View source for Three-dimensional space

{{short description|Geometric model of the physical space}}
{{For|a broader, less mathematical treatment related to this topic|Space}}
{{redir|Three-dimensional||3D (disambiguation)}}
{{more footnotes|date=April 2016}}
[[File:Coord planes color.svg|right|thumb|300px|A representation of a three-dimensional [[Cartesian coordinate system]] with the ''x''-axis pointing towards the observer.]]
{{General geometry |expanded=3d}}
'''Three-dimensional space''' (also: '''3-space''' or, rarely, '''tri-dimensional space''') is a geometric setting in which three values (called [[parameter]]s) are required to determine the position of an element (i.e., [[Point (mathematics)|point]]). This is the informal meaning of the term [[dimension]].

In [[physics]] and [[mathematics]], a [[Euclidean vector|sequence]] of {{math|''n''}} [[Real number|numbers]] can be understood as a location in {{math|''n''}}-dimensional space. When {{math|1=''n'' = 3}}, the set of all such locations is called {{anchor|3D EUCLIDEAN SPACE}}'''three-dimensional [[Euclidean space]]'''. It is commonly represented by the symbol {{math|ℝ<sup>3</sup>}}. This serves as a three-parameter model of the physical [[universe]] (that is, the spatial part, without considering time) in which all known [[matter]] exists. However, this space is only one example of a large variety of spaces in three dimensions called [[3-manifold]]s. In this classical example, when the three values refer to measurements in different directions ([[Coordinate system|coordinates]]), any three directions can be chosen, provided that [[Vector space|vectors]] in these directions do not all lie in the same [[Two-dimensional space|2-space]] ([[plane (geometry)|plane]]). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms ''[[width]]'', ''[[height]]'', ''[[Elevation|depth]]'', and ''[[length]]''.

==In Euclidean geometry==

===Coordinate systems===
{{main|Coordinate system}}
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 {{math|''x'', ''y''}}, and {{math|''z''}}. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of [[real number]]s, 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.<ref name="Hughes">{{cite book|last1=Hughes-Hallett|first1=Deborah|last2=McCallum|first2=William G.|last3=Gleason|first3=Andrew M.|title=Calculus : Single and Multivariable|date=2013|publisher=John wiley|isbn=978-0470-88861-2|edition=6}}</ref>

Other popular methods of describing the location of a point in three-dimensional space include [[cylindrical coordinates]] and [[spherical coordinates]], though there are an infinite number of possible methods. See [[Euclidean space]].

Below are images of the above-mentioned systems.

<gallery>
Image:Coord XYZ.svg|[[Cartesian coordinate system]]
Image:Cylindrical Coordinates.svg|[[Cylindrical coordinate system]]
Image:Spherical Coordinates (Colatitude, Longitude).svg|[[Spherical coordinate system]]
</gallery>

===Lines and planes===

Two distinct points always determine a (straight) [[Line (geometry)|line]]. Three distinct points are either [[collinear]] or determine a unique plane. Four distinct points can either be collinear, [[coplanar]] or determine the entire space.

Two distinct lines can either intersect, be [[Parallel line|parallel]] or be [[Skew lines|skew]]. Two parallel lines, or [[two intersecting lines]], lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.

Two distinct planes can either meet in a common line or are parallel (do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.

A line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line.

A [[hyperplane]] is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of cartesian coordinates, the points of a hyperplane satisfy a single [[linear equation]], so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations, each representing a plane having this line as a common intersection.

[[Varignon's theorem]] states that the midpoints of any quadrilateral in ℝ<sup>3</sup> form a [[parallelogram]], and so, are coplanar.

===Spheres and balls===
{{main|Sphere}}
[[File:Sphere wireframe 10deg 6r.svg|right|thumb|A [[3D projection#Perspective projection|perspective projection]] of a sphere onto two dimensions]]
A [[sphere]] in 3-space (also called a '''2-sphere''' because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance {{math|''r''}} from a central point {{mvar|P}}. The solid enclosed by the sphere is called a [[Ball (mathematics)|'''ball''']] (or, more precisely a '''3-ball'''). The volume of the ball is given by

:<math>V = \frac{4}{3}\pi r^{3}</math>.

Another type of sphere arises from a 4-ball, whose three-dimensional surface is the '''3-sphere''': points equidistant to the origin of the euclidean space {{math|ℝ<sup>4</sup>}}. If a point has coordinates, {{math|''P''(''x'', ''y'', ''z'', ''w'')}}, then {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''w''<sup>2</sup> = 1}} characterizes those points on the unit 3-sphere centered at the origin.

===Polytopes===
{{main|Polyhedron}}
In three dimensions, there are nine regular polytopes: the five convex [[Platonic solid]]s and the four nonconvex [[Kepler-Poinsot polyhedra]].

{| class=wikitable
|+ Regular polytopes in three dimensions
|- align=center
!Class
!colspan=5|[[Platonic solid]]s
!colspan=4|[[Kepler-Poinsot polyhedra]]
|-
![[Polyhedral symmetry|Symmetry]]
![[Tetrahedral symmetry|T<sub>d</sub>]]
!colspan=2|[[Octahedral symmetry|O<sub>h</sub>]]
!colspan=6|[[Icosahedral symmetry|I<sub>h</sub>]]
|-
![[Coxeter group]]
!A<sub>3</sub>, [3,3]
!colspan=2|B<sub>3</sub>, [4,3]
!colspan=6|H<sub>3</sub>, [5,3]
|- align=center
![[Symmetry order|Order]]
|24
|colspan=2|48
|colspan=6|120
|- align=center
![[Regular polyhedron|Regular<br>polyhedron]]
|[[File:Tetrahedron.svg|50px]]<br>[[Tetrahedron|{3,3}]]
|[[File:Hexahedron.svg|50px]]<br>[[Cube|{4,3}]]
|[[File:Octahedron.svg|50px]]<br>[[Octahedron|{3,4}]]
|[[File:POV-Ray-Dodecahedron.svg|50px]]<br>[[Dodecahedron|{5,3}]]
|[[File:Icosahedron.svg|50px]]<br>[[Icosahedron|{3,5}]]
|[[File:SmallStellatedDodecahedron.jpg|50px]]<br>[[Small stellated dodecahedron|{5/2,5}]]
|[[File:GreatDodecahedron.jpg|50px]]<br>[[Great dodecahedron|{5,5/2}]]
|[[File:GreatStellatedDodecahedron.jpg|50px]]<br>[[Great stellated dodecahedron|{5/2,3}]]
|[[File:GreatIcosahedron.jpg|50px]]<br>[[Great icosahedron|{3,5/2}]]
|}

===Surfaces of revolution===
{{main|Surface of revolution}}
A [[Surface (topology)|surface]] generated by revolving a plane [[curve]] about a fixed line in its plane as an axis is called a [[surface of revolution]]. The plane curve is called the ''generatrix'' of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.

Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular [[Cone (mathematics)|cone]] with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, the surface of revolution is a circular [[Cylinder (geometry)|cylinder]].

===Quadric surfaces===
{{main|Quadric surface}}
In analogy with the [[conic section]]s, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely,
:<math>Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0,</math>
where {{math|''A'', ''B'', ''C'', ''F'', ''G'', ''H'', ''J'', ''K'', ''L''}} and {{math|''M''}} are real numbers and not all of {{math|''A'', ''B'', ''C'', ''F'', ''G''}} and {{math|''H''}} are zero is called a '''quadric surface'''.<ref name=Brannan>{{harvnb|Brannan|Esplen|Gray|1999|pages=34–5}}</ref>

There are six types of [[degeneracy (mathematics)|non-degenerate]] quadric surfaces:
# [[Ellipsoid]]
# [[Hyperboloid of one sheet]]
# [[Hyperboloid of two sheets]]
# [[Elliptic cone]]
# [[Elliptic paraboloid]]
# [[Hyperbolic paraboloid]]

The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane {{pi}} and all the lines of {{math|ℝ<sup>3</sup>}} through that conic that are normal to {{pi}}).<ref name=Brannan /> Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.

Both the hyperboloid of one sheet and the hyperbolic paraboloid are [[ruled surface]]s, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family.<ref>{{harvnb|Brannan|Esplen|Gray|1999|pages=41–2}}</ref> Each family is called a [[regulus (geometry)|regulus]].

==In linear algebra==
Another 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.

===Dot product, angle, and length===
{{main|Dot product}}
A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in {{math|ℝ<sup>3</sup>}} can be represented by an ordered triple of real numbers. These numbers are called the '''components''' of the vector.

The dot product of two vectors {{math|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>''3''</sub>]}} and {{math|1='''B''' = [''B''<sub>1</sub>, ''B''<sub>2</sub>, ''B''<sub>''3''</sub>]}} is defined as:<ref>{{harvnb|Anton|1994|page=133}}</ref>

:<math>\mathbf{A}\cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3.</math>

The magnitude of a vector {{math|'''A'''}} is denoted by {{math|{{!}}{{!}}'''A'''{{!}}{{!}}}}. The dot product of a vector {{math|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>''3''</sub>]}} with itself is
:<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2 = A_1^2 + A_2^2 + A_3^2,</math>
which gives
: <math> \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A} = \sqrt{A_1^2 + A_2^2 + A_3^2},</math>
the formula for the [[Euclidean length]] of the vector.

Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors {{math|'''A'''}} and {{math|'''B'''}} is given by<ref>{{harvnb|Anton|1994|page=131}}</ref>
:<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,</math>
where {{math|''θ''}} is the [[angle]] between {{math|'''A'''}} and {{math|'''B'''}}.

===Cross product===
{{main|Cross product}}
The [[cross product]] or '''vector product''' is a [[binary operation]] on two [[Euclidean vector|vector]]s in three-dimensional [[Euclidean space|space]] and is denoted by the symbol  ×. The cross product '''a''' × '''b''' of the vectors '''a''' and '''b''' is a vector that is [[perpendicular]] to both and therefore [[Normal (geometry)|normal]] to the plane containing them. It has many applications in mathematics, [[physics]], and [[engineering]].

The space and product form an [[algebra over a field]], which is neither [[commutative]] nor [[associative]], but is a [[Lie algebra]] with the cross product being the Lie bracket.

One can in ''n'' dimensions take the product of {{nowrap|''n'' − 1}} vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and [[Seven-dimensional cross product|seven dimensions]].<ref name=Massey2>{{cite journal |title=Cross products of vectors in higher dimensional Euclidean spaces |author=WS Massey |year=1983 |jstor=2323537|quote=If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space. |pages=697–701 |journal=The American Mathematical Monthly |volume=90 |issue=10 |ref=harv |doi=10.2307/2323537}}</ref>
[[File:Cross product vector.svg|thumb|right|The cross-product in respect to a right-handed coordinate system]]

==In calculus==
{{main|vector calculus}}

===Gradient, divergence and curl===
In a rectangular coordinate system, the gradient is given by

:<math>\nabla f = \frac{\partial f}{\partial x} \mathbf{i} +
\frac{\partial f}{\partial y}  \mathbf{j} +
\frac{\partial f}{\partial z} \mathbf{k}</math>

The divergence of a [[continuously differentiable]] [[vector field]] '''F''' = ''U'' '''i''' + ''V'' '''j''' + ''W'' '''k''' is equal to the [[scalar (mathematics)|scalar]]-valued function:

:<math>\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F}
=\frac{\partial U}{\partial x}
+\frac{\partial V}{\partial y}
+\frac{\partial W}{\partial z
}. </math>

Expanded in [[Cartesian coordinate system|Cartesian coordinates]] (see [[Del in cylindrical and spherical coordinates]] for [[Spherical coordinate system|spherical]] and [[Cylindrical coordinate system|cylindrical]] coordinate representations), the curl ∇ × '''F''' is, for '''F''' composed of [''F''<sub>x</sub>, ''F''<sub>y</sub>, ''F''<sub>z</sub>]:

:<math>\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\  \\
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 \\  F_x & F_y & F_z \end{vmatrix}</math>

where '''i''', '''j''', and '''k''' are the [[unit vector]]s for the ''x''-, ''y''-, and ''z''-axes, respectively. This expands as follows:<ref>Arfken, p. 43.</ref>

:<math>\left(\frac{\partial F_z}{\partial y}  - \frac{\partial F_y}{\partial z}\right) \mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}</math>

===Line integrals, surface integrals, and volume integrals===
For some [[scalar field]] ''f'' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R''', the line integral along a [[piecewise smooth]] [[curve]] ''C'' ⊂  ''U'' is defined as
:<math>\int\limits_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.</math>
where '''r''': [a, b] → ''C'' is an arbitrary [[bijective]] [[parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C'' and <math>a < b</math>.

For a [[vector field]] '''F''' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>, the line integral along a [[piecewise smooth]] [[curve]] ''C'' ⊂ ''U'', in the direction of '''r''', is defined as

:<math>\int\limits_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.</math>

where · is the [[dot product]] and '''r''': [a, b] → ''C'' is a [[bijective]] [[parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C''.

A [[surface integral]]  is a generalization of [[multiple integral]]s to integration over [[Surface (topology)|surface]]s. It can be thought of as the [[double integral]] analog of the [[line integral]]. To find an explicit formula for the surface integral, we need to [[Coordinate system|parameterize]] the surface of interest, ''S'', by considering a system of [[curvilinear coordinates]] on ''S'', like the [[Geographic coordinate system|latitude and longitude]] on a [[sphere]]. Let such a parameterization be '''x'''(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|plane]]. Then, the surface integral is given by

:<math>
\iint_{S} f \,\mathrm dS
= \iint_{T} f(\mathbf{x}(s, t)) \left\|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right\| \mathrm ds\, \mathrm dt
</math>
where the expression between bars on the right-hand side is the [[Magnitude (mathematics)|magnitude]] of the [[cross product]] of the [[partial derivative]]s of '''x'''(''s'', ''t''), and is known as the surface [[volume element|element]]. Given a vector field '''v''' on ''S'', that is a function that assigns to each '''x''' in ''S'' a vector '''v'''('''x'''), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.

A [[volume integral]] refers to an [[integral]] over a 3-[[dimension]]al domain.

It can also mean a [[multiple integral|triple integral]] within a region ''D'' in '''R'''<sup>3</sup> of a [[function (mathematics)|function]] <math>f(x,y,z),</math> and is usually written as:

:<math>\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.</math>

===Fundamental theorem of line integrals===
{{main|Fundamental theorem of line integrals}}
The [[fundamental theorem of line integrals]], says that a [[line integral]] through a [[gradient]] field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let <math> \varphi : U \subseteq \mathbb{R}^n \to \mathbb{R}</math>. Then

:<math> \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\varphi(\mathbf{r})\cdot d\mathbf{r}. </math>

===Stokes' theorem===
{{main|Stokes' theorem}}
[[Stokes' theorem]] relates the [[surface integral]] of the [[Curl (mathematics)|curl]] of a [[vector field]] F over a surface Σ in Euclidean three-space to the [[line integral]] of the vector field over its boundary ∂Σ:

:<math> \iint_{\Sigma} \nabla \times \mathbf{F} \cdot \mathrm{d}\mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d} \mathbf{r}. </math>

===Divergence theorem===
{{main|Divergence theorem}}
Suppose {{mvar|V}} is a subset of <math>\mathbb{R}^n</math> (in the case of {{math|''n'' {{=}} 3, ''V''}} represents a volume in 3D space) which is [[compact space|compact]] and has a [[piecewise]] [[smooth surface|smooth boundary]] {{mvar|S}} (also indicated with {{math|∂''V'' {{=}} ''S''&thinsp;}}). If {{math|'''F'''}} is a continuously differentiable vector field defined on a neighborhood of {{mvar|V}}, then the [[divergence theorem]] says:<ref name=spiegel>{{cite book |author1=M. R. Spiegel |author2=S. Lipschutz |author3=D. Spellman | title = Vector Analysis | edition = 2nd | series = Schaum’s Outlines | publisher = McGraw Hill | location = USA | year = 2009 | isbn = 978-0-07-161545-7 }}</ref>

:{{oiint
| preintegral = <math>\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV=</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>(\mathbf{F}\cdot\mathbf{n})\,dS .</math>
}}

The left side is a [[volume integral]] over the volume {{mvar|V}}, the right side is the [[surface integral]] over the boundary of the volume {{mvar|V}}. The closed manifold {{math|∂''V''}} is quite generally the boundary of {{mvar|V}} oriented by outward-pointing [[Surface normal|normals]], and {{math|'''n'''}} is the outward pointing unit normal field of the boundary {{math|∂''V''}}. ({{math|''d'''''S'''}} may be used as a shorthand for {{math|'''n'''''dS''}}.)

==In topology==
[[File:WikipediaGlobeOnePiece.stl|thumb|upright=1.2|[[Wikipedia]]'s globe logo in 3-D]]
Three-dimensional space has a number of topological 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>

In [[differential geometry]] the generic three-dimensional spaces are [[3-manifold]]s, which locally resemble <math>{\mathbb{R}}^3</math>.

==In finite geometry==
Many ideas of dimension can be tested with [[finite geometry]]. The simplest instance is [[PG(3,2)]] which has [[Fano plane]]s as its 2-dimensional subspaces. It is an instance of [[Galois geometry]], a study of [[projective geometry]] using [[finite field]]s. Thus, for any Galois field GF(''q''), there is a [[projective space]] PG(3,''q'') of three dimensions. For example, any three [[skew lines]] in PG(3,''q'') are contained in exactly one [[regulus (geometry)|regulus]].<ref>[[Albrecht Beutelspacher]] & Ute Rosenbaum (1998) ''Projective Geometry'', page 72, [[Cambridge University Press]]  {{ISBN|0-521-48277-1}}</ref>

==See also==
* [[Dimensional analysis]]
* [[Distance from a point to a plane]]
* {{section link|Skew lines|Distance}}
* [[Graph of a function of two variables|Three-dimensional graph]]

==Notes==
{{reflist}}

==References==
* {{citation|first=Howard|last=Anton|title=Elementary Linear Algebra|edition=7th|year=1994|publisher=John Wiley & Sons|isbn=978-0-471-58742-2}}
* [[George B. Arfken|Arfken, George B.]] and Hans J. Weber. ''Mathematical Methods For Physicists'', Academic Press; 6 edition (June 21, 2005). {{isbn|978-0-12-059876-2}}.
* {{citation|first1=David A.|last1=Brannan|first2=Matthew F.|last2=Esplen|first3=Jeremy J.|last3=Gray|title=Geometry|year=1999|publisher=Cambridge University Press|isbn=978-0-521-59787-6}}

==External links==
{{Wikiquote}}
{{commons category|3D}}
* {{wiktionary-inline|three-dimensional}}
* {{MathWorld |title=Four-Dimensional Geometry |id=Four-DimensionalGeometry}}
* [http://www.numbertheory.org/book/cha8.pdf Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry] Keith Matthews from [[University of Queensland]], 1991
{{Dimension topics}}

[[Category:Euclidean solid geometry|*]]
[[Category:Analytic geometry]]
[[Category:Multi-dimensional geometry]]
[[Category:3 (number)]]