Contents 1 Overview 2 History 3 Important concepts in geometry 3.1 Axioms 3.2 Points 3.3 Lines 3.4 Planes 3.5 Angles 3.6 Curves 3.7 Surfaces 3.8 Manifolds 3.9 Topologies and metrics 3.10 Compass and straightedge constructions 3.11 Dimension 3.12 Symmetry 3.13 Non-Euclidean geometry 4 Contemporary geometry 4.1 Euclidean geometry
4.2 Differential geometry
4.3
5 Applications 5.1 Art 5.2 Architecture 5.3 Physics 5.4 Other fields of mathematics 6 See also 6.1 Lists 6.2 Related topics 6.3 Other fields 7 Notes 8 Sources 9 Further reading 10 External links Overview Contemporary geometry has many subfields:
History Main article: History of geometry A European and an
The earliest recorded beginnings of geometry can be traced to ancient
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
An illustration of Euclid's parallel postulate See also: Euclidean geometry
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.[31] The study of the angles of a triangle or of angles in a
unit circle forms the basis of trigonometry.[41]
In differential geometry and calculus, the angles between plane curves
or space curves or surfaces can be calculated using the
derivative.[42][43]
Curves
Main article:
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.[47] In differential geometry[45] and topology,[37] 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.[46] 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.[37] In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.[45] Manifolds are used extensively in physics, including in general relativity and string theory[48] Topologies and metrics Main article: Topology Visual checking of the
A topology is a mathematical structure on a set that tells how
elements of the set relate spatially to each other.[37] The best-known
examples of topologies come from metrics, which are ways of measuring
distances between points.[49] 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
The 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.
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's
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.[51] This dominant view was overturned by the
revolutionary discovery of non-
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.
Quintic Calabi–Yau threefold The field of algebraic geometry is the modern incarnation of the
The 421polytope, orthogonally projected into the E8
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-
The
An important area of application is number theory. In ancient Greece
the
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
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).
( 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
"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
^ a b c d e
Sources Boyer, C. B. (1991) [1989]. A History of
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
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