109-433: In geometry , a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere . In more modern usage, the length d {\displaystyle d} of a diameter is also called the diameter. In this sense one speaks of
218-536: A code point in Unicode at U+2300 ⌀ DIAMETER SIGN , in the Miscellaneous Technical set. It should not be confused with several other characters (such as U+00D8 Ø LATIN CAPITAL LETTER O WITH STROKE or U+2205 ∅ EMPTY SET ) that resemble it but have unrelated meanings. It has the compose sequence Compose d i . The diameter of
327-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; 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 without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as
436-673: A hypercube or a set of scattered points. The diameter or metric diameter of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if S {\displaystyle S} is the subset and if ρ {\displaystyle \rho } is the metric , the diameter is diam ( S ) = sup x , y ∈ S ρ ( x , y ) . {\displaystyle \operatorname {diam} (S)=\sup _{x,y\in S}\rho (x,y).} If
545-456: A metaphorical , rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance , which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have
654-451: A metric space is an ordered pair ( M , d ) where M is a set and d is a metric on M , i.e., a function d : M × M → R {\displaystyle d\,\colon M\times M\to \mathbb {R} } satisfying the following axioms for all points x , y , z ∈ M {\displaystyle x,y,z\in M} : If
763-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to
872-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes
981-751: A "structure-preserving" map is one that fully preserves the distance function: It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry . One perhaps non-obvious example of an isometry between spaces described in this article is the map f : ( R 2 , d 1 ) → ( R 2 , d ∞ ) {\displaystyle f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })} defined by f ( x , y ) = ( x + y , x − y ) . {\displaystyle f(x,y)=(x+y,x-y).} If there
1090-399: A broader and more flexible way. This was important for the growing field of functional analysis. Mathematicians like Hausdorff and Stefan Banach further refined and expanded the framework of metric spaces. Hausdorff introduced topological spaces as a generalization of metric spaces. Banach's work in functional analysis heavily relied on the metric structure. Over time, metric spaces became
1199-399: A central part of modern mathematics . They have influenced various fields including topology , geometry , and applied mathematics . Metric spaces continue to play a crucial role in the study of abstract mathematical concepts. A distance function is enough to define notions of closeness and convergence that were first developed in real analysis . Properties that depend on the structure of
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#17327877717701308-493: A characterization of metrizability in terms of other topological properties, without reference to metrics. Convergence of sequences in Euclidean space is defined as follows: Convergence of sequences in a topological space is defined as follows: In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for topological properties of metric spaces: while they can be defined in
1417-489: A circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric . Jung's theorem provides more general inequalities relating the diameter to the radius. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría ) 'land measurement'; from γῆ ( gê ) 'earth, land' and μέτρον ( métron ) 'a measure')
1526-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,
1635-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In
1744-400: A metric space are referred to as metric properties . Every metric space is also a topological space , and some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really topological properties . For any point x in a metric space M and any real number r > 0 , the open ball of radius r around x is defined to be
1853-434: A metric space by measuring distances the same way we would in M . Formally, the induced metric on A is a function d A : A × A → R {\displaystyle d_{A}:A\times A\to \mathbb {R} } defined by d A ( x , y ) = d ( x , y ) . {\displaystyle d_{A}(x,y)=d(x,y).} For example, if we take
1962-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,
2071-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry
2180-404: A natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds , normed vector spaces , and graphs . In abstract algebra , the p -adic numbers arise as elements of the completion of a metric structure on the rational numbers . Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces . Many of
2289-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in
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#17327877717702398-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received
2507-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to
2616-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies
2725-400: A purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis. Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges. To make this precise: a sequence ( x n ) in a metric space M is Cauchy if for every ε > 0 there
2834-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. 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. Hilbert , in his work on creating
2943-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c. 1900 , with
3052-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in
3161-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of
3270-412: A way of measuring distances between them. Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways. For example, weak solutions to differential equations typically live in a completion (a Sobolev space ) rather than the original space of nice functions for which
3379-487: Is uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for all points x and y in M 1 such that d ( x , y ) < δ {\displaystyle d(x,y)<\delta } , we have d 2 ( f ( x ) , f ( y ) ) < ε . {\displaystyle d_{2}(f(x),f(y))<\varepsilon .} The only difference between this definition and
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3488-549: Is K - Lipschitz if d 2 ( f ( x ) , f ( y ) ) ≤ K d 1 ( x , y ) for all x , y ∈ M 1 . {\displaystyle d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\text{for all}}\quad x,y\in M_{1}.} Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of
3597-775: Is Lebesgue's number lemma , which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover. Unlike in the case of topological spaces or algebraic structures such as groups or rings , there is no single "right" type of structure-preserving function between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} and ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} are two metric spaces. The words "function" and "map" are used interchangeably. One interpretation of
3706-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing
3815-410: Is not a topological property, since R {\displaystyle \mathbb {R} } is complete but the homeomorphic space (0, 1) is not. This notion of "missing points" can be made precise. In fact, every metric space has a unique completion , which is a complete space that contains the given space as a dense subset. For example, [0, 1] is the completion of (0, 1) , and
3924-415: Is a neighborhood of x (informally, it contains all points "close enough" to x ) if it contains an open ball of radius r around x for some r > 0 . An open set is a set which is a neighborhood of all its points. It follows that the open balls form a base for a topology on M . In other words, the open sets of M are exactly the unions of open balls. As in any topology, closed sets are
4033-540: Is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model
4142-559: Is a continuous bijection whose inverse is also continuous; if there is a homeomorphism between M 1 and M 2 , they are said to be homeomorphic . Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties. For example, R {\displaystyle \mathbb {R} } is unbounded and complete, while (0, 1) is bounded but not complete. A function f : M 1 → M 2 {\displaystyle f\,\colon M_{1}\to M_{2}}
4251-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in
4360-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood
4469-482: Is an integer N such that for all m , n > N , d ( x m , x n ) < ε . By the triangle inequality, any convergent sequence is Cauchy: if x m and x n are both less than ε away from the limit, then they are less than 2ε away from each other. If the converse is true—every Cauchy sequence in M converges—then M is complete. Euclidean spaces are complete, as is R 2 {\displaystyle \mathbb {R} ^{2}} with
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4578-435: Is an isometry between the spaces M 1 and M 2 , they are said to be isometric . Metric spaces that are isometric are essentially identical . On the other end of the spectrum, one can forget entirely about the metric structure and study continuous maps , which only preserve topological structure. There are several equivalent definitions of continuity for metric spaces. The most important are: A homeomorphism
4687-496: Is bounded. To see this, start with a finite cover by r -balls for some arbitrary r . Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D . By the triangle inequality, the diameter of the whole space is at most D + 2 r . The converse does not hold: an example of a metric space that is bounded but not totally bounded is R 2 {\displaystyle \mathbb {R} ^{2}} (or any other infinite set) with
4796-418: Is defined by d 1 ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = | x 2 − x 1 | + | y 2 − y 1 | {\displaystyle d_{1}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{2}-x_{1}|+|y_{2}-y_{1}|} and can be thought of as
4905-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are
5014-442: Is measured by a function called a metric or distance function . Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry . The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane . A metric may correspond to
5123-437: 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. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given
5232-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study
5341-477: Is often abbreviated DIA , dia , d , {\displaystyle {\text{DIA}},{\text{dia}},d,} or ∅ . {\displaystyle \varnothing .} The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of n {\displaystyle n} -dimensional (convex or non-convex) object, such as
5450-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,
5559-446: Is the same as the diameter of its convex hull . In medical terminology concerning a lesion or in geology concerning a rock, the diameter of an object is the least upper bound of the set of all distances between pairs of points in the object. In differential geometry , the diameter is an important global Riemannian invariant . In planar geometry , a diameter of a conic section is typically defined as any chord which passes through
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#17327877717705668-484: The width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers . For a curve of constant width such as the Reuleaux triangle , the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse , the standard terminology is different. A diameter of an ellipse is any chord passing through
5777-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs
5886-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c. 1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated
5995-538: The Heine–Cantor theorem states that if M 1 is compact, then every continuous map is uniformly continuous. In other words, uniform continuity cannot distinguish any non-topological features of compact metric spaces. A Lipschitz map is one that stretches distances by at most a bounded factor. Formally, given a real number K > 0 , the map f : M 1 → M 2 {\displaystyle f\,\colon M_{1}\to M_{2}}
6104-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c. 1230 – c. 1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first
6213-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with
6322-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of
6431-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,
6540-476: The conic's centre ; such diameters are not necessarily of uniform length, except in the case of the circle, which has eccentricity e = 0. {\displaystyle e=0.} The symbol or variable for diameter, ⌀ , is sometimes used in technical drawings or specifications as a prefix or suffix for a number (e.g. "⌀ 55 mm"), indicating that it represents diameter. Photographic filter thread sizes are often denoted in this way. The symbol has
6649-409: The diameter rather than a diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r . {\displaystyle r.} For a convex shape in the plane , the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and
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#17327877717706758-468: The surface of the Earth as a set of points. We can measure the distance between two such points by the length of the shortest path along the surface , " as the crow flies "; this is particularly useful for shipping and aviation. We can also measure the straight-line distance between two points through the Earth's interior; this notion is, for example, natural in seismology , since it roughly corresponds to
6867-574: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing
6976-496: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into
7085-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by
7194-486: The Euclidean metric and its subspace the interval (0, 1) with the induced metric are homeomorphic but have very different metric properties. Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called metrizable and are particularly well-behaved in many ways: in particular, they are paracompact Hausdorff spaces (hence normal ) and first-countable . The Nagata–Smirnov metrization theorem gives
7303-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in
7412-436: The basic notions of mathematical analysis , including balls , completeness , as well as uniform , Lipschitz , and Hölder continuity , can be defined in the setting of metric spaces. Other notions, such as continuity , compactness , and open and closed sets , can be defined for metric spaces, but also in the even more general setting of topological spaces . To see the utility of different notions of distance, consider
7521-428: The centre of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter is called the major axis . The word "diameter" is derived from Ancient Greek : διάμετρος ( diametros ), "diameter of a circle", from διά ( dia ), "across, through" and μέτρον ( metron ), "measure". It
7630-495: The complements of open sets. Sets may be both open and closed as well as neither open nor closed. This topology does not carry all the information about the metric space. For example, the distances d 1 , d 2 , and d ∞ defined above all induce the same topology on R 2 {\displaystyle \mathbb {R} ^{2}} , although they behave differently in many respects. Similarly, R {\displaystyle \mathbb {R} } with
7739-412: The concept of angle and distance, finite geometry that omits continuity , 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 of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry
7848-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c. 287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of
7957-407: The differential equation actually makes sense. A metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart. The least such r is called the diameter of M . The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r . Every totally bounded space
8066-479: The discrete metric no longer remembers that the set is a plane, but treats it just as an undifferentiated set of points. All of these metrics make sense on R n {\displaystyle \mathbb {R} ^{n}} as well as R 2 {\displaystyle \mathbb {R} ^{2}} . Given a metric space ( M , d ) and a subset A ⊆ M {\displaystyle A\subseteq M} , we can consider A to be
8175-414: The discrete metric. Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space. There are several equivalent definitions of compactness in metric spaces: One example of a compact space is the closed interval [0, 1] . Compactness is important for similar reasons to completeness: it makes it easy to find limits. Another important tool
8284-1066: The distance function d ( x , y ) = | y − x | {\displaystyle d(x,y)=|y-x|} given by the absolute difference form a metric space. Many properties of metric spaces and functions between them are generalizations of concepts in real analysis and coincide with those concepts when applied to the real line. The Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} can be equipped with many different metrics. The Euclidean distance familiar from school mathematics can be defined by d 2 ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . {\displaystyle d_{2}((x_{1},y_{1}),(x_{2},y_{2}))={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.} The taxicab or Manhattan distance
8393-774: The distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article. The maximum , L ∞ {\displaystyle L^{\infty }} , or Chebyshev distance is defined by d ∞ ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = max { | x 2 − x 1 | , | y 2 − y 1 | } . {\displaystyle d_{\infty }((x_{1},y_{1}),(x_{2},y_{2}))=\max\{|x_{2}-x_{1}|,|y_{2}-y_{1}|\}.} This distance does not have an easy explanation in terms of paths in
8502-413: The empty set as a special case, assigning it a diameter of 0 , {\displaystyle 0,} which corresponds to taking the codomain of ρ {\displaystyle \rho } to be the set of nonnegative reals. For any solid object or set of scattered points in n {\displaystyle n} -dimensional Euclidean space , the diameter of the object or set
8611-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits
8720-515: The field of non-euclidean geometry through the use of the Cayley-Klein metric . The idea of an abstract space with metric properties was addressed in 1906 by René Maurice Fréchet and the term metric space was coined by Felix Hausdorff in 1914. Fréchet's work laid the foundation for understanding convergence , continuity , and other key concepts in non-geometric spaces. This allowed mathematicians to study functions and sequences in
8829-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c. 355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as
8938-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between
9047-1168: The formula d ∞ ( p , q ) ≤ d 2 ( p , q ) ≤ d 1 ( p , q ) ≤ 2 d ∞ ( p , q ) , {\displaystyle d_{\infty }(p,q)\leq d_{2}(p,q)\leq d_{1}(p,q)\leq 2d_{\infty }(p,q),} which holds for every pair of points p , q ∈ R 2 {\displaystyle p,q\in \mathbb {R} ^{2}} . A radically different distance can be defined by setting d ( p , q ) = { 0 , if p = q , 1 , otherwise. {\displaystyle d(p,q)={\begin{cases}0,&{\text{if }}p=q,\\1,&{\text{otherwise.}}\end{cases}}} Using Iverson brackets , d ( p , q ) = [ p ≠ q ] {\displaystyle d(p,q)=[p\neq q]} In this discrete metric , all distinct points are 1 unit apart: none of them are close to each other, and none of them are very far away from each other either. Intuitively,
9156-542: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning
9265-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including
9374-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived
9483-481: The length of time it takes for seismic waves to travel between those two points. The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts. Like many fundamental mathematical concepts,
9592-469: The metric ρ {\displaystyle \rho } is viewed here as having codomain R {\displaystyle \mathbb {R} } (the set of all real numbers ), this implies that the diameter of the empty set (the case S = ∅ {\displaystyle S=\varnothing } ) equals − ∞ {\displaystyle -\infty } ( negative infinity ). Some authors prefer to treat
9701-430: The metric d is unambiguous, one often refers by abuse of notation to "the metric space M ". By taking all axioms except the second, one can show that distance is always non-negative: 0 = d ( x , x ) ≤ d ( x , y ) + d ( y , x ) = 2 d ( x , y ) {\displaystyle 0=d(x,x)\leq d(x,y)+d(y,x)=2d(x,y)} Therefore
9810-531: The metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with Wasserstein metrics on spaces of measures ) or the degree of difference between two objects (for example, the Hamming distance between two strings of characters, or the Gromov–Hausdorff distance between metric spaces themselves). Formally,
9919-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of
10028-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,
10137-473: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Metric (mathematics) In mathematics , a metric space is a set together with a notion of distance between its elements , usually called points . The distance
10246-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of
10355-543: The other metrics described above. Two examples of spaces which are not complete are (0, 1) and the rationals, each with the metric induced from R {\displaystyle \mathbb {R} } . One can think of (0, 1) as "missing" its endpoints 0 and 1. The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in R {\displaystyle \mathbb {R} } (for example, its successive decimal approximations). These examples show that completeness
10464-514: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During
10573-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of
10682-428: The plane, but it still satisfies the metric space axioms. It can be thought of similarly to the number of moves a king would have to make on a chess board to travel from one point to another on the given space. In fact, these three distances, while they have distinct properties, are similar in some ways. Informally, points that are close in one are close in the others, too. This observation can be quantified with
10791-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself 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
10900-472: The real line. Arthur Cayley , in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective space. His distance was given by logarithm of a cross ratio . Any projectivity leaving the conic stable also leaves the cross ratio constant, so isometries are implicit. This method provides models for elliptic geometry and hyperbolic geometry , and Felix Klein , in several publications, established
11009-407: The real numbers are the completion of the rationals. Since complete spaces are generally easier to work with, completions are important throughout mathematics. For example, in abstract algebra, the p -adic numbers are defined as the completion of the rationals under a different metric. Completion is particularly common as a tool in functional analysis . Often one has a set of nice functions and
11118-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid
11227-415: The second axiom can be weakened to If x ≠ y , then d ( x , y ) ≠ 0 {\textstyle {\text{If }}x\neq y{\text{, then }}d(x,y)\neq 0} and combined with the first to make d ( x , y ) = 0 ⟺ x = y {\textstyle d(x,y)=0\iff x=y} . The real numbers with
11336-482: The set of points that are strictly less than distance r from x : B r ( x ) = { y ∈ M : d ( x , y ) < r } . {\displaystyle B_{r}(x)=\{y\in M:d(x,y)<r\}.} This is a natural way to define a set of points that are relatively close to x . Therefore, a set N ⊆ M {\displaystyle N\subseteq M}
11445-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry
11554-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and
11663-410: The two-dimensional sphere S as a subset of R 3 {\displaystyle \mathbb {R} ^{3}} , the Euclidean metric on R 3 {\displaystyle \mathbb {R} ^{3}} induces the straight-line metric on S described above. Two more useful examples are the open interval (0, 1) and the closed interval [0, 1] thought of as subspaces of
11772-413: The ε–δ definition of continuity is the order of quantifiers: the choice of δ must depend only on ε and not on the point x . However, this subtle change makes a big difference. For example, uniformly continuous maps take Cauchy sequences in M 1 to Cauchy sequences in M 2 . In other words, uniform continuity preserves some metric properties which are not purely topological. On the other hand,
11881-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in
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