Three-dimensional space

three-dimensional3Dthree dimensions3-D3-dimensionalspatial3-spacethree dimensional3D spaceEuclidean 3-space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).wikipedia
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Euclidean space

EuclideanspaceEuclidean vector space
, the set of all such locations is called three-dimensional Euclidean space. The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×.
Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

Dimension

dimensionsdimensionalone-dimensional
This is the informal meaning of the term dimension. A volume integral refers to an integral over a 3-dimensional domain.
The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

3-manifold

3-manifoldsthree-manifoldthree-dimensional manifold
However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds.
A topological space X is a 3-manifold if it is a second-countable Hausdorff space and if every point in X has a neighbourhood that is homeomorphic to Euclidean 3-space.

Vector space

vectorvector spacesvectors
In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane).
In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.

Plane (geometry)

planeplanarplanes
In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane).
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.

Universe

physical worldThe Universeuniverses
. 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.
On the average, space is observed to be very nearly flat (with a curvature close to zero), meaning that Euclidean geometry is empirically true with high accuracy throughout most of the Universe.

Parallel (geometry)

parallelparallel linesparallelism
Two distinct lines can either intersect, be parallel or be skew.
By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel.

Platonic solid

Platonic solidsPlatonicregular polyhedra
In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.
In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates.
Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions).

Tetrahedron

tetrahedraltetrahedra{3,3}
The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.

Elevation

hightopographic elevationelevated
Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and length.
The term elevation is mainly used when referring to points on the Earth's surface, while altitude or geopotential height is used for points above the surface, such as an aircraft in flight or a spacecraft in orbit, and depth is used for points below the surface.

Line (geometry)

linestraight linelines
Two distinct points always determine a (straight) line.
In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes.

Line–line intersection

intersectintersectionintersections
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.
In three-dimensional Euclidean geometry, if two lines are not in the same plane they are called skew lines and have no point of intersection.

Cone

conicalconesconic
If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex (apex) the point of intersection.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

Cross product

vector cross productvector productcross-product
The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×.
In mathematics, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space and is denoted by the symbol \times.

Sphere

sphericalhemisphereglobose
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 To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere.
Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, in a process called sphere eversion.

Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesian
Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of [F x, F y, F z ]:
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).

Integral

integrationintegral calculusdefinite integral
A volume integral refers to an integral over a 3-dimensional domain.
In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

Dot product

scalar productdotinner product
where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.
For instance, in three-dimensional space, the dot product of vectors and is:

Linear algebra

linearlinear algebraiclinear-algebraic
Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial.
In this new (at that time) geometry, now called Cartesian geometry, points are represented by Cartesian coordinates, which are sequences of three real numbers (in the case of the usual three-dimensional space).

Curvilinear coordinates

curvilinearcurvilinear coordinate systemcurvilinear coordinate
To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere.
For now, consider 3-D space.

Divergence theorem

Gauss's theoremGauss theoremdivergent-free
is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:
represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary S (also indicated with

Projective space

projectiveprojective coordinatesprojective ''n''-space
Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions.
Such statements are suggested by the study of perspective, which may be considered as a central projection of the three dimensional space onto a plane (see Pinhole camera model).

Perpendicular

perpendicularlyPerpendicularitynormal
The cross product a × b of the vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them.
Up to three lines in three-dimensional space can be pairwise perpendicular, as exemplified by the x, y, and z axes of a three-dimensional Cartesian coordinate system.

Parameter

parametersparametricargument
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).