 1991 , "Ionia and the Pythagoreans" p. 43)
* ^ Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews
(1998).
Fractal geometry in digital imaging.
Academic Press . p. 1.
ISBN 0-12-703970-8
* ^ A B (Staal 1999 )
* ^ A B O\'Connor, John J. ; Robertson, Edmund F. , "Al-Sabi Thabit
ibn Qurra al-Harrani", MacTutor History of
Mathematics
Mathematics archive ,
University of St Andrews
University of St Andrews .
* ^ Lamb, Evelyn (2015-11-08). "By Solving the Mysteries of
Shape-Shifting Spaces, Mathematician Wins $3-Million Prize".
Scientific American. Retrieved 2016-08-29.
* ^ A B Tabak, John (2014). Geometry: the language of space and
form. Infobase Publishing. p. xiv. ISBN 081604953X .
* ^ A B Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent
curriculum. American educator, 26(2), 1-18.
* ^ J. Friberg, "Methods and traditions of Babylonian mathematics.
Plimpton 322, Pythagorean triples, and the Babylonian triangle
parameter equations", Historia Mathematica, 8, 1981, pp. 277—318.
* ^ Neugebauer, Otto (1969) . The Exact Sciences in Antiquity (2
ed.).
Dover Publications . ISBN 978-0-486-22332-2 . Chap. IV
"Egyptian
Mathematics
Mathematics and Astronomy", pp. 71–96.
* ^ (Boyer 1991 , "Egypt" p. 19)
* ^ Ossendrijver, Mathieu (29 Jan 2016). "Ancient Babylonian
astronomers calculated Jupiter’s position from the area under a
time-velocity graph". Science. 351 (6272): 482–484. PMID 26823423 .
doi :10.1126/science.aad8085 . Retrieved 29 January 2016.
* ^ Depuydt, Leo (1 January 1998). "Gnomons at Meroë and Early
Trigonometry". The Journal of Egyptian Archaeology. 84: 171–180.
JSTOR
JSTOR 3822211 . doi :10.2307/3822211 – via JSTOR.
* ^ Slayman, Andrew (May 27, 1998). "Neolithic Skywatchers".
Archaeology Magazine Archive.
* ^ Eves, Howard, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0 .
* ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by
Hippasus of Metapontum". The Annals of Mathematics.
* ^ James R. Choike (1980). "The Pentagram and the Discovery of an
Irrational Number". The Two-Year College
Mathematics
Mathematics Journal.
* ^ (Boyer 1991 , "The Age of Plato and Aristotle" p. 92)
* ^ (Boyer 1991 , "
Euclid
Euclid of Alexandria" p. 119)
* ^ (Boyer 1991 , "
Euclid
Euclid of Alexandria" p. 104)
* ^ Howard Eves, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible
, has been more widely used...."
* ^ O'Connor, J.J.; Robertson, E.F. (February 1996). "A history of
calculus".
University of St Andrews
University of St Andrews . Retrieved 2007-08-07.
* ^
Pythagorean triples are triples of integers ( a , b , c )
{displaystyle (a,b,c)} with the property: a 2 + b 2 = c
2 {displaystyle a^{2}+b^{2}=c^{2}} . Thus, 3 2 + 4 2
= 5 2 {displaystyle 3^{2}+4^{2}=5^{2}} , 8 2 + 15 2
= 17 2 {displaystyle 8^{2}+15^{2}=17^{2}} , 12 2 + 35
2 = 37 2 {displaystyle 12^{2}+35^{2}=37^{2}} etc.
* ^ (Cooke 2005 , p. 198): "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."
* ^ (Hayashi 2005 , p. 371)
* ^ A B (Hayashi 2003 , pp. 121–122)
* ^ R. Rashed (1994), The development of Arabic mathematics:
between arithmetic and algebra, p. 35
London
London
* ^ Boyer (1991). "The Arabic Hegemony". A History of Mathematics.
pp. 241–242. Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote
an
Algebra that went beyond that of al-Khwarizmi to include equations
of third degree. Like his
Arab
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."
* ^ O\'Connor, John J. ; Robertson, Edmund F. , "Al-Mahani",
MacTutor History of
Mathematics
Mathematics archive ,
University of St Andrews
University of St Andrews .
* ^ O\'Connor, John J. ; Robertson, Edmund F. , "Omar Khayyam",
MacTutor History of
Mathematics
Mathematics archive ,
University of St Andrews
University of St Andrews .
* ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996),
"Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of
Arabic Science , Vol. 2, p. 447–494 ,
Routledge
Routledge ,
London
London and New
York:
"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the
most considerable contribution to this branch of geometry whose
importance came to be completely recognized only in the 19th century.
In essence, their propositions concerning the properties of
quadrangles which they considered, assuming that some of the angles of
these figures were acute of obtuse, embodied the first few theorems of
the hyperbolic and the elliptic geometries. Their other proposals
showed that various geometric statements were equivalent to the
Euclidean postulate V. It is extremely important that these scholars
established the mutual connection between this postulate and the sum
of the angles of a triangle and a quadrangle. By their works on the
theory of parallel lines
Arab
Arab mathematicians directly influenced the
relevant investigations of their European counterparts. The first
European attempt to prove the postulate on parallel lines – made by
Witelo, the Polish scientists of the 13th century, while revising Ibn
al-Haytham's
Book of Optics
Book of Optics (Kitab al-Manazir) – was undoubtedly
prompted by Arabic sources. The proofs put forward in the 14th century
by the Jewish scholar Levi ben Gerson, who lived in southern France,
and by the above-mentioned
Alfonso
Alfonso from Spain directly border on Ibn
al-Haytham's demonstration. Above, we have demonstrated that
Pseudo-Tusi's Exposition of
Euclid
Euclid had stimulated both J. Wallis's and
G. Saccheri's studies of the theory of parallel lines." * ^ A B C D
E
Euclid's Elements
Euclid's Elements – All thirteen books in one volume, Based on
Heath's translation, Green Lion Press ISBN 1-888009-18-7 .
* ^ Clark, Bowman L. (Jan 1985). "Individuals and Points". Notre
Dame Journal of Formal Logic. 26 (1): 61–75. doi
:10.1305/ndjfl/1093870761 . Retrieved 29 August 2016.
* ^ Gerla, G., 1995, "Pointless Geometries" in Buekenhout, F.,
Kantor, W. eds., Handbook of incidence geometry: buildings and
foundations. North-Holland: 1015–31.
* ^ John Casey (1885) Analytic
Geometry
Geometry of the Point, Line, Circle,
and Conic Sections, link from
Internet Archive
Internet Archive .
* ^ Buekenhout, Francis (1995), Handbook of Incidence Geometry:
Buildings and Foundations, Elsevier B.V.
* ^ "geodesic – definition of geodesic in English from the Oxford
dictionary".