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| {{merge to|space|discuss=Talk:Three-dimensional space#Merger proposal|date=November 2013}}
| | Jayson Berryhill is how I'm known as and my wife doesn't like it at all. Credit authorising is exactly where my primary income comes from. Kentucky is where I've usually been residing. To perform lacross is the factor I love most of all.<br><br>My blog ... psychic phone readings ([http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ click this site]) |
| {{redir|Three-dimensional||3D (disambiguation)}}
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| [[Image:Coord planes color.svg|right|thumb|300px|Three-dimensional [[Cartesian coordinate system]] with the ''x''-axis pointing towards the observer]]
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| {{General geometry}}
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| '''Three-dimensional space''' is a geometric 3-parameters model of the physical [[universe]] (without considering time) in which all known [[matter]] exists. These three dimensions can be labeled by a combination of three chosen from the terms ''[[length]]'', ''[[width]]'', ''[[height]]'', ''[[Elevation|depth]]'', and ''[[breadth]]''. Any three directions can be chosen, provided that they do not all lie in the same [[plane (geometry)|plane]].
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| In [[physics]] and [[mathematics]], a [[Euclidean vector|sequence]] of ''n'' [[Real number|numbers]] can be understood as a location in ''n''-dimensional space. When ''n'' = 3, the set of all such locations is called '''3-dimensional Euclidean space'''. It is commonly represented by the symbol <math>\scriptstyle{\mathbb{R}}^3</math>. This space is only one example of a great variety of spaces in three dimensions called [[3-manifold]]s.
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| == Details ==
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| In mathematics, [[analytic geometry]] (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three [[coordinate axis|coordinate axes]] are given, usually each perpendicular to the other two at the [[Origin (mathematics)|origin]], 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 (mathematics)|origin]] measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.
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| Other popular methods of describing the location of a point in three-dimensional space include [[cylindrical coordinates]] and [[spherical coordinates]], though there is an infinite number of possible methods. See [[Euclidean space]].
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| Another mathematical way of viewing three-dimensional space is found in [[linear algebra]], where the idea of independence is crucial. Space has three dimensions because the length of a [[cuboid|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 [[coordinate vector|vector]]s. In this view, space-time is four-dimensional because the location of a point in time is independent of its location in space.
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| Three-dimensional space has a number of properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a [[knot theory|knot]] in a piece of string.<ref>{{cite book |first1=Dale |last1=Rolfsen |title=Knots and Links |publisher=Publish or Perish |location=Berkeley, California |year=1976 |isbn=0-914098-16-0}}</ref> Many of the laws of physics, such as the various [[inverse square law]]s, depend on dimension three.<ref>{{cite book |first1=Brian |last1=Greene |title=The Fabric of the Cosmos |publisher=[[Random House]] |location=New York |year=2003 |isbn=0-375-72720-5}}</ref>
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| The understanding of three-dimensional space in humans is thought to be learned during infancy using [[visual perception#Unconscious inference|unconscious inference]], and is closely related to [[hand-eye coordination]]. The visual ability to perceive the world in three dimensions is called [[depth perception]].
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| With the space <math>\scriptstyle{\mathbb{R}}^3</math>, the topologists locally model all other [[3-manifold]]s.
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| In physics, our three-dimensional space is viewed as embedded in four-dimensional [[space-time]], called [[Minkowski space]] (see [[special relativity]]). The idea behind space-time is that time is [[hyperbolic-orthogonal]] to each of the three spatial dimensions.
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| ==Geometry==
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| ===Polytopes===
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| {{main|Polyhedron}}
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| In three dimensions, there are nine regular polytopes: the five convex [[Platonic solid]]s and the four nonconvex [[Kepler-Poinsot polyhedra]].
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| {| class=wikitable
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| |+ Regular polytopes in three dimensions
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| |- align=center
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| !Class
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| !colspan=5|[[Platonic solid]]s
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| !colspan=4|[[Kepler-Poinsot polyhedra]]
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| |-
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| ![[Polyhedral symmetry|Symmetry]]
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| ![[Tetrahedral symmetry|T<sub>d</sub>]]
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| !colspan=2|[[Octahedral symmetry|O<sub>h</sub>]]
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| !colspan=6|[[Icosahedral symmetry|I<sub>h</sub>]]
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| |-
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| ![[Coxeter group]]
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| !A<sub>3</sub>
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| !colspan=2|BC<sub>3</sub>
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| !colspan=6|H<sub>3</sub>
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| |- align=center
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| ![[Symmetry order|Order]]
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| |24
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| |colspan=2|48
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| |colspan=6|120
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| |- align=center
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| ![[Regular polyhedron|Regular<br>polyhedron]]
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| |[[File:Tetrahedron.svg|50px]]<br>[[Tetrahedron|{3,3}]]
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| |[[File:Hexahedron.svg|50px]]<br>[[Cube|{4,3}]]
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| |[[File:Octahedron.svg|50px]]<br>[[Octahedron|{3,4}]]
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| |[[File:POV-Ray-Dodecahedron.svg|50px]]<br>[[Dodecahedron|{5,3}]]
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| |[[File:Icosahedron.svg|50px]]<br>[[Icosahedron|{3,5}]]
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| |[[File:SmallStellatedDodecahedron.jpg|50px]]<br>[[Small stellated dodecahedron|{5/2,5}]]
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| |[[File:GreatDodecahedron.jpg|50px]]<br>[[Great dodecahedron|{5,5/2}]]
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| |[[File:GreatStellatedDodecahedron.jpg|50px]]<br>[[Great stellated dodecahedron|{5/2,3}]]
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| |[[File:GreatIcosahedron.jpg|50px]]<br>[[Great icosahedron|{3,5/2}]]
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| |}
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| ===Hypersphere===
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| {{main|Sphere}}
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| [[Image:Sphere wireframe 10deg 6r.svg|right|thumb|A two-dimensional [[3D projection#Perspective projection|perspective projection]] of a sphere]]
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| A [[hypersphere]] in 3-space (also called a '''2-sphere''' because its surface is 2-dimensional) consists of the set of all points in 3-space at a fixed distance ''r'' from a central point P. The volume enclosed by this surface is:
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| <math>V = \frac{4}{3}\pi r^{3}</math>
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| Another hypersphere, but having a three-dimensional surface is the '''3-sphere''': points equidistant to the origin of the euclidean space <math>\mathbb{R}^4</math> at distance one. If any position is <math>P=(x,y,z,t)</math>, then <math>x^2+y^2+z^2+t^2=1</math> characterize a point in the 3-sphere.
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| ===Orthogonality===
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| In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: up/down (altitude), north/south (latitude), and east/west (longitude). These pairs of directions are mutually [[orthogonal]]: They are at right angles to each other. In mathematical terms, they lie on three [[coordinate axes]], usually labelled ''x'', ''y'', and ''z''. The [[z-buffer]] in computer graphics refers to this ''z''-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.
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| ===Coordinate systems===
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| {{main|Coordinate system}}
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| In mathematics, [[analytic geometry]] (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three [[coordinate axis|coordinate axes]] are given, each perpendicular to the other two at the [[Origin (mathematics)|origin]], 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 (mathematics)|origin]] measured along the given axis, which is equal to the distance of that point from the plane determined by the other two 2 axes.
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| Other popular methods of describing the location of a point in three-dimensional space include [[cylindrical coordinates]] and [[spherical coordinates]], though there is an infinite number of possible methods. See [[Euclidean space]].
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| Below are images of the above-mentioned systems.
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| <gallery>
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| Image:Coord XYZ.svg|[[Cartesian coordinate system]]
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| Image:Cylindrical Coordinates.svg|[[Cylindrical coordinate system]]
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| Image:Spherical Coordinates (Colatitude, Longitude).svg|[[Spherical coordinate system]]
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| </gallery>
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| ==See also==
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| * [[3-manifold]]s
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| * [[Dimensional analysis]]
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| * [[Three-dimensional graph]]
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| * [[Two-dimensional space]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{commons category|3D}}
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| * {{wiktionary-inline|three-dimensional}}
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| * {{MathWorld |title=Four-Dimensional Geometry |id=Four-DimensionalGeometry}}
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| * [http://www.numbertheory.org/book/cha8.pdf Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry] Keith Matthews from [[University of Queensland]], 1991
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| {{Dimension topics}}
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| [[Category:Euclidean solid geometry|*]]
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| [[Category:Analytic geometry]]
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| [[Category:Multi-dimensional geometry]]
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Jayson Berryhill is how I'm known as and my wife doesn't like it at all. Credit authorising is exactly where my primary income comes from. Kentucky is where I've usually been residing. To perform lacross is the factor I love most of all.
My blog ... psychic phone readings (click this site)