In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions and physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves, the dimension of a space Space 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 the boundless four-dimensional continuum known as spacetime. In mathematics one examines ' or object is informally defined as the minimum number of coordinates In geometry, a coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary needed to specify each point In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one within it.^[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A 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. On the other hand, there are surfaces, such as the Klein bottle, that cannot be such as a plane In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of or the surface of a cylinder A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since or sphere A 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. This distance r is known as the radius of the sphere. The 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 latitude Latitude, usually denoted by the Greek letter phi gives the location of a place on Earth (or other planetary body) north or south of the equator. Lines of Latitude are the imaginary horizontal lines shown running east-to-west (or west to east) on maps (particularly so in the Mercator projection) that run either north or south of the equator and its longitude Longitude , identified by the Greek letter lambda (λ), is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement. Constant longitude is represented by lines running from north to south. The line of longitude (meridian) that passes through the Royal Observatory, Greenwich, in England,). 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. 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 of trigonal trapezohedron. The cube is dual to the, 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 structure Structure is a fundamental if sometimes intangible notion referring to the recognition, observation, nature, and stability of patterns and relationships of entities. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic fields, the concept of structure is now often an essential of all space (cf. volume Volume is how much three-dimensional space a substance or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of) as well as the spatial constitution of objects within —structures that have correlations with both particle and field In physics and chemistry, wave–particle duality is the concept that all energy exhibits both wave-like and particle-like properties. Being a central concept of quantum mechanics, this duality addresses the inadequacy of classical concepts like "particle" and "wave" in fully describing the behavior of quantum-scale objects conceptions, interact according to relative properties of mass In physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: Inertial mass, active gravitational mass and passive gravitational mass. In everyday usage, mass is often taken to mean weight, but in scientific use, they refer to different properties, and which are fundamentally mathematical in description. Physical theories that incorporate time Time is "a nonspatial continuum in which events occur in apparently irreversible succession from the past through the present to the future." It is used to sequence events, to quantify the durations of events and the intervals between them, and to quantify and measure the motions of objects and other changes. Time is quantified in, such as general relativity General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1915. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time, or spacetime, are said to work in 4-dimensional "spacetime In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. According to certain Euclidean space perceptions, the universe has three", (defined as a Minkowski space In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime). Modern theories tend to be "higher-dimensional" including quantum field Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized (represented) by an infinite number of dynamical degrees of freedom, that is, fields and (in a condensed matter context) many-body systems. It is the natural and quantitative language of particle physics and and string String theory is a developing theory in particle physics which attempts to reconcile quantum mechanics and general relativity. String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines , possessing only the dimension of length, but not height or width. The theory theories. The state-space of quantum mechanics Quantum 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 scales. In advanced topics of QM, some of these is an infinite-dimensional 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.

The concept of dimension is not restricted to physical objects. High-dimensional spaces In mathematics, an n-dimensional space is a topological space whose dimension is n . The archetypical example is n-dimensional Euclidean space, which describes Euclidean geometry in n dimensions. n-dimensional spaces with large values of n are sometimes called high-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints. The configuration space of a typical system has the structure of a manifold; for this reason it is also called the configuration manifold such as in Lagrangian Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Italian mathematician Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the or Hamiltonian mechanics Hamiltonian 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 recourse to Lagrangian mechanics using symplectic; 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 circle In 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. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere in the plane can be specified by two Cartesian coordinates 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 but one can make do with a single coordinate (the polar coordinate In 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 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. These can be handy to know about.

The dimension of Euclidean n-space In 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. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity Eⁿ is n. When trying to generalize to other types of spaces, one is faced with the question “what makes Eⁿ n-dimensional?" One answer is that to cover a fixed ball in Eⁿ by small balls of radius ε, one needs on the order of ε⁻ⁿ such small balls. This observation leads to the definition of the Minkowski dimension In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in a metric space and its more sophisticated variant, the Hausdorff dimension In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional vector space equals n. This means, for example the Hausdorff dimension of a point is zero, the. But there are also other answers to that question. For example, one may observe that the boundary of a ball in Eⁿ looks locally like E^{n − 1} 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 or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have. While these notions agree on Eⁿ, they turn out to be different when one looks at more general spaces.

A tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron, 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 cubical cells. The tesseract is one of the six convex regular 4- 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 Higher dimension as a term in mathematics most commonly refers to any number of spatial dimensions greater than three goes back to René Descartes René Descartes , (31 March 1596 – 11 February 1650), also known as Renatus Cartesius (Latinized form; adjectival form: "Cartesian"), was a French philosopher, mathematician, physicist, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the "Father of Modern Philosophy", and much of, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley Arthur Cayley was a British mathematician. He helped found the modern British school of pure mathematics, William Rowan Hamilton Sir 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. His greatest contribution is perhaps the reformulation of Newtonian mechanics,, Ludwig Schläfli Ludwig Schläfli was a Swiss geometer and complex analyst (at the time called function theory) 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. Although his ideas have become so and Bernhard Riemann Georg Friedrich Bernhard Riemann (German pronunciation: [ˈʁiːman]; September 17, 1826 – July 20, 1866) 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 Habilitation is the highest academic qualification a person can achieve by his own pursuit in certain European and Asian countries. Earned after obtaining a research doctorate , the habilitation requires the candidate to write a professorial thesis based on independent scholarly accomplishments, reviewed by and defended before an academic, Schlafi's 1852 Theorie der vielfachen Kontinuität, Hamilton's 1843 discovery of the quaternions 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. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two and the construction of the Cayley Algebra In mathematics, the octonions are a nonassociative and noncommutative extension of the quaternions. Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley–Dickson construction. The octonion algebra is often denoted O, or in blackboard bold by 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

The dimension of a 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 numbers, rational numbers, or even more general fields is the number of vectors in any basis In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others. In other words, a basis is a linearly independent spanning set, or more simply put a "coordinate system& 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 = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.

Manifolds

A connected In 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. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component topological manifold is locally homeomorphic to Euclidean n-space, and the number 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 topology, is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.

Lebesgue covering dimension

For any normal topological space X, the Lebesgue covering dimension of X is defined to be n if n is the smallest integer 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 n + 1 elements. In this case we write dim X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the dimension of X is said to be infinite, and we write dim X = ∞. Note also that we say X has dimension −1, i.e. dim X = −1 if and only if 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

An inductive definition of dimension can be created as follows. Consider a discrete set 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 n+1-dimensional object by dragging an 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 (n + 1)-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.

Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values.^[3] The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, 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.

In physics

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, 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 and Cartesian coordinate system.)

Time

A temporal 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 are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).

The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space.

Additional dimensions

Theories such as string theory and M-theory predict that physical space in general has in fact 10 and 11 dimensions, respectively. The extra dimensions are spatial. We perceive only three spatial dimensions, and no physical experiments have confirmed the reality of additional 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.

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 fiction texts often mention the concept of dimension, when really referring to parallel universes, 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 spatial 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 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 Flatland 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's “The Appendix and the Spectacles” (1928) and Murray Leinster'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. Heinlein'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. Nourse's "Tiger by the Tail" and "The Universe Between," both 1951. Another reference would be Madeleine L'Engle's novel "A Wrinkle In Time" (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.

Philosophy

In 1783, Kant 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."^[4]

More dimensions

• Dimension of an algebraic variety
• Exterior dimension
• Hurst exponent
• Isoperimetric dimension
• Kaplan-Yorke dimension
• Lebesgue covering dimension
• Lyapunov dimension
• Metric dimension
• Pointwise dimension
• Poset dimension
• q-dimension; especially:
  • Information dimension (corresponding to q = 1)
  • Correlation dimension (corresponding to q = 2)
• Vector space dimension / Hamel dimension

See also

• Degrees of freedom
• Dimension (data warehouse) and dimension tables
• Fractal dimension
• Hyperspace (disambiguation page)
• Space-filling curve

A list of topics indexed by dimension

• Zero dimensions:
  • Point
  • Zero-dimensional space
  • Integer
• One dimension:
  • Line
  • Graph (combinatorics)
  • Real number
• Two dimensions:
  • Complex number
  • Cartesian coordinate system
  • List of uniform tilings
  • Surface
• Three dimensions
  • Platonic solid
  • Stereoscopy (3-D imaging)
  • Euler angles
  • 3-manifold
  • Knot (mathematics)
• Four dimensions:
  • Spacetime
  • Fourth spatial dimension
  • Convex regular 4-polytope
  • Quaternion
  • 4-manifold
• High-dimensional topics from mathematics:
  • Octonion
  • Vector space
  • Manifold
  • Calabi-Yau spaces
• High-dimensional topics from physics:
  • Kaluza-Klein theory
  • String theory
  • M-theory
• Infinitely many dimensions:
  • Hilbert space
  • Function space

References

1. ^ Curious About Astronomy
2. ^ MathWorld: Dimension
3. ^ Fractal Dimension, Boston University Department of Mathematics and Statistics
4. ^ Prolegomena, § 12

Further reading

• Edwin A. Abbott, (1884) Flatland: A Romance of Many Dimensions, Public Domain. Online version with ASCII approximation of illustrations at Project Gutenberg.
• Thomas Banchoff, (1996) Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition, Freeman.
• Clifford A. Pickover, (1999) Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Oxford University Press.
• Rudy Rucker, (1984) The Fourth Dimension, Houghton-Mifflin.

Categories: Physics | Dimension | Dimension theory | Abstract algebra | Algebra | Linear algebra

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