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N-dimensional space

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	Topics N-dimensional space Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , an n-dimensional space is a topological spaceTopological space Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.... whose dimensionDimension In common usage, a dimension is a parameter or measurement required to define the characteristics of an object—i.e.... is n (where n is a fixed natural numberFacts About Natural number In mathematics, a natural number is either a positive integer or a non-negative integer .... ). The archetypical example is n-dimensional Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... , which describes Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... in n dimensions. Many familiar geometric objects can be generalized to any number of dimensions. For example, the two-dimensional triangleTriangle A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line seg... and the three-dimensional tetrahedronTetrahedron A tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex.... can be seen as specific instances of the n-dimensional simplexSimplex In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle.... . Also, the circleCircle In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed poi... and the sphereSphere A sphere is a perfectly symmetrical geometrical object.... can be seen as specific instances of the n-dimensional hypersphereHypersphere In mathematics, a hypersphere is a sphere which has dimension 3 or higher.... . More generally, an n-dimensional manifold is a space that locally looks like n-dimensional Euclidean space, but whose global structure may be non-Euclidean. There are also notions of dimension (such as Hausdorff dimensionHausdorff dimension In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space.... in topologyTopology Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation ; these are ... and Kodaira dimension in algebraic geometryAlgebraic geometry Overview Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative... ) that apply to even more general spaces. Sometimes it is convenient in science to describe an object with n degrees of freedomDegrees of freedom (physics and chemistry) Summary *Degrees of freedom* is a general term used in explaining dependence on parameters, and implying the possibility of counti... as if it were a point in some n-dimensional space. For example, classical mechanicsClassical mechanics Classical mechanics is used to describe the motion of macroscopic objects, from projectiles to parts of machinery, as well a... describes the three-dimensional position and momentum of a point particle as a point in 6-dimensional phase spacePhase space In mathematics and physics, phase space is the space in which all possible states of a system are represented, with each pos... . Rotation Regular polygons with odd numbers of sides are not as easily generalized to higher dimensions, however, unless they are triangles. This is because they exhibit rotational symmetryRotational symmetry Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.... , as do many spirals, swastikas, and other two-dimensional geometric designs. RotationRotation Overview Rotation is the movement of an object in a circular motion.... is motion of points in a circular (not spherical) path. A point cannot trace out a spherical path as it moves through space, so rotation is inherently a two-dimensions-at-a-time phenomenon. The sine and cosine functions, which describe circular motion, are a pair of functions that cannot readily be expanded to form a triple. A matrixMatrix (mathematics) In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities tha... , however, is very easily generalized to any natural number of dimensions. A rotation matrixRotation matrix A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but n... , which is a type of square matrix, requires at least two rows and at least two columns. When a rotation matrix has an odd number of rows and columns, and describes rotation around a coordinate axis, one row and one column are uninvolved. That row and that column contain all zero entries except for a single one. (See the 3 x 3 matrix below.) This is consistent with the fact that three-dimensional objects rotate around a linear axis, and the points that are on that axis rotate "in place," moving in a circle with a radius of zero, as it were. This 3 × 3 matrix represents rotation around the x-axis: where is the roll angle. Analogously, this 4 × 4 matrix represents rotation around the wx-plane: where is one of six angles that are analogous to the roll angle. By examining rotation matrices of various sizes, several things can be inferred: First, that linear axes of rotation can be found in any odd number of dimensions, including three. Secondly, that objects in four dimensions can rotate around a two-dimensional axis, in which case the points that are in that plane rotate "in place." Thirdly, objects in four dimensions can also rotate at two different speeds at the same time, since a two-dimensional axis of rotation offers enough space for rotation to begin at a speed that is different from that of the rotation that is already taking place. More generally, objects in an even number of dimensions can rotate at up to half that number of different speeds simultaneously. This is a 4 × 4 matrix that represents rotation at two speeds at once: See also n-dimensional calculus facet (mathematics)Facet (mathematics) A facet* of an n-dimensional simplex is one of its-dimensional "faces."... fourth dimensionFourth dimension The concept of a fourth dimension is one that is often described in considering its physical implications; that is, we know ... four-vectorFacts About Four-vector In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space, whose components t... vector decompositionVector decomposition Vector decomposition refers to decomposing a vector of Rn into several vectors, each linearly independent.... coordinate spaceCoordinate space In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-... kissing number problemKissing number problem In geometry, the kissing number is the maximum number of spheres of radius 1 that can simultaneously touch the unit sphere i...