higher-dimensional space

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OR:

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...
and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the dimension of a mathematical space (or object) is informally defined as the minimum number of
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
needed to specify any point within it. Thus, a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, such as the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
of a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...
or
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
are required to locate a point on the surface of a sphere. A
two-dimensional Euclidean space In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
is a two-dimensional space on the plane. The inside of a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the onl ...
, a cylinder or a sphere is
three-dimensional Three-dimensional space (also: 3D space, 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). This is the informal ...
(3D) because three coordinates are needed to locate a point within these spaces. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classica ...
,
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
and
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
are different categories and refer to
absolute space and time Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Before Newton A version of the concept of absolute space (in the sense of a pre ...
. That conception of the world is a
four-dimensional space A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
but not the one that was found necessary to describe
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
. The four dimensions (4D) of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an
observer An observer is one who engages in observation or in watching an experiment. Observer may also refer to: Computer science and information theory * In information theory, any system which receives information from an object * State observer in co ...
.
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
first approximates the universe without
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
; the
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...
describe spacetime with matter and gravity. 10 dimensions are used to describe
superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string theo ...
(6D
hyperspace In science fiction, hyperspace (also known as nulspace, subspace, overspace, jumpspace and similar terms) is a concept relating to higher dimensions as well as parallel universes and a faster-than-light (FTL) method of interstellar travel. ...
+ 4D), 11 dimensions can describe
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such ...
and
M-theory M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten ...
(7D hyperspace + 4D), and the state-space of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
is an infinite-dimensional
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
. The concept of dimension is not restricted to physical objects. s frequently occur in mathematics and the sciences. They may be
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s or more general parameter spaces or configuration spaces such as in Lagrangian or
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''moment ...
; these are abstract spaces, independent of the physical space in which we live.

# In mathematics

In mathematics, the dimension of an object is, roughly speaking, the number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of a point that moves on this object. In other words, the dimension is the number of independent
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s or
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
that are needed for defining the position of a point that is constrained to be on the object. For example, the dimension of a point is zero; the dimension of a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two, etc. The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded. For example, a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
, such as a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve. This is independent from the fact that a curve cannot be embedded in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension lower than two, unless it is a line. The dimension of Euclidean -space is . When trying to generalize to other types of spaces, one is faced with the question "what makes -dimensional?" One answer is that to cover a fixed
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...
in by small balls of radius , one needs on the order of such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
, but there are also other answers to that question. For example, the boundary of a ball in looks locally like and this leads to the notion of the
inductive dimension In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
. While these notions agree on , they turn out to be different when one looks at more general spaces. A
tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of ...
is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract ''has four dimensions''", mathematicians usually express this as: "The tesseract ''has dimension 4''", or: "The dimension of the tesseract ''is'' 4" or: 4D. Although the notion of higher dimensions goes back to
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Math ...
, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
,
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ...
,
Ludwig Schläfli Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
. Riemann's 1854
Habilitationsschrift Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
, Schläfli's 1852 ''Theorie der vielfachen Kontinuität'', and Hamilton's discovery of the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a qua ...
s and
John T. Graves John Thomas Graves (4 December 1806 – 29 March 1870) was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions in October 1843 and then discov ...
' discovery of the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s in 1843 marked the beginning of higher-dimensional geometry. The rest of this section examines some of the more important mathematical definitions of dimension.

## Vector spaces

The dimension of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of a basis) is often referred to as the ''Hamel dimension'' or ''algebraic dimension'' to distinguish it from other notions of dimension. For the non-
free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
case, this generalizes to the notion of the
length of a module In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. ...
.

## Manifolds

The uniquely defined dimension of every connected topological
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
can be calculated. A connected topological manifold is
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Pr ...
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to Euclidean -space, in which the number is the manifold's dimension. For connected
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s, the dimension is also the dimension of the tangent vector space at any point. In
geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated ...
, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases are simplified by having extra space in which to "work"; and the cases and are in some senses the most difficult. This state of affairs was highly marked in the various cases of the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by ...
, in which four different proof methods are applied.

### Complex dimension

The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
, it is sometimes useful in the study of
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...
to work over the
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
instead. A complex number (''x'' + ''iy'') has a
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
''x'' and an
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
''y'', in which x and y are both real numbers; hence, the complex dimension is half the real dimension. Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, becomes a
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
of one complex dimension.

## Varieties

The dimension of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...
may be defined in various equivalent ways. The most intuitive way is probably the dimension of the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at any
Regular point of an algebraic variety In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
. Another intuitive way is to define the dimension as the number of
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyp ...
s that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety. An
algebraic set Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains $V_0\subsetneq V_1\subsetneq \cdots \subsetneq V_d$ of sub-varieties of the given algebraic set (the length of such a chain is the number of "$\subsetneq$"). Each variety can be considered as an
algebraic stack In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's rep ...
, and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Specifically, if ''V'' is a variety of dimension ''m'' and ''G'' is an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
of dimension ''n'' acting on ''V'', then the
quotient stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. T ...
'V''/''G''has dimension ''m'' − ''n''.

## Krull dimension

The
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
of a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is the maximal length of chains of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s in it, a chain of length ''n'' being a sequence $\mathcal_0\subsetneq \mathcal_1\subsetneq \cdots \subsetneq\mathcal_n$ of prime ideals related by inclusion. It is strongly related to the dimension of an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring of the polynomials on the variety. For an
algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...
, the dimension as
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
is finite if and only if its Krull dimension is 0.

## Topological spaces

For any
normal topological space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
, the Lebesgue covering dimension of is defined to be the smallest
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''n'' for which the following holds: any
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
has an open refinement (a second open cover in which each element is a subset of an element in the first cover) such that no point is included in more than elements. In this case dim . For a manifold, this coincides with the dimension mentioned above. If no such integer exists, then the dimension of is said to be infinite, and one writes dim . Moreover, has dimension −1, i.e. dim if and only if is empty. This definition of covering dimension can be extended from the class of normal spaces to all
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...
s merely by replacing the term "open" in the definition by the term "functionally open". An
inductive dimension In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
may be defined inductively as follows. Consider a
discrete set ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a ''new direction'', one obtains a 2-dimensional object. In general one obtains an ()-dimensional object by dragging an -dimensional object in a ''new'' direction. The inductive dimension of a topological space may refer to the ''small inductive dimension'' or the ''large inductive dimension'', and is based on the analogy that, in the case of metric spaces, balls have -dimensional boundary (topology), boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover, the boundary of a discrete set of points is the empty set, and therefore the empty set can be taken to have dimension -1. Similarly, for the class of
CW complexes A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...
, the dimension of an object is the largest for which the -skeleton is nontrivial. Intuitively, this can be described as follows: if the original space can be continuously deformed into a collection of higher-dimensional triangles joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.

## Hausdorff dimension

The
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
is useful for studying structurally complicated sets, especially
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
s. The Hausdorff dimension is defined for all
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s and, unlike the dimensions considered above, can also have non-integer real values.Fractal Dimension
, Boston University Department of Mathematics and Statistics
The
box dimension A box ( plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and ca ...
or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is mea ...
s that work for highly irregular sets and attain non-integer positive real values.

## Hilbert spaces

Every
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For exam ...
, and any two such bases for a particular space have the same
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's
Hamel dimension In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dis ...
is finite, and in this case the two dimensions coincide.

# In physics

## Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward and forward is just as the name of the direction implies; ''i.e.'', moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See
Space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
and
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
.)

## Time

A temporal dimension, or time dimension, is a dimension of time. Time is often referred to as the " fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move in one direction. The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classica ...
are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the
laws of thermodynamics The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various parame ...
(we perceive time as flowing in the direction of increasing
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
). The best-known treatment of time as a dimension is Poincaré and
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
(and extended to
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...
), which treats perceived space and time as components of a four-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
, known as
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, and in the special, flat case as
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
. Time is different from other spatial dimensions as time operates in all spatial dimensions. Time operates in the first, second and third as well as theoretical spatial dimensions such as a fourth spatial dimension. Time is not however present in a single point of absolute infinite singularity as defined as a
geometric point This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...
, as an infinitely small point can have no change and therefore no time. Just as when an object moves through positions in space, it also moves through positions in time. In this sense the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
moving any object to change is ''time''.

In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt to unify the four
fundamental forces In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electro ...
by introducing
extra dimensions In physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions a ...
/
hyperspace In science fiction, hyperspace (also known as nulspace, subspace, overspace, jumpspace and similar terms) is a concept relating to higher dimensions as well as parallel universes and a faster-than-light (FTL) method of interstellar travel. ...
. Most notably,
superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string theo ...
requires 10 spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called
M-theory M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten ...
which subsumes five previously distinct superstring theories. Supergravity theory also promotes 11D spacetime = 7D hyperspace + 4 common dimensions. To date, no direct experimental or observational evidence is available to support the existence of these extra dimensions. If hyperspace exists, it must be hidden from us by some physical mechanism. One well-studied possibility is that the extra dimensions may be "curled up" at such tiny scales as to be effectively invisible to current experiments. Limits on the size and other properties of extra dimensions are set by particle experiments such as those at the
Large Hadron Collider The Large Hadron Collider (LHC) is the world's largest and highest-energy particle collider. It was built by the European Organization for Nuclear Research (CERN) between 1998 and 2008 in collaboration with over 10,000 scientists and hundre ...
. In 1921, Kaluza–Klein theory presented 5D including an extra dimension of space. At the level of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, Kaluza–Klein theory unifies
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
with gauge interactions, based on the realization that gravity propagating in small, compact extra dimensions is equivalent to gauge interactions at long distances. In particular when the geometry of the extra dimensions is trivial, it reproduces
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
. However at sufficiently high energies or short distances, this setup still suffers from the same pathologies that famously obstruct direct attempts to describe
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the ...
. Therefore, these models still require a
UV completion In theoretical physics, ultraviolet completion, or UV completion, of a quantum field theory is the passing from a lower energy quantum field theory to a more general quantum field theory above a threshold value known as the cutoff. In partic ...
, of the kind that string theory is intended to provide. In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a Calabi–Yau manifold. Thus Kaluza-Klein theory may be considered either as an incomplete description on its own, or as a subset of string theory model building. In addition to small and curled up extra dimensions, there may be extra dimensions that instead aren't apparent because the matter associated with our visible universe is localized on a subspace. Thus the extra dimensions need not be small and compact but may be
large extra dimensions In particle physics and string theory (M-theory), the ADD model, also known as the model with large extra dimensions (LED), is a model framework that attempts to solve the hierarchy problem. (''Why is the force of gravity so weak compared to the e ...
.
D-brane In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polch ...
s are dynamical extended objects of various dimensionalities predicted by string theory that could play this role. They have the property that open string excitations, which are associated with gauge interactions, are confined to the
brane In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime accord ...
by their endpoints, whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, or "the bulk". This could be related to why gravity is exponentially weaker than the other forces, as it effectively dilutes itself as it propagates into a higher-dimensional volume. Some aspects of brane physics have been applied to
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
. For example, brane gas cosmology attempts to explain why there are three dimensions of space using topological and thermodynamic considerations. According to this idea it would be since three is the largest number of spatial dimensions in which strings can generically intersect. If initially there are many windings of strings around compact dimensions, space could only expand to macroscopic sizes once these windings are eliminated, which requires oppositely wound strings to find each other and annihilate. But strings can only find each other to annihilate at a meaningful rate in three dimensions, so it follows that only three dimensions of space are allowed to grow large given this kind of initial configuration. Extra dimensions are said to be universal if all fields are equally free to propagate within them.

# In computer graphics and spatial data

Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, including illustration software,
Computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve ...
, and
Geographic information systems A geographic information system (GIS) is a type of database containing geographic data (that is, descriptions of phenomena for which location is relevant), combined with software tools for managing, analyzing, and visualizing those data. In a ...
. Different vector systems use a wide variety of data structures to represent shapes, but almost all are fundamentally based on a set of
geometric primitive In vector computer graphics, CAD systems, and geographic information systems, geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that ...
s corresponding to the spatial dimensions: * Point (0-dimensional), a single coordinate in a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
. * Line or Polyline (1-dimensional), usually represented as an ordered list of points sampled from a continuous line, whereupon the software is expected to
interpolate In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
the intervening shape of the line as straight or curved line segments. * Polygon (2-dimensional), usually represented as a line that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior. * Surface (3-dimensional), represented using a variety of strategies, such as a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
consisting of connected polygon faces. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior. Frequently in these systems, especially GIS and
Cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, a representation of a real-world phenomena may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This ''dimensional generalization'' correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood, but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines).

# In literature

Science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and futuristic concepts such as advanced science and technology, space exploration, time travel, parallel univ ...
texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones. One of the most heralded science fiction stories regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novella ''
Flatland ''Flatland: A Romance of Many Dimensions'' is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dim ...
'' by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described ''Flatland'' as "The best introduction one can find into the manner of perceiving dimensions." The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in
Miles J. Breuer Miles John Breuer (January 3, 1889 – October 14, 1945) was an American physician and science fiction writer of Czech origin. Although he had published elsewhere since the early 20th century, he is considered the part of the first generation ...
's ''The Appendix and the Spectacles'' (1928) and
Murray Leinster Murray Leinster (June 16, 1896 – June 8, 1975) was a pen name of William Fitzgerald Jenkins, an American writer of genre fiction, particularly of science fiction. He wrote and published more than 1,500 short stories and articles, 14 movie ...
's ''The Fifth-Dimension Catapult'' (1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include
Robert A. Heinlein Robert Anson Heinlein (; July 7, 1907 – May 8, 1988) was an American science fiction author, aeronautical engineer, and naval officer. Sometimes called the "dean of science fiction writers", he was among the first to emphasize scientific accu ...
's '' —And He Built a Crooked House'' (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract;
Alan E. Nourse Alan Edward Nourse (August 11, 1928 – July 19, 1992) was an American science fiction writer and physician. He wrote both juvenile and adult science fiction, as well as nonfiction works about medicine and science. His SF works sometimes focus ...
's ''Tiger by the Tail'' and ''The Universe Between'' (both 1951); and
The Ifth of Oofth
' (1957) by
Walter Tevis Walter Stone Tevis (February 28, 1928 – August 9, 1984) was an American novelist and short story writer. Three of his six novels were adapted into major films: ''The Hustler'', ''The Color of Money'' and ''The Man Who Fell to Earth''. A fourth, ...
. Another reference is
Madeleine L'Engle Madeleine L'Engle DStJ (; November 29, 1918 – September 6, 2007) was an American writer of fiction, non-fiction, poetry, and young adult fiction, including ''A Wrinkle in Time'' and its sequels: ''A Wind in the Door'', '' A Swiftly Tilting Pla ...
's novel ''
A Wrinkle In Time ''A Wrinkle in Time'' is a young adult science fantasy novel written by American author Madeleine L'Engle. First published in 1962, the book won the Newbery Medal, the Sequoyah Book Award, the Lewis Carroll Shelf Award, and was runner-up for ...
'' (1962), which uses the fifth dimension as a way for "tesseracting the universe" or "folding" space in order to move across it quickly. The fourth and fifth dimensions are also a key component of the book ''
The Boy Who Reversed Himself ''The Boy Who Reversed Himself'' (1986) is a science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and futuristic concepts such as advanced scie ...
'' by
William Sleator William Warner Sleator III (February 13, 1945 – August 3, 2011), known as William Sleator, was an American science fiction author who wrote primarily young adult novels but also wrote for younger readers. His books typically deal with adolesce ...
.

# In philosophy

Immanuel Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ...
, in 1783, wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition ''a priori'' because it is apodictically (demonstrably) certain." "Space has Four Dimensions" is a short story published in 1846 by German philosopher and
experimental psychologist Experimental psychology refers to work done by those who apply experimental methods to psychological study and the underlying processes. Experimental psychologists employ human participants and animal subjects to study a great many topics, in ...
Gustav Fechner Gustav Theodor Fechner (; ; 19 April 1801 – 18 November 1887) was a German physicist, philosopher, and experimental psychologist. A pioneer in experimental psychology and founder of psychophysics (techniques for measuring the mind), he inspired ...
under the
pseudonym A pseudonym (; ) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true name ( orthonym). This also differs from a new name that entirely or legally replaces an individu ...
"Dr. Mises". The protagonist in the tale is a shadow who is aware of and able to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time. The story bears a strong similarity to the "
Allegory of the Cave The Allegory of the Cave, or Plato's Cave, is an allegory presented by the Greek philosopher Plato in his work '' Republic'' (514a–520a) to compare "the effect of education ( παιδεία) and the lack of it on our nature". It is written ...
" presented in
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's '' The Republic'' ( 380 BC). Simon Newcomb wrote an article for the ''Bulletin of the American Mathematical Society'' in 1898 entitled "The Philosophy of Hyperspace".
Linda Dalrymple Henderson Linda Dalrymple Henderson (born 1948) is a historian of art whose research involves the connections between modern art, science and technology, and the occult. She is the David Bruton, Jr. Centennial Professor in Art History at the University of T ...
coined the term "hyperspace philosophy", used to describe writing that uses higher dimensions to explore
metaphysical Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art. Examples of "hyperspace philosophers" include
Charles Howard Hinton Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction works titled ''Scientific Romances''. He was interested in higher dimensions, particularly the fourth dimension. He is known for coining t ...
, the first writer, in 1888, to use the word "tesseract";. and the Russian
esotericist Western esotericism, also known as esotericism, esoterism, and sometimes the Western mystery tradition, is a term scholars use to categorise a wide range of loosely related ideas and movements that developed within Western society. These ideas ...
P. D. Ouspensky Pyotr Demianovich Ouspenskii (known in English as Peter D. Ouspensky; rus, Пётр Демья́нович Успе́нский, Pyotr Demyánovich Uspénskiy; 5 March 1878 – 2 October 1947) was a Russian esotericist known for his expositions ...
.