In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
dimension of a
spaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
or
objectIn mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
is informally defined as the minimum number of coordinates needed to specify any
pointIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
within it. Thus a
lineThe notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A
surfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
such as a
planeIn mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
or the surface of a
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
or
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and its
longitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
). The inside of a
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
In physical terms,
dimension refers to the constituent
structureStructure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...
of all space (cf.
volumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
) and its position in time (perceived as a scalar dimension along the
t-axis), as well as the spatial constitution of objects within – structures that have correlations with both particle and field conceptions, interact according to relative properties of
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, and which are fundamentally mathematical in description. These or other axes may be referenced to uniquely identify a point or structure in its attitude and relationship to other objects and occurrences. Physical theories that incorporate
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, such as
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, are said to work in 4-dimensional "
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
", (defined as a
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
). Modern theories tend to be "higher-dimensional" including
quantum fieldQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and
stringString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
theories. The state-space of
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is an infinite-dimensional
function spaceIn mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
.
The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as
configuration space- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
s such as in
LagrangianLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
or
Hamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
; these are abstract spaces, independent of the physical space we live in.
In mathematics
In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the
unit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This
intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.
The dimension of
Euclidean
{{redirect|0d|0D}}
{{Other uses}}
In physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
dimension of a
spaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
or
objectIn mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
is informally defined as the minimum number of coordinates needed to specify any
pointIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
within it. Thus a
lineThe notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A
surfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
such as a
planeIn mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
or the surface of a
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
or
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and its
longitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
). The inside of a
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
In physical terms,
dimension refers to the constituent
structureStructure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...
of all space (cf.
volumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
) and its position in time (perceived as a scalar dimension along the
t-axis), as well as the spatial constitution of objects within – structures that have correlations with both particle and field conceptions, interact according to relative properties of
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, and which are fundamentally mathematical in description. These or other axes may be referenced to uniquely identify a point or structure in its attitude and relationship to other objects and occurrences. Physical theories that incorporate
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, such as
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, are said to work in 4-dimensional "
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
", (defined as a
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
). Modern theories tend to be "higher-dimensional" including
quantum fieldQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and
stringString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
theories. The state-space of
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is an infinite-dimensional
function spaceIn mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
.
The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as
configuration space- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
s such as in
LagrangianLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
or
Hamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
; these are abstract spaces, independent of the physical space we live in.
In mathematics
In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the
unit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This
intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.
The dimension of
Euclidean
{{redirect|0d|0D}}
{{Other uses}}
In physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
dimension of a
spaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
or
objectIn mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
is informally defined as the minimum number of coordinates needed to specify any
pointIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
within it. Thus a
lineThe notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A
surfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
such as a
planeIn mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
or the surface of a
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
or
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and its
longitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
). The inside of a
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
In physical terms,
dimension refers to the constituent
structureStructure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...
of all space (cf.
volumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
) and its position in time (perceived as a scalar dimension along the
t-axis), as well as the spatial constitution of objects within – structures that have correlations with both particle and field conceptions, interact according to relative properties of
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, and which are fundamentally mathematical in description. These or other axes may be referenced to uniquely identify a point or structure in its attitude and relationship to other objects and occurrences. Physical theories that incorporate
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, such as
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, are said to work in 4-dimensional "
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
", (defined as a
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
). Modern theories tend to be "higher-dimensional" including
quantum fieldQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and
stringString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
theories. The state-space of
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is an infinite-dimensional
function spaceIn mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
.
The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as
configuration space- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
s such as in
LagrangianLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
or
Hamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
; these are abstract spaces, independent of the physical space we live in.
In mathematics
In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the
unit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This
intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.
The dimension of
Euclidean {{mathIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
{{math|
En }}is {{math|
n}}. When trying to generalize to other types of spaces, one is faced with the question “what makes {{math|
En }} {{math|
n}}-dimensional?" One answer is that to cover a fixed ball in {{math|
En }} by small balls of radius {{math|
ε}}, one needs on the order of {{math|
ε-n}} such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the
Hausdorff dimensionthumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
. But there are also other answers to that question. For example, one may observe that the boundary of a ball in {{math|
En }} looks locally like {{math|
En-1 }} and this leads to the notion of the
inductive dimensionIn the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind or the large inductive dimension Ind...
. While these notions agree on {{math|
En }}, they turn out to be different when one looks at more general spaces.
A
tesseractIn geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog 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 6 square faces, the hypersurface of the tesseract consists of 8...
is an example of a four-dimensional object. Whereas outside of 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".
Although the notion of higher dimensions goes back to
René DescartesRené Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of
Arthur CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
,
William Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
,
Ludwig SchläfliLudwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...
and
Bernhard RiemannGeorg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
. Riemann's 1854 Habilitationsschrift, Schlafi's 1852
Theorie der vielfachen Kontinuität, Hamilton's 1843 discovery of the quaternions and the construction of the
Cayley AlgebraIn mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
marked the beginning of higher-dimensional geometry.
The rest of this section examines some of the more important mathematical definitions of the dimensions.
Dimension of a vector space
{{Main|Dimension (vector space)}}
The dimension of a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
is the number of vectors in any
basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the
cardinality of a basis) is often referred to as the
Hamel dimension or
algebraic dimension to distinguish it from other notions of dimension.
Manifolds
A
connectedIn mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
topological
manifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is locally homeomorphic to Euclidean {{math|
n}}-space, and the number {{math|
n}} is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold.
The theory of manifolds, in the field of
geometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, is characterized by the way dimensions 1 and 2 are relatively elementary, the
high-dimensional cases {{nowrap|{{math|
n > 4}}}} are simplified by having extra space in which to "work"; and the cases {{math|
n = 3}} and {{math|4}} are in some senses the most difficult. This state of affairs was highly marked in the various cases of the
Poincaré conjectureIn mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, where four different proof methods are applied.
Lebesgue covering dimension
{{Main|Lebesgue covering dimension}}
For any normal topological space {{math|
X}}, the Lebesgue covering dimension of {{math|
X}} is defined to be n if
n is the smallest
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
for which the following holds: any open cover has an open refinement (a second open cover where each element is a subset of an element in the first cover) such that no point is included in more than {{math|
n + 1}} elements. In this case dim {{math|
X {{=}}
n}}. For {{math|
X}} a manifold, this coincides with the dimension mentioned above. If no such integer {{math|
n}} exists, then the dimension of {{math|
X}} is said to be infinite, and one writes dim {{math|
X {{=}} ∞}}. Moreover, {{math|
X}} has dimension −1, i.e. dim {{math|
X {{=}} −1}} if and only if {{math|
X}} is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "
functionally open".
Inductive dimension
{{Main|Inductive dimension}}
An inductive definition of dimension can be created as follows. Consider a
discrete setIn topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.In particular, in a Euclidean space ,...
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 ({{math|
n + 1}})-dimensional object by dragging an {{math|
n}} dimensional object in a
new direction.
The inductive dimension of a topological space could refer to the
small inductive dimension or the
large inductive dimension, and is based on the analogy that {{nowrap|({{math|
n + 1}})-dimensional}} balls have {{math|
n}} dimensional
boundariesIn topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
, permitting an inductive definition based on the dimension of the boundaries of open sets.
Hausdorff dimension
{{Main|Hausdorff dimension}}
For sets which are of a complicated structure, especially
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s, the
Hausdorff dimensionthumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
is useful. The Hausdorff dimension is defined for all
metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and, unlike the Hamel dimension, can also attain non-integer real values. The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of
fractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
s that work for highly irregular sets and attain non-integer positive real values. Fractals have been found useful to describe many natural objects and phenomenon.
Hilbert spaces
Every
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
admits an
orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
, and any two such bases for a particular space have the same
cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the above dimensions coincide.
Spatial dimensions
Classical physics theories describe three physical dimensions: from a particular point in
spaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
, 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 combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
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
SpaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
and
Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
.)
| Number of dimensions |
Example co-ordinate systems |
| 1 In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 1, the set of all such locations is called 1-dimensional Euclidean space.-Polytopes:...
|
Number lineIn basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...
|
AngleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
|
|
| 2 - Details :Bi-dimensional space is a geometric model of the planar projection of the physical universe in which we live.The two dimensions are commonly called length and width .Both directions lies in the same plane....
|
CartesianA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
(2-dimensional) |
PolarIn mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....
|
Latitude and longitudeA geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...
|
|
| 3 Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
|
CartesianA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
(3-dimensional) |
CylindricalA cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...
|
.svg.png)
SphericalIn mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...
|
|
Time
A
temporal dimension is a dimension of time. Time is often referred to as the "
fourth dimensionIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
" 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 directionThe arrow of time, or time’s arrow, is a term coined in 1927 by the British astronomer Arthur Eddington to describe the "one-way direction" or "asymmetry" of time...
.
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 mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
are
symmetric with respect to timeT Symmetry is the symmetry of physical laws under a time reversal transformation: T: t \mapsto -t.Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.Time asymmetries...
, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as
chargeIn physics, C-symmetry means the symmetry of physical laws under a charge-conjugation transformation. Electromagnetism, gravity and the strong interaction all obey C-symmetry, but weak interactions violate C-symmetry.-Charge reversal in electromagnetism:...
and
parityIn physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
) are reversed. In these models, the perception of time flowing in one direction is an artifact of the
laws of thermodynamicsThe four laws of thermodynamics summarize its most important facts. They define fundamental physical quantities, such as temperature, energy, and entropy, in order to describe thermodynamic systems. They also describe the transfer of energy as heat and work in thermodynamic processes...
(we perceive time as flowing in the direction of increasing
entropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
).
The best-known treatment of time as a dimension is
PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
and
EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
(and extended to
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
), which treats perceived space and time as components of a four-dimensional
manifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, known as
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, and in the special, flat case as
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
.
Additional dimensions
Theories such as
string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and
M-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
posit that physical space has 10 and 11 dimensions, respectively. These
extra dimensionsSeveral speculative physical theories have introduced extra dimensions of space for various reasons:*Kaluza-Klein theory introduces extra dimensions to explain the fundamental forces other than gravity ....
are said to be spatial. However, we perceive only three spatial dimensions and, to date, no experimental or observational evidence is available to confirm the existence of these extra dimensions. A possible explanation that has been suggested is that space acts as if it were "curled up" in the extra dimensions on a subatomic scale, possibly at the quark/string level of scale or below.
An analysis of results from the
Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....
in December 2010 severely constrains theories with large extra dimensions.
Networks and dimension
Some complex networks are characterized by fractal dimensions. The concept of dimension can be generalized to include networks embedded in space. The dimension characterize their spatial constraints.
Literature
Perhaps the most basic way in which the word
dimension is used in literature is as a hyperbolic synonym for
feature,
attribute,
aspect, or
magnitude. Frequently the hyperbole is quite literal as in
he's so 2-dimensional, meaning that one can see at a glance what he
is. This contrasts with 3-dimensional objects which have an interior that is hidden from view, and a back that can only be seen with further examination.
Science fictionScience fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...
texts often mention the concept of dimension, when really referring to
parallel universesA parallel universe or alternative reality is a hypothetical self-contained separate reality coexisting with one's own. A specific group of parallel universes is called a "multiverse", although this term can also be used to describe the possible parallel universes that constitute reality...
, alternate universes, or other 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 novellas regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novel
FlatlandFlatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously as "A Square", Abbott used the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture...
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. BreuerMiles John Breuer was an American physician and science fiction writer. He was part of the first generation of writers to appear regularly in the pulp science fiction magazines, publishing his first story, "The Man with the Strange Head", in the January 1927 issue of Amazing Stories...
's
The Appendix and the Spectacles (1928) and
Murray LeinsterMurray Leinster was a nom de plume of William Fitzgerald Jenkins, an award-winning American writer of science fiction and alternate history...
's
The Fifth-Dimension Catapult (1931); and appeared irregularly in science fiction by the 1940s. Some of the classic stories involving other dimensions include
Robert A. HeinleinRobert Anson Heinlein was an American science fiction writer. Often called the "dean of science fiction writers", he was one of the most influential and controversial authors of the genre. He set a standard for science and engineering plausibility and helped to raise the genre's standards of...
's 1941
—And He Built a Crooked House, in which a California architect designs a house based on a three-dimensional projection of a tesseract, and
Alan E. NourseAlan Edward Nourse was an American science fiction author and physician. He wrote both juvenile and adult science fiction, as well as nonfiction works about medicine and science. His SF works generally focused on medicine and/or psionics.-Biography:Alan Nourse was born August 11, 1928 to...
's
Tiger by the Tail and
The Universe Between, both from 1951. Another reference would be
Madeleine L'EngleMadeleine L'Engle was an American writer best known for her young-adult fiction, particularly the Newbery Medal-winning A Wrinkle in Time and its sequels A Wind in the Door, A Swiftly Tilting Planet, Many Waters, and An Acceptable Time...
's novel
A Wrinkle In TimeA Wrinkle in Time is a science fantasy novel by Madeleine L'Engle, first published in 1962. The story revolves around a young girl whose father, a government scientist, has gone missing after working on a mysterious project called a tesseract. The book won a Newbery Medal, Sequoyah Book Award, and...
(1962) which uses the 5th Dimension as a way for "tesseracting the universe", or in a better sense, "folding" space in half to move across it quickly.
The fourth and fifth dimensions were also a key component of the book
The Boy Who Reversed HimselfThe Boy Who Reversed Himself is a science fiction novel by William Sleator. The novel deals with an exploration into other dimensions, and provides a journey into the world beyond our own.- Plot summary:...
, by
William SleatorWilliam Warner Sleator III , 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 adolescents coming across a peculiar phenomenon related to an element of theoretical science, then...
.
Philosophy
In 1783,
KantImmanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
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."
See also
- Degrees of freedom
Degrees of freedom can mean:* Degrees of freedom , independent displacements and/or rotations that specify the orientation of the body or system...
- Dimension (data warehouse)
In a data warehouse, a dimension is a data element that categorizes each item in a data set into non-overlapping regions. A data warehouse dimension provides the means to "slice and dice" data in a data warehouse. Dimensions provide structured labeling information to otherwise unordered numeric...
and dimension tableIn data warehousing, a dimension table is one of the set of companion tables to a fact table.The fact table contains business facts or measures and foreign keys which refer to candidate keys in the dimension tables....
s
- Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
- Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
- Hyperspace
Hyperspace may refer to:*in mathematics and general science** A Euclidean space of dimension greater than three, see fourth dimension and higher dimensions ** Space of any dimension, whether lower, higher, or...
(disambiguation page)
- Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...
- Intrinsic dimension
In signal processing of multidimensional signals, for example in computer vision, the intrinsic dimension of the signal describes how many variables are needed to represent the signal. For a signal of N variables, its intrinsic dimension M satisfies 0 ≤ M ≤ N.Usually the intrinsic dimension...
A list of topics indexed by dimension
- Zero dimensions:
- Point
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
- Zero-dimensional space
In mathematics, a zero-dimensional topological space is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space...
- Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
- One dimension:
- Line
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
- Graph
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
(combinatorics)
- Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
- Two dimensions:
- Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
- Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
- List of uniform tilings
- Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
- Three dimensions
- Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
- Stereoscopy
Stereoscopy refers to a technique for creating or enhancing the illusion of depth in an image by presenting two offset images separately to the left and right eye of the viewer. Both of these 2-D offset images are then combined in the brain to give the perception of 3-D depth...
(3-D imaging)
- Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...
- 3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
- Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
- Four dimensions:
- Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
- Fourth spatial dimension
- Convex regular 4-polytope
In mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
- Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
- 4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
- High-dimensional topics from mathematics:
- Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
- Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
- Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
- Calabi–Yau spaces
- High-dimensional topics from physics:
- Kaluza–Klein theory
In physics, Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921. It was proposed by the mathematician Theodor Kaluza who extended general relativity to a five-dimensional spacetime...
- String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
- M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
- Infinitely many dimensions:
- Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
- Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
Further reading
- Edwin A. Abbott, (1884) Flatland: A Romance of Many Dimensions
Flatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously as "A Square", Abbott used the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture...
, Public Domain. Online version with ASCII approximation of illustrations at Project GutenbergProject Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". Founded in 1971 by Michael S. Hart, it is the oldest digital library. Most of the items in its collection are the full texts of public domain books...
.
- Thomas Banchoff
Thomas Francis Banchoff is an American mathematicianspecializing in geometry. He is a professor at Brown University, where he has taught since 1967...
, (1996) Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman.
- Clifford A. Pickover
Clifford A. Pickover is an American author, editor, and columnist in the fields of science, mathematics, and science fiction, and is employed at the IBM Thomas J. Watson Research Center in Yorktown, New York.- Biography :He received his Ph.D...
, (1999) Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press.
- Rudy Rucker
Rudolf von Bitter Rucker is an American mathematician, computer scientist, science fiction author, and philosopher, and is one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known for the novels in the Ware Tetralogy, the first two of...
, (1984) The Fourth Dimension, Houghton-Mifflin.
- Michio Kaku
is an American theoretical physicist, the Henry Semat Professor of Theoretical Physics in the City College of New York of City University of New York, the co-founder of string field theory, and a "communicator" and "popularizer" of science...
, (1994) Hyperspace, a Scientific Odyssey Through the 10th DimensionHyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension is a book by Michio Kaku, a theoretical physicist from the City College of New York. It focuses on Kaku's studies of higher dimensions referred to as hyperspace...
, Oxford University Press.
{{Dimension topics}}