Geometry

Geometry (from

Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics.^[4] Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

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

The earliest recorded beginnings of geometry can be traced to ancient

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).^[31] This was a necessary precursor to the development of calculus and a precise quantitative science of physics.^[32] The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661).^[33] Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.^[34]

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

The following are some of the most important concepts in geometry.[3]^[37][38]

Axioms

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",

With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes.

However, there are modern geometries in which points are not primitive objects, or even without points.^[46][47] One of the oldest such geometries is Whitehead's point-free geometry, formulated by Alfred North Whitehead in 1919–1920.

Lines

Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself".^[45] 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,^[48] 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.^[49] In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.^[50]

Planes

In Euclidean geometry a

In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.^[45] The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.^[55]

In

A

A

A

Manifolds are used extensively in physics, including in general relativity and string theory.^[62]

Measures: length, area, and volume

Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.^[63]

In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.^[64]

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.

Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.^[72]

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

The theme of symmetry in geometry is nearly as old as the science of geometry itself.^[77] Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers^[78] and were investigated in detail before the time of Euclid.^[41] 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.^[79] 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.^[80] 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.^[81] 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.^[82] Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,^[83][84] the latter in Lie theory and Riemannian geometry.^[85][86]

A different type of symmetry is the principle of duality in projective geometry, among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and the result is an equally true theorem.^[87] A similar and closely related form of duality exists between a vector space and its dual space.^[88]

Contemporary geometry

Euclidean geometry

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

Differential geometry

Differential geometry uses techniques of calculus and linear algebra to study problems in geometry.^[96] It has applications in physics,^[97] econometrics,^[98] and bioinformatics,^[99] among others.

In particular, differential geometry is of importance to

This section is an excerpt from Non-Euclidean geometry.[edit]

In

metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries

that have also been called non-Euclidean geometry.

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.^[104] 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.^[105]

Algebraic geometry

Algebraic geometry has applications in many areas, including cryptography^[109] and string theory.^[110]

Complex geometry

Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces.^{[115][116][117]} 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.^[118][119] The primary objects of study in complex geometry are

Discrete geometry is a subject that has close connections with convex geometry.^{[120][121][122]} It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc.^[123][124] It shares many methods and principles with combinatorics.

Computational geometry

Geometric group theory uses large-scale geometric techniques to study finitely generated groups.^[127] 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.^[128]

Geometric group theory often revolves around the

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.^[131]

Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure.^[132] These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.^[133]

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.^[134]

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

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.^[141]

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
• Lists of mathematics topics

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
• Geometric space
• List of interactive geometry software

Other applications

• Molecular geometry

Notes

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• Hayashi, Takao (2003). "Indian Mathematics". In Grattan-Guinness, Ivor (ed.). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vol. 1. Baltimore, MD: The
  ISBN 978-0-8018-7396-6
  .
• Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.). The Blackwell Companion to Hinduism. Oxford:
  ISBN 978-1-4051-3251-0
  .
• Nikolai I. Lobachevsky (2010). Pangeometry. Heritage of European Mathematics Series. Vol. 4. translator and editor: A. Papadopoulos. European Mathematical Society.

Further reading

• Zbl 1364.00004
  .
• ISBN 978-0-7139-9634-0
  .

External links

Wikibooks has more on the topic of: Geometry

Library resources about Geometry

Resources in your library

• "Geometry" . Encyclopædia Britannica. Vol. 11 (11th ed.). 1911. pp. 675–736.
• 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