Three-dimensional space is a geometric model of the physical universe The universe is commonly defined as the totality of everything that exists, including all physical matter and energy, the planets, stars, galaxies, and the contents of intergalactic space, although this usage may differ with the context . The term universe may be used in slightly different contextual senses, denoting such concepts as the cosmos, in which we live. The three dimensions are commonly called length, width, and depth (or height), although any three mutually perpendicular directions can serve as the three dimensions.

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In physics, our three-dimensional space is viewed as embedded in 4-dimensional In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two space-time, called Minkowski space In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime (see special relativity Special relativity (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies"). The idea behind space-time is that time is hyperbolic-orthogonal In plane geometry, two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola. Two particular hyperbolas are frequently used in the plane: to each of the three spatial dimensions.

In mathematics, analytic geometry Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers, scalars or variables to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other are given, each perpendicular to the other two at the origin In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect. In Euclidean geometry, the origin may be chosen freely as any convenient, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect. In Euclidean geometry, the origin may be chosen freely as any convenient measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative and spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane, though there are an infinite number of possible methods. See Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity.

Another mathematical way of viewing three-dimensional space is found in linear algebra Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps or linear transformations and are often represented by matrices. Linear, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three dimensional because every point in space can be described by a linear combination of three independent vectors In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers. In this view, space-time is four dimensional because the location of a point in time is independent of its location in space.

Three-dimensional space has a number of properties that distinguish it from spaces of other dimension numbers. For example, at least 3 dimensions are required to tie a knot In mathematics, knot theory is the area of topology that studies mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together to prevent it from becoming undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional in a piece of string.[1] Many of the laws of physics, such as the various inverse square laws In physics, an inverse-square law is any physical law stating that some physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity, depend on dimension three.[2]

The understanding of three-dimensional space in humans is thought to be learned during infancy using unconscious inference Visual perception is the ability to interpret information and surroundings from the effects of visible light reaching the eye. The resulting perception is also known as eyesight, sight, or vision . The various physiological components involved in vision are referred to collectively as the visual system, and are the focus of much research in, and is closely related to hand-eye coordination Eye–hand coordination is the coordinated control of eye movement with hand movement, and the processing of visual input to guide reaching and grasping along with the use of proprioception of the hands to guide the eyes. It has been studied in activities as diverse as tea making, the movement of solid objects such as wooden blocks, sporting. The visual ability to perceive the world in three dimensions is called depth perception Depth perception arises from a variety of depth cues. These are typically classified into binocular cues that require input from both eyes and monocular cues that require the input from just one eye. Binocular cues include stereopsis, yielding depth from binocular vision through exploitation of parallax. Monocular cues include size: distant.

See also

References

  1. ^ Dale Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976, ISBN 0-914098-16-0
  2. ^ Brian Greene, The Fabric of the Cosmos, Random House, New York, 2003, ISBN 0-375-72720-5
Dimension In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two - (category Categories: Physics | Geometry | Manifolds | Theoretical physics )
Dimensional spaces One In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 1, the set of all such locations is called 1-dimensional Euclidean space · Two In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called 2-dimensional Euclidean space · Three · Four In mathematics, the fourth dimension, or a four-dimensional space, is an abstract concept derived by generalizing the rules of three-dimensional space. It has been studied by mathematicians and philosophers for almost two hundred years, both for its own interest and for the insights it offered into mathematics and related fields · Five · Six Six-dimensional space is a term used to describe any space that has six dimensions, i.e. six degrees of freedom, and which needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but the ones that are of most interest are simpler ones which are models of some aspect of our environment · Seven · Eight In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional Euclidean space · Nine · n-dimensions · Spacetime In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. According to certain Euclidean space perceptions, the universe has three · Projective space In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively · Hyperplane In geometry, a hyperplane of an n-dimensional space V is a "flat" subset of dimension n − 1, or equivalently, of codimension 1 in V; it may therefore be referred to as an -flat of V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies
Polytopes In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions . Some theories further generalise the idea to include such things as unbounded polytopes (apeirotopes and and Shapes Simplex In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron. A single point may be · Hypercube In geometry, a hypercube is an n-dimensional analogue of a square and a cube (n = 3). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length · Hyperrectangle In geometry, an orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals · Demihypercube In geometry, demihypercubes are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n-1)-demicubes and 2n (n-1)-simplex facets are formed in place of the deleted vertices · Cross-polytope In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope consist of all permutations of . The cross-polytope is the convex hull of its vertices. (Note: some authors define a cross-polytope only as the boundary of this region.) · n-sphere In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. It is an n-dimensional
Concepts and mathematics Cartesian coordinates A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length · Linear algebra Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps or linear transformations and are often represented by matrices. Linear · Geometric algebra In mathematical physics, geometric algebra is a geometric interpretation of nondegenerate Clifford algebras, and a geometric algebra is a nondegenerate Clifford algebra, interpreted geometrically. Geometric algebra provides an alternative to 3-dimensional vector calculus , which, unlike vector calculus, generalizes directly to higher dimensions · Conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces. In more than two dimensions, conformal geometry may refer either to the study of conformal transformations of "flat" spaces (such as Euclidean · Reflection In mathematics, a reflection is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three- · Rotation In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming. A rotation and the above-mentioned transformations · Plane of rotation · Space Mathematical spaces often form a hierarchy, i.e., one space may inherit all the characteristics of a parent space. For instance, all inner product spaces are also normed vector spaces, because the inner product induces a norm on the inner product space such that: · Fractal dimension In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal dimensions are the Rényi dimension, the Hausdorff · Multiverse The multiverse is the hypothetical set of multiple possible universes (including the one unique universe we are pretty sure we consistently inhabit) that together comprise everything that physically exists: the entirety of space and time, all forms of matter, energy and momentum, and the physical laws and constants that govern them. The term was

Categories: Euclidean solid geometry Euclidean solid geometry is the traditional solid geometry of three-dimensional space. See also computer graphics and 3D imaging | Analytic geometry

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The Lives of the Most Eminent Literary and Scientific Men were five volumes of Dionysius Lardner’s 133-volume Cabinet Cyclopaedia . Aimed at the self-educating middle class, this encyclopedia was written during the 19th-century literary revolution in Britain that encouraged more people to read. The Lives formed part of the Cabinet of Biography
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