# Geometry

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Geometry
> Markdown URL: https://mediated.wiki/source/Geometry.md
> Source: https://en.wikipedia.org/wiki/Geometry
> Source revision: 1354533348
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Branch of mathematics

For other uses, see [Geometry (disambiguation)](/source/Geometry_(disambiguation)).

Geometry Projecting a sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional Point One-dimensional Line Line segment Ray Curve Length Two-dimensional Surface Plane Area Polygon Triangle Centers Altitude Hypotenuse Pythagorean theorem Circular Hyperbolic Spherical Quadrilateral Parallelogram Square Rectangle Rhombus Rhomboid Trapezoid Kite Circle Radius Diameter Circumference Disk Area Three-dimensional Surface area Volume Polyhedron Platonic Solid Tetrahedron cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Solid of revolution Sphere Great circle Cylinder Cone Four - other-dimensional 4-polytope Simplex 5-cell Hypercube Tesseract n-sphere Hypersphere Geometers by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov v t e

Part of a series on Mathematics History Index Areas Number theory Geometry Algebra Calculus and Analysis Discrete mathematics Logic Probability and Statistics Relationship with sciences Physics Chemistry Geosciences Computation Biology Linguistics Economics Philosophy Education Mathematics Portal v t e

**Geometry**[a][1] is a branch of [mathematics](/source/Mathematics) concerned with properties of space such as the distance, shape, size, and relative position of figures.[2] Geometry is, along with [arithmetic](/source/Arithmetic), one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a *[geometer](/source/List_of_geometers)*. Until the 19th century, geometry was almost exclusively devoted to [Euclidean geometry](/source/Euclidean_geometry),[b] which includes the notions of [point](/source/Point_(geometry)), [line](/source/Line_(geometry)), [plane](/source/Plane_(geometry)), [distance](/source/Distance), [angle](/source/Angle), [surface](/source/Surface_(mathematics)), and [curve](/source/Curve), as fundamental concepts.[3]

Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, [architecture](/source/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](/source/Wiles's_proof_of_Fermat's_Last_Theorem) of [Fermat's Last Theorem](/source/Fermat's_Last_Theorem), a problem that was stated in terms of [elementary arithmetic](/source/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 [Carl Friedrich Gauss](/source/Carl_Friedrich_Gauss)'s [Theorema Egregium](/source/Theorema_Egregium) ("remarkable theorem") that asserts roughly that the [Gaussian curvature](/source/Gaussian_curvature) of a surface is independent from any specific [embedding](/source/Embedding) in a [Euclidean space](/source/Euclidean_space). This implies that surfaces can be studied *intrinsically*, that is, as stand-alone spaces, and has been expanded into the theory of [manifolds](/source/Manifold) and [Riemannian geometry](/source/Riemannian_geometry). Later in the 19th century, it appeared that geometries without the [parallel postulate](/source/Parallel_postulate) ([non-Euclidean geometries](/source/Non-Euclidean_geometries)) can be developed without introducing any contradiction. The geometry that underlies [general relativity](/source/General_relativity) is a famous application of non-Euclidean geometry.

Since the late 19th century, 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](/source/Differential_geometry), [algebraic geometry](/source/Algebraic_geometry), [computational geometry](/source/Computational_geometry), [algebraic topology](/source/Algebraic_topology), [discrete geometry](/source/Discrete_geometry) (also known as *combinatorial geometry*), etc.—or on the properties of Euclidean spaces that are disregarded—[projective geometry](/source/Projective_geometry) that consider only alignment of points but not distance and parallelism, [affine geometry](/source/Affine_geometry) that omits the concept of angle and distance, [finite geometry](/source/Finite_geometry) that omits [continuity](/source/Continuity_(mathematics)), and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional [space](/source/Space) of the physical world and its [model](/source/Model) provided by Euclidean geometry; presently a **geometric space**, or simply a *space* is a [mathematical structure](/source/Mathematical_structure) on which some geometry is defined.

## History

Main article: [History of geometry](/source/History_of_geometry)

A [European](/source/Ethnic_groups_in_Europe) and an [Arab](/source/Arab) practicing geometry in the 15th century

The earliest recorded beginnings of geometry can be traced to ancient [Mesopotamia](/source/Mesopotamia) and [Egypt](/source/Ancient_Egypt) in the 2nd millennium BC.[5][6] 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](/source/Surveying), [construction](/source/Construction), [astronomy](/source/Astronomy), and various crafts. The earliest known texts on geometry are the [Egyptian](/source/Egyptian_mathematics) [Rhind Papyrus](/source/Rhind_Mathematical_Papyrus) (2000–1800 BC) and [Moscow Papyrus](/source/Moscow_Mathematical_Papyrus) (c. 1890 BC), and the [Babylonian clay tablets](/source/Babylonian_mathematics), such as [Plimpton 322](/source/Plimpton_322) (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or [frustum](/source/Frustum).[7] Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented [trapezoid](/source/Trapezoid) procedures for computing Jupiter's position and [motion](/source/Displacement_(vector)) within time-velocity space. These geometric procedures anticipated the [Oxford Calculators](/source/Oxford_Calculators), including the [mean speed theorem](/source/Mean_speed_theorem), by 14 centuries.[8] South of Egypt the [ancient Nubians](/source/Nubia) established a system of geometry including early versions of sun clocks.[9][10]

In the 7th century BC, the [Greek](/source/Greek_mathematics) mathematician [Thales of Miletus](/source/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's theorem](/source/Thales's_theorem).[11] [Pythagoras](/source/Pythagoras) established the [Pythagorean School](/source/Pythagoreans), which is credited with the first proof of the [Pythagorean theorem](/source/Pythagorean_theorem),[12] though the statement of the theorem has a long history.[13][14] [Eudoxus](/source/Eudoxus_of_Cnidus) (408–c. 355 BC) developed the [method of exhaustion](/source/Method_of_exhaustion), which allowed the calculation of areas and volumes of curvilinear figures,[15] as well as a theory of ratios that avoided the problem of [incommensurable magnitudes](/source/Incommensurable_magnitudes), which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose *[Elements](/source/Euclid's_Elements)*, widely considered the most successful and influential textbook of all time,[16] introduced [mathematical rigor](/source/Mathematical_rigor) through the [axiomatic method](/source/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.[17] 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.[18] [Archimedes](/source/Archimedes) (c. 287–212 BC) of [Syracuse, Italy](/source/Syracuse%2C_Italy) used the method of exhaustion to calculate the [area](/source/Area) under the arc of a [parabola](/source/Parabola) with the [summation of an infinite series](/source/Series_(mathematics)), and gave remarkably accurate approximations of [pi](/source/Pi).[19] He also studied the [spiral](/source/Archimedes_spiral) bearing his name and obtained formulas for the [volumes](/source/Volume) of [surfaces of revolution](/source/Surface_of_revolution).

*Woman teaching geometry*. Illustration at the beginning of a medieval translation of [Euclid's Elements](/source/Euclid's_Elements), c. 1310.

[Indian](/source/Indian_mathematics) mathematicians also made many important contributions in geometry. The *[Shatapatha Brahmana](/source/Shatapatha_Brahmana)* (3rd century BC) contains rules for ritual geometric constructions that are similar to the *[Sulba Sutras](/source/Shulba_Sutras)*.[20] According to ([Hayashi 2005](#CITEREFHayashi2005), 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](/source/Pythagorean_triples),[c] which are particular cases of [Diophantine equations](/source/Diophantine_equations).[21] In the [Bakhshali manuscript](/source/Bakhshali_manuscript), there are 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."[22] [Aryabhata](/source/Aryabhata)'s *[Aryabhatiya](/source/Aryabhatiya)* (499) includes the computation of areas and volumes. [Brahmagupta](/source/Brahmagupta) wrote his astronomical work *[*Brāhmasphuṭasiddhānta*](/source/Brahmasphutasiddhanta)* in 628. Chapter 12, containing 66 [Sanskrit](/source/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).[23] In the latter section, he stated his famous theorem on the diagonals of a [cyclic quadrilateral](/source/Cyclic_quadrilateral). Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of [Heron's formula](/source/Heron's_formula)), as well as a complete description of [rational triangles](/source/Rational_triangle) (*i.e.* triangles with rational sides and rational areas).[23]

In the [Middle Ages](/source/Middle_Ages), [mathematics in medieval Islam](/source/Mathematics_in_medieval_Islam) contributed to the development of geometry, especially [algebraic geometry](/source/Algebraic_geometry).[24][25] [Al-Mahani](/source/Al-Mahani) (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[26] [Thābit ibn Qurra](/source/Th%C4%81bit_ibn_Qurra) (known as Thebit in [Latin](/source/Latin)) (836–901) dealt with [arithmetic](/source/Arithmetic) operations applied to [ratios](/source/Ratio) of geometrical quantities, and contributed to the development of [analytic geometry](/source/Analytic_geometry).[27] [Omar Khayyam](/source/Omar_Khayyam) (1048–1131) found geometric solutions to [cubic equations](/source/Cubic_equation).[28] The theorems of [Ibn al-Haytham](/source/Ibn_al-Haytham) (Alhazen), Omar Khayyam and [Nasir al-Din al-Tusi](/source/Nasir_al-Din_al-Tusi) on [quadrilaterals](/source/Quadrilateral), including the [Lambert quadrilateral](/source/Lambert_quadrilateral) and [Saccheri quadrilateral](/source/Saccheri_quadrilateral), were part of a line of research on the [parallel postulate](/source/Parallel_postulate) continued by later European geometers, including [Vitello](/source/Vitello) (c. 1230 – c. 1314), [Gersonides](/source/Gersonides) (1288–1344), Alfonso, [John Wallis](/source/John_Wallis), and [Giovanni Girolamo Saccheri](/source/Giovanni_Girolamo_Saccheri), that by the 19th century led to the discovery of [hyperbolic geometry](/source/Hyperbolic_geometry).[29]

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with [coordinates](/source/Coordinate_system) and [equations](/source/Equation), by [René Descartes](/source/Ren%C3%A9_Descartes) (1596–1650) and [Pierre de Fermat](/source/Pierre_de_Fermat) (1601–1665).[30] This was a necessary precursor to the development of [calculus](/source/Calculus) and a precise quantitative science of [physics](/source/Physics).[31] The second geometric development of this period was the systematic study of [projective geometry](/source/Projective_geometry) by [Girard Desargues](/source/Girard_Desargues) (1591–1661).[32] Projective geometry studies properties of shapes which are unchanged under [projections](/source/Projection_(linear_algebra)) and [sections](/source/Section_(fiber_bundle)), especially as they relate to [artistic perspective](/source/Perspective_(graphical)).[33]

Two developments in geometry in the 19th century changed the way it had been studied previously.[34] These were the discovery of [non-Euclidean geometries](/source/Non-Euclidean_geometry) by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of [symmetry](/source/Symmetry) as the central consideration in the [Erlangen programme](/source/Erlangen_programme) of [Felix Klein](/source/Felix_Klein) (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were [Bernhard Riemann](/source/Bernhard_Riemann) (1826–1866), working primarily with tools from [mathematical analysis](/source/Mathematical_analysis), and introducing the [Riemann surface](/source/Riemann_surface), and [Henri Poincaré](/source/Henri_Poincar%C3%A9), the founder of [algebraic topology](/source/Algebraic_topology) and the geometric theory of [dynamical systems](/source/Dynamical_system). As a consequence of these major changes in the conception of geometry, the concept of "[space](/source/Space_(mathematics))" became something rich and varied, and the natural background for theories as different as [complex analysis](/source/Complex_analysis) and [classical mechanics](/source/Classical_mechanics).[35]

## Main concepts

The following are some of the most important concepts in geometry.[3][36]

### Axioms

An illustration of Euclid's [parallel postulate](/source/Parallel_postulate)

See also: [Euclidean geometry](/source/Euclidean_geometry) and [Axiom](/source/Axiom)

[Euclid](/source/Euclid) took an abstract approach to geometry in his [Elements](/source/Euclid's_Elements),[37] one of the most influential books ever written.[38] Euclid introduced certain [axioms](/source/Axiom), or [postulates](/source/Postulate), expressing primary or self-evident properties of points, lines, and planes.[39] 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](/source/Synthetic_geometry)* geometry.[40] At the start of the 19th century, the discovery of [non-Euclidean geometries](/source/Non-Euclidean_geometries) by [Nikolai Ivanovich Lobachevsky](/source/Nikolai_Ivanovich_Lobachevsky) (1792–1856), [János Bolyai](/source/J%C3%A1nos_Bolyai) (1802–1860), [Carl Friedrich Gauss](/source/Carl_Friedrich_Gauss) (1777–1855) and others[41] led to a revival of interest in this discipline, and in the 20th century, [David Hilbert](/source/David_Hilbert) (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.[42]

### Spaces and subspaces

#### Points

Main article: [Point (geometry)](/source/Point_(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",[43] or in [synthetic geometry](/source/Synthetic_geometry). In modern mathematics, they are generally defined as [elements](/source/Element_(set_theory)) of a [set](/source/Set_(mathematics)) called [space](/source/Space_(mathematics)), which is itself [axiomatically](/source/Axiomatically) defined.

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.[44][45] One of the oldest such geometries is [Whitehead's point-free geometry](/source/Whitehead's_point-free_geometry), formulated by [Alfred North Whitehead](/source/Alfred_North_Whitehead) in 1919–1920.

#### Lines

Main article: [Line (geometry)](/source/Line_(geometry))

[Euclid](/source/Euclid) described a line as "breadthless length" which "lies equally with respect to the points on itself".[43] 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](/source/Analytic_geometry), a line in the plane is often defined as the set of points whose coordinates satisfy a given [linear equation](/source/Linear_equation),[46] but in a more abstract setting, such as [incidence geometry](/source/Incidence_geometry), a line may be an independent object, distinct from the set of points which lie on it.[47] In differential geometry, a [geodesic](/source/Geodesic) is a generalization of the notion of a line to [curved spaces](/source/Manifold).[48]

#### Planes

Main article: [Euclidean plane](/source/Euclidean_plane)

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely;[43] 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](/source/Surface_(topology)) without reference to distances or angles;[49] it can be studied as an [affine space](/source/Affine_space), where collinearity and ratios can be studied but not distances;[50] it can be studied as the [complex plane](/source/Complex_plane) using techniques of [complex analysis](/source/Complex_analysis);[51] and so on.

#### Curves

Main article: [Curve (geometry)](/source/Curve_(geometry))

A [curve](/source/Curve_(geometry)) is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called [plane curves](/source/Plane_curve) and those in 3-dimensional space are called [space curves](/source/Space_curve).[52]

In topology, a curve is defined by a function from an interval of the real numbers to another space.[49] In differential geometry, the same definition is used, but the defining function is required to be differentiable.[53] Algebraic geometry studies [algebraic curves](/source/Algebraic_curve), which are defined as [algebraic varieties](/source/Algebraic_varieties) of [dimension](/source/Dimension_of_an_algebraic_variety) one.[54]

#### Surfaces

Main article: [Surface (mathematics)](/source/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 *x*2 + *y*2 + *z*2 − *r*2 = 0).

A [surface](/source/Surface_(mathematics)) is a two-dimensional object, such as a sphere or paraboloid.[55] In [differential geometry](/source/Differential_geometry)[53] and [topology](/source/Topology),[49] surfaces are described by two-dimensional 'patches' (or [neighborhoods](/source/Neighborhood_(topology))) that are assembled by [diffeomorphisms](/source/Diffeomorphism) or [homeomorphisms](/source/Homeomorphism), respectively. In algebraic geometry, surfaces are described by [polynomial equations](/source/Polynomial_equation).[54]

#### Solids

Main article: [Solid geometry](/source/Solid_geometry)

In [Euclidean space](/source/Euclidean_space), a ball is the volume bounded by a sphere.

A [solid](/source/Solid_(mathematics)) is a three-dimensional object bounded by a closed surface; for example, a [ball](/source/Ball_(mathematics)) is the volume bounded by a sphere.

#### Manifolds

Main article: [Manifold](/source/Manifold)

A [manifold](/source/Manifold) is a generalization of the concepts of curve and surface. In [topology](/source/Topology), a manifold is a [topological space](/source/Topological_space) where every point has a [neighborhood](/source/Neighborhood_(topology)) that is [homeomorphic](/source/Homeomorphism) to Euclidean space.[49] In [differential geometry](/source/Differential_geometry), a [differentiable manifold](/source/Differentiable_manifold) is a space where each neighborhood is [diffeomorphic](/source/Diffeomorphism) to Euclidean space.[53]

Manifolds are used extensively in physics, including in [general relativity](/source/General_relativity) and [string theory](/source/String_theory).[56]

### Angles

Main article: [Angle](/source/Angle)

Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.

[Euclid](/source/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.[43] In modern terms, an angle is the figure formed by two [rays](/source/Ray_(geometry)), called the *sides* of the angle, sharing a common endpoint, called the *[vertex](/source/Vertex_(geometry))* of the angle.[57] The size of an angle is formalized as an [angular measure](/source/Angular_measure).

In [Euclidean geometry](/source/Euclidean_geometry), angles are used to study [polygons](/source/Polygon) and [triangles](/source/Triangle), as well as forming an object of study in their own right.[43] The study of the angles of a triangle or of angles in a [unit circle](/source/Unit_circle) forms the basis of [trigonometry](/source/Trigonometry).[58]

In [differential geometry](/source/Differential_geometry) and [calculus](/source/Calculus), the angles between [plane curves](/source/Plane_curve) or [space curves](/source/Space_curve) or [surfaces](/source/Surface_(geometry)) can be calculated using the [derivative](/source/Derivative_(calculus)).[59][60]

### Measures: length, area, and volume

Main articles: [Length](/source/Length), [Area](/source/Area), and [Volume](/source/Volume)

See also: [Area § List of formulas](/source/Area#List_of_formulas), and [Volume § Volume formulas](/source/Volume#Volume_formulas)

[Length](/source/Length), [area](/source/Area), and [volume](/source/Volume) describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.

In [Euclidean geometry](/source/Euclidean_geometry) and [analytic geometry](/source/Analytic_geometry), the length of a line segment can often be calculated by the [Pythagorean theorem](/source/Pythagorean_theorem).[61]

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](/source/Area#List_of_formulas) and [formulas for volume](/source/Volume#Formulas) of various geometric objects. In [calculus](/source/Calculus), area and volume can be defined in terms of [integrals](/source/Integral), such as the [Riemann integral](/source/Riemann_integral)[62] or the [Lebesgue integral](/source/Lebesgue_integral).[63]

Other geometrical measures include the [curvature](/source/Curvature) and [compactness](/source/Compactness_measure).

#### Metrics and measures

Main articles: [Metric (mathematics)](/source/Metric_(mathematics)) and [Measure (mathematics)](/source/Measure_(mathematics))

Visual checking of the [Pythagorean theorem](/source/Pythagorean_theorem) for the (3, 4, 5) [triangle](/source/Triangle) as in the [Zhoubi Suanjing](/source/Zhoubi_Suanjing) 500–200 BC. The Pythagorean theorem is a consequence of the [Euclidean metric](/source/Euclidean_metric).

The concept of length or distance can be generalized, leading to the idea of [metrics](/source/Metric_space).[64] For instance, the [Euclidean metric](/source/Euclidean_metric) measures the distance between points in the [Euclidean plane](/source/Euclidean_plane), while the [hyperbolic metric](/source/Hyperbolic_metric) measures the distance in the [hyperbolic plane](/source/Hyperbolic_plane). Other important examples of metrics include the [Lorentz metric](/source/Lorentz_metric) of [special relativity](/source/Special_relativity) and the semi-[Riemannian metrics](/source/Riemannian_metric) of [general relativity](/source/General_relativity).[65]

In a different direction, the concepts of length, area and volume are extended by [measure theory](/source/Measure_theory), which studies methods of assigning a size or *measure* to [sets](/source/Set_(mathematics)), where the measures follow rules similar to those of classical area and volume.[66]

### Congruence and similarity

Main articles: [Congruence (geometry)](/source/Congruence_(geometry)) and [Similarity (geometry)](/source/Similarity_(geometry))

[Congruence](/source/Congruence_(geometry)) and [similarity](/source/Similarity_(geometry)) are concepts that describe when two shapes have similar characteristics.[67] 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.[68] [Hilbert](/source/Hilbert), in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by [axioms](/source/Axiom).

Congruence and similarity are generalized in [transformation geometry](/source/Transformation_geometry), which studies the properties of geometric objects that are preserved by different kinds of transformations.[69]

### Compass and straightedge constructions

Main article: [Compass and straightedge constructions](/source/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](/source/Compass_(drafting)) and [straightedge](/source/Ruler).[d] 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](/source/Neusis_construction), parabolas and other curves, or mechanical devices, were found.

### Rotation and orientation

Main articles: [Rotation (geometry)](/source/Rotation_(geometry)) and [Orientation (geometry)](/source/Orientation_(geometry))

The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space.

### Dimension

For broader coverage of this topic, see [Dimension (mathematics)](/source/Dimension_(mathematics)).

The [Koch snowflake](/source/Koch_snowflake), with [fractal dimension](/source/Fractal_dimension)=log4/log3 and [topological dimension](/source/Topological_dimension)=1

Traditional geometry allowed dimensions 1 (a [line](/source/Line_(geometry)) or curve), 2 (a [plane](/source/Plane_(mathematics)) or surface), and 3 (our ambient world conceived of as [three-dimensional space](/source/Three-dimensional_space)). Furthermore, mathematicians and physicists have used [higher dimensions](/source/Higher_dimension) for nearly two centuries.[70] One example of a mathematical use for higher dimensions is the [configuration space](/source/Configuration_space_(physics)) of a physical system, which has a dimension equal to the system's [degrees of freedom](/source/Degrees_of_freedom). For instance, the configuration of a screw can be described by five coordinates.[71]

In [general topology](/source/General_topology), the concept of dimension has been extended from [natural numbers](/source/Natural_number), to infinite dimension ([Hilbert spaces](/source/Hilbert_space), for example) and positive [real numbers](/source/Real_number) (in [fractal geometry](/source/Fractal_geometry)).[72] In [algebraic geometry](/source/Algebraic_geometry), the [dimension of an algebraic variety](/source/Dimension_of_an_algebraic_variety) has received a number of apparently different definitions, which are all equivalent in the most common cases.[73]

### Symmetry

Main article: [Symmetry](/source/Symmetry)

A [tiling](/source/Order-3_bisected_heptagonal_tiling) of the [hyperbolic plane](/source/Hyperbolic_geometry)

The theme of [symmetry](/source/Symmetry) in geometry is nearly as old as the science of geometry itself.[74] Symmetric shapes such as the [circle](/source/Circle), [regular polygons](/source/Regular_polygon) and [platonic solids](/source/Platonic_solid) held deep significance for many ancient philosophers[75] and were investigated in detail before the time of Euclid.[39] Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of [Leonardo da Vinci](/source/Leonardo_da_Vinci), [M. C. Escher](/source/M._C._Escher), and others.[76] In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. [Felix Klein](/source/Felix_Klein)'s [Erlangen program](/source/Erlangen_program) proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation [group](/source/Group_(mathematics)), determines what geometry *is*.[77] Symmetry in classical [Euclidean geometry](/source/Euclidean_geometry) is represented by [congruences](/source/Congruence_(geometry)) and rigid motions, whereas in [projective geometry](/source/Projective_geometry) an analogous role is played by [collineations](/source/Collineation), [geometric transformations](/source/Geometric_transformation) that take straight lines into straight lines.[78] However it was in the new geometries of Bolyai and Lobachevsky, Riemann, [Clifford](/source/William_Kingdon_Clifford) and Klein, and [Sophus Lie](/source/Sophus_Lie) that Klein's idea to 'define a geometry via its [symmetry group](/source/Symmetry_group)' found its inspiration.[79] Both discrete and continuous symmetries play prominent roles in geometry, the former in [topology](/source/Topology) and [geometric group theory](/source/Geometric_group_theory),[80][81] the latter in [Lie theory](/source/Lie_theory) and [Riemannian geometry](/source/Riemannian_geometry).[82][83]

A different type of symmetry is the principle of [duality](/source/Duality_(projective_geometry)) in [projective geometry](/source/Projective_geometry), among other fields. This meta-phenomenon can roughly be described as follows: in any [theorem](/source/Theorem), exchange *point* with *plane*, *join* with *meet*, *lies in* with *contains*, and the result is an equally true theorem.[84] A similar and closely related form of duality exists between a [vector space](/source/Vector_space) and its [dual space](/source/Dual_space).[85]

## Contemporary geometry

### Euclidean geometry

Main article: [Euclidean geometry](/source/Euclidean_geometry)

[Euclidean geometry](/source/Euclidean_geometry) is geometry in its classical sense.[86] As it models the space of the physical world, it is used in many scientific areas, such as [mechanics](/source/Mechanics), [astronomy](/source/Astronomy), [crystallography](/source/Crystallography),[87] and many technical fields, such as [engineering](/source/Engineering),[88] [architecture](/source/Architecture),[89] [geodesy](/source/Geodesy),[90] [aerodynamics](/source/Aerodynamics),[91] and [navigation](/source/Navigation).[92] The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as [points](/source/Point_(geometry)), [lines](/source/Line_(geometry)), [planes](/source/Plane_(mathematics)), [angles](/source/Angle), [triangles](/source/Triangle), [congruence](/source/Congruence_(geometry)), [similarity](/source/Similarity_(geometry)), [solid figures](/source/Solid_figure), [circles](/source/Circle), and [analytic geometry](/source/Analytic_geometry).[93]

#### Euclidean vectors

Main article: [Euclidean vector](/source/Euclidean_vector)

Euclidean vectors are used for a myriad of applications in physics and engineering, such as [position](/source/Position_(geometry)), [displacement](/source/Displacement_(geometry)), [deformation](/source/Deformation_(physics)), [velocity](/source/Velocity), [acceleration](/source/Acceleration), [force](/source/Force), etc.

### Differential geometry

[Differential geometry](/source/Differential_geometry) uses tools from [calculus](/source/Calculus) to study problems involving curvature.

Main article: [Differential geometry](/source/Differential_geometry)

[Differential geometry](/source/Differential_geometry) uses techniques of [calculus](/source/Calculus) and [linear algebra](/source/Linear_algebra) to study problems in geometry.[94] It has applications in [physics](/source/Physics),[95] [econometrics](/source/Econometrics),[96] and [bioinformatics](/source/Bioinformatics),[97] among others.

In particular, differential geometry is of importance to [mathematical physics](/source/Mathematical_physics) due to [Albert Einstein](/source/Albert_Einstein)'s [general relativity](/source/General_relativity) postulation that the [universe](/source/Universe) is [curved](/source/Curvature).[98] Differential geometry can either be *intrinsic* (meaning that the spaces it considers are [smooth manifolds](/source/Smooth_manifold) whose geometric structure is governed by a [Riemannian metric](/source/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).[99]

#### Non-Euclidean geometry

Main article: [non-Euclidean geometry](/source/Non-Euclidean_geometry)

[Non-Euclidean geometry](/source/Non-Euclidean_geometry) consists of two geometries based on [axioms](/source/Axiom) closely related to those that specify [Euclidean geometry](/source/Euclidean_geometry). As Euclidean geometry lies at the intersection of [metric geometry](/source/Metric_geometry) and [affine geometry](/source/Affine_geometry), non-Euclidean geometry arises by either replacing the [parallel postulate](/source/Parallel_postulate) with an alternative, or consideration of [quadratic forms](/source/Quadratic_form) other than the [definite quadratic forms](/source/Definite_quadratic_form) associated with [metric geometry](/source/Metric_geometry). In the former case, one obtains [hyperbolic geometry](/source/Hyperbolic_geometry) and [elliptic geometry](/source/Elliptic_geometry), the traditional non-Euclidean geometries. When [isotropic quadratic forms](/source/Isotropic_quadratic_form) are admitted, then there are affine planes associated with the [planar algebras](/source/Non-Euclidean_geometry#Planar_algebras), which give rise to [kinematic geometries](/source/Non-Euclidean_geometry#Kinematic_geometries) that have also been called non-Euclidean geometry.

### Topology

Main article: [Topology](/source/Topology)

A thickening of the [trefoil knot](/source/Trefoil_knot)

Topology is the field concerned with the properties of [continuous mappings](/source/Continuous_mapping),[100] and can be considered a generalization of Euclidean geometry.[101] In practice, topology often means dealing with large-scale properties of spaces, such as [connectedness](/source/Connectedness) and [compactness](/source/Compact_(topology)).[49]

The field of topology, which saw massive development in the 20th century, is in a technical sense a type of [transformation geometry](/source/Transformation_geometry), in which transformations are [homeomorphisms](/source/Homeomorphism).[102] This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include [geometric topology](/source/Geometric_topology), [differential topology](/source/Differential_topology), [algebraic topology](/source/Algebraic_topology) and [general topology](/source/General_topology).[103]

### Algebraic geometry

Main article: [Algebraic geometry](/source/Algebraic_geometry)

Quintic [Calabi–Yau threefold](/source/Calabi%E2%80%93Yau_manifold)

Algebraic geometry is fundamentally the study by means of [algebraic](/source/Algebra) methods of some geometrical shapes, called [algebraic sets](/source/Algebraic_set), and defined as common [zeros](/source/Zero_of_a_function) of [multivariate polynomials](/source/Multivariate_polynomial).[104] Algebraic geometry became an autonomous subfield of geometry c. 1900, with a theorem called [Hilbert's Nullstellensatz](/source/Hilbert's_Nullstellensatz) that establishes a strong correspondence between algebraic sets and [ideals](/source/Ideal_(ring_theory)) of [polynomial rings](/source/Polynomial_ring). This led to a parallel development of algebraic geometry, and its algebraic counterpart, called [commutative algebra](/source/Commutative_algebra).[105] From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by [Alexander Grothendieck](/source/Alexander_Grothendieck) of [scheme theory](/source/Scheme_theory), which allows using [topological methods](/source/Algebraic_topology), including [cohomology theories](/source/Cohomology_theory) in a purely algebraic context.[105] Scheme theory allowed to solve many difficult problems not only in geometry, but also in [number theory](/source/Number_theory). [Wiles' proof of Fermat's Last Theorem](/source/Wiles'_proof_of_Fermat's_Last_Theorem) is a famous example of a long-standing problem of [number theory](/source/Number_theory) whose solution uses scheme theory and its extensions such as [stack theory](/source/Stack_(mathematics)). One of seven [Millennium Prize problems](/source/Millennium_Prize_problems), the [Hodge conjecture](/source/Hodge_conjecture), is a question in algebraic geometry.[106]

Algebraic geometry has applications in many areas, including [cryptography](/source/Cryptography)[107] and [string theory](/source/String_theory).[108]

### Complex geometry

Main article: [Complex geometry](/source/Complex_geometry)

[Complex geometry](/source/Complex_geometry) studies the nature of geometric structures modelled on, or arising out of, the [complex plane](/source/Complex_plane).[109][110][111] Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of [several complex variables](/source/Several_complex_variables), and has found applications to [string theory](/source/String_theory) and [mirror symmetry](/source/Mirror_symmetry_(string_theory)).[112]

Complex geometry first appeared as a distinct area of study in the work of [Bernhard Riemann](/source/Bernhard_Riemann) in his study of [Riemann surfaces](/source/Riemann_surface).[113][114][115] Work in the spirit of Riemann was carried out by the [Italian school of algebraic geometry](/source/Italian_school_of_algebraic_geometry) in the early 1900s. Contemporary treatment of complex geometry began with the work of [Jean-Pierre Serre](/source/Jean-Pierre_Serre), who introduced the concept of [sheaves](/source/Sheaf_(mathematics)) to the subject, and illuminated the relations between complex geometry and algebraic geometry.[116][117] The primary objects of study in complex geometry are [complex manifolds](/source/Complex_manifold), [complex algebraic varieties](/source/Complex_algebraic_varieties), and [complex analytic varieties](/source/Complex_analytic_varieties), and [holomorphic vector bundles](/source/Holomorphic_vector_bundles) and [coherent sheaves](/source/Coherent_sheaves) over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and [Calabi–Yau manifolds](/source/Calabi%E2%80%93Yau_manifold), and these spaces find uses in string theory. In particular, [worldsheets](/source/Worldsheet) of strings are modelled by Riemann surfaces, and [superstring theory](/source/Superstring_theory) predicts that the extra 6 dimensions of 10 dimensional [spacetime](/source/Spacetime) may be modelled by Calabi–Yau manifolds.

### Discrete geometry

Main article: [Discrete geometry](/source/Discrete_geometry)

Discrete geometry includes the study of various [sphere packings](/source/Sphere_packing).

[Discrete geometry](/source/Discrete_geometry) is a subject that has close connections with [convex geometry](/source/Convex_geometry).[118][119][120] 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](/source/Sphere_packing), [triangulations](/source/Triangulation_(geometry)), the Kneser-Poulsen conjecture, etc.[121][122] It shares many methods and principles with [combinatorics](/source/Combinatorics).

### Computational geometry

Main article: [Computational geometry](/source/Computational_geometry)

[Computational geometry](/source/Computational_geometry) deals with [algorithms](/source/Algorithm) and their [implementations](/source/Implementation_(computer_science)) for manipulating geometrical objects. Important problems historically have included the [travelling salesman problem](/source/Travelling_salesman_problem), [minimum spanning trees](/source/Minimum_spanning_tree), [hidden-line removal](/source/Hidden-line_removal), and [linear programming](/source/Linear_programming).[123]

Although being a young area of geometry, it has many applications in [computer vision](/source/Computer_vision), [image processing](/source/Image_processing), [computer-aided design](/source/Computer-aided_design), [medical imaging](/source/Medical_imaging), etc.[124]

### Geometric group theory

Main article: [Geometric group theory](/source/Geometric_group_theory)

The Cayley graph of the [free group](/source/Free_group) on two generators *a* and *b*

Groups have been understood as geometric objects since [Klein's Erlangen programme](/source/Erlangen_program). [Geometric group theory](/source/Geometric_group_theory) studies [group actions](/source/Group_action) on objects that are regarded as geometric (significantly, isometric actions on [metric spaces](/source/Metric_space)) to study [finitely generated groups](/source/Finitely_generated_group), often involving large-scale geometric techniques[125] and borrowing from topology, geometry, dynamics and analysis.[126] It had a significant impact on [low-dimensional topology](/source/Low-dimensional_topology), a celebrated result being Agol's proof of the [virtually Haken conjecture](/source/Virtually_Haken_conjecture) that combines [Perelman geometrization](/source/Geometrization_conjecture) with [cubulation](/source/Cubical_complex) techniques.[127]

Group actions on their [Cayley graphs](/source/Cayley_graph) are foundational examples of isometric group actions. Other major topics include [quasi-isometries](/source/Quasi-isometry), [Gromov-hyperbolic groups](/source/Gromov-hyperbolic_group) and their generalizations ([relatively](/source/Relatively_hyperbolic_group) and [acylindrically hyperbolic groups](/source/Acylindrically_hyperbolic_group)), [free groups](/source/Free_group) and [their automorphisms](/source/Out(Fn)), [groups acting on trees](/source/Bass%E2%80%93Serre_theory), various notions of nonpositive curvature for groups ([CAT(0) groups](/source/CAT(0)_group), [Dehn functions](/source/Dehn_function), [automaticity](/source/Automatic_group)...), [right angled Artin groups](/source/Right_angled_Artin_group), and topics close to [combinatorial group theory](/source/Combinatorial_group_theory) such as [small cancellation theory](/source/Small_cancellation_theory) and algorithmic problems (e.g. the [word](/source/Word_problem_for_groups), [conjugacy](/source/Conjugacy_problem), and [isomorphism problems](/source/Group_isomorphism_problem)). Other group-theoretic topics like [mapping class groups](/source/Mapping_class_group_of_a_surface), [property (T)](/source/Kazhdan's_property_(T)), [solvability](/source/Solvable_group), [amenability](/source/Amenable_group) and [lattices in Lie groups](/source/Lattice_(discrete_subgroup)) are sometimes regarded as strongly geometric as well.[125][128][129][130]

### Convex geometry

Main article: [Convex geometry](/source/Convex_geometry)

[Convex geometry](/source/Convex_geometry) investigates [convex](/source/Convex_set) shapes in the Euclidean space and its more abstract analogues, often using techniques of [real analysis](/source/Real_analysis) and [discrete mathematics](/source/Discrete_mathematics).[131] It has close connections to [convex analysis](/source/Convex_analysis), [optimization](/source/Optimization) and [functional analysis](/source/Functional_analysis) and important applications in [number theory](/source/Number_theory).

Convex geometry dates back to antiquity.[131] [Archimedes](/source/Archimedes) gave the first known precise definition of convexity. The [isoperimetric problem](/source/Isoperimetric_problem), a recurring concept in convex geometry, was studied by the Greeks as well, including [Zenodorus](/source/Zenodorus_(mathematician)). Archimedes, [Plato](/source/Plato), [Euclid](/source/Euclid), and later [Kepler](/source/Kepler) and [Coxeter](/source/Coxeter) all studied [convex polytopes](/source/Convex_polytope) 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](/source/Gaussian_curvature), [algorithms](/source/Algorithms), [tilings](/source/Tiling_(geometry)) and [lattices](/source/Lattice_(group)).

## Applications

Geometry has found applications in many fields, some of which are described below.

### Art

Main article: [Mathematics and art](/source/Mathematics_and_art)

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](/source/Perspective_(graphical)) showed that there is more to geometry than just the metric properties of figures: perspective is the origin of [projective geometry](/source/Projective_geometry).[132]

Artists have long used concepts of [proportion](/source/Proportionality_(mathematics)) in design. [Vitruvius](/source/Vitruvius) developed a complicated theory of *ideal proportions* for the human figure.[133] These concepts have been used and adapted by artists from [Michelangelo](/source/Michelangelo) to modern comic book artists.[134]

The [golden ratio](/source/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.[135]

[Tilings](/source/Tiling_(geometry)), or tessellations, have been used in art throughout history. [Islamic art](/source/Islamic_art) makes frequent use of tessellations, as did the art of [M. C. Escher](/source/M._C._Escher).[136] Escher's work also made use of [hyperbolic geometry](/source/Hyperbolic_geometry).

[Cézanne](/source/C%C3%A9zanne) advanced the theory that all images can be built up from the [sphere](/source/Sphere), the [cone](/source/Cone), and the [cylinder](/source/Cylinder). This is still used in art theory today, although the exact list of shapes varies from author to author.[137][138]

### Architecture

Main articles: [Mathematics and architecture](/source/Mathematics_and_architecture) and [Architectural geometry](/source/Architectural_geometry)

Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.[139][140] Applications of geometry to architecture include the use of [projective geometry](/source/Projective_geometry) to create [forced perspective](/source/Forced_perspective),[141] the use of [conic sections](/source/Conic_section) in constructing domes and similar objects,[89] the use of [tessellations](/source/Tessellations),[89] and the use of symmetry.[89]

### Physics

Main article: [Mathematical physics](/source/Mathematical_physics)

The field of [astronomy](/source/Astronomy), especially as it relates to mapping the positions of [stars](/source/Star) and [planets](/source/Planet) on the [celestial sphere](/source/Celestial_sphere) and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.[142]

[Riemannian geometry](/source/Riemannian_geometry) and [pseudo-Riemannian](/source/Pseudo-Riemannian) geometry are used in [general relativity](/source/General_relativity).[143] [String theory](/source/String_theory) makes use of several variants of geometry,[144] as does [quantum information theory](/source/Quantum_information_theory).[145]

### Other fields of mathematics

The Pythagoreans discovered that the sides of a triangle could have [incommensurable](/source/Commensurability_(mathematics)) lengths.

[Calculus](/source/Calculus) was strongly influenced by geometry.[30] For instance, the introduction of [coordinates](/source/Coordinates) by [René Descartes](/source/Ren%C3%A9_Descartes) and the concurrent developments of [algebra](/source/Algebra) marked a new stage for geometry, since geometric figures such as [plane curves](/source/Plane_curve) could now be represented [analytically](/source/Analytic_geometry) in the form of functions and equations. This played a key role in the emergence of [infinitesimal calculus](/source/Infinitesimal_calculus) in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.[146][147]

Another important area of application is [number theory](/source/Number_theory).[148] In [ancient Greece](/source/Ancient_Greece) the [Pythagoreans](/source/Pythagoreans) considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.[149] Since the 19th century, geometry has been used for solving problems in number theory, for example through the [geometry of numbers](/source/Geometry_of_numbers) or, more recently, [scheme theory](/source/Scheme_theory), which is used in [Wiles's proof of Fermat's Last Theorem](/source/Wiles's_proof_of_Fermat's_Last_Theorem).[150]

## See also

- [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics)

Main category: [Geometry](https://en.wikipedia.org/wiki/Category:Geometry)

**Lists**

- [List of geometers](/source/List_of_geometers) - [Category:Algebraic geometers](https://en.wikipedia.org/wiki/Category:Algebraic_geometers) - [Category:Differential geometers](https://en.wikipedia.org/wiki/Category:Differential_geometers) - [Category:Geometers](https://en.wikipedia.org/wiki/Category:Geometers) - [Category:Topologists](https://en.wikipedia.org/wiki/Category:Topologists)

- [List of formulas in elementary geometry](/source/List_of_formulas_in_elementary_geometry)

- [List of geometry topics](/source/List_of_geometry_topics)

- [List of important publications in geometry](/source/List_of_important_publications_in_mathematics#Geometry)

- [Lists of mathematics topics](/source/Lists_of_mathematics_topics)

**Related topics**

- [Descriptive geometry](/source/Descriptive_geometry)

- *[Flatland](/source/Flatland)*, a book written by [Edwin Abbott Abbott](/source/Edwin_Abbott_Abbott) about two- and [three-dimensional space](/source/Three-dimensional_space), to understand the concept of four dimensions

- [List of interactive geometry software](/source/List_of_interactive_geometry_software)

**Other applications**

- [Molecular geometry](/source/Molecular_geometry)

## Notes

1. **[^](#cite_ref-1)** (from [Ancient Greek](/source/Ancient_Greek_language) [γεωμετρία](https://en.wiktionary.org/wiki/%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1#Ancient_Greek)*(*geōmetría*)* 'land measurement'; from [γῆ](https://en.wiktionary.org/wiki/%CE%B3%E1%BF%86#Ancient_Greek)*(*gê*)* 'earth, land' and [μέτρον](https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%84%CF%81%CE%BF%CE%BD#Ancient_Greek)*(*métron*)* 'a measure')

1. **[^](#cite_ref-4)** Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. In the 19th century and later, this was challenged by the development of [hyperbolic geometry](/source/Hyperbolic_geometry) by [Lobachevsky](/source/Nikolai_Lobachevsky) and other [non-Euclidean geometries](/source/Non-Euclidean_geometries) by [Gauss](/source/Gauss) and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of [Desargues](/source/Desargues) in the 17th century, all the way back to the implicit use of [spherical geometry](/source/Spherical_geometry) to understand the [Earth's geodesy](/source/Geodesy) and to navigate the oceans since antiquity.

1. **[^](#cite_ref-23)** 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.

1. **[^](#cite_ref-73)** The ancient Greeks had some constructions using other instruments.

## References

1. **[^](#cite_ref-2)** ["Geometry - Formulas, Examples | Plane and Solid Geometry"](https://cuemath.com/geometry). *Cuemath*. Retrieved 31 August 2023.

1. **[^](#cite_ref-Risi2015_3-0)** Vincenzo De Risi (2015). [*Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age*](https://books.google.com/books?id=1m11BgAAQBAJ&pg=PA1). Birkhäuser. pp. 1–. [ISBN](/source/ISBN_(identifier)) [978-3-319-12102-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-12102-4). [Archived](https://web.archive.org/web/20210220094741/https://books.google.com/books?id=1m11BgAAQBAJ&pg=PA1) from the original on 20 February 2021. Retrieved 14 September 2019.

1. ^ [***a***](#cite_ref-Tabak_2014_xiv_5-0) [***b***](#cite_ref-Tabak_2014_xiv_5-1) Tabak, John (2014). [*Geometry: the language of space and form*](https://archive.org/details/geometrylanguage0000taba). Infobase Publishing. p. xiv. [ISBN](/source/ISBN_(identifier)) [978-0-8160-4953-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8160-4953-0).

1. **[^](#cite_ref-Meyer2006_6-0)** Walter A. Meyer (2006). [*Geometry and Its Applications*](https://books.google.com/books?id=ez6H5Ho6E3cC). Elsevier. [ISBN](/source/ISBN_(identifier)) [978-0-08-047803-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-047803-6). [Archived](https://web.archive.org/web/20210901183207/https://books.google.com/books?id=ez6H5Ho6E3cC) from the original on 1 September 2021. Retrieved 14 September 2019.

1. **[^](#cite_ref-7)** Friberg, Jöran (1981). ["Methods and traditions of Babylonian mathematics"](https://doi.org/10.1016%2F0315-0860%2881%2990069-0). *Historia Mathematica*. **8** (3): 277–318. [doi](/source/Doi_(identifier)):[10.1016/0315-0860(81)90069-0](https://doi.org/10.1016%2F0315-0860%2881%2990069-0).

1. **[^](#cite_ref-8)** [Neugebauer, Otto](/source/Otto_E._Neugebauer) (1969) [1957]. ["Chap. IV Egyptian Mathematics and Astronomy"](https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA71). *The Exact Sciences in Antiquity* (2 ed.). [Dover Publications](/source/Dover_Publications). pp. 71–96. [ISBN](/source/ISBN_(identifier)) [978-0-486-22332-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-22332-2). [Archived](https://web.archive.org/web/20200814151056/https://books.google.com/books?id=JVhTtVA2zr8C) from the original on 14 August 2020. Retrieved 27 February 2021..

1. **[^](#cite_ref-Boyer_1991_loc=Egypt_p._19_9-0)** ([Boyer 1991](#CITEREFBoyer1991), "Egypt" p. 19)

1. **[^](#cite_ref-10)** Ossendrijver, Mathieu (29 January 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph". *Science*. **351** (6272): 482–484. [Bibcode](/source/Bibcode_(identifier)):[2016Sci...351..482O](https://ui.adsabs.harvard.edu/abs/2016Sci...351..482O). [doi](/source/Doi_(identifier)):[10.1126/science.aad8085](https://doi.org/10.1126%2Fscience.aad8085). [PMID](/source/PMID_(identifier)) [26823423](https://pubmed.ncbi.nlm.nih.gov/26823423). [S2CID](/source/S2CID_(identifier)) [206644971](https://api.semanticscholar.org/CorpusID:206644971).

1. **[^](#cite_ref-11)** Depuydt, Leo (1 January 1998). "Gnomons at Meroë and Early Trigonometry". *The Journal of Egyptian Archaeology*. **84**: 171–180. [doi](/source/Doi_(identifier)):[10.2307/3822211](https://doi.org/10.2307%2F3822211). [JSTOR](/source/JSTOR_(identifier)) [3822211](https://www.jstor.org/stable/3822211).

1. **[^](#cite_ref-12)** Slayman, Andrew (27 May 1998). ["Neolithic Skywatchers"](https://archive.archaeology.org/online/news/nubia.html). *Archaeology Magazine Archive*. [Archived](https://web.archive.org/web/20110605234044/http://www.archaeology.org/online/news/nubia.html) from the original on 5 June 2011. Retrieved 17 April 2011.

1. **[^](#cite_ref-Boyer_1991_loc=Ionia_and_the_Pythagoreans_p._43_13-0)** ([Boyer 1991](#CITEREFBoyer1991), "Ionia and the Pythagoreans" p. 43)

1. **[^](#cite_ref-14)** Eves, Howard, *[An Introduction to the History of Mathematics](https://archive.org/details/introductiontohi0000eves)*, Saunders, 1990, [ISBN](/source/ISBN_(identifier)) [0-03-029558-0](https://en.wikipedia.org/wiki/Special:BookSources/0-03-029558-0).

1. **[^](#cite_ref-15)** Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". *Classics in the History of Greek Mathematics*. Annals of Mathematics; Boston Studies in the Philosophy of Science. Vol. 240. Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University. pp. 211–231. [doi](/source/Doi_(identifier)):[10.1007/978-1-4020-2640-9_11](https://doi.org/10.1007%2F978-1-4020-2640-9_11). [ISBN](/source/ISBN_(identifier)) [978-90-481-5850-8](https://en.wikipedia.org/wiki/Special:BookSources/978-90-481-5850-8). [JSTOR](/source/JSTOR_(identifier)) [1969021](https://www.jstor.org/stable/1969021). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

1. **[^](#cite_ref-16)** James R. Choike (1980). ["The Pentagram and the Discovery of an Irrational Number"](https://www.tandfonline.com/doi/abs/10.1080/00494925.1980.11972468). *The Two-Year College Mathematics Journal*. **11** (5): 312–316. [doi](/source/Doi_(identifier)):[10.2307/3026893](https://doi.org/10.2307%2F3026893). [JSTOR](/source/JSTOR_(identifier)) [3026893](https://www.jstor.org/stable/3026893). [Archived](https://web.archive.org/web/20220909203418/https://www.tandfonline.com/doi/abs/10.1080/00494925.1980.11972468) from the original on 9 September 2022. Retrieved 9 September 2022.

1. **[^](#cite_ref-17)** ([Boyer 1991](#CITEREFBoyer1991), "The Age of Plato and Aristotle" p. 92)

1. **[^](#cite_ref-18)** ([Boyer 1991](#CITEREFBoyer1991), "Euclid of Alexandria" p. 119)

1. **[^](#cite_ref-Boyer_1991_loc=Euclid_of_Alexandria_p._104_19-0)** ([Boyer 1991](#CITEREFBoyer1991), "Euclid of Alexandria" p. 104)

1. **[^](#cite_ref-20)** [Howard Eves](/source/Howard_Eves), *[An Introduction to the History of Mathematics](https://archive.org/details/introductiontohi0000eves)*, Saunders, 1990, [ISBN](/source/ISBN_(identifier)) [0-03-029558-0](https://en.wikipedia.org/wiki/Special:BookSources/0-03-029558-0) p. 141: "No work, except [The Bible](/source/The_Bible), has been more widely used...."

1. **[^](#cite_ref-21)** O'Connor, J.J.; Robertson, E.F. (February 1996). ["A history of calculus"](https://web.archive.org/web/20070715191704/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html). [University of St Andrews](/source/University_of_St_Andrews). Archived from [the original](http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html) on 15 July 2007. Retrieved 7 August 2007.

1. **[^](#cite_ref-Staal_1999_22-0)** [Staal, Frits](/source/Frits_Staal) (1999). "Greek and Vedic Geometry". *Journal of Indian Philosophy*. **27** (1–2): 105–127. [doi](/source/Doi_(identifier)):[10.1023/A:1004364417713](https://doi.org/10.1023%2FA%3A1004364417713). [S2CID](/source/S2CID_(identifier)) [170894641](https://api.semanticscholar.org/CorpusID:170894641).

1. **[^](#cite_ref-cooke198_24-0)** ([Cooke 2005](#CITEREFCooke2005), 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."

1. **[^](#cite_ref-hayashi2005-371_25-0)** ([Hayashi 2005](#CITEREFHayashi2005), p. 371)

1. ^ [***a***](#cite_ref-hayashi2003-p121-122_26-0) [***b***](#cite_ref-hayashi2003-p121-122_26-1) ([Hayashi 2003](#CITEREFHayashi2003), pp. 121–122)

1. **[^](#cite_ref-27)** Rāshid, Rushdī (1994). [*The development of Arabic mathematics : between arithmetic and algebra*](https://archive.org/details/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/page/n43/mode/2up). Boston Studies in the Philosophy of Science. Vol. 156. p. 35. [doi](/source/Doi_(identifier)):[10.1007/978-94-017-3274-1](https://doi.org/10.1007%2F978-94-017-3274-1). [ISBN](/source/ISBN_(identifier)) [978-0-7923-2565-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7923-2565-9). [OCLC](/source/OCLC_(identifier)) [29181926](https://search.worldcat.org/oclc/29181926).

1. **[^](#cite_ref-28)** ([Boyer 1991](#CITEREFBoyer1991), "The Arabic Hegemony" pp. 241–242) "Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an *Algebra* that went beyond that of al-Khwarizmi to include equations of third degree. Like his 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."".

1. **[^](#cite_ref-29)** O'Connor, John J.; [Robertson, Edmund F.](/source/Edmund_F._Robertson) ["Al-Mahani"](https://mathshistory.st-andrews.ac.uk/Biographies/Al-Mahani.html). *[MacTutor History of Mathematics Archive](/source/MacTutor_History_of_Mathematics_Archive)*. [University of St Andrews](/source/University_of_St_Andrews).

1. **[^](#cite_ref-ReferenceA_30-0)** O'Connor, John J.; [Robertson, Edmund F.](/source/Edmund_F._Robertson) ["Al-Sabi Thabit ibn Qurra al-Harrani"](https://mathshistory.st-andrews.ac.uk/Biographies/Thabit.html). *[MacTutor History of Mathematics Archive](/source/MacTutor_History_of_Mathematics_Archive)*. [University of St Andrews](/source/University_of_St_Andrews).

1. **[^](#cite_ref-31)** O'Connor, John J.; [Robertson, Edmund F.](/source/Edmund_F._Robertson) ["Omar Khayyam"](https://mathshistory.st-andrews.ac.uk/Biographies/Khayyam.html). *[MacTutor History of Mathematics Archive](/source/MacTutor_History_of_Mathematics_Archive)*. [University of St Andrews](/source/University_of_St_Andrews).

1. **[^](#cite_ref-32)** Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., *[Encyclopedia of the History of Arabic Science](/source/Encyclopedia_of_the_History_of_Arabic_Science)*, Vol. 2, pp. 447–494 [470], [Routledge](/source/Routledge), 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 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](/source/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 from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that *Pseudo-Tusi's Exposition of Euclid* had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."

1. ^ [***a***](#cite_ref-Boyer2012_33-0) [***b***](#cite_ref-Boyer2012_33-1) [Carl B. Boyer](/source/Carl_Benjamin_Boyer) (2012). [*History of Analytic Geometry*](https://books.google.com/books?id=2T4i5fXZbOYC). Courier Corporation. [ISBN](/source/ISBN_(identifier)) [978-0-486-15451-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-15451-0). [Archived](https://web.archive.org/web/20191226215605/https://books.google.com/books?id=2T4i5fXZbOYC) from the original on 26 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Edwards2012_34-0)** C. H. Edwards Jr. (2012). [*The Historical Development of the Calculus*](https://books.google.com/books?id=ilrlBwAAQBAJ&pg=PA95). Springer Science & Business Media. p. 95. [ISBN](/source/ISBN_(identifier)) [978-1-4612-6230-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-6230-5). [Archived](https://web.archive.org/web/20191229201529/https://books.google.com/books?id=ilrlBwAAQBAJ&pg=PA95) from the original on 29 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-FieldGray2012_35-0)** [Judith V. Field](/source/Judith_V._Field); Jeremy Gray (2012). [*The Geometrical Work of Girard Desargues*](https://books.google.com/books?id=zSvSBwAAQBAJ&pg=PA43). Springer Science & Business Media. p. 43. [ISBN](/source/ISBN_(identifier)) [978-1-4613-8692-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4613-8692-6). [Archived](https://web.archive.org/web/20191227054645/https://books.google.com/books?id=zSvSBwAAQBAJ&pg=PA43) from the original on 27 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Wylie2011_36-0)** C. R. Wylie (2011). [*Introduction to Projective Geometry*](https://books.google.com/books?id=VVvGc8kaajEC). Courier Corporation. [ISBN](/source/ISBN_(identifier)) [978-0-486-14170-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-14170-1). [Archived](https://web.archive.org/web/20191228051716/https://books.google.com/books?id=VVvGc8kaajEC) from the original on 28 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Gray2011_37-0)** Jeremy Gray (2011). [*Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century*](https://books.google.com/books?id=3UeSCvazV0QC). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-0-85729-060-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-85729-060-1). [Archived](https://web.archive.org/web/20191207041658/https://books.google.com/books?id=3UeSCvazV0QC) from the original on 7 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Bayro-Corrochano2018_38-0)** Eduardo Bayro-Corrochano (2018). [*Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing*](https://books.google.com/books?id=SSVhDwAAQBAJ&pg=PA4). Springer. p. 4. [ISBN](/source/ISBN_(identifier)) [978-3-319-74830-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-74830-6). [Archived](https://web.archive.org/web/20191228052142/https://books.google.com/books?id=SSVhDwAAQBAJ&pg=PA4) from the original on 28 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Kline1990_39-0)** Morris Kline (1990). [*Mathematical Thought From Ancient to Modern Times: Volume 3*](https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010). US: Oxford University Press. pp. 1010–. [ISBN](/source/ISBN_(identifier)) [978-0-19-506137-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-506137-6). [Archived](https://web.archive.org/web/20210901183204/https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010) from the original on 1 September 2021. Retrieved 14 September 2019.

1. **[^](#cite_ref-Katz2000_40-0)** [Victor J. Katz](/source/Victor_J._Katz) (2000). [*Using History to Teach Mathematics: An International Perspective*](https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45). Cambridge University Press. pp. 45–. [ISBN](/source/ISBN_(identifier)) [978-0-88385-163-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-163-0). [Archived](https://web.archive.org/web/20210901183205/https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45) from the original on 1 September 2021. Retrieved 14 September 2019.

1. **[^](#cite_ref-Berlinski2014_41-0)** [David Berlinski](/source/David_Berlinski) (2014). [*The King of Infinite Space: Euclid and His Elements*](https://archive.org/details/kingofinfinitesp00davi). Basic Books. [ISBN](/source/ISBN_(identifier)) [978-0-465-03863-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-03863-3).

1. ^ [***a***](#cite_ref-Hartshorne2013_42-0) [***b***](#cite_ref-Hartshorne2013_42-1) [Robin Hartshorne](/source/Robin_Hartshorne) (2013). [*Geometry: Euclid and Beyond*](https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29). Springer Science & Business Media. pp. 29–. [ISBN](/source/ISBN_(identifier)) [978-0-387-22676-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-22676-7). [Archived](https://web.archive.org/web/20210901183205/https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29) from the original on 1 September 2021. Retrieved 14 September 2019.

1. **[^](#cite_ref-HerbstFujita2017_43-0)** Pat Herbst; Taro Fujita; Stefan Halverscheid; Michael Weiss (2017). [*The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective*](https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20). Taylor & Francis. pp. 20–. [ISBN](/source/ISBN_(identifier)) [978-1-351-97353-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-351-97353-3). [Archived](https://web.archive.org/web/20210901183206/https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20) from the original on 1 September 2021. Retrieved 14 September 2019.

1. **[^](#cite_ref-Yaglom2012_44-0)** [I. M. Yaglom](/source/Isaak_Yaglom) (2012). [*A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity*](https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6). Springer Science & Business Media. pp. 6–. [ISBN](/source/ISBN_(identifier)) [978-1-4612-6135-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-6135-3). [Archived](https://web.archive.org/web/20210901183221/https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6) from the original on 1 September 2021. Retrieved 14 September 2019.

1. **[^](#cite_ref-Holme2010_45-0)** Audun Holme (2010). [*Geometry: Our Cultural Heritage*](https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254). Springer Science & Business Media. pp. 254–. [ISBN](/source/ISBN_(identifier)) [978-3-642-14441-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-14441-7). [Archived](https://web.archive.org/web/20210901183209/https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254) from the original on 1 September 2021. Retrieved 14 September 2019.

1. ^ [***a***](#cite_ref-EuclidAll_46-0) [***b***](#cite_ref-EuclidAll_46-1) [***c***](#cite_ref-EuclidAll_46-2) [***d***](#cite_ref-EuclidAll_46-3) [***e***](#cite_ref-EuclidAll_46-4) *Euclid's Elements – All thirteen books in one volume*, Based on Heath's translation, Green Lion Press [ISBN](/source/ISBN_(identifier)) [1-888009-18-7](https://en.wikipedia.org/wiki/Special:BookSources/1-888009-18-7).

1. **[^](#cite_ref-47)** Gerla, G. (1995). ["Pointless Geometries"](https://web.archive.org/web/20110717210751/http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf) (PDF). In Buekenhout, F.; Kantor, W. (eds.). *Handbook of incidence geometry: buildings and foundations*. North-Holland. pp. 1015–1031. Archived from [the original](http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf) (PDF) on 17 July 2011.

1. **[^](#cite_ref-48)** Clark, Bowman L. (January 1985). ["Individuals and Points"](https://doi.org/10.1305%2Fndjfl%2F1093870761). *Notre Dame Journal of Formal Logic*. **26** (1): 61–75. [doi](/source/Doi_(identifier)):[10.1305/ndjfl/1093870761](https://doi.org/10.1305%2Fndjfl%2F1093870761).

1. **[^](#cite_ref-49)** [John Casey](/source/John_Casey_(mathematician)) (1885). [*Analytic Geometry of the Point, Line, Circle, and Conic Sections*](https://archive.org/details/cu31924001520455).

1. **[^](#cite_ref-50)** Francis Buekenhout, ed. (1995). *Handbook of incidence geometry : buildings and foundations*. Amsterdam: Elsevier. [ISBN](/source/ISBN_(identifier)) [978-0-444-88355-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-444-88355-1). [OCLC](/source/OCLC_(identifier)) [162589397](https://search.worldcat.org/oclc/162589397).

1. **[^](#cite_ref-51)** ["geodesic – definition of geodesic in English from the Oxford dictionary"](https://web.archive.org/web/20160715034047/http://www.oxforddictionaries.com/definition/english/geodesic). [OxfordDictionaries.com](/source/OxfordDictionaries.com). Archived from [the original](https://www.oxforddictionaries.com/definition/english/geodesic) on 15 July 2016. Retrieved 20 January 2016.

1. ^ [***a***](#cite_ref-Munkres_52-0) [***b***](#cite_ref-Munkres_52-1) [***c***](#cite_ref-Munkres_52-2) [***d***](#cite_ref-Munkres_52-3) [***e***](#cite_ref-Munkres_52-4) [Munkres, James R.](/source/James_Munkres) (2000). *Topology*. Vol. 2 (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. [ISBN](/source/ISBN_(identifier)) [0-13-181629-2](https://en.wikipedia.org/wiki/Special:BookSources/0-13-181629-2). [OCLC](/source/OCLC_(identifier)) [42683260](https://search.worldcat.org/oclc/42683260).

1. **[^](#cite_ref-53)** [Szmielew, Wanda](/source/Wanda_Szmielew) (1983). [*From Affine to Euclidean Geometry*](https://books.google.com/books?id=xDJPAQAAIAAJ). Springer. [ISBN](/source/ISBN_(identifier)) [978-90-277-1243-1](https://en.wikipedia.org/wiki/Special:BookSources/978-90-277-1243-1). [Archived](https://web.archive.org/web/20230301145204/https://books.google.com/books?id=xDJPAQAAIAAJ) from the original on 1 March 2023. Retrieved 9 September 2022.

1. **[^](#cite_ref-54)** [Ahlfors, Lars V.](/source/Lars_Ahlfors) (1979). [*Complex analysis : an introduction to the theory of analytic functions of one complex variable*](https://books.google.com/books?id=2MRuus-5GGoC) (3rd ed.). New York: McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [9780070006577](https://en.wikipedia.org/wiki/Special:BookSources/9780070006577). [OCLC](/source/OCLC_(identifier)) [4036464](https://search.worldcat.org/oclc/4036464). [Archived](https://web.archive.org/web/20230301145208/https://books.google.com/books?id=2MRuus-5GGoC) from the original on 1 March 2023. Retrieved 9 September 2022.

1. **[^](#cite_ref-55)** Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.

1. ^ [***a***](#cite_ref-Carmo_56-0) [***b***](#cite_ref-Carmo_56-1) [***c***](#cite_ref-Carmo_56-2) Carmo, Manfredo Perdigão do (1976). [*Differential geometry of curves and surfaces*](https://books.google.com/books?id=1v0YAQAAIAAJ). Vol. 2. Englewood Cliffs, N.J.: Prentice-Hall. [ISBN](/source/ISBN_(identifier)) [0-13-212589-7](https://en.wikipedia.org/wiki/Special:BookSources/0-13-212589-7). [OCLC](/source/OCLC_(identifier)) [1529515](https://search.worldcat.org/oclc/1529515). [Archived](https://web.archive.org/web/20230301145145/https://books.google.com/books?id=1v0YAQAAIAAJ) from the original on 1 March 2023. Retrieved 9 September 2022.

1. ^ [***a***](#cite_ref-mumford_57-0) [***b***](#cite_ref-mumford_57-1) [Mumford, David](/source/David_Mumford) (1999). *The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians* (2nd ed.). [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). [ISBN](/source/ISBN_(identifier)) [978-3-540-63293-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-63293-1). [Zbl](/source/Zbl_(identifier)) [0945.14001](https://zbmath.org/?format=complete&q=an:0945.14001).

1. **[^](#cite_ref-58)** Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." [ISBN](/source/ISBN_(identifier)) [978-0-321-57056-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-321-57056-7).

1. **[^](#cite_ref-59)** [Yau, Shing-Tung](/source/Shing-Tung_Yau); Nadis, Steve (2010). *The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions*. Basic Books. [ISBN](/source/ISBN_(identifier)) [978-0-465-02023-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-02023-2).

1. **[^](#cite_ref-60)** Sidorov, L.A. (2001) [1994]. ["Angle"](https://www.encyclopediaofmath.org/index.php?title=Angle). *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*. [EMS Press](/source/European_Mathematical_Society).

1. **[^](#cite_ref-61)** [Gelʹfand, I. M.](/source/Israel_Gelfand) (2001). [*Trigonometry*](https://books.google.com/books?id=ZCYtwHFVZHgC). Mark E. Saul. Boston: Birkhäuser. pp. 1–20. [ISBN](/source/ISBN_(identifier)) [0-8176-3914-4](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-3914-4). [OCLC](/source/OCLC_(identifier)) [41355833](https://search.worldcat.org/oclc/41355833). [Archived](https://web.archive.org/web/20230301145230/https://books.google.com/books?id=ZCYtwHFVZHgC) from the original on 1 March 2023. Retrieved 10 September 2022.

1. **[^](#cite_ref-Stewart_62-0)** [Stewart, James](/source/James_Stewart_(mathematician)) (2012). *Calculus: Early Transcendentals*, 7th ed., Brooks Cole Cengage Learning. [ISBN](/source/ISBN_(identifier)) [978-0-538-49790-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-538-49790-9)

1. **[^](#cite_ref-63)** [Jost, Jürgen](/source/J%C3%BCrgen_Jost) (2002). *Riemannian Geometry and Geometric Analysis*. Berlin: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [978-3-540-42627-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42627-1)..

1. **[^](#cite_ref-Cannon2017_64-0)** [James W. Cannon](/source/James_W._Cannon) (2017). [*Geometry of Lengths, Areas, and Volumes*](https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11). American Mathematical Soc. p. 11. [ISBN](/source/ISBN_(identifier)) [978-1-4704-3714-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4704-3714-5). [Archived](https://web.archive.org/web/20191231135911/https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11) from the original on 31 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Strang1991_65-0)** [Gilbert Strang](/source/Gilbert_Strang) (1991). [*Calculus*](https://books.google.com/books?id=OisInC1zvEMC). SIAM. [ISBN](/source/ISBN_(identifier)) [978-0-9614088-2-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9614088-2-4). [Archived](https://web.archive.org/web/20191224134500/https://books.google.com/books?id=OisInC1zvEMC) from the original on 24 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Bear2002_66-0)** H. S. Bear (2002). [*A Primer of Lebesgue Integration*](https://books.google.com/books?id=__AmiGnEEewC). Academic Press. [ISBN](/source/ISBN_(identifier)) [978-0-12-083971-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-083971-1). [Archived](https://web.archive.org/web/20191225032645/https://books.google.com/books?id=__AmiGnEEewC) from the original on 25 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-67)** [Dmitri Burago](/source/Dmitri_Burago), [Yu D Burago](/source/Yuri_Dmitrievich_Burago), [[Sergei Ivanov (mathematician)|]], *A Course in Metric Geometry*, American Mathematical Society, 2001, [ISBN](/source/ISBN_(identifier)) [0-8218-2129-6](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-2129-6).

1. **[^](#cite_ref-68)** [Wald, Robert M.](/source/Robert_Wald) (1984). [*General Relativity*](/source/General_Relativity_(book)). University of Chicago Press. [ISBN](/source/ISBN_(identifier)) [978-0-226-87033-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-87033-5).

1. **[^](#cite_ref-Tao2011_69-0)** [Terence Tao](/source/Terence_Tao) (2011). [*An Introduction to Measure Theory*](https://books.google.com/books?id=HoGDAwAAQBAJ). American Mathematical Soc. [ISBN](/source/ISBN_(identifier)) [978-0-8218-6919-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-6919-2). [Archived](https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ) from the original on 27 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Libeskind2008_70-0)** Shlomo Libeskind (2008). [*Euclidean and Transformational Geometry: A Deductive Inquiry*](https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255). Jones & Bartlett Learning. p. 255. [ISBN](/source/ISBN_(identifier)) [978-0-7637-4366-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7637-4366-6). [Archived](https://web.archive.org/web/20191225090248/https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255) from the original on 25 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Freitag2013_71-0)** Mark A. Freitag (2013). [*Mathematics for Elementary School Teachers: A Process Approach*](https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614). Cengage Learning. p. 614. [ISBN](/source/ISBN_(identifier)) [978-0-618-61008-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-618-61008-2). [Archived](https://web.archive.org/web/20191228123854/https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614) from the original on 28 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Martin2012_72-0)** George E. Martin (2012). [*Transformation Geometry: An Introduction to Symmetry*](https://books.google.com/books?id=gevlBwAAQBAJ). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4612-5680-9](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-5680-9). [Archived](https://web.archive.org/web/20191207041210/https://books.google.com/books?id=gevlBwAAQBAJ) from the original on 7 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Blacklock2018_74-0)** Mark Blacklock (2018). [*The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle*](https://books.google.com/books?id=nrNSDwAAQBAJ). Oxford University Press. [ISBN](/source/ISBN_(identifier)) [978-0-19-875548-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-875548-7). [Archived](https://web.archive.org/web/20191227145318/https://books.google.com/books?id=nrNSDwAAQBAJ) from the original on 27 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Joly1895_75-0)** Charles Jasper Joly (1895). [*Papers*](https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62). The Academy. pp. 62–. [Archived](https://web.archive.org/web/20191227195202/https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62) from the original on 27 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Temam2013_76-0)** Roger Temam (2013). [*Infinite-Dimensional Dynamical Systems in Mechanics and Physics*](https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367). Springer Science & Business Media. p. 367. [ISBN](/source/ISBN_(identifier)) [978-1-4612-0645-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-0645-3). [Archived](https://web.archive.org/web/20191224015857/https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367) from the original on 24 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-JacobLam1994_77-0)** Bill Jacob; [Tsit-Yuen Lam](/source/Tsit_Yuen_Lam) (1994). [*Recent Advances in Real Algebraic Geometry and Quadratic Forms: Proceedings of the RAGSQUAD Year, Berkeley, 1990–1991*](https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111). American Mathematical Soc. p. 111. [ISBN](/source/ISBN_(identifier)) [978-0-8218-5154-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-5154-8). [Archived](https://web.archive.org/web/20191228124040/https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111) from the original on 28 December 2019. Retrieved 18 September 2019.

1. **[^](#cite_ref-Stewart2008_78-0)** [Ian Stewart](/source/Ian_Stewart_(mathematician)) (2008). [*Why Beauty Is Truth: A History of Symmetry*](https://books.google.com/books?id=6akF1v7Ds3MC). Basic Books. p. 14. [ISBN](/source/ISBN_(identifier)) [978-0-465-08237-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-08237-7). [Archived](https://web.archive.org/web/20191225201454/https://books.google.com/books?id=6akF1v7Ds3MC) from the original on 25 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Alexey2009_79-0)** Stakhov Alexey (2009). [*Mathematics Of Harmony: From Euclid To Contemporary Mathematics And Computer Science*](https://books.google.com/books?id=3k7ICgAAQBAJ&pg=PA144). World Scientific. p. 144. [ISBN](/source/ISBN_(identifier)) [978-981-4472-57-9](https://en.wikipedia.org/wiki/Special:BookSources/978-981-4472-57-9). [Archived](https://web.archive.org/web/20191229132952/https://books.google.com/books?id=3k7ICgAAQBAJ&pg=PA144) from the original on 29 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Hahn1998_80-0)** Werner Hahn (1998). [*Symmetry as a Developmental Principle in Nature and Art*](https://books.google.com/books?id=wzhqDQAAQBAJ). World Scientific. [ISBN](/source/ISBN_(identifier)) [978-981-02-2363-2](https://en.wikipedia.org/wiki/Special:BookSources/978-981-02-2363-2). [Archived](https://web.archive.org/web/20200101082827/https://books.google.com/books?id=wzhqDQAAQBAJ) from the original on 1 January 2020. Retrieved 23 September 2019.

1. **[^](#cite_ref-Cantwell2002_81-0)** Brian J. Cantwell (2002). [*Introduction to Symmetry Analysis*](https://books.google.com/books?id=76RS2ZQ0UyUC&pg=PR34). Cambridge University Press. p. 34. [ISBN](/source/ISBN_(identifier)) [978-1-139-43171-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-139-43171-2). [Archived](https://web.archive.org/web/20191227012548/https://books.google.com/books?id=76RS2ZQ0UyUC&pg=PR34) from the original on 27 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-RosenfeldWiebe2013_82-0)** B. Rosenfeld; Bill Wiebe (2013). [*Geometry of Lie Groups*](https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA158). Springer Science & Business Media. pp. 158ff. [ISBN](/source/ISBN_(identifier)) [978-1-4757-5325-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4757-5325-7). [Archived](https://web.archive.org/web/20191224193157/https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA158) from the original on 24 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Pesic2007_83-0)** Peter Pesic (2007). [*Beyond Geometry: Classic Papers from Riemann to Einstein*](https://books.google.com/books?id=Z67x6IOuOUAC). Courier Corporation. [ISBN](/source/ISBN_(identifier)) [978-0-486-45350-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-45350-7). [Archived](https://web.archive.org/web/20210901183221/https://books.google.com/books?id=Z67x6IOuOUAC) from the original on 1 September 2021. Retrieved 23 September 2019.

1. **[^](#cite_ref-Kaku2012_84-0)** [Michio Kaku](/source/Michio_Kaku) (2012). [*Strings, Conformal Fields, and Topology: An Introduction*](https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151). Springer Science & Business Media. p. 151. [ISBN](/source/ISBN_(identifier)) [978-1-4684-0397-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4684-0397-8). [Archived](https://web.archive.org/web/20191224015822/https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151) from the original on 24 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-BestvinaSageev2014_85-0)** Mladen Bestvina; Michah Sageev; [Karen Vogtmann](/source/Karen_Vogtmann) (2014). [*Geometric Group Theory*](https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132). American Mathematical Soc. p. 132. [ISBN](/source/ISBN_(identifier)) [978-1-4704-1227-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4704-1227-2). [Archived](https://web.archive.org/web/20191229224624/https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132) from the original on 29 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Steeb1996_86-0)** W-H. Steeb (1996). [*Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra*](https://books.google.com/books?id=rZBIDQAAQBAJ). World Scientific Publishing Company. [ISBN](/source/ISBN_(identifier)) [978-981-310-503-4](https://en.wikipedia.org/wiki/Special:BookSources/978-981-310-503-4). [Archived](https://web.archive.org/web/20191226205450/https://books.google.com/books?id=rZBIDQAAQBAJ) from the original on 26 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Misner2005_87-0)** [Charles W. Misner](/source/Charles_W._Misner) (2005). [*Directions in General Relativity: Volume 1: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner*](https://books.google.com/books?id=zpGZwmTJZIUC&pg=PA272). Cambridge University Press. p. 272. [ISBN](/source/ISBN_(identifier)) [978-0-521-02139-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-02139-5). [Archived](https://web.archive.org/web/20191226063925/https://books.google.com/books?id=zpGZwmTJZIUC&pg=PA272) from the original on 26 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Dowling1917_88-0)** Linnaeus Wayland Dowling (1917). [*Projective Geometry*](https://archive.org/details/cu31924001523897). McGraw-Hill book Company, Incorporated. p. [10](https://archive.org/details/cu31924001523897/page/n29).

1. **[^](#cite_ref-Gierz2006_89-0)** G. Gierz (2006). [*Bundles of Topological Vector Spaces and Their Duality*](https://books.google.com/books?id=2ml6CwAAQBAJ&pg=PA252). Springer. p. 252. [ISBN](/source/ISBN_(identifier)) [978-3-540-39437-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-39437-2). [Archived](https://web.archive.org/web/20191227123430/https://books.google.com/books?id=2ml6CwAAQBAJ&pg=PA252) from the original on 27 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-ButtsBrown2012_90-0)** Robert E. Butts; J.R. Brown (2012). [*Constructivism and Science: Essays in Recent German Philosophy*](https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127). Springer Science & Business Media. pp. 127–. [ISBN](/source/ISBN_(identifier)) [978-94-009-0959-5](https://en.wikipedia.org/wiki/Special:BookSources/978-94-009-0959-5). [Archived](https://web.archive.org/web/20210901183207/https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127) from the original on 1 September 2021. Retrieved 20 September 2019.

1. **[^](#cite_ref-91)** [*Science*](https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181). Moses King. 1886. pp. 181–. [Archived](https://web.archive.org/web/20191227013042/https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181) from the original on 27 December 2019. Retrieved 20 September 2019.

1. **[^](#cite_ref-Abbot2013_92-0)** W. Abbot (2013). [*Practical Geometry and Engineering Graphics: A Textbook for Engineering and Other Students*](https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6). Springer Science & Business Media. pp. 6–. [ISBN](/source/ISBN_(identifier)) [978-94-017-2742-6](https://en.wikipedia.org/wiki/Special:BookSources/978-94-017-2742-6). [Archived](https://web.archive.org/web/20191225201450/https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6) from the original on 25 December 2019. Retrieved 20 September 2019.

1. ^ [***a***](#cite_ref-HerseyHersey2001_93-0) [***b***](#cite_ref-HerseyHersey2001_93-1) [***c***](#cite_ref-HerseyHersey2001_93-2) [***d***](#cite_ref-HerseyHersey2001_93-3) George L. Hersey (2001). [*Architecture and Geometry in the Age of the Baroque*](https://books.google.com/books?id=F1Tl9ok-7_IC). University of Chicago Press. [ISBN](/source/ISBN_(identifier)) [978-0-226-32783-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-32783-9). [Archived](https://web.archive.org/web/20191225141623/https://books.google.com/books?id=F1Tl9ok-7_IC) from the original on 25 December 2019. Retrieved 20 September 2019.

1. **[^](#cite_ref-VanícekKrakiwsky2015_94-0)** P. Vanícek; E.J. Krakiwsky (2015). [*Geodesy: The Concepts*](https://books.google.com/books?id=1Mz-BAAAQBAJ). Elsevier. p. 23. [ISBN](/source/ISBN_(identifier)) [978-1-4832-9079-9](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4832-9079-9). [Archived](https://web.archive.org/web/20191231233050/https://books.google.com/books?id=1Mz-BAAAQBAJ) from the original on 31 December 2019. Retrieved 20 September 2019.

1. **[^](#cite_ref-CummingsMorton2015_95-0)** Russell M. Cummings; Scott A. Morton; William H. Mason; David R. McDaniel (2015). [*Applied Computational Aerodynamics*](https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449). Cambridge University Press. p. 449. [ISBN](/source/ISBN_(identifier)) [978-1-107-05374-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-05374-8). [Archived](https://web.archive.org/web/20210901183207/https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449) from the original on 1 September 2021. Retrieved 20 September 2019.

1. **[^](#cite_ref-Williams1998_96-0)** Roy Williams (1998). [*Geometry of Navigation*](https://books.google.com/books?id=yNzf7OKGLxIC). Horwood Pub. [ISBN](/source/ISBN_(identifier)) [978-1-898563-46-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-898563-46-4). [Archived](https://web.archive.org/web/20191207041213/https://books.google.com/books?id=yNzf7OKGLxIC) from the original on 7 December 2019. Retrieved 20 September 2019.

1. **[^](#cite_ref-Schmidt,_W._2002_97-0)** Schmidt, W.; Houang, R.; Cogan, Leland S. (2002). ["A Coherent Curriculum: The Case of Mathematics"](https://www.nifdi.org/research/journal-of-di/volume-4-no-1-winter-2004/454-a-coherent-curriculum-the-case-of-mathematics/file.html). *The American Educator*. **26** (2): 10–26. [S2CID](/source/S2CID_(identifier)) [118964353](https://api.semanticscholar.org/CorpusID:118964353).

1. **[^](#cite_ref-Walschap2015_98-0)** Gerard Walschap (2015). [*Multivariable Calculus and Differential Geometry*](https://books.google.com/books?id=cXPyCQAAQBAJ). De Gruyter. [ISBN](/source/ISBN_(identifier)) [978-3-11-036954-0](https://en.wikipedia.org/wiki/Special:BookSources/978-3-11-036954-0). [Archived](https://web.archive.org/web/20191227012551/https://books.google.com/books?id=cXPyCQAAQBAJ) from the original on 27 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Flanders2012_99-0)** Harley Flanders (2012). [*Differential Forms with Applications to the Physical Sciences*](https://books.google.com/books?id=U_GLN1eOKaMC). Courier Corporation. [ISBN](/source/ISBN_(identifier)) [978-0-486-13961-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-13961-6). [Archived](https://web.archive.org/web/20210901183207/https://books.google.com/books?id=U_GLN1eOKaMC) from the original on 1 September 2021. Retrieved 23 September 2019.

1. **[^](#cite_ref-MarriottSalmon2000_100-0)** Paul Marriott; Mark Salmon (2000). [*Applications of Differential Geometry to Econometrics*](https://books.google.com/books?id=1Jjm4I5tqkUC). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-0-521-65116-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-65116-5). [Archived](https://web.archive.org/web/20210901183207/https://books.google.com/books?id=1Jjm4I5tqkUC) from the original on 1 September 2021. Retrieved 23 September 2019.

1. **[^](#cite_ref-HePetoukhov2011_101-0)** Matthew He; Sergey Petoukhov (2011). [*Mathematics of Bioinformatics: Theory, Methods and Applications*](https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106). John Wiley & Sons. p. 106. [ISBN](/source/ISBN_(identifier)) [978-1-118-09952-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-118-09952-0). [Archived](https://web.archive.org/web/20191227163605/https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106) from the original on 27 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Dirac2016_102-0)** P.A.M. Dirac (2016). [*General Theory of Relativity*](https://books.google.com/books?id=qkWPDAAAQBAJ). Princeton University Press. [ISBN](/source/ISBN_(identifier)) [978-1-4008-8419-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-8419-3). [Archived](https://web.archive.org/web/20191226205400/https://books.google.com/books?id=qkWPDAAAQBAJ) from the original on 26 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-AyJost2017_103-0)** Nihat Ay; Jürgen Jost; Hông Vân Lê; Lorenz Schwachhöfer (2017). [*Information Geometry*](https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185). Springer. p. 185. [ISBN](/source/ISBN_(identifier)) [978-3-319-56478-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-56478-4). [Archived](https://web.archive.org/web/20191224015858/https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185) from the original on 24 December 2019. Retrieved 23 September 2019.

1. **[^](#cite_ref-Crossley2011_104-0)** Martin D. Crossley (2011). [*Essential Topology*](https://books.google.com/books?id=QhCgVrLHlLgC). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-85233-782-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85233-782-7). [Archived](https://web.archive.org/web/20191228094221/https://books.google.com/books?id=QhCgVrLHlLgC) from the original on 28 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-NashSen1988_105-0)** Charles Nash; Siddhartha Sen (1988). [*Topology and Geometry for Physicists*](https://books.google.com/books?id=nnnNCgAAQBAJ). Elsevier. p. 1. [ISBN](/source/ISBN_(identifier)) [978-0-08-057085-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-057085-3). [Archived](https://web.archive.org/web/20191226215609/https://books.google.com/books?id=nnnNCgAAQBAJ) from the original on 26 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-Martin1996_106-0)** George E. Martin (1996). [*Transformation Geometry: An Introduction to Symmetry*](https://books.google.com/books?id=KW4EwONsQJgC). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-0-387-90636-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90636-2). [Archived](https://web.archive.org/web/20191222024915/https://books.google.com/books?id=KW4EwONsQJgC) from the original on 22 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-May1999_107-0)** J. P. May (1999). [*A Concise Course in Algebraic Topology*](https://books.google.com/books?id=g8SG03R1bpgC). University of Chicago Press. [ISBN](/source/ISBN_(identifier)) [978-0-226-51183-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-51183-2). [Archived](https://web.archive.org/web/20191223144107/https://books.google.com/books?id=g8SG03R1bpgC) from the original on 23 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-AHartshorne2013_108-0)** Robin Hartshorne (2013). [*Algebraic Geometry*](https://books.google.com/books?id=7z4mBQAAQBAJ). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4757-3849-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4757-3849-0). [Archived](https://web.archive.org/web/20191227163607/https://books.google.com/books?id=7z4mBQAAQBAJ) from the original on 27 December 2019. Retrieved 24 September 2019.

1. ^ [***a***](#cite_ref-Dieudonne1985_109-0) [***b***](#cite_ref-Dieudonne1985_109-1) [Jean Dieudonné](/source/Jean_Dieudonn%C3%A9) (1985). [*History of Algebraic Geometry*](https://books.google.com/books?id=_uhlf38jOrgC). Translated by Judith D. Sally. CRC Press. [ISBN](/source/ISBN_(identifier)) [978-0-412-99371-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-412-99371-8). [Archived](https://web.archive.org/web/20191225090149/https://books.google.com/books?id=_uhlf38jOrgC) from the original on 25 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-CarlsonCarlson2006_110-0)** James Carlson; James A. Carlson; Arthur Jaffe; Andrew Wiles (2006). [*The Millennium Prize Problems*](https://books.google.com/books?id=7wJIPJ80RdUC). American Mathematical Soc. [ISBN](/source/ISBN_(identifier)) [978-0-8218-3679-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-3679-8). [Archived](https://web.archive.org/web/20160530210333/https://books.google.com/books?id=7wJIPJ80RdUC) from the original on 30 May 2016. Retrieved 24 September 2019.

1. **[^](#cite_ref-HoweLauter2017_111-0)** Everett W. Howe; [Kristin E. Lauter](/source/Kristin_Lauter); [Judy L. Walker](/source/Judy_L._Walker) (2017). [*Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016*](https://books.google.com/books?id=bPM-DwAAQBAJ). Springer. [ISBN](/source/ISBN_(identifier)) [978-3-319-63931-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-63931-4). [Archived](https://web.archive.org/web/20191227195203/https://books.google.com/books?id=bPM-DwAAQBAJ) from the original on 27 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-MarinoThaddeus2008_112-0)** Marcos Marino; Michael Thaddeus; Ravi Vakil (2008). [*Enumerative Invariants in Algebraic Geometry and String Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6–11, 2005*](https://books.google.com/books?id=mb1qCQAAQBAJ). Springer. [ISBN](/source/ISBN_(identifier)) [978-3-540-79814-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-79814-9). [Archived](https://web.archive.org/web/20191227041441/https://books.google.com/books?id=mb1qCQAAQBAJ) from the original on 27 December 2019. Retrieved 24 September 2019.

1. **[^](#cite_ref-113)** [Huybrechts, Daniel](/source/Daniel_Huybrechts) (2005). [*Complex geometry : an introduction*](https://books.google.com/books?id=eZPCfJlHkXMC). Berlin: Springer. [ISBN](/source/ISBN_(identifier)) [9783540266877](https://en.wikipedia.org/wiki/Special:BookSources/9783540266877). [OCLC](/source/OCLC_(identifier)) [209857590](https://search.worldcat.org/oclc/209857590). [Archived](https://web.archive.org/web/20230301145147/https://books.google.com/books?id=eZPCfJlHkXMC) from the original on 1 March 2023. Retrieved 10 September 2022.

1. **[^](#cite_ref-114)** Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.

1. **[^](#cite_ref-115)** Wells, R. O. Jr. (2008). [*Differential analysis on complex manifolds*](https://books.google.com/books?id=aZXAs9Vu14cC). Graduate Texts in Mathematics. Vol. 65. O. García-Prada (3rd ed.). New York: Springer-Verlag. [doi](/source/Doi_(identifier)):[10.1007/978-0-387-73892-5](https://doi.org/10.1007%2F978-0-387-73892-5). [ISBN](/source/ISBN_(identifier)) [9780387738918](https://en.wikipedia.org/wiki/Special:BookSources/9780387738918). [OCLC](/source/OCLC_(identifier)) [233971394](https://search.worldcat.org/oclc/233971394). [Archived](https://web.archive.org/web/20230301145230/https://books.google.com/books?id=aZXAs9Vu14cC) from the original on 1 March 2023. Retrieved 9 September 2022.

1. **[^](#cite_ref-116)** Hori, K., [Thomas, R.](/source/Richard_Thomas_(mathematician)), [Katz, S.](/source/Sheldon_Katz), [Vafa, C.](/source/Cumrun_Vafa), [Pandharipande, R.](/source/Rahul_Pandharipande), Klemm, A., ... & [Zaslow, E.](/source/Eric_Zaslow) (2003). Mirror symmetry (Vol. 1). American Mathematical Soc.

1. **[^](#cite_ref-117)** Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.

1. **[^](#cite_ref-118)** Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.

1. **[^](#cite_ref-119)** [Donaldson, S. K.](/source/Simon_Donaldson) (2011). *Riemann surfaces*. Oxford: Oxford University Press. [ISBN](/source/ISBN_(identifier)) [978-0-19-154584-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-154584-9). [OCLC](/source/OCLC_(identifier)) [861200296](https://search.worldcat.org/oclc/861200296).

1. **[^](#cite_ref-120)** [Serre, J. P.](/source/Jean-Pierre_Serre) (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.

1. **[^](#cite_ref-121)** [Serre, J. P.](/source/Jean-Pierre_Serre) (1956). Géométrie algébrique et géométrie analytique. In Annales de l'Institut Fourier (vol. 6, pp. 1–42).

1. **[^](#cite_ref-Matoušek2013_122-0)** [Jiří Matoušek](/source/Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)) (2013). [*Lectures on Discrete Geometry*](https://books.google.com/books?id=K0fhBwAAQBAJ). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4613-0039-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4613-0039-7). [Archived](https://web.archive.org/web/20191227013417/https://books.google.com/books?id=K0fhBwAAQBAJ) from the original on 27 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Zong2006_123-0)** Chuanming Zong (2006). [*The Cube – A Window to Convex and Discrete Geometry*](https://books.google.com/books?id=Ola6htFUQ1IC). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-0-521-85535-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-85535-8). [Archived](https://web.archive.org/web/20191223150837/https://books.google.com/books?id=Ola6htFUQ1IC) from the original on 23 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Gruber2007_124-0)** Peter M. Gruber (2007). [*Convex and Discrete Geometry*](https://books.google.com/books?id=bSZKAAAAQBAJ). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-3-540-71133-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-71133-9). [Archived](https://web.archive.org/web/20191224175343/https://books.google.com/books?id=bSZKAAAAQBAJ) from the original on 24 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-DevadossO'Rourke2011_125-0)** [Satyan L. Devadoss](/source/Satyan_Devadoss); [Joseph O'Rourke](/source/Joseph_O'Rourke_(professor)) (2011). [*Discrete and Computational Geometry*](https://books.google.com/books?id=InJL6iAaIQQC). Princeton University Press. [ISBN](/source/ISBN_(identifier)) [978-1-4008-3898-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-3898-1). [Archived](https://web.archive.org/web/20191227194659/https://books.google.com/books?id=InJL6iAaIQQC) from the original on 27 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Bezdek2010_126-0)** [Károly Bezdek](/source/K%C3%A1roly_Bezdek) (2010). [*Classical Topics in Discrete Geometry*](https://books.google.com/books?id=Tov0d9VMOfMC). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4419-0600-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4419-0600-7). [Archived](https://web.archive.org/web/20191228051643/https://books.google.com/books?id=Tov0d9VMOfMC) from the original on 28 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-PreparataShamos2012_127-0)** [Franco P. Preparata](/source/Franco_P._Preparata); [Michael I. Shamos](/source/Michael_Ian_Shamos) (2012). [*Computational Geometry: An Introduction*](https://books.google.com/books?id=_p3eBwAAQBAJ). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4612-1098-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-1098-6). [Archived](https://web.archive.org/web/20191228093733/https://books.google.com/books?id=_p3eBwAAQBAJ) from the original on 28 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-GuYau2008_128-0)** Xianfeng David Gu; Shing-Tung Yau (2008). [*Computational Conformal Geometry*](https://books.google.com/books?id=4FDvAAAAMAAJ). International Press. [ISBN](/source/ISBN_(identifier)) [978-1-57146-171-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-57146-171-1). [Archived](https://web.archive.org/web/20191224054942/https://books.google.com/books?id=4FDvAAAAMAAJ) from the original on 24 December 2019. Retrieved 25 September 2019.

1. ^ [***a***](#cite_ref-Löh2017_129-0) [***b***](#cite_ref-Löh2017_129-1) Clara Löh (2017). [*Geometric Group Theory: An Introduction*](https://books.google.com/books?id=1AxEDwAAQBAJ). Springer. [ISBN](/source/ISBN_(identifier)) [978-3-319-72254-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-72254-2). [Archived](https://web.archive.org/web/20191229132923/https://books.google.com/books?id=1AxEDwAAQBAJ) from the original on 29 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-130)** Bestvina, Mladen; Sageev, Michah; Vogtmann, Karen (24 December 2014). ["Geometric Group Theory"](https://www.ams.org/books/pcms/021/). *American Mathematical Society*. Retrieved 5 May 2025.

1. **[^](#cite_ref-131)** Agol, Ian (2013). ["The virtual Haken Conjecture"](https://www.math.uni-bielefeld.de/documenta/vol-18/33.html). *Doc. Math*. **18**. With an appendix by Ian Agol, Daniel Groves, and Jason Manning: 1045–1087. [doi](/source/Doi_(identifier)):[10.4171/dm/421](https://doi.org/10.4171%2Fdm%2F421). [MR](/source/MR_(identifier)) [3104553](https://mathscinet.ams.org/mathscinet-getitem?mr=3104553). [S2CID](/source/S2CID_(identifier)) [255586740](https://api.semanticscholar.org/CorpusID:255586740).

1. **[^](#cite_ref-Wise2012_132-0)** Daniel T. Wise (2012). [*From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry*](https://books.google.com/books?id=GsTW5oQhRPkC). American Mathematical Soc. [ISBN](/source/ISBN_(identifier)) [978-0-8218-8800-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-8800-1). [Archived](https://web.archive.org/web/20191228115647/https://books.google.com/books?id=GsTW5oQhRPkC) from the original on 28 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-133)** Margalit, Dan; Clay, Matt, eds. (11 July 2017). [*Office Hours with a Geometric Group Theorist*](https://academic.oup.com/princeton-scholarship-online/book/13467?login=false). Princeton University Press. [doi](/source/Doi_(identifier)):[10.23943/princeton/9780691158662.001.0001](https://doi.org/10.23943%2Fprinceton%2F9780691158662.001.0001). [ISBN](/source/ISBN_(identifier)) [978-0-691-15866-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-15866-2).

1. **[^](#cite_ref-134)** [Druţu, Cornelia](/source/Cornelia_Dru%C8%9Bu); [Kapovich, Michael](/source/Michael_Kapovich) (28 March 2018). [*Geometric Group Theory*](https://www.ams.org/books/coll/063/). Colloquium Publications. Vol. 63. American Mathematical Society. [doi](/source/Doi_(identifier)):[10.1090/coll/063](https://doi.org/10.1090%2Fcoll%2F063). [ISBN](/source/ISBN_(identifier)) [978-1-4704-1104-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4704-1104-6). Retrieved 8 December 2024.

1. ^ [***a***](#cite_ref-Meurant2014_135-0) [***b***](#cite_ref-Meurant2014_135-1) Gerard Meurant (2014). [*Handbook of Convex Geometry*](https://books.google.com/books?id=M2viBQAAQBAJ). Elsevier Science. [ISBN](/source/ISBN_(identifier)) [978-0-08-093439-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-093439-6). [Archived](https://web.archive.org/web/20210901183208/https://books.google.com/books?id=M2viBQAAQBAJ) from the original on 1 September 2021. Retrieved 24 September 2019.

1. **[^](#cite_ref-Richter-Gebert2011_136-0)** Jürgen Richter-Gebert (2011). [*Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry*](https://books.google.com/books?id=F_NP8Kub2XYC). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-3-642-17286-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-17286-1). [Archived](https://web.archive.org/web/20191229224621/https://books.google.com/books?id=F_NP8Kub2XYC) from the original on 29 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Elam2001_137-0)** Kimberly Elam (2001). [*Geometry of Design: Studies in Proportion and Composition*](https://books.google.com/books?id=JXIEz2XYnp8C). Princeton Architectural Press. [ISBN](/source/ISBN_(identifier)) [978-1-56898-249-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56898-249-6). [Archived](https://web.archive.org/web/20191231135913/https://books.google.com/books?id=JXIEz2XYnp8C) from the original on 31 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Guigar2004_138-0)** Brad J. Guigar (2004). [*The Everything Cartooning Book: Create Unique And Inspired Cartoons For Fun And Profit*](https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82). Adams Media. pp. 82–. [ISBN](/source/ISBN_(identifier)) [978-1-4405-2305-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4405-2305-2). [Archived](https://web.archive.org/web/20191227032210/https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82) from the original on 27 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Livio2008_139-0)** Mario Livio (2008). [*The Golden Ratio: The Story of PHI, the World's Most Astonishing Number*](https://books.google.com/books?id=bUARfgWRH14C&pg=PA166). Crown/Archetype. p. 166. [ISBN](/source/ISBN_(identifier)) [978-0-307-48552-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-307-48552-6). [Archived](https://web.archive.org/web/20191230093236/https://books.google.com/books?id=bUARfgWRH14C&pg=PA166) from the original on 30 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-EmmerSchattschneider2007_140-0)** Michele Emmer; [Doris Schattschneider](/source/Doris_Schattschneider) (2007). [*M. C. Escher's Legacy: A Centennial Celebration*](https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107). Springer. p. 107. [ISBN](/source/ISBN_(identifier)) [978-3-540-28849-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-28849-7). [Archived](https://web.archive.org/web/20191222200130/https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107) from the original on 22 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-CapitoloSchwab2004_141-0)** Robert Capitolo; Ken Schwab (2004). [*Drawing Course 101*](https://archive.org/details/drawingcourse1010000capi). Sterling Publishing Company, Inc. p. [22](https://archive.org/details/drawingcourse1010000capi/page/22). [ISBN](/source/ISBN_(identifier)) [978-1-4027-0383-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4027-0383-6).

1. **[^](#cite_ref-Gelineau2011_142-0)** Phyllis Gelineau (2011). [*Integrating the Arts Across the Elementary School Curriculum*](https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55). Cengage Learning. p. 55. [ISBN](/source/ISBN_(identifier)) [978-1-111-30126-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-111-30126-2). [Archived](https://web.archive.org/web/20191207041800/https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55) from the original on 7 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-CeccatoHesselgren2016_143-0)** Cristiano Ceccato; Lars Hesselgren; Mark Pauly; Helmut Pottmann, Johannes Wallner (2016). [*Advances in Architectural Geometry 2010*](https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6). Birkhäuser. p. 6. [ISBN](/source/ISBN_(identifier)) [978-3-99043-371-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-99043-371-3). [Archived](https://web.archive.org/web/20191225201452/https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6) from the original on 25 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Pottmann2007_144-0)** Helmut Pottmann (2007). [*Architectural geometry*](https://books.google.com/books?id=bIceAQAAIAAJ). Bentley Institute Press. [ISBN](/source/ISBN_(identifier)) [978-1-934493-04-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-934493-04-5). [Archived](https://web.archive.org/web/20191224030536/https://books.google.com/books?id=bIceAQAAIAAJ) from the original on 24 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-MoffettFazio2003_145-0)** Marian Moffett; Michael W. Fazio; Lawrence Wodehouse (2003). [*A World History of Architecture*](https://books.google.com/books?id=IFMohetegAcC&pg=PT371). Laurence King Publishing. p. 371. [ISBN](/source/ISBN_(identifier)) [978-1-85669-371-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85669-371-4). [Archived](https://web.archive.org/web/20191227145458/https://books.google.com/books?id=IFMohetegAcC&pg=PT371) from the original on 27 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-GreenGreen1985_146-0)** Robin M. Green; Robin Michael Green (1985). [*Spherical Astronomy*](https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1). Cambridge University Press. p. 1. [ISBN](/source/ISBN_(identifier)) [978-0-521-31779-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-31779-5). [Archived](https://web.archive.org/web/20191221211420/https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1) from the original on 21 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Alekseevskiĭ2008_147-0)** Dmitriĭ Vladimirovich Alekseevskiĭ (2008). [*Recent Developments in Pseudo-Riemannian Geometry*](https://books.google.com/books?id=K6-TgxMKu4QC). European Mathematical Society. [ISBN](/source/ISBN_(identifier)) [978-3-03719-051-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-03719-051-7). [Archived](https://web.archive.org/web/20191228115649/https://books.google.com/books?id=K6-TgxMKu4QC) from the original on 28 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-YauNadis2010_148-0)** Shing-Tung Yau; Steve Nadis (2010). [*The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions*](https://books.google.com/books?id=M40Ytp8Os_gC). Basic Books. [ISBN](/source/ISBN_(identifier)) [978-0-465-02266-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-02266-3). [Archived](https://web.archive.org/web/20191224015855/https://books.google.com/books?id=M40Ytp8Os_gC) from the original on 24 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-149)** Bengtsson, Ingemar; [Życzkowski, Karol](/source/Karol_%C5%BByczkowski) (2017). [*Geometry of Quantum States: An Introduction to Quantum Entanglement*](/source/Geometry_of_Quantum_States) (2nd ed.). [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [978-1-107-02625-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-02625-4). [OCLC](/source/OCLC_(identifier)) [1004572791](https://search.worldcat.org/oclc/1004572791).

1. **[^](#cite_ref-FlandersPrice2014_150-0)** Harley Flanders; Justin J. Price (2014). [*Calculus with Analytic Geometry*](https://books.google.com/books?id=5abiBQAAQBAJ). Elsevier Science. [ISBN](/source/ISBN_(identifier)) [978-1-4832-6240-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4832-6240-6). [Archived](https://web.archive.org/web/20191224175037/https://books.google.com/books?id=5abiBQAAQBAJ) from the original on 24 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-RogawskiAdams2015_151-0)** Jon Rogawski; Colin Adams (2015). [*Calculus*](https://books.google.com/books?id=OWeZBgAAQBAJ). W. H. Freeman. [ISBN](/source/ISBN_(identifier)) [978-1-4641-7499-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4641-7499-5). [Archived](https://web.archive.org/web/20200101083409/https://books.google.com/books?id=OWeZBgAAQBAJ) from the original on 1 January 2020. Retrieved 25 September 2019.

1. **[^](#cite_ref-Lozano-Robledo2019_152-0)** Álvaro Lozano-Robledo (2019). [*Number Theory and Geometry: An Introduction to Arithmetic Geometry*](https://books.google.com/books?id=ESiODwAAQBAJ). American Mathematical Soc. [ISBN](/source/ISBN_(identifier)) [978-1-4704-5016-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4704-5016-8). [Archived](https://web.archive.org/web/20191227145316/https://books.google.com/books?id=ESiODwAAQBAJ) from the original on 27 December 2019. Retrieved 25 September 2019.

1. **[^](#cite_ref-Sangalli2009_153-0)** Arturo Sangalli (2009). [*Pythagoras' Revenge: A Mathematical Mystery*](https://archive.org/details/pythagorasreveng0000sang). Princeton University Press. p. [57](https://archive.org/details/pythagorasreveng0000sang/page/57). [ISBN](/source/ISBN_(identifier)) [978-0-691-04955-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-04955-7).

1. **[^](#cite_ref-CornellSilverman2013_154-0)** Gary Cornell; [Joseph H. Silverman](/source/Joseph_H._Silverman); [Glenn Stevens](/source/Glenn_H._Stevens) (2013). [*Modular Forms and Fermat's Last Theorem*](https://books.google.com/books?id=jD3TBwAAQBAJ). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4612-1974-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-1974-3). [Archived](https://web.archive.org/web/20191230181409/https://books.google.com/books?id=jD3TBwAAQBAJ) from the original on 30 December 2019. Retrieved 25 September 2019.

### Sources

- [Boyer, C.B.](/source/Carl_Benjamin_Boyer) (1991) [1989]. [*A History of Mathematics*](https://archive.org/details/historyofmathema00boye) (Second edition, revised by [Uta C. Merzbach](/source/Uta_Merzbach) ed.). New York: Wiley. [ISBN](/source/ISBN_(identifier)) [978-0-471-54397-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-54397-8).

- Cooke, Roger (2005). *The History of Mathematics*. New York: Wiley-Interscience. [ISBN](/source/ISBN_(identifier)) [978-0-471-44459-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-44459-6).

- 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 [Johns Hopkins University Press](/source/Johns_Hopkins_University_Press). pp. 118–130. [ISBN](/source/ISBN_(identifier)) [978-0-8018-7396-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8018-7396-6).

- Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.). *The Blackwell Companion to Hinduism*. Oxford: [Basil Blackwell](/source/Basil_Blackwell). pp. 360–375. [ISBN](/source/ISBN_(identifier)) [978-1-4051-3251-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4051-3251-0).

## Further reading

- [Jay Kappraff](/source/Jay_Kappraff) (2014). [*A Participatory Approach to Modern Geometry*](http://www.worldscientific.com/worldscibooks/10.1142/8952). World Scientific Publishing. [doi](/source/Doi_(identifier)):[10.1142/8952](https://doi.org/10.1142%2F8952). [ISBN](/source/ISBN_(identifier)) [978-981-4556-70-5](https://en.wikipedia.org/wiki/Special:BookSources/978-981-4556-70-5). [Zbl](/source/Zbl_(identifier)) [1364.00004](https://zbmath.org/?format=complete&q=an:1364.00004).

- Nikolai I. Lobachevsky (2010). *Pangeometry*. Heritage of European Mathematics Series. Vol. 4. translator and editor: A. Papadopoulos. European Mathematical Society.

- [Leonard Mlodinow](/source/Leonard_Mlodinow) (2002). *Euclid's Window – The Story of Geometry from Parallel Lines to Hyperspace* (UK ed.). Allen Lane. [ISBN](/source/ISBN_(identifier)) [978-0-7139-9634-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7139-9634-0).

## External links

**Geometry**  at Wikipedia's [sister projects](https://en.wikipedia.org/wiki/Wikipedia:Wikimedia_sister_projects)

- [Definitions](https://en.wiktionary.org/wiki/Special:Search/Geometry) from Wiktionary
- [Media](https://commons.wikimedia.org/wiki/Category:Geometry) from Commons
- [Quotations](https://en.wikiquote.org/wiki/Geometry) from Wikiquote
- [Texts](https://en.wikisource.org/wiki/Special:Search/Geometry) from Wikisource
- [Textbooks](https://en.wikibooks.org/wiki/Geometry) from Wikibooks
- [Resources](https://en.wikiversity.org/wiki/Geometry) from Wikiversity

Wikibooks has more on the topic of: ***[Geometry](https://en.wikibooks.org/wiki/Special:Search/Geometry)***

[Library resources](https://en.wikipedia.org/wiki/Wikipedia:The_Wikipedia_Library) about
 **Geometry**

- [Resources in your library](https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Geometry)

- ["Geometry"](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Geometry). *[Encyclopædia Britannica](/source/Encyclop%C3%A6dia_Britannica_Eleventh_Edition)*. Vol. 11 (11th ed.). 1911. pp. 675–736.

- A [geometry](https://en.wikiversity.org/wiki/Geometry) course from [Wikiversity](https://en.wikiversity.org/wiki/)

- [*Unusual Geometry Problems*](http://www.8foxes.com/)

- [*The Math Forum* – Geometry](http://mathforum.org/library/topics/geometry/) [Archived](https://web.archive.org/web/20220128062957/http://mathforum.org/library/topics/geometry/) 28 January 2022 at the [Wayback Machine](/source/Wayback_Machine) - [*The Math Forum* – K–12 Geometry](http://mathforum.org/geometry/k12.geometry.html) [Archived](https://web.archive.org/web/20080415225526/http://mathforum.org/geometry/k12.geometry.html) 15 April 2008 at the [Wayback Machine](/source/Wayback_Machine) - [*The Math Forum* – College Geometry](http://mathforum.org/geometry/coll.geometry.html) [Archived](https://web.archive.org/web/20080415055232/http://mathforum.org/geometry/coll.geometry.html) 15 April 2008 at the [Wayback Machine](/source/Wayback_Machine) - [*The Math Forum* – Advanced Geometry](http://mathforum.org/advanced/geom.html) [Archived](https://web.archive.org/web/20080416182158/http://mathforum.org/advanced/geom.html) 16 April 2008 at the [Wayback Machine](/source/Wayback_Machine)

- [Nature Precedings – *Pegs and Ropes Geometry at Stonehenge*](http://precedings.nature.com/documents/2153/version/1/)

- [*The Mathematical Atlas* – Geometric Areas of Mathematics](https://web.archive.org/web/20060906203141/http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html)

- ["4000 Years of Geometry"](https://web.archive.org/web/20071004174210/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618), lecture by Robin Wilson given at [Gresham College](/source/Gresham_College), 3 October 2007 (available for MP3 and MP4 download as well as a text file) - [Finitism in Geometry](http://plato.stanford.edu/entries/geometry-finitism/) at the Stanford Encyclopedia of Philosophy

- [The Geometry Junkyard](http://www.ics.uci.edu/~eppstein/junkyard/topic.html)

- [Interactive geometry reference with hundreds of applets](http://www.mathopenref.com)

- [Dynamic Geometry Sketches (with some Student Explorations)](https://web.archive.org/web/20090321024112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm)

- [Geometry classes](http://www.khanacademy.org/?video=ca-geometry--area--pythagorean-theorem#california-standards-test-geometry) at [Khan Academy](/source/Khan_Academy)

v t e Geometry History Timeline Lists of geometry topics Foundations of geometry Outline of geometry Euclidean geometry Convex Discrete Plane geometry Solid Affine Trigonometry Spherical Generalized Fundamental Concepts (Euclidean) Point Line Plane Space Distance Angle Parallel Perpendicular Triangle Circle Polygon Congruence Similarity Euclid's Elements Euclidean space Coordinate system Dimension 2D 3D Non-Euclidean geometry Non-Euclidean space Elliptic Hyperbolic Spherical Other kinds of geometry Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Absolute Analytic Complex Computational Conformal Constructive Discrete Finite Information Non-Archimedean Noncommutative Ordered Projective Spectral Tropical Metric Lists Lists of geometry topics Combinatorial computational geometry topics Differential geometry topics Formulas in elementary geometry Formulas in Riemannian geometry Geometry topics Knot theory topics Numerical computational geometry topics Polygons Shapes Outline of geometry Glossaries Algebraic geometry Arithmetic and diophantine geometry Classical algebraic geometry Differential geometry and topology Riemannian and metric geometry Symplectic geometry Related Geometric algebra Geometric measure theory Geometric analysis Geometric group theory Manifold Vector calculus Geometry of numbers Lattice theory Elliptic curve Foundations of geometry Axiomatic system Categories: Geometry Trigonometry Topology Category:History of geometry

v t e Major mathematics areas History Timeline Future Lists Glossary Foundations Category theory Information theory Mathematical logic Philosophy of mathematics Set theory Type theory Algebra Abstract Boolean Clifford Commutative Elementary Field theory Group theory Homological Lie Linear Multilinear Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis Measure theory Discrete Combinatorics Discrete geometry Graph theory Matroid theory Order theory Geometry Algebraic Affine Analytic Arithmetic Complex Computational Convex Differential Discrete Euclidean Finite Information Projective Number theory Algebraic Analytic Arithmetic Diophantine geometry Topology General Algebraic Differential Geometric Homotopy theory Knot theory Applied Engineering mathematics Mathematical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical geology Mathematical linguistics Mathematical physics Mathematical psychology Mathematical sociology Mathematical statistics Probability Statistics Systems science Control theory Game theory Operations research Computational Computer science Theory of computation Computational complexity theory Numerical analysis Optimization Computer algebra Related topics Mathematicians lists Informal mathematics Films about mathematicians Recreational mathematics Mathematics and art Mathematics education Number Large Mathematics portal Category Commons WikiProject

Authority control databases International GND National United States France BnF data Japan Czech Republic Korea Israel Other Encyclopedia of Modern Ukraine İslâm Ansiklopedisi Yale LUX

---
Adapted from the Wikipedia article [Geometry](https://en.wikipedia.org/wiki/Geometry) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Geometry?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.