DimensionFrom Wikipedia, the free encyclopediaContents1 Dimension 11.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Dimension of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Krull dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.5 Lebesgue covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.6 Inductive dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.7 Hausdor dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.8 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Spatial dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Additional dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Networks and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 In literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 In philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 More dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7.1 Topics by dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Extended real number line 92.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Measure and integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Order and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Interval (mathematics) 123.1 Notations for intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.1 Including or excluding endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Innite endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.3 Integer intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Classication of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Intervals of the extended real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Properties of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Dyadic intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6.1 Multi-dimensional intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6.2 Complex intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Line (geometry) 174.1 Denitions versus descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.1 Cartesian plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.3 Vector equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.4 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.5 Types of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Point (geometry) 265.1 Points in Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Dimension of a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2.1 Vector space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.2 Topological dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.3 Hausdor dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28CONTENTS iii5.3 Geometry without points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 Point masses and the Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 305.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Chapter 1DimensionThis article is about dimensions of space. For the dimension of a quantity, see Dimensional analysis. For other uses,see Dimension (disambiguation).In physics and mathematics, the dimension of a mathematical space (or object) is informally dened as the minimumFrom left to right: the square, the cube and the tesseract. The two-dimensional (2d) square is bounded by one-dimensional (1d) lines;the three-dimensional (3d) cube by two-dimensional areas; and the four-dimensional (4d) tesseract by three-dimensional volumes.For display on a two-dimensional surface such as a screen, the 3d cube and 4d tesseract require projection.The rst four spatial dimensions.number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because onlyone coordinate is needed to specify a point on it for example, the point at 5 on a number line. A surface such as aplane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify apoint on it for example, both a latitude and longitude are required to locate a point on the surface of a sphere. Theinside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a pointwithin these spaces.12 CHAPTER 1. DIMENSIONIn classical mechanics, space and time are dierent categories and refer to absolute space and time. That conceptionof the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. Thefour dimensions of spacetime consist of events that are not absolutely dened spatially and temporally, but rather areknown relative to the motion of an observer.Minkowski space rst approximates the universe without gravity; thepseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions areused to describe string theory, and the state-space of quantum mechanics is an innite-dimensional function space.The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathe-matics and the sciences. They may be parameter spaces or conguration spaces such as in Lagrangian or Hamiltonianmechanics; these are abstract spaces, independent of the physical space we live in.1.1 In mathematicsIn mathematics, the dimension of an object is an intrinsic property independent of the space in which the object isembedded. For example, a point on the unit circle in the plane can be specied by two Cartesian coordinates, buta single polar coordinate (the angle) would be sucient, so the circle is 1-dimensional even though it exists in the2-dimensional plane. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimensiondiers from its common usages.The dimension of Euclidean n-space Enis n. When trying to generalize to other types of spaces, one is faced withthe question what makes Enn-dimensional?" One answer is that to cover a xed ball in Enby small balls of radius ,one needs on the order of nsuch small balls. This observation leads to the denition of the Minkowski dimensionand its more sophisticated variant, the Hausdor dimension, but there are also other answers to that question. Forexample, the boundary of a ball in Enlooks locally like En1 and this leads to the notion of the inductive dimension.While these notions agree on En, they turn out to be dierent when one looks at more general spaces.Atesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the termdimensionis as in: A tesseract has four dimensions", mathematicians usually express this as: The tesseract has dimension 4",or: The dimension of the tesseract is 4.Although the notion of higher dimensions goes back to Ren Descartes, substantial development of a higher-dimensionalgeometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schliand Bernhard Riemann. Riemanns 1854 Habilitationsschrift, Schlis 1852 Theorie der vielfachen Kontinuitt,Hamiltons 1843 discovery of the quaternions and the construction of the Cayley algebra marked the beginning ofhigher-dimensional geometry.The rest of this section examines some of the more important mathematical denitions of the dimensions.1.1.1 Dimension of a vector spaceMain article: Dimension (vector space)The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinatesnecessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hameldimension or algebraic dimension to distinguish it from other notions of dimension.1.1.2 ManifoldsA connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called themanifolds dimension. One can show that this yields a uniquely dened dimension for every connected topologicalmanifold.For connected dierentiable manifolds, the dimension is also the dimension of the tangent vector space at any point.In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively ele-mentary, the high-dimensional cases n > 4 are simplied by having extra space in which to work"; and the casesn = 3 and 4 are in some senses the most dicult. This state of aairs was highly marked in the various cases of thePoincar conjecture, where four dierent proof methods are applied.1.1. IN MATHEMATICS 31.1.3 VarietiesMain article: Dimension of an algebraic varietyThe dimension of an algebraic variety may be dened in various equivalent ways. The most intuitive way is probablythe dimension of the tangent space at any regular point. Another intuitive way is to dene the dimension as the numberof hyperplanes that are needed in order to have an intersection with the variety that is reduced to a nite number ofpoints (dimension zero). This denition is based on the fact that the intersection of a variety with a hyperplane reducesthe dimension by one unless if the hyperplane contains the variety.An algebraic set being a nite union of algebraic varieties, its dimension is the maximum of the dimensions of itscomponents. It is equal to the maximal length of the chainsV0V1. . . Vd of sub-varieties of the givenalgebraic set (the length of such a chain is the number of " ").Each variety can be considered as an algebraic stack, and its dimension as variety agrees with its dimension as stack.There are however many stacks which do not correspond to varieties, and some of these have negative dimension.Specically, if V is a variety of dimension mand G is an algebraic group of dimension n acting on V, then the quotientstack [V/G] has dimension mn.[3]1.1.4 Krull dimensionMain article: Krull dimensionThe Krull dimension of a commutative ring is the maximal length of chains of prime ideals in it, a chain of length nbeing a sequence P0 P1 . . . Pn of prime ideals related by inclusion. It is strongly related to the dimensionof an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring ofthe polynomials on the variety.For an algebra over a eld, the dimension as vector space is nite if and only if its Krull dimension is 0.1.1.5 Lebesgue covering dimensionMain article: Lebesgue covering dimensionFor any normal topological space X, the Lebesgue covering dimension of X is dened to be n if n is the smallest integerfor which the following holds: any open cover has an open renement (a second open cover where each element is asubset of an element in the rst cover) such that no point is included in more than n + 1 elements. In this case dimX = n.For X a manifold, this coincides with the dimension mentioned above.If no such integer n exists, then thedimension of X is said to be innite, and one writes dim X = . Moreover, X has dimension 1, i.e. dim X = 1 ifand only if X is empty. This denition of covering dimension can be extended from the class of normal spaces to allTychono spaces merely by replacing the term open in the denition by the term "functionally open".1.1.6 Inductive dimensionMain article: Inductive dimensionAn inductive denition of dimension can be created as follows. Consider a discrete set of points (such as a nitecollection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. Ingeneral one obtains an (n + 1)-dimensional object by dragging an n-dimensional object in a new direction.The inductive dimension of a topological space may refer to the small inductive dimension or the large inductivedimension, and is based on the analogy that (n + 1)-dimensional balls have n-dimensional boundaries, permitting aninductive denition based on the dimension of the boundaries of open sets.4 CHAPTER 1. DIMENSION1.1.7 Hausdor dimensionMain article: Hausdor dimensionFor structurally complicated sets, especially fractals, the Hausdor dimension is useful. The Hausdor dimension isdened for all metric spaces and, unlike the dimensions considered above, can also attain non-integer real values.[4]The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more denitions offractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have beenfound useful to describe many natural objects and phenomena.[5][6]1.1.8 Hilbert spacesEvery Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the samecardinality. This cardinality is called the dimension of the Hilbert space. This dimension is nite if and only ifthe spaces Hamel dimension is nite, and in this case the above dimensions coincide.1.2 In physics1.2.1 Spatial dimensionsClassical physics theories describe three physical dimensions: from a particular point in space, the basic directions inwhich we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressedin terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upwardand forward is just as the name of the direction implies;i.e., moving in a linear combination of up and forward.In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes threedimensions. (See Space and Cartesian coordinate system.)1.2.2 TimeA temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension" for this reason,but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change.It is perceived dierently from the three spatial dimensions in that there is only one of it, and that we cannot movefreely in time but subjectively move in one direction.The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it.The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics aretypically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, theperception of time owing in one direction is an artifact of the laws of thermodynamics (we perceive time as owingin the direction of increasing entropy).The best-known treatment of time as a dimension is Poincar and Einstein's special relativity (and extended to generalrelativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime,and in the special, at case as Minkowski space.1.2.3 Additional dimensionsIn physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attemptto unify the four fundamental forces by introducing more dimensions. Most notably, superstring theory requires 10spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called M-theorywhich subsumes ve previously distinct superstring theories.To date, no experimental or observational evidence isavailable to conrmthe existence of these extra dimensions. If extra dimensions exist, they must be hidden fromus bysome physical mechanism. One well-studied possibility is that the extra dimensions may be curled up at such tinyscales as to be eectively invisible to current experiments. Limits on the size and other properties of extra dimensionsare set by particle experiments such as those at the Large Hadron Collider.[7]1.3. NETWORKS AND DIMENSION 5At the level of quantumeld theory, KaluzaKlein theory unies gravity with gauge interactions, based on the realiza-tion that gravity propagating in small, compact extra dimensions is equivalent to gauge interactions at long distances.In particular when the geometry of the extra dimensions is trivial, it reproduces electromagnetism. However at su-ciently high energies or short distances, this setup still suers from the same pathologies that famously obstruct directattempts to describe quantum gravity. Therefore, these models still require a UV completion, of the kind that stringtheory is intended to provide. Thus Kaluza-Klein theory may be considered either as an incomplete description onits own, or as a subset of string theory model building.In addition to small and curled up extra dimensions, there may be extra dimensions that instead aren't apparentbecause the matter associated with our visible universe is localized on a (3 + 1)-dimensional subspace. Thus the extradimensions need not be small and compact but may be large extra dimensions. D-branes are dynamical extendedobjects of various dimensionalities predicted by string theory that could play this role. They have the property thatopen string excitations, which are associated with gauge interactions, are conned to the brane by their endpoints,whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, orthe bulk. This could be related to why gravity is exponentially weaker than the other forces, as it eectively dilutesitself as it propagates into a higher-dimensional volume.Some aspects of brane physics have been applied to cosmology. For example, brane gas cosmology[8][9] attempts toexplain why there are three dimensions of space using topological and thermodynamic considerations. According tothis idea it would be because three is the largest number of spatial dimensions where strings can generically intersect.If initially there are lots of windings of strings around compact dimensions, space could only expand to macroscopicsizes once these windings are eliminated, which requires oppositely wound strings to nd each other and annihilate.But strings can only nd each other to annihilate at a meaningful rate in three dimensions, so it follows that only threedimensions of space are allowed to grow large given this kind of initial conguration.Extra dimensions are said to be universal if all elds are equally free to propagate within them.1.3 Networks and dimensionSome complex networks are characterized by fractal dimensions.[10] The concept of dimension can be generalized toinclude networks embedded in space.[11] The dimension characterize their spatial constraints.1.4 In literatureMain article: Fourth dimension in literatureScience ction texts often mention the concept of dimension when referring to parallel or alternate universes or otherimagined planes of existence. This usage is derived fromthe idea that to travel to parallel/alternate universes/planes ofexistence one must travel in a direction/dimension besides the standard ones. In eect, the other universes/planes arejust a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension,not the standard ones.One of the most heralded science ction stories regarding true geometric dimensionality, and often recommended asa starting point for those just starting to investigate such matters, is the 1884 novella Flatland by Edwin A. Abbott.Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as The best introduction onecan nd into the manner of perceiving dimensions.The idea of other dimensions was incorporated into many early science ction stories, appearing prominently, forexample, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-DimensionCatapult (1931); and appeared irregularly in science ction by the 1940s. Classic stories involving other dimensionsinclude Robert A. Heinlein's And He Built a Crooked House (1941), in which a California architect designs a housebased on a three-dimensional projection of a tesseract; and Alan E. Nourse's Tiger by the Tail and The UniverseBetween (both 1951). Another reference is Madeleine L'Engle's novel A Wrinkle In Time (1962), which uses the fthdimension as a way for tesseracting the universe or folding space in order to move across it quickly. The fourthand fth dimensions were also a key component of the book The Boy Who Reversed Himself by William Sleator.6 CHAPTER 1. DIMENSION1.5 In philosophyImmanuel Kant, in 1783, wrote: That everywhere space (which is not itself the boundary of another space) has threedimensions and that space in general cannot have more dimensions is based on the proposition that not more thanthree lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but restsimmediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain.[12]Space has Four Dimensions is a short story published in 1846 by German philosopher and experimental psychologistGustav Fechner under the pseudonym Dr. Mises. The protagonist in the tale is a shadow who is aware of and ableto communicate with other shadows, but who is trapped on a two-dimensional surface.According to Fechner, thisshadow-man would conceive of the third dimension as being one of time.[13] The story bears a strong similarity tothe "Allegory of the Cave" presented in Plato's The Republic (c. 380 BC).Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled The Phi-losophy of Hyperspace.[14] Linda Dalrymple Henderson coined the term hyperspace philosophy, used to describewriting that uses higher dimensions to explore metaphysical themes, in her 1983 thesis about the fourth dimensionin early-twentieth-century art.[15] Examples of hyperspace philosophers include Charles Howard Hinton, the rstwriter, in 1888, to use the word tesseract";[16] and the Russian esotericist P. D. Ouspensky.1.6 More dimensionsDegrees of freedom in mechanics / physics and chemistry / statistics1.7 See also1.7.1 Topics by dimensionZeroPointZero-dimensional spaceIntegerOneLineGraph (combinatorics)Real numberTwoComplex numberCartesian coordinate systemList of uniform tilingsSurfaceThreePlatonic solidStereoscopy (3-D imaging)1.8. REFERENCES 7Euler angles3-manifoldKnotsFourSpacetimeFourth spatial dimensionConvex regular 4-polytopeQuaternion4-manifoldFourth dimension in artFourth dimension in literatureHigher dimensionsin mathematicsOctonionVector spaceManifoldCalabiYau spacesCurse of dimensionalityin physicsKaluzaKlein theoryString theoryM-theoryInniteHilbert spaceFunction space1.8 References[1] Curious About Astronomy. Curious.astro.cornell.edu. Retrieved 2014-03-03.[2] MathWorld: Dimension. Mathworld.wolfram.com. 2014-02-27. Retrieved 2014-03-03.[3] Fantechi, Barbara (2001), Stacks for everybody (PDF), European Congress of Mathematics Volume I, Progr. Math. 201,Birkhuser, pp. 349359[4] Fractal Dimension, Boston University Department of Mathematics and Statistics[5] Bunde, Armin; Havlin, Shlomo, eds. (1991). Fractals and Disordered Systems. Springer.[6] Bunde, Armin; Havlin, Shlomo, eds. (1994). Fractals in Science. Springer.[7] CMS Collaboration, Search for Microscopic Black Hole Signatures at the Large Hadron Collider (arxiv.org)[8] Brandenberger, R., Vafa, C., Superstrings in the early universe8 CHAPTER 1. DIMENSION[9] Scott Watson, Brane Gas Cosmology (pdf).[10] Song, Chaoming; Havlin, Shlomo; Makse, Hernn A. (2005). Self-similarity of complex networks. Nature 433 (7024).arXiv:cond-mat/0503078v1. Bibcode:2005Natur.433..392S. doi:10.1038/nature03248.[11] Daqing, Li; Kosmidis, Kosmas; Bunde, Armin; Havlin, Shlomo (2011). Dimension of spatially embedded networks.Nature Physics 7 (6). Bibcode:2011NatPh...7..481D. doi:10.1038/nphys1932.[12] Prolegomena, 12[13] Bancho, Thomas F. (1990). From Flatland to Hypergraphics: Interacting with Higher Dimensions. InterdisciplinaryScience Reviews 15 (4): 364. doi:10.1179/030801890789797239.[14] Newcomb, Simon (1898). The Philosophy of Hyperspace. Bulletin of the American Mathematical Society 4 (5):187.doi:10.1090/S0002-9904-1898-00478-0.[15] Kruger, Runette (2007). Art in the Fourth Dimension: Giving Form to Form The Abstract Paintings of Piet Mondrian(PDF). Spaces of Utopia: an Electronic Journal (5): 11.[16] Pickover, Cliord A. (2009), Tesseract, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in theHistory of Mathematics, Sterling Publishing Company, Inc., p. 282, ISBN 9781402757969.1.9 Further readingKatta G Murty, Systems of Simultaneous Linear Equations (Chapter 1 of Computational and AlgorithmicLinear Algebra and n-Dimensional Geometry, World Scientic Publishing: 2014 (ISBN 978-981-4366-62-5).Edwin A. Abbott, Flatland: A Romance of Many Dimensions (1884) (Public domain: Online version withASCII approximation of illustrations at Project Gutenberg).Thomas Bancho, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Sec-ond Edition, W. H. Freeman and Company: 1996.Cliord A. Pickover, Surng through Hyperspace: Understanding Higher Universes in Six Easy Lessons, OxfordUniversity Press: 1999.Rudy Rucker, The Fourth Dimension, Houghton-Miin: 1984.Michio Kaku, Hyperspace, a Scientic Odyssey Through the 10th Dimension, Oxford University Press: 1994.1.10 External linksCopeland, Ed (2009). Extra Dimensions. Sixty Symbols. Brady Haran for the University of Nottingham.Chapter 2Extended real number linePositive innity redirects here. For the band, see Positive Innity.In mathematics, the anely extended real number system is obtained from the real number system R by addingtwo elements: + and (read as positive innity and negative innity respectively). These new elements are notreal numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially inthe theory of measure and integration. The anely extended real number system is denoted R or [, +] or R U{, +}.When the meaning is clear from context, the symbol + is often written simply as .2.1 Motivation2.1.1 LimitsWe often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) getsvery big in some sense. For example, consider the functionf(x) = x2.The graph of this function has a horizontal asymptote at y = 0. Geometrically, as we move farther and farther to theright along the x-axis, the value of 1/x2approaches 0. This limiting behavior is similar to the limit of a function at areal number, except that there is no real number to which x approaches.By adjoining the elements + and to R, we allow a formulation of a limit at innity with topological propertiessimilar to those for R.To make things completely formal, the Cauchy sequences denition of R allows us to dene + as the set of allsequences of rationals which, for any K>0, from some point on exceed K. We can dene similarly.2.1.2 Measure and integrationIn measure theory, it is often useful to allow sets which have innite measure and integrals whose value may beinnite.Such measures arise naturally out of calculus.For example, in assigning a measure to R that agrees with the usuallength of intervals, this measure must be larger than any nite real number. Also, when considering innite integrals,such as1dxx910 CHAPTER 2. EXTENDED REAL NUMBER LINEthe value innity arises. Finally, it is often useful to consider the limit of a sequence of functions, such asfn(x) ={2n(1 nx), if 0 x 1n0, if1n< x 1Without allowing functions to take on innite values, such essential results as the monotone convergence theorem andthe dominated convergence theorem would not make sense.2.2 Order and topological propertiesThe anely extended real number system turns into a totally ordered set by dening a + for all a.Thisorder has the desirable property that every subset has a supremum and an inmum: it is a complete lattice.This induces the order topology on R. In this topology, a set U is a neighborhood of + if and only if it contains aset {x : x > a} for some real number a, and analogously for the neighborhoods of . R is a compact Hausdorspace homeomorphic to the unit interval [0, 1]. Thus the topology is metrizable, corresponding (for a given home-omorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metricon R.With this topology the specially dened limits for x tending to + and , and the specially dened concepts oflimits equal to + and , reduce to the general topological denitions of limits.2.3 Arithmetic operationsThe arithmetic operations of R can be partially extended to R as follows:a += ++ a = +, a = a = + a = , a = +a () = a = , a (0, +]a () = a = , a [, 0)a= 0, a Ra= , a (0, +)a= , a (, 0)For exponentiation, see Exponentiation#Limits of powers. Here, "a + " means both "a + (+)" and "a ()",while "a " means both "a (+)" and "a + ()".The expressions , 0 () and / (called indeterminate forms) are usually left undened. These rulesare modeled on the laws for innite limits. However, in the context of probability or measure theory, 0 () isoften dened as 0.The expression 1/0 is not dened either as + or , because although it is true that whenever f(x) 0 for acontinuous function f(x) it must be the case that 1/f(x) is eventually contained in every neighborhood of the set {,+}, it is not true that 1/f(x) must tend to one of these points. An example is f(x) = (sin x)/x (as x goes to innity).(The modulus | 1/f(x) |, nevertheless, does approach +.)2.4 Algebraic propertiesWith these denitions R is not a eld, nor a ring, and not even a group or semigroup.However, it still has severalconvenient properties:2.5. MISCELLANEOUS 11a + (b + c) and (a + b) + c are either equal or both undened.a + b and b + a are either equal or both undened.a (b c) and (a b) c are either equal or both undened.a b and b a are either equal or both undeneda (b + c) and (a b) + (a c) are equal if both are dened.if a b and if both a + c and b + c are dened, then a + c b + c.if a b and c > 0 and both a c and b c are dened, then a c b c.In general, all laws of arithmetic are valid in R as long as all occurring expressions are dened.2.5 MiscellaneousSeveral functions can be continuously extended to Rby taking limits. For instance, one denes exp() = 0, exp(+)= +, ln(0) = , ln(+) = + etc.Some discontinuities may additionally be removed. For example, the function 1/x2can be made continuous (undersome denitions of continuity) by setting the value to + for x = 0, and 0 for x = + and x = . The function1/x can not be made continuous because the function approaches as x approaches 0 from below, and + as xapproaches 0 from above.Compare the real projective line, which does not distinguish between + and . As a result, on one hand a functionmay have limit on the real projective line, while in the anely extended real number system only the absolute valueof the function has a limit, e.g. in the case of the function 1/x at x = 0. On the other handlimx f(x) and limx+ f(x)correspond on the real projective line to only a limit from the right and one from the left, respectively, with the fulllimit only existing when the two are equal.Thus exand arctan(x) cannot be made continuous at x = on the realprojective line.2.6 See alsoReal projective line, which adds a single, unsigned innity to the real number line.Division by zeroExtended complex planeImproper integralSeries (mathematics)2.7 ReferencesAliprantis, Charalambos D.; Burkinshaw, Owen (1998), Principles of Real Analysis (3rd ed.), San Diego, CA:Academic Press, Inc., p. 29, ISBN 0-12-050257-7, MR 1669668David W. Cantrell, Anely Extended Real Numbers, MathWorld.Chapter 3Interval (mathematics)This article is about intervals of real numbers and other totally ordered sets. For the most general denition, seepartially ordered set. For other uses, see Interval (disambiguation).In mathematics, an (real) interval is a set of real numbers with the property that any number that lies between twonumbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 x 1 is an intervalwhich contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all realnumbers R , the set of all negative real numbers, and the empty set.Real intervals play an important role in the theory of integration, because they are the simplest sets whose size ormeasure or length is easy to dene. The concept of measure can then be extended to more complicated sets ofreal numbers, leading to the Borel measure and eventually to the Lebesgue measure.Intervals are central to interval arithmetic, a general numerical computing technique that automatically providesguaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations,and arithmetic roundo.Intervals are likewise dened on an arbitrary totally ordered set, such as integers or rational numbers. The notationof integer intervals is considered in the special section below.3.1 Notations for intervalsThe interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called theendpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be usedas a separator, to avoid ambiguity.3.1.1 Including or excluding endpointsTo indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be eitherreplaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11. Thus, inset builder notation,(a, b) = ]a, b[ = {x R| a < x < b},[a, b) = [a, b[ = {x R| a x < b},(a, b] = ]a, b] = {x R| a < x b},[a, b] = [a, b] = {x R| a x b}.Note that (a, a), [a, a), and (a, a] each represents the empty set, whereas [a, a] denotes the set {a}. When a > b, allfour notations are usually taken to represent the empty set.Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation(a, b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry123.2. TERMINOLOGY 13and linear algebra, or (sometimes) a complex number in algebra. That is why Bourbaki introduced the notation ]a,b[ to denote the open interval.[1] The notation [a, b] too is occasionally used for ordered pairs, especially in computerscience.Some authors use ]a, b[ to denote the complement of the interval (a, b); namely, the set of all real numbers that areeither less than or equal to a, or greater than or equal to b.3.1.2 Innite endpointsIn both styles of notation, one may use an innite endpoint to indicate that there is no bound in that direction.Specically, one may use a = or b = + (or both). For example, (0, +) is the set of positive real numbers alsowritten +, and (, +) is the set of real numbers .The extended real number line includes and + as elements. The notations [, b] , [, b) , [a, +] , and (a,+] may be used in this context. For example (, +] means the extended real numbers excluding only .3.1.3 Integer intervalsThe notation [a .. b] when a and b are integers, or {a .. b}, or just a .. b is sometimes used to indicate the intervalof all integers between a and b, including both. This notation is used in some programming languages; in Pascal, forexample, it is used to formally dene a subrange type, most frequently used to specify lower and upper bounds ofvalid indices of an array.An integer interval that has a nite lower or upper endpoint always includes that endpoint. Therefore, the exclusionof endpoints can be explicitly denoted by writing a .. b 1 , a + 1 .. b , or a + 1 .. b 1. Alternate-bracket notationslike [a .. b) or [a .. b[ are rarely used for integer intervals.3.2 TerminologyAn open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greaterthan 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example[0,1] means greater than or equal to 0 and less than or equal to 1.A degenerate interval is any set consisting of a single real number. Some authors include the empty set in thisdenition. A real interval that is neither empty nor degenerate is said to be proper, and has innitely many elements.An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smallerthan or larger than all its elements. An interval is said to be bounded if it is both left- and right-bounded; and is saidto be unbounded otherwise.Intervals that are bounded at only one end are said to be half-bounded. The emptyset is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are alsocommonly known as nite intervals.Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute dierence betweenthe endpoints) is nite. The diameter may be called the length, width, measure, or size of the interval. The size ofunbounded intervals is usually dened as +, and the size of the empty interval may be dened as 0 or left undened.The centre (midpoint) of bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length |a b|/2. These concepts are undened for empty or unbounded intervals.An interval is said to be left-open if and only if it has no minimum(an element that is smaller than all other elements);right-open if it has no maximum; and open if it has both properties. The interval [0,1) = {x | 0 x < 1}, for example,is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negativereals, for example, is a right-open but not left-open interval. The open intervals coincide with the open sets of thereal line in its standard topology.An interval is said to be left-closed if it has a minimumelement, right-closed if it has a maximum, and simply closedif it has both. These denitions are usually extended to include the empty set and to the (left- or right-) unboundedintervals, so that the closed intervals coincide with closed sets in that topology.The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which arenot endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented14 CHAPTER 3. INTERVAL (MATHEMATICS)with its nite endpoints.For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains Xand does not properly contain any other interval that also contains X.3.3 Classication of intervalsThe intervals of real numbers can be classied into eleven dierent types, listed below; where a and b are real numbers,with a < b :empty:[b, a] = (a, a) = [a, a) = (a, a] = {} = degenerate:[a, a] = {a}proper and bounded:(a, b) = {x| a < x < b}[a, b] = {x| a x b}[a, b) = {x| a x < b}(a, b] = {x| a < x b}left-bounded and right-unbounded:(a, ) = {x| x > a}[a, ) = {x| x a}left-unbounded and right-bounded:(, b) = {x| x < b}(, b] = {x| x b}unbounded at both ends:(, +) = R3.3.1 Intervals of the extended real lineIn some contexts, an interval may be dened as a subset of the extended real numbers, the set of all real numbersaugmented with and +.In this interpretation, the notations [, b] , [, b) , [a, +] , and (a, +] are all meaningful and distinct. Inparticular, (, +) denotes the set of all ordinary real numbers, while [, +] denotes the extended reals.This choice aects some of the above denitions and terminology. For instance, the interval (, +) = R is closedin the realm of ordinary reals, but not in the realm of the extended reals.3.4 Properties of intervalsThe intervals are precisely the connected subsets of R . It follows that the image of an interval by any continuousfunction is also an interval. This is one formulation of the intermediate value theorem.The intervals are also the convex subsets of R . The interval enclosure of a subset X R is also the convex hull ofX .The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if andonly if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other(e.g., (a, b) [b, c] = (a, c] ).If R is viewed as a metric space, its open balls are the open bounded sets (c + r, c r), and its closed balls are theclosed bounded sets [c + r, c r].3.5. DYADIC INTERVALS 15Any element x of an interval I denes a partition of I into three disjoint intervals I1, I2, I3: respectively, the elementsof I that are less than x, the singleton [x, x] = {x} , and the elements that are greater than x. The parts I1 and I3 areboth non-empty (and have non-empty interiors) if and only if x is in the interior of I. This is an interval version ofthe trichotomy principle.3.5 Dyadic intervalsA dyadic interval is a bounded real interval whose endpoints arej2n andj+12n, where j and n are integers. Dependingon the context, either endpoint may or may not be included in the interval.Dyadic intervals have the following properties:The length of a dyadic interval is always an integer power of two.Each dyadic interval is contained in exactly one dyadic interval of twice the length.Each dyadic interval is spanned by two dyadic intervals of half the length.If two open dyadic intervals overlap, then one of them is a subset of the other.The dyadic intervals consequently have a structure that reects that of an innite binary tree.Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh renement, multigridmethods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for p = 2).[2]3.6 Generalizations3.6.1 Multi-dimensional intervalsIn many contexts, ann -dimensional interval is dened as a subset of Rnthat is the Cartesian product of n intervals,I= I1I2 In , one on each coordinate axis.For n = 2 , this generally denes a rectangle whose sides are parallel to the coordinate axes; for n = 3 , it denes anaxis-aligned rectangular box.A facet of such an interval I is the result of replacing any non-degenerate interval factor Ik by a degenerate intervalconsisting of a nite endpoint of Ik . The faces of I comprise I itself and all faces of its facets. The corners of Iare the faces that consist of a single point of Rn.3.6.2 Complex intervalsIntervals of complex numbers can be dened as regions of the complex plane, either rectangular or circular.[3]3.7 Topological algebraIntervals can be associated with points of the plane and hence regions of intervals can be associated with regions ofthe plane. Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R Rof real numbers with itself. Often it is assumed that y > x. For purposes of mathematical structure, this restrictionis discarded,[4] and reversed intervals where y x < 0 are allowed. Then the collection of all intervals [x,y] can beidentied with the topological ring formed by the direct sum of R with itself where addition and multiplication aredened component-wise.The direct sum algebra (R R, +, ) has two ideals, { [x,0] :x R } and { [0,y] :y R }. The identity elementof this algebra is the condensed interval [1,1].If interval [x,y] is not in one of the ideals, then it has multiplicativeinverse [1/x, 1/y]. Endowed with the usual topology, the algebra of intervals forms a topological ring. The group ofunits of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component ofthis group is quadrant I.16 CHAPTER 3. INTERVAL (MATHEMATICS)Every interval can be considered a symmetric interval around its midpoint. In a reconguration published in 1956by M Warmus, the axis of balanced intervals [x, x] is used along with the axis of intervals [x,x] that reduce to apoint.Instead of the direct sum R R , the ring of intervals has been identied[5] with the split-complex numberplane by M. Warmus and D. H. Lehmer through the identicationz = (x + y)/2 + j (x y)/2.This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicativestructure having some analogies to ordinary complex arithmetic, such as polar decomposition.3.8 See alsoInequalityInterval graphInterval nite element3.9 References[1] http://hsm.stackexchange.com/a/193[2] Kozyrev, Sergey (2002). Wavelet theory as p-adic spectral analysis. IzvestiyaRAN. Ser. Mat. 66 (2): 149158.doi:10.1070/IM2002v066n02ABEH000381. Retrieved 2012-04-05.[3] Complex interval arithmetic and its applications, Miodrag Petkovi, Ljiljana Petkovi, Wiley-VCH, 1998, ISBN 978-3-527-40134-5[4] Kaj Madsen (1979) Review of Interval analysis in the extended interval space by Edgar Kaucher from MathematicalReviews[5] D. H. Lehmer (1956) Review of Calculus of Approximations from Mathematical ReviewsT. Sunaga, Theory of interval algebra and its application to numerical analysis, In: Research Association ofApplied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 2946(547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp.126143.3.10 External linksA Lucid Interval by Brian Hayes: An American Scientist article provides an introduction.Interval Notation BasicsInterval computations websiteInterval computations research centersInterval Notation by George Beck, Wolfram Demonstrations Project.Weisstein, Eric W., Interval, MathWorld.Chapter 4Line (geometry)Not to be confused with Curved line.The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e.,4 3 21 0 43 2 1 12341234yxy=2.0x+1y=0.5x1y=0.5x+1The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross they-axis at the same place).1718 CHAPTER 4. LINE (GEOMETRY)A representation of one line segment.having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the seventeenthcentury, lines were dened like this: The [straight or curved] line is the rst species of quantity, which has only onedimension, namely length, without any width nor depth, and is nothing else than the ow or run of the point which[] will leave from its imaginary moving some vestige in length, exempt of any width. [] The straight line is thatwhich is equally extended between its points[1]Euclid described a line as breadthless length which lies equally with respect to the points on itself"; he intro-duced several postulates as basic unprovable properties from which he constructed the geometry, which is now calledEuclidean geometry to avoid confusion with other geometries which have been introduced since the end of nineteenthcentury (such as non-Euclidean, projective and ane geometry).In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometryis described. For instance, in analytic geometry, a line in the plane is often dened as the set of points whosecoordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may bean independent object, distinct from the set of points which lie on it.When a geometry is described by a set of axioms, the notion of a line is usually left undened (a so-called primitiveobject). The properties of lines are then determined by the axioms which refer to them. One advantage to thisapproach is the exibility it gives to users of the geometry. Thus in dierential geometry a line may be interpretedas a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vectorspace (all linear combinations of two independent vectors). This exibility also extends beyond mathematics and, forexample, permits physicists to think of the path of a light ray as being a line.A line segment is a part of a line that is bounded by two distinct end points and contains every point on the linebetween its end points. Depending on how the line segment is dened, either of the two end points may or may notbe part of the line segment.Two or more line segments may have some of the same relationships as lines, such asbeing parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do notintersect or are collinear.4.1 Denitions versus descriptionsAll denitions are ultimately circular in nature since they depend on concepts which must themselves have denitions,a dependence which can not be continued indenitely without returning to the starting point.To avoid this viciouscircle certain concepts must be taken as primitive concepts; terms which are given no denition.[2] In geometry, itis frequently the case that the concept of line is taken as a primitive.[3] In those situations where a line is a denedconcept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept isa primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy.In a non-axiomatic or simplied axiomatic treatment of geometry, the concept of a primitive notion may be tooabstract to be dealt with. In this circumstance it is possible that a description or mental image of a primitive notionis provided to give a foundation to build the notion on which would formally be based on the (unstated) axioms.Descriptions of this type may be referred to, by some authors, as denitions in this informal style of presentation.These are not true denitions and could not be used in formal proofs of statements. The denition of line in EuclidsElements falls into this category.[4] Even in the case where a specic geometry is being considered (for example,Euclidean geometry), there is no generally accepted agreement among authors as to what an informal description ofa line should be when the subject is not being treated formally.4.2 RayGiven a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part iscalled a ray (or half-line) and the point A is called its initial point. The point A is considered to be a member of the4.3. EUCLIDEAN GEOMETRY 19ray.[5] Intuitively, a ray consists of those points on a line passing through A and proceeding indenitely, starting at A,in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise denitionis required.Given distinct points A and B, they determine a unique ray with initial point A. As two points dene a unique line,this ray consists of all the points between A and B (including A and B) and all the points C on the line through A and Bsuch that B is between A and C.[6] This is, at times, also expressed as the set of all points C such that A is not betweenB and C.[7] A point D, on the line determined by A and B but not in the ray with initial point A determined by B, willdetermine another ray with initial point A. With respect to the AB ray, the AD ray is called the opposite ray.A B CRayThus, we would say that two dierent points, A and B, dene a line and a decomposition of this line into the disjointunion of an open segment (A, B) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the leftof A on the line AB). These are not opposite rays since they have dierent initial points.In Euclidean geometry two rays with a common endpoint form an angle.The denition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only forgeometries for which this notion exists, typically Euclidean geometry or ane geometry over an ordered eld.Onthe other hand, rays do not exist in projective geometry nor in a geometry over a non-ordered eld, like the complexnumbers or any nite eld.In topology, a ray in a space X is a continuous embedding R+ X. It is used to dene the important concept of endof the space.4.3 Euclidean geometrySee also: Euclidean geometryWhen geometry was rst formalised by Euclid in the Elements, he dened a general line (straight or curved) to bebreadthless length with a straight line being a line which lies evenly with the points on itself.[8] These denitionsserve little purpose since they use terms which are not, themselves, dened. In fact, Euclid did not use these denitionsin this work and probably included them just to make it clear to the reader what was being discussed. In moderngeometry, a line is simply taken as an undened object with properties given by axioms,[9] but is sometimes denedas a set of points obeying a linear relationship when some other fundamental concept is left undened.In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclids original axioms contained variousaws which have been corrected by modern mathematicians),[10] a line is stated to have certain properties which relateit to other lines and points. For example, for any two distinct points, there is a unique line containing them, and anytwo distinct lines intersect in at most one point.[11] In two dimensions, i.e., the Euclidean plane, two lines which donot intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are containedin a plane, or skew if they are not.Any collection of nitely many lines partitions the plane into convex polygons (possibly unbounded); this partition isknown as an arrangement of lines.4.3.1 Cartesian planeMain article: Linear equationLines in a Cartesian plane or, more generally, in ane coordinates, can be described algebraically by linear equations.In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:20 CHAPTER 4. LINE (GEOMETRY)y= mx + bwhere:m is the slope or gradient of the line.b is the y-intercept of the line.x is the independent variable of the function y = f(x).The slope of the line through points A(x, y) and B(x, y), when x x, is given by m = (y y)/(x x) and theequation of this line can be written y = m(x x) + y.In R2, every line L (including vertical lines) is described by a linear equation of the formL = {(x, y) | ax + by= c}with xed real coecients a, b and c such that a and b are not both zero. Using this form, vertical lines correspondto the equations with b = 0.There are many variant ways to write the equation of a line which can all be converted fromone to another by algebraicmanipulation. These forms (see Linear equation for other forms) are generally named by the type of information (data)about the line that is needed to write down the form. Some of the important data of a line is its slope, x-intercept,known points on the line and y-intercept.The equation of the line passing through two dierent points P0= (x0, y0) and P1= (x1, y1) may be written as(y y0)(x1x0) = (y1y0)(x x0)If x0 x1, this equation may be rewritten asy= (x x0)y1y0x1x0+ y0ory= xy1y0x1x0+x1y0x0y1x1x0.In three dimensions, lines can not be described by a single linear equation, so they are frequently described byparametric equations:x = x0 + aty= y0 + btz= z0 + ctwhere:x, y, and z are all functions of the independent variable t which ranges over the real numbers.(x0, y0, z0) is any point on the line.a, b, and c are related to the slope of the line, such that the vector (a, b, c) is parallel to the line.They may also be described as the simultaneous solutions of two linear equationsa1x + b1y + c1z d1= 0a2x + b2y + c2z d2= 0such that (a1, b1, c1) and (a2, b2, c2) are not proportional (the relations a1=ta2, b1=tb2, c1=tc2 imply t = 0).This follows since in three dimensions a single linear equation typically describes a plane and a line is what is commonto two distinct intersecting planes.4.3. EUCLIDEAN GEOMETRY 21Normal formThe normal segment for a given line is dened to be the line segment drawn from the origin perpendicular to the line.This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of astraight line on the plane is given by:y sin + xcos p = 0,where is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x axis to thissegment), and p is the (positive) length of the normal segment. The normal form can be derived from the generalform by dividing all of the coecients by|c|ca2+ b2.This form is also called the Hesse normal form,[12] after the German mathematician Ludwig Otto Hesse.Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two niteparameters, and p, to be specied. Note that if p > 0, then is uniquely dened modulo 2. On the other hand,if the line is through the origin (c = 0, p = 0), one drops the |c|/(c) term to compute sin and cos, and is onlydened modulo .4.3.2 Polar coordinatesIn polar coordinates on the Euclidean plane the slope-intercept form of the equation of a line is expressed as:r =mr cos + bsin ,where m is the slope of the line and b is the y-intercept. When = 0 the graph will be undened. The equation canbe rewritten to eliminate discontinuities in this manner:r sin = mr cos + b.In polar coordinates on the Euclidean plane, the intercept form of the equation of a line that is non-horizontal, non-vertical, and does not pass through pole may be expressed as,r =1cos xo+sin yowhere xo and yo represent the x and y intercepts respectively. The above equation is not applicable for vertical andhorizontal lines because in these cases one of the intercepts does not exist. Moreover, it is not applicable on linespassing through the pole since in this case, both x and y intercepts are zero (which is not allowed here since xo andyo are denominators). A vertical line that doesn't pass through the pole is given by the equationr cos = xo.22 CHAPTER 4. LINE (GEOMETRY)Similarly, a horizontal line that doesn't pass through the pole is given by the equationr sin = yo.The equation of a line which passes through the pole is simply given as: = mwhere m is the slope of the line.4.3.3 Vector equationThe vector equation of the line through points A and B is given by r = OA + AB (where is a scalar).If a is vector OA and b is vector OB, then the equation of the line can be written: r = a + (b a).A ray starting at point A is described by limiting . One ray is obtained if 0, and the opposite ray comes from 0.4.3.4 Euclidean spaceIn three-dimensional space, a rst degree equation in the variables x, y, and z denes a plane, so two such equations,provided the planes they give rise to are not parallel, dene a line which is the intersection of the planes. Moregenerally, in n-dimensional space n1 rst-degree equations in the n coordinate variables dene a line under suitableconditions.In more general Euclidean space, Rn(and analogously in every other ane space), the line L passing through twodierent points a and b (considered as vectors) is the subsetL = {(1 t) a + t b | t R}The direction of the line is from a (t = 0) to b (t = 1), or in other words, in the direction of the vector b a. Dierentchoices of a and b can yield the same line.Collinear pointsMain article: CollinearityThree points are said to be collinear if they lie on the same line. Three points usually determine a plane, but in thecase of three collinear points this does not happen.In ane coordinates, in n-dimensional space the points X=(x1, x2, ..., x), Y=(y1, y2, ..., y), and Z=(z1, z2, ..., z)are collinear if the matrix1 x1x2. . . xn1 y1y2. . . yn1 z1z2. . . znhas a rank less than 3. In particular, for three points in the plane (n = 2), the above matrix is square and the pointsare collinear if and only if its determinant is zero.Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of pointsequals the slope between any other pair of points (in which case the slope between the remaining pair of points will4.3. EUCLIDEAN GEOMETRY 23equal the other slopes). By extension, k points in a plane are collinear if and only if any (k1) pairs of points havethe same pairwise slopes.In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinear-ity between three points by:[13][14]The points a, b and c are collinear if and only if d(x,a) = d(c,a) and d(x,b) = d(c,b) implies x=c.However there are other notions of distance (such as the Manhattan distance) for which this property is not true.In the geometries where the concept of a line is a primitive notion, as may be the case in some synthetic geometries,other methods of determining collinearity are needed.4.3.5 Types of linesIn a sense,[15] all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart fromone another. However, lines may play special roles with respect to other objects in the geometry and be divided intotypes according to that relationship. For instance, with respect to a conic (a circle, ellipse, parabola, or hyperbola),lines can be:tangent lines, which touch the conic at a single point;secant lines, which intersect the conic at two points and pass through its interior;exterior lines, which do not meet the conic at any point of the Euclidean plane; ora directrix, whose distance from a point helps to establish whether the point is on the conic.In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other linesthat may or not be parallel to each other.For more general algebraic curves, lines could also be:i-secant lines, meeting the curve in i points counted without multiplicity, orasymptotes, which a curve approaches arbitrarily closely without touching it.With respect to triangles we have:the Euler line,the Simson lines, andcentral lines.For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints ofthe two diagonals.For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pairof lines, we have the Pappus line.Parallel lines are lines in the same plane that never cross. Intersecting lines share a single point in common. Coinci-dental lines coincide with each otherevery point that is on either one of them is also on the other.Perpendicular lines are lines that intersect at right angles.In three-dimensional space, skew lines are lines that are not in the same plane and thus do not intersect each other.24 CHAPTER 4. LINE (GEOMETRY)4.4 Projective geometryMain article: Projective geometryIn many models of projective geometry, the representation of a line rarely conforms to the notion of the straightcurve as it is visualised in Euclidean geometry. In elliptic geometry we see a typical example of this.[16] In thespherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametricallyopposite points identied. In a dierent model of elliptic geometry, lines are represented by Euclidean planes passingthrough the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, twopoints determining a unique line) that make them suitable representations for lines in this geometry.4.5 GeodesicsMain article: geodesicsThe shortness and straightness of a line, interpreted as the property that the distance along the line between anytwo of its points is minimized (see triangle inequality), can be generalized and leads to the concept of geodesics inmetric spaces.4.6 See alsoLine coordinatesLine segmentCurveLocusDistance from a point to a lineDistance between two linesAne functionIncidence (geometry)Plane (geometry)Rectilinear4.7 Notes[1] In (rather old) French: La ligne est la premire espece de quantit, laquelle a tant seulement une dimension savoirlongitude, sans aucune latitude ni profondit, & n'est autre chose que le ux ou coulement du poinct, lequel [] laisserade son mouvement imaginaire quelque vestige en long, exempt de toute latitude. [] La ligne droicte est celle qui estgalement estendu entre ses poincts. Pages 7 and 8 of Les quinze livres des lments gomtriques d'Euclide Megarien,traduits de Grec en Franois, & augmentez de plusieurs gures & demonstrations, avec la corrections des erreurs commisess autres traductions, by Pierre Mardele, Lyon, MDCXLV (1645).[2] Coxeter 1969, pg. 4[3] Faber 1983, pg. 95[4] Faber 1983, pg. 95[5] On occasion we may consider a ray without its initial point.Such rays are called open rays, in contrast to the typical raywhich would be said to be closed.4.8. REFERENCES 25[6] Wylie, Jr. 1964, pg. 59, Denition 3[7] Pedoe 1988, pg. 2[8] Faber, Appendix A, p. 291.[9] Faber, Part III, p. 95.[10] Faber, Part III, p. 108.[11] Faber, Appendix B, p. 300.[12] Bcher, Maxime (1915), Plane Analytic Geometry: With Introductory Chapters on the Dierential Calculus, H. Holt, p. 44.[13] Alessandro Padoa, Un nouveau systme de dnitions pour la gomtrie euclidienne, International Congress of Mathe-maticians, 1900[14] Bertrand Russell, The Principles of Mathematics, p.410[15] Technically, the collineation group acts transitively on the set of lines.[16] Faber, Part III, p. 108.4.8 ReferencesCoxeter, H.S.M (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 0-471-18283-4Faber, Richard L. (1983). Foundations of Euclidean and Non-Euclidean Geometry. NewYork: Marcel Dekker.ISBN 0-8247-1748-1.Pedoe, Dan (1988), Geometry: A Comprehensive Course, Mineola, NY: Dover, ISBN 0-486-65812-0Wylie, Jr., C. R. (1964), Foundations of Geometry, New York: McGraw-Hill, ISBN 0-07-072191-24.9 External linksHazewinkel, Michiel, ed. (2001), Line (curve)", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Line, MathWorld.Equations of the Straight Line at Cut-the-KnotCitizendiumChapter 5Point (geometry)In modern mathematics, a point refers usually to an element of some set called a space.More specically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Beinga primitive notion means that a point cannot be dened in terms of previously dened objects. That is, a point isdened only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not haveany length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a pointis meant to capture the notion of a unique location in Euclidean space.5.1 Points in Euclidean geometryPoints, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclidoriginally dened the point as that which has no part. In two-dimensional Euclidean space, a point is representedby an ordered pair (x, y) of numbers, where the rst number conventionally represents the horizontal and is oftendenoted by x, and the second number conventionally represents the vertical and is often denoted by y.This idea iseasily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z)with the additional third number representing depth and often denoted by z. Further generalizations are representedby an ordered tuplet of n terms, (a1, a2, , an) where n is the dimension of the space in which the point is located.Many constructs within Euclidean geometry consist of an innite collection of points that conform to certain ax-ioms. This is usually represented by a set of points; As an example, a line is an innite set of points of the formL={(a1,a2,...an)|a1c1+a2c2+...ancn=d} , where c1 through cn and d are constants and n is the dimension of the space.Similar constructions exist that dene the plane, line segment and other related concepts. By the way, a degenerateline segment consists of only one point.In addition to dening points and constructs related to points, Euclid also postulated a key idea about points; heclaimed that any two points can be connected by a straight line. This is easily conrmed under modern expansionsof Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all thegeometric concepts of the time.However, Euclids postulation of points was neither complete nor denitive, as heoccasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of pointson the line or the existence of specic points. In spite of this, modern expansions of the system serve to remove theseassumptions.5.2 Dimension of a pointThere are several inequivalent denitions of dimension in mathematics. In all of the common denitions, a point is0-dimensional.265.2. DIMENSION OF A POINT 2712-2-11 2 -2 -1A nite set of points (blue) in two-dimensional Euclidean space.5.2.1 Vector space dimensionMain article: Dimension (vector space)The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consistingof a single point (which must be the zero vector 0), there is no linearly independent subset.The zero vector is notitself linearly independent, because there is a non trivial linear combination making it zero:1 0 = 0 .5.2.2 Topological dimensionMain article: Lebesgue covering dimensionThe topological dimension of a topological space X is dened to be the minimum value of n, such that every niteopen cover Aof X admits a nite open cover B of X which renes Ain which no point is included in more than n+1elements. If no such minimal n exists, the space is said to be of innite covering dimension.A point is zero-dimensional with respect to the covering dimension because every open cover of the space has arenement consisting of a single open set.28 CHAPTER 5. POINT (GEOMETRY)5.2.3 Hausdor dimensionLet X be a metric space. If S X and d [0, ), the d-dimensional Hausdor content of S is the inmum of theset of numbers 0 such that there is some (indexed) collection of balls {B(xi, ri) : i I} covering S with ri > 0for each i I that satisesiI rdi< .The Hausdor dimension of X is dened bydimH(X) := inf{d 0 : CdH(X) = 0}.A point has Hausdor dimension 0 because it can be covered by a single ball of arbitrarily small radius.5.3 Geometry without pointsAlthough the notion of a point is generally considered fundamental in mainstream geometry and topology, there aresome systems that forgo it, e.g. noncommutative geometry and pointless topology. A pointless or pointfree spaceis dened not as a set, but via some structure (algebraic or logical respectively) which looks like a well-known functionspace on the set: an algebra of continuous functions or an algebra of sets respectively. More precisely, such structuresgeneralize well-known spaces of functions in a way that the operation take a value at this point may not be dened.Afurther tradition starts fromsome books of A. N. Whitehead in which the notion of region is assumed as a primitivetogether with the one of inclusion or connection.5.4 Point masses and the Dirac delta functionMain article: Dirac delta functionOften in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especiallycommon in classical electromagnetism, where electrons are idealized as points with non-zero charge). The Diracdelta function, or function, is (informally) a generalized function on the real number line that is zero everywhereexcept at zero, with an integral of one over the entire real line.[1][2][3] The delta function is sometimes thought of asan innitely high, innitely thin spike at the origin, with total area one under the spike, and physically represents anidealized point mass or point charge.[4] It was introduced by theoretical physicist Paul Dirac. In the context of signalprocessing it is often referred to as the unit impulse symbol (or function).[5] Its discrete analog is the Kroneckerdelta function which is usually dened on a nite domain and takes values 0 and 1.5.5 See alsoAccumulation pointAne spaceBoundary pointCritical pointCuspFoundations of geometryPosition (geometry)PointwiseSingular point of a curve5.6. REFERENCES 295.6 References[1] Dirac 1958, 15 The function, p. 58[2] Gel'fand & Shilov 1968, Volume I, 1.1, 1.3[3] Schwartz 1950, p. 3[4] Arfken & Weber 2000, p. 84[5] Bracewell 1986, Chapter 5Clarke, Bowman, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 6175.De Laguna, T., 1922, Point, line and surface as sets of solids, The Journal of Philosophy 19: 44961.Gerla, G., 1995, "Pointless Geometries" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry:buildings and foundations. North-Holland: 101531.Whitehead A. N., 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press.2nd ed., 1925.--------, 1920. The Concept of Nature.Cambridge Univ.Press.2004 paperback, Prometheus Books.Beingthe 1919 Tarner Lectures delivered at Trinity College.--------, 1979 (1929). Process and Reality. Free Press.5.7 External linksDenition of Point with interactive appletPoints denition pages, with interactive animations that are also useful in a classroom setting. Math OpenReferencePoint at PlanetMath.org.Weisstein, Eric W., Point, MathWorld.30 CHAPTER 5. POINT (GEOMETRY)5.8 Text and image sources, contributors, and licenses5.8.1 Text Dimension Source: https://en.wikipedia.org/wiki/Dimension?oldid=675219095 Contributors: AxelBoldt, BF, Bryan Derksen, Zundark,The Anome, Josh Grosse, Vignaux, XJaM, Stevertigo, Frecklefoot, Patrick, Boud, Michael Hardy, Isomorphic, Menchi, Ixfd64, Kalki,Delirium, Looxix~enwiki, William M. 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Peterson, MagnaMopus, SteveG23, Footballrocks41237, Pthag, Thomas Howey, Jim.belk, Loadmaster, Waggers,Mets501, Michael Greiner, Daviddaniel37, Autonova, Fredil Yupigo, Iridescent, Kiwi8, Mulder416sBot, Courcelles, Rccarman, Tantha-las39, KyraVixen, Randalllin, 345Kai, Requestion, MarsRover, WeggeBot, Myasuda, FilipeS, Equendil, Vectro, Cydebot, Hanju, Tawker-bot4, Xantharius, Dragonare82, Arb, JamesAM, Thijs!bot, Epbr123, Mbell, Mojo Hand, SomeStranger, The Hybrid, Nick Number,Heroeswithmetaphors, Escarbot, Stannered, AntiVandalBot, Mrbip, Salgueiro~enwiki, Kent Witham, The man stephen, MatthewFennell,Vanished user s4irtj34tivkj12erhskj46thgdg, Beaumont, Cynwolfe, Acroterion, Bongwarrior, VoABot II, Alvatros~enwiki, Rafuki 33,Wikidudeman, JamesBWatson, TheChard, Stijn Vermeeren, Indon, Illspirit, David Eppstein, Fang 23, Spellmaster, JoergenB, DerHexer,Patstuart, DukeTwicep, Greenguy1090, Torrr, Gasheadsteve, Leyo, J.delanoy, Captain panda, Enchepilon, Elizabethrhodes, EscapingLife,Inimino, Maurice Carbonaro, DRTORMEY, Good-afternun!, It Is Me Here, Thomlm, TomasBat, Han Solar de Harmonics, SemblaceII,CardinalDan, Hamzabahaa, VolkovBot, Seldon1, JohnBlackburne, Orthologist, LokiClock, Jacroe, Philip Trueman, Mdmkolbe, TXiK-iBoT, Gaara144, Vipinhari, Technopat, Aodessey, Anonymous Dissident, Immorality, Berchin, Ferengi, LariAnn, Gbaor, Andyo2000,Jmath666, Wenli, HX Aeternus, Wolfrock, Enviroboy, Mike4ty4, Symane, Rybu, BriEnBest, YonaBot, Racer x124, Pelyukhno.erik,Oysterguitarist, Harry~enwiki, SuperLightningKick, Techman224, OKBot, Anchor Link Bot, JL-Bot, Mr. Granger, Martarius, ClueBot,WurmWoode, The Thing That Should Not Be, Gamehero, Roal AT, DragonBot, Kitsunegami, Excirial, D1a4l2s3y5, Grb 1991, Cute loli-con, Puceron, Jennonpress, SchreiberBike, ChrisHodgesUK, Qwfp, Alousybum, TimothyRias, Rror, Egyptianboy15223, NellieBly, Me,Myself, and I, Addbot, Leszek Jaczuk, NjardarBot, Joonojob, Sillyfolkboy, NerdBoy1392, Ranjitvohra, Renatokeshet, Numbo3-bot,Tide rolls, Lightbot, Yobot, Tester999, Tohd8BohaithuGh1, Fraggle81, Sarrus, Zenquin, AnomieBOT, Andrewrp, AdjustShift, Mate-rialscientist, Hunnjazal, 90 Auto, Citation bot, Maxis ftw, Nagoshi1, Staysfresh, Danny955, MeerkatNerd1, Anna Frodesiak, Mlpearc,AbigailAbernathy, Srich32977, Charvest, Natural Cut, Sonoluminesence, Currydump, FrescoBot, Inc ru, PhysicsExplorer, Sae1962, Rty-coon, Drew R. Smith, RandomDSdevel, Pinethicket, I dream of horses, Edderso, Jonesey95, MastiBot, , December21st2012Freak,AGiorgio08, SkyMachine, Double sharp, Sheogorath, Lotje, Chrisjameshull, 4, Diannaa, Jesse V., Xnn, Distortiondude, Mean as cus-tard, Alison22, Pokdhjdj, J36miles, Envirodan, Ovizelu, RA0808, Scleria, Slightsmile, Wikipelli, Hhhippo, Traxs7, Medeis, StudyLak-shan, Sarapaxton, D.Lazard, SpikeTD, Markshutter, Ewa5050, Jay-Sebastos, L Kensington, Bomazi, BioPupil, RockMagnetist, 28bot,Isocli, Khestwol, ClueBot NG, Wcherowi, Thekk2007, Lanthanum-138, O.Koslowski, MerlIwBot, Bibcode Bot, Loves Labour Lost,Snaevar-bot, Nospildoh, Bereziny, Jxuan, Rahul.quara, Naeem21, Ownedroad9, Balance of paradox, Brat162, ChrisGualtieri, Kelvinsong,Uevboweburvkuwbekl, RobertAnderson1432, Hillbillyholiday, Theeverst87, Titz69, Awesome2013, Wamiq, Penitence, Dez Moines,I3roly, Anonymous-232, Brandon Ernst, Kylejaylee, Chuluojun, Prachi.apomr, Loraof, BakedLikaBiscuit, Inkanyamba, Knight victor,Srednuas Lenoroc and Anonymous: 481 Extended real number line Source:https://en.wikipedia.org/wiki/Extended_real_number_line?oldid=671639548 Contributors:Axel-Boldt, B4hand, Patrick, Michael Hardy, TakuyaMurata, Loren Rosen, Revolver, Rbraunwa, Dysprosia, Fibonacci, Robbot, Tobias Berge-mann, Giftlite, Dbenbenn, Inter, Stevenmattern, Paul August, RoyBoy, Kevin Lamoreau, Eric Kvaalen, Keenan Pepper, ABCD, Sligocki,Schapel, David W. Cantrell, Linas, Georgia guy, Isnow, DVdm, RussBot, Trovatore, Saric, Cojoco, SmackBot, Eskimbot, Skylarkk,Ser Amantio di Nicolao, Mr Stephen, EdC~enwiki, Wjejskenewr, Zero sharp, CRGreathouse, FilipeS, Xantharius, Epbr123, Dugwiki,P64, David Eppstein, STBotD, TXiKiBoT, A4bot, DumZiBoT, SilvonenBot, Addbot, Some jerk on the Internet, LaaknorBot, Yobot,Boleyn3, HRoestBot, Thinking of England, Schubi87, Distortiondude, Sheeana, Rnddim, ZroBot, SporkBot, Brad7777, Limit-theorem,Connorpark and Anonymous: 44 Interval (mathematics)Source: https://en.wikipedia.org/wiki/Interval_(mathematics)?oldid=674865208Contributors: Zundark,EdPoor, JeLuF, Patrick, Michael Hardy, Andres, Bjcairns, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, OlivierM, McKay, Robbot,Ruinia, Ojigiri~enwiki, Ambarish, Tobias Bergemann, Alan Liefting, Tosha, Giftlite, Markus Krtzsch, MSGJ, Markus Kuhn, GarethWyn, Jorge Stol, Gazibara, Sam Hocevar, PhotoBox, Mormegil, Paul August, Rgdboer, Liberatus, EmilJ, Jpgordon, Teorth, Jsoulie,Jumbuck, Burn, Krellion, Oleg Alexandrov, Simetrical, Mindmatrix, Ikescs, OdedSchramm, Rejnal, Theboywonder, Dpr, Dpv, Salixalba, Mathbot, Greg321, AttishOculus, Gurch, Dmitry-kazakov, Chobot, YurikBot, Hairy Dude, RussBot, Michael Slone, Stephenb,Gaius Cornelius, PaulGarner, Scilicet, GrinBot~enwiki, JJL, SmackBot, Incnis Mrsi, Tom Petek, SmartGuy Old, Silly rabbit, Nbarth,DHN-bot~enwiki, Cybercobra, SashatoBot, Cronholm144, Amine Brikci N, Pet-ro, Eassin, Ylloh, CRGreathouse, Woudloper, CBM, HeWho Is, Thijs!bot, Epbr123, Jojan, Pjvpjv, Marek69, NERIUM, Opelio, JAnDbot, Martinkunev, Nyq, Albmont, Email4mobile, DavidEppstein, Anaxial, J.delanoy, Pharaoh of the Wizards, Extransit, Redrad, NewEnglandYankee, Cometstyles, DavidCBryant, Hulten, The-NewPhobia, Pleasantville, Philip Trueman, TXiKiBoT, Enviroboy, Insanity Incarnate, Dmcq, Monty845, Cowlinator, Quietbritishjim,SieBot, ToePeu.bot, Nathan B. Kitchen, Ezh, Lightmouse, Skeptical scientist, Msrasnw, Anchor Link Bot, ClueBot, Rumping, Marino-slo,The Thing That Should Not Be, Idleloop~enwiki, CptCutLess, Otolemur crassicaudatus, Excirial, Bremerenator, Hans Adler, Schreiber-Bike, Ottawa4ever, Kikos, SoxBot III, Knopfkind, SilvonenBot, Addbot, Mammothx, LaaknorBot, Glane23, AndersBot, DavidBParker,5.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 31Jasper Deng, 5 albert square, Tide rolls, Zorrobot, Luckas-bot, Yobot, Vs64vs, JorgeFierro, Allent511, AnomieBOT, Qdinar, AdjustShift,RandomAct, Flewis, Xelnx, ArthurBot, Pownuk, LilHelpa, Obersachsebot, Capricorn42, SteveWoolf, TonyHagale, Charvest, Aghajan-pour, Shadowjams, LucienBOT, Calmer Waters, ItsZippy, Sumone10154, DARTH SIDIOUS 2, Woogee, Hyarmendacil, Noommos,Jowa fan, EmausBot, K6ka, John Cline, Quondum, Wayne Slam, Sassospicco, Mayur, Wikiloop, Bean49Bot, DASHBotAV, 28bot, Clue-Bot NG, Jack Greenmaven, Wcherowi, Tideat, Widr, Vibhijain, Helpful Pixie Bot, , Webclient101, Stephan Kulla, I am One ofMany, Theopolito, Ginsuloft and Anonymous: 204 Line (geometry) Source: https://en.wikipedia.org/wiki/Line_(geometry)?oldid=670299951 Contributors: Zundark, XJaM, William Av-ery, Stevertigo, Patrick, Liftarn, Tango, Dcljr, AugPi, Dcoetzee, Furrykef, SEWilco, Mazin07, Fredrik, Altenmann, Mikalaari, Stew-artadcock, Carnildo, Tobias Bergemann, Giftlite, BenFrantzDale, Tom harrison, No Guru, Cantus, Python eggs, Lakefall~enwiki, HPadleckas, Tomruen, Zfr, Neutrality, Zeman, PhotoBox, Ta bu shi da yu, Discospinster, Rich Farmbrough, Paul August, El C, Edwin-stearns, Ascorbic, Rgdboer, Longhair, La goutte de pluie, James Foster, Obradovic Goran, Zaraki~enwiki, Tsirel, Alansohn, AnthonyAppleyard, Free Bear, Qrc, Andrewpmk, Burn, Velella, Frankman, HenkvD, Max Naylor, Oleg Alexandrov, Gatewaycat, Linas, Btyner,Enzo Aquarius, Juan Marquez, Ravidreams, PlatypeanArchcow, Mathbot, Wars, Fresheneesz, Srleer, CiaPan, Rija, DVdm, Algebraist,Wavelength, Splash, David R. 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