Quantum Enigma

 

 

 

Quantum Enigma is an enjoyable, expansive, and clear look at quantum theory, its various interpretations, and the philosophical questions that arise in conjunction with it. The authors focus on the role consciousness plays in resolving quantum-level situations, and what the ultimate significance of this might be. For anyone intrigued by quantum physics, this book provides clear insight into the key issues of the field, though there is little new for those already well-versed in the topic.

Pros

• Lucid look at the key elements of basic quantum theory
• Written with emphasis on understanding among nonscientists - No mathematics.
• Consciousness' role in quantum theory addressed without straying to metaphysical realms.

Cons

• Some examples result in oversimplification
• Fictional "dialogue" segments are contrived & somewhat cheesy

Description

• Oxford University Press, 2006, hardcover, 17 chapters (+ index), 211 pages (w/ index)
• Authored by University of California Santa Cruz physics professors Bruce Rosenblum & Fred Kuttner
• Written in an non-technical style that makes it easily accessible to laymen. No mathematics!
• Presents all background necessary for "understanding" the topic (as well as quantum theory can be)

Guide Review - Quantum Enigma: Physics Encounters Consciousness

Quantum Enigma: Physics Encounters Consciousness addresses the issue of the "skeleton in the closet" of quantum physics: in short, that quantum phenomena ultimately results in the conclusion that observational choice (and thus consciousness) is related to the outcome of a natural event. In presenting the fundamentals of quantum mechanics, the authors address:

• Classical physics
• Development of quantum mechanics
• Schroedinger's equation
• How technology derived from quantum mechanics fuels the economy
• The Copenhagen (and other) interpretations of quantum theory
• Wave particle duality
• Schroedinger's cat paradox
• Einstein-Podolsky-Rosen (EPR) paradox
• Bell's theorem & Bell's inequality

With this foundation established, quantum theory shows that every property of nature, at the most fundamental level, is represented by a wave function of quantum probability. This wave function "collapses" into a single state upon observation, but experiments make it clear that prior to the observation, the reality of the situation is the probability wave. In certain highly specialized quantum situations, the choice of experiment actually determines what will be observed. As the authors make clear, the point of the book is not to resolve the enigma, but merely to bring it to light. This book is intended to motivate philosophical inquiry into the topic, not promote an agenda. In this respect, it does an outstanding job, and avoids the trap of pseudo-mysticism that many such works fall prey to.

 

Blogger Ref Link http://www.p2pfoundation.net/Multi-Dimensional_Science

 

Topological Dimensions

To some extent, topology may have great relevance to my evolving project along with the Hilbert Space. http://www.p2pfoundation.net/Multi-Dimensional_Science

RS





topology, branch of mathematics, sometimes referred to as “rubber sheet geometry,” in which two objects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while disallowing tearing apart or gluing together parts. The main topics of interest in topology are the properties that remain unchanged by such continuous deformations. Topology, while similar to geometry, differs from geometry in that geometrically equivalent objects often share numerically measured quantities, such as lengths or angles, while topologically equivalent objects resemble each other in a more qualitative sense.

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                                                 The area of topology dealing with abstract   objects is referred to as general, or point-set, topology. General topology overlaps with another important area of topology called algebraic topology. These areas of specialization form the two major subdisciplines of topology that developed during its relatively modern history.

Basic concepts of general topology

Simply connected

           In some cases, the objects considered in topology are ordinary objects residing in three- (or lower-) dimensional space. For example, a simple loop in a plane and the boundary edge of a square in a plane are topologically equivalent, as may be observed by imagining the loop as a rubber band that can be stretched to fit tightly around the square. On the other hand, the surface of a sphere is not topologically equivalent to a torus, the surface of a solid doughnut ring. To see this, note that any small loop lying on a fixed sphere may be continuously shrunk, while being kept on the sphere, to any arbitrarily small diameter. An object possessing this property is said to be simply connected, and the property of being simply connected is indeed a property retained under a continuous deformation. However, some loops on a torus cannot be shrunk, as shown in the figure.
Many results of topology involve objects as simple as those mentioned above. The importance of topology as a branch of mathematics, however, arises from its more general consideration of objects contained in higher-dimensional spaces or even abstract objects that are sets of elements of a very general nature. To facilitate this generalization, the notion of topological equivalence must be clarified.

Topological equivalence

The motions associated with a continuous deformation from one object to another occur in the context of some surrounding space, called the ambient space of the deformation. When a continuous deformation from one object to another can be performed in a particular ambient space, the two objects are said to be isotopic with respect to that space. For example, consider an object that consists of a circle and an isolated point inside the circle. Let a second object consist of a circle and an isolated point outside the circle, but in the same plane as the circle. In a two-dimensional ambient space these two objects cannot be continuously deformed into each other because it would require cutting the circles open to allow the isolated points to pass through. However, if three-dimensional space serves as the ambient space, a continuous deformation can be performed—simply lift the isolated point out of the plane and reinsert it on the other side of the circle to accomplish the task. Thus, these two objects are isotopic with respect to three-dimensional space, but they are not isotopic with respect to two-dimensional space.


The notion of objects being isotopic with respect to a larger ambient space provides a definition of extrinsic topological equivalence, in the sense that the space in which the objects are embedded plays a role. The example above motivates some interesting and entertaining extensions. One might imagine a pebble trapped inside a spherical shell. In three-dimensional space the pebble cannot be removed without cutting a hole through the shell, but by adding an abstract fourth dimension it can be removed without any such surgery. Similarly, a closed loop of rope that is tied as a trefoil, or overhand, knot (see figure) in three-dimensional space can be untied in an abstract four-dimensional space.

Homeomorphism

An intrinsic definition of topological equivalence (independent of any larger ambient space) involves a special type of function known as a homeomorphism. A function h is a homeomorphism, and objects X and Y are said to be homeomorphic, if and only if the function satisfies the following conditions.

• (1) h is a one-to-one correspondence between the elements of X and Y;
• (2) h is continuous: nearby points of X are mapped to nearby points of Y and distant points of X are mapped to distant points of Y—in other words, “neighbourhoods” are preserved;
• (3) there exists a continuous inverse function h⁻¹: thus, h⁻¹h(x) = x for all x ∊ X and hh⁻¹(y) = y for all y ∊ Y—in other words, there exists a function that “undoes” (is the inverse of) the homeomorphism, so that for any x in X or any y in Y the original value can be restored by combining the two functions in the proper order.

The notion of two objects being homeomorphic provides the definition of intrinsic topological equivalence and is the generally accepted meaning of topological equivalence. Two objects that are isotopic in some ambient space must also be homeomorphic. Thus, extrinsic topological equivalence implies intrinsic topological equivalence.

Topological structure

In its most general setting, topology involves objects that are abstract sets of elements. To discuss properties such as continuity of functions between such abstract sets, some additional structure must be imposed on them.

Topological space

One of the most basic structural concepts in topology is to turn a set X into a topological space by specifying a collection of subsets T of X. Such a collection must satisfy three axioms: (1) the set X itself and the empty set are members of T, (2) the intersection of any finite number of sets in T is in T, and (3) the union of any collection of sets in T is in T. The sets in T are called open sets and T is called a topology on X. For example, the real number line becomes a topological space when its topology is specified as the collection of all possible unions of open intervals—such as (−5, 2), (1/2, π), (0, √2), …. (An analogous process produces a topology on a metric space.) Other examples of topologies on sets occur purely in terms of set theory. For example, the collection of all subsets of a set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X. A given topological space gives rise to other related topological spaces. For example, a subset A of a topological space X inherits a topology, called the relative topology, from X when the open sets of A are taken to be the intersections of A with open sets of X. The tremendous variety of topological spaces provides a rich source of examples to motivate general theorems, as well as counterexamples to demonstrate false conjectures. Moreover, the generality of the axioms for a topological space permit mathematicians to view many sorts of mathematical structures, such as collections of functions in analysis, as topological spaces and thereby explain associated phenomena in new ways.


A topological space may also be defined by an alternative set of axioms involving closed sets, which are complements of open sets. In early consideration of topological ideas, especially for objects in n-dimensional Euclidean space, closed sets had arisen naturally in the investigation of convergence of infinite sequences (see infinite series). It is often convenient or useful to assume extra axioms for a topology in order to establish results that hold for a significant class of topological spaces but not for all topological spaces. One such axiom requires that two distinct points should belong to disjoint open sets. A topological space satisfying this axiom has come to be called a Hausdorff space.

Continuity

An important attribute of general topological spaces is the ease of defining continuity of functions. A function f mapping a topological space X into a topological space Y is defined to be continuous if, for each open set V of Y, the subset of X consisting of all points p for which f(p) belongs to V is an open set of X. Another version of this definition is easier to visualize, as shown in the figure. A function f from a topological space X to a topological space Y is continuous at p ∊ X if, for any neighbourhood V of f(p), there exists a neighbourhood U of p such that f(U) ⊆ V. These definitions provide important generalizations of the usual notion of continuity studied in analysis and also allow for a straightforward generalization of the notion of homeomorphism to the case of general topological spaces. Thus, for general topological spaces, invariant properties are those preserved by homeomorphisms.

Algebraic topology

The idea of associating algebraic objects or structures with topological spaces arose early in the history of topology. The basic incentive in this regard was to find topological invariants associated with different structures. The simplest example is the Euler characteristic, which is a number associated with a surface. In 1750 the Swiss mathematician Leonhard Euler proved the polyhedral formula V – E + F = 2, or Euler characteristic, which relates the numbers V and E of vertices and edges, respectively, of a network that divides the surface of a polyhedron (being topologically equivalent to a sphere) into F simply connected faces. This simple formula motivated many topological results once it was generalized to the analogous Euler-Poincaré characteristic χ = V – E + F = 2 – 2g for similar networks on the surface of a g-holed torus. Two homeomorphic surfaces will have the same Euler-Poincaré characteristic, and so two surfaces with different Euler-Poincaré characteristics cannot be topologically equivalent. However, the primary algebraic objects used in algebraic topology are more intricate and include such structures as abstract groups, vector spaces, and sequences of groups. Moreover, the language of algebraic topology has been enhanced by the introduction of category theory, in which very general mappings translate topological spaces and continuous functions between them to the associated algebraic objects and their natural mappings, which are called homomorphisms.

Fundamental group

A very basic algebraic structure called the fundamental group of a topological space was among the algebraic ideas studied by the French mathematician Henri Poincaré in the late 19th century. This group essentially consists of curves in the space that are combined by an operation arising in a geometric way. While this group was well understood even in the early days of algebraic topology for compact two-dimensional surfaces, some questions related to it still remain unanswered, especially for certain compact manifolds, which generalize surfaces to higher dimensions.



The most famous of these questions, called the Poincaré conjecture, asks if a compact three-dimensional manifold with trivial fundamental group is necessarily homeomorphic to the three-dimensional sphere (the set of points in four-dimensional space that are equidistant from the origin), as is known to be true for the two-dimensional case. Much research in algebraic topology has been related in some way to this conjecture since it was posed by Poincaré in 1904. One such research effort concerned a conjecture on the geometrization of three-dimensional manifolds that was posed in the 1970s by the American mathematician William Thurston. Thurston’s conjecture implies the Poincaré conjecture, and in recognition of his work toward proving these conjectures, the Russian mathematician Grigori Perelman was awarded a Fields Medal at the 2006 International Congress of Mathematicians.
The fundamental group is the first of what are known as the homotopy groups of a topological space. These groups, as well as another class of groups called homology groups, are actually invariant under mappings called homotopy retracts, which include homeomorphisms. Homotopy theory and homology theory are among the many specializations within algebraic topology.

Differential topology

Many tools of algebraic topology are well-suited to the study of manifolds. In the field of differential topology an additional structure involving “smoothness,” in the sense of differentiability (see analysis: Formal definition of the derivative), is imposed on manifolds. Since early investigation in topology grew from problems in analysis, many of the first ideas of algebraic topology involved notions of smoothness. Results from differential topology and geometry have found application in modern physics.

Knot theory

Another branch of algebraic topology that is involved in the study of three-dimensional manifolds is knot theory, the study of the ways in which knotted copies of a circle can be embedded in three-dimensional space. Knot theory, which dates back to the late 19th century, gained increased attention in the last two decades of the 20th century when its potential applications in physics, chemistry, and biomedical engineering were recognized.

History of topology

Mathematicians associate the emergence of topology as a distinct field of mathematics with the 1895 publication of Analysis Situs by the Frenchman Henri Poincaré, although many topological ideas had found their way into mathematics during the previous century and a half. The Latin phrase analysis situs may be translated as “analysis of position” and is similar to the phrase geometria situs, meaning “geometry of position,” used in 1735 by the Swiss mathematician Leonhard Euler to describe his solution to the Königsberg bridge problem. Euler’s work on this problem also is cited as the beginning of graph theory, the study of networks of vertices connected by edges, which shares many ideas with topology.


During the 19th century two distinct movements developed that would ultimately produce the sibling specializations of algebraic topology and general topology. The first was characterized by attempts to understand the topological aspects of surfacelike objects that arise by combining elementary shapes, such as polygons or polyhedra. One early contributor to combinatorial topology, as this subject was eventually called, was the German mathematician Johann Listing, who published Vorstudien zur Topologie (1847; “Introductory Studies in Topology”), which is often cited as the first print occurrence of the term topology. In 1851 the German mathematician Bernhard Riemann considered surfaces related to complex number theory and, hence, utilized combinatorial topology as a tool for analyzing functions. The German geometers August Möbius and Felix Klein published works on “one-sided” surfaces in 1858 and 1882, respectively. Möbius’s example, now known as the Möbius strip, may be constructed by gluing together the ends of a long rectangular strip of paper that has been given a half twist. Surfaces containing subsets homeomorphic to the Möbius strip are called nonorientable surfaces and play an important role in the classification of two-dimensional surfaces. Klein provided an example of a one-sided surface that is closed, that is, without any one-dimensional boundaries. This example, now called the Klein bottle, cannot exist in three-dimensional space without intersecting itself and, thus, was of interest to mathematicians who previously had considered surfaces only in three-dimensional space.
Work by many mathematicians, including the four mentioned above, preceded the 1895 publication of Analysis Situs, in which Poincaré established a basic context for using algebraic ideas in combinatorial topology. Combinatorial topology continued to be developed, especially by the German-born American mathematician Max Dehn and the Danish mathematician Poul Heegaard, who jointly presented one of the first classification theorems for two-dimensional surfaces in 1907. Soon thereafter the importance of associating algebraic structures with topological objects was clearly established by, for example, the Dutch mathematician L.E.J. Brouwer and his fixed point theorem. Although the phrase algebraic topology was first used somewhat later in 1936 by the Russian-born American mathematician Solomon Lefschetz, research in this major area of topology was well under way much earlier in the 20th century.
Simultaneous with the early development of combinatorial topology, 19th-century analysts, such as the French mathematician Augustin Cauchy and the German mathematician Karl Weierstrass, investigated Fourier series, in which sequences of functions converged to other functions in a sense similar to convergence of sequences of points in space. From another point of view, mathematicians such as the German Georg Cantor and the French Émile Borel studied the relationship between Fourier series and set theory. Two initiatives arose from these efforts: establishing a rigorous mathematical setting for major problems of analysis and providing a general setting for mathematical ideas related to convergence of sequences. In 1899 the German mathematician David Hilbert proposed an axiomatic setting for general geometry beyond what the ancient Greeks had considered. In 1905 the French mathematician Maurice Fréchet proposed a consistent scheme of axioms for convergence in an abstract set and also axioms for a metric space, which is a set supplied with a distance function (or “metric”). In 1910 Hilbert suggested axioms for neighbourhoods of points in an abstract set, thereby generalizing properties of small disks centred at points in the plane. Finally, the German mathematician Felix Hausdorff in his Grundzüge der Mengenlehre (1914; “Elements of Set Theory”) proposed the foundational axiomatic relationships among the metric, limit, and neighbourhood approaches for general spaces (see Hausdorff space). Although it was not until 1925 that the Russian mathematician Pavel Alexandrov introduced the modern axioms for a topology on an abstract set, the field of general topology was born in Hausdorff’s work.
During the period up to the 1960s, research in the field of general topology flourished and settled many important questions. The notion of dimension and its meaning for general topological spaces was satisfactorily addressed with the introduction of an inductive theory of dimension. Compactness, a property that generalizes closed and bounded subsets of n-dimensional Euclidean space, was successfully extended to topological spaces through a definition involving “covers” of a space by collections of open sets, and many problems involving compactness were solved during this period. The metrization problem, which sought a topological description of the spaces for which the topology could be induced by a metric, was settled following considerable work on the notion of paracompactness, a property that generalizes compactness.


Source Reference 

 

 

The Mathematics of the Paranormal

13.3.13

                                               

James D. Stein. The Paranormal Equation: A New Scientific Perspective on Remote Viewing, Clairvoyance, and Other Inexplicable Phenomena. New Page Books, 2013.

Dr Stein is a professor of mathematics and he believes that there are truths about the natural world that nominally fall within the domain of science that might be impossible for science to discover. He thinks that some as yet undiscovered natural laws might account for some of what appear to be paranormal phenomena.

Unlike some scientists and mathematicians who have studied such things, though, he is not fooled by the tricks of magicians, or the psychic researchers who resort to cheating and fiddle their results. He gives as one of his examples a BBC television show in which the physicist John Taylor appeared as a "scientific hatchet man" during a performance by the Israeli "psychic" and entertainer Uri Geller. Geller did his fork-bending act, followed by telepathy, followed by apparently mending broken watches just by holding them in his hands. Taylor was baffled by these tricks, and thought Geller had psychic powers, even though, like other skilled and experienced magicians, he could make his tricks seem inexplicable.

One of the first problems he considers is how to decide what is meant by paranormal or supernatural phenomena. This is rather difficult. In the case of ghosts, for example, Stein points out that there is no scientific framework in which one can prove that there is no such thing as a ghost. If one could provide scientific proof of the reality of ghosts, they would then be regarded as natural rather than supernatural. It seems to me, though, that this introduces the idea that they could have some sort of subtle physical existence, independent of the people who see them.

Stein, though, says that a law which is a law that cannot be discovered would legitimately be supernatural. "It would be a principle governing behaviour in the Universe which would be beyond our ability, or the ability of any sentient creature, to discover."

As Dr Stein is a mathematician, he gives us a discussion on arithmetic, a subject with which most of us have some acquaintance, and he shows us that some problems with numbers are not as simple as they look. He includes two interesting examples.

The Goldbach Conjecture states that every even number is the sum of two prime numbers. For example, 84 is the sum of prime numbers 41 and 43. Mathematicians have been trying to prove or disprove this for about 300 years, without success. It could, of course, be proved false by finding an even number which is not the sum of two primes, but as the natural numbers are an infinite series, this is not practicable.

To test the Collatz Conjecture, pick any number. If it is even, divide by 2. If it is odd, triple it and add 1. Apply these rules to the result and keep going, and you will always eventually end up with 1. Mathematicians have so far failed to establish whether this conjecture is decidable.

Dr Stein describes some of the counterintuitive results of experiments in quantum physics, and his belief that the universe is infinite. In his concluding remarks, he writes: "The time may come when we have made enough measurements to conjecture a relationship involving information transfer that has all the earmarks of paranormality". This book provides interesting ideas on attempts to discover how some kinds of apparently paranormal phenomena might be compatible with scientific principles. -- John Harney

 

 

 

 

Dimension (mathematics and physics)

As already seen the fourth dimension entry to this blog, and the concept of dimensions can play an important part in the theoretical aspects of parapsychology. What follows is just a general article on the subject with no  reference to psi phenomena per se. However, the evolving project of  Multi-Dimensional Science may have relevance.

 

http://www.p2pfoundation.net/Multi-Dimensional_Science

 

 

RS

 

 

 

From Wikipedia, the free encyclopedia

(Redirected from N-dimensional space)

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Not to be confused with Dimension of a physical quantity. For other uses, see Dimension (disambiguation).

From left to right, the square, the cube, and the tesseract. The square is bounded by 1-dimensional lines, the cube by 2-dimensional areas, and the tesseract by 3-dimensional volumes. A projection of the cube is given since it is viewed on a two-dimensional screen. The same applies to the tesseract, which additionally can only be shown as a projection even in three-dimensional space.

A diagram showing the first four spatial dimensions.

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.^[1][2] Thus a line 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 surface such as a plane or the surface of a cylinder or sphere 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 and its longitude). The inside of a cube, 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 of all space (cf. volume) 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 correlate with both particle and field conceptions, interact according to relative properties of mass—and 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 time, such as general relativity, are said to work in 4-dimensional "spacetime", (defined as a Minkowski space). Modern theories tend to be "higher-dimensional" including quantum field and string theories. The state-space of quantum mechanics is an infinite-dimensional function space.
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 spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

Contents 1 In mathematics 1.1 Dimension of a vector space 1.2 Manifolds 1.3 Varieties 1.4 Krull dimension 1.5 Lebesgue covering dimension 1.6 Inductive dimension 1.7 Hausdorff dimension 1.8 Hilbert spaces 2 In physics 2.1 Spatial dimensions 2.2 Time 2.3 Additional dimensions 3 Networks and dimension 4 Literature 5 Philosophy 6 More dimensions 7 See also 7.1 A list of topics indexed by dimension 8 References 9 Further reading

[edit] In mathematics

In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. For example, a point on the unit circle 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 the mathematical notion of dimension differs from its common usages.
The dimension of Euclidean n-space 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 and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball in Eⁿ looks locally like E^n-1 and this leads to the notion of the inductive dimension. While these notions agree on Eⁿ , they turn out to be different when one looks at more general spaces.
A tesseract 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 Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schlafi's 1852 Theorie der vielfachen Kontinuität, Hamilton's 1843 discovery of the quaternions and the construction of the Cayley Algebra marked the beginning of higher-dimensional geometry.
The rest of this section examines some of the more important mathematical definitions of the dimensions.

[edit] Dimension of a vector space

Main article: Dimension (vector space)

The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.

[edit] Manifolds

A connected 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.
For connected differentiable manifolds the dimension is also the dimension of the tangent vector space at any point
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.

[edit] Varieties

Main article: Dimension of an algebraic variety

The dimension of an algebraic variety may be defined in various equivalent ways. The most intuitive way is probably the dimension of the tangent space at any regular point. Another intuitive way is to define the dimension as the number of hyperplanes that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety.
An algebraic set being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains of sub varieties of the given variety (the length of such a chain is the number of "").

[edit] Krull dimension

Main article: Krull dimension

The Krull dimension of a commutative ring is the maximal length of prime ideals in it. It is strongly related to the dimension of an algebraic variety, because of a natural correspondence between sub varieties and prime ideals of ring of the polynomials on the variety.
For an algebra over a field, the dimension as vector space is finite if and only if its Krull dimension is 0.

[edit] Lebesgue covering dimension

Main article: 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 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 one writes dim X = ∞. Moreover, 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".

[edit] Inductive dimension

Main article: 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 may 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.

[edit] Hausdorff dimension

Main article: Hausdorff dimension

For structurally complicated sets, 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. Fractals have been found useful to describe many natural objects and phenomena.^[4][5]

[edit] 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.

[edit] In physics

[edit] 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.)

Number of dimensions	Example co-ordinate systems
1	Number line Angle
2	Cartesian (2-dimensional) Polar Latitude and longitude
3	Cartesian (3-dimensional) Cylindrical Spherical

[edit] 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.

[edit] Additional dimensions

In physics, three dimensions of space and one of time is the accepted norm. There are theories that try to unify different forces and such—these theories require more dimensions. Superstring theory, M-theory and Bosonic string theory respectively posit that physical space has 10, 11 and 26 dimensions. These extra dimensions 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 Collider in December 2010 severely constrains theories with large extra dimensions.^[6]

[edit] Networks and dimension

Some complex networks are characterized by fractal dimensions.^[7] The concept of dimension can be generalized to include networks embedded in space.^[8] The dimension characterize their spatial constraints.

[edit] Literature

Perhaps the most basic way 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 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 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. Classic stories involving other dimensions include Robert A. Heinlein's —And He Built a Crooked House (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract, and Alan E. Nourse's Tiger by the Tail and The Universe Between (both 1951). Another reference is 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 to move across it quickly. The fourth and fifth dimensions were also a key component of the book The Boy Who Reversed Himself, by William Sleator.

[edit] 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."^[9]

[edit] More dimensions

• Degrees of freedom (mechanics)
• Degrees of freedom (physics and chemistry)
• Degrees of freedom (statistics)
• 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

[edit] See also

• Dimension (data warehouse) and dimension tables
• Dimensional analysis
• Fractal dimension
• Hyperspace (disambiguation page)
• Space-filling curve
• Intrinsic dimension
• Demonstrating Size (dimension)

[edit] 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
  • Curse of dimensionality
• High-dimensional topics from physics:
  • Kaluza–Klein theory
  • String theory
  • M-theory
• Infinitely many dimensions:
  • Hilbert space
  • Function space

[edit] References

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2010)

1. ^ Curious About Astronomy
2. ^ MathWorld: Dimension
3. ^ Fractal Dimension, Boston University Department of Mathematics and Statistics
4. ^ S. Havlin, A. Bunde (1991). Fractals and Disordered Systems. Springer. http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php.
5. ^ S. Havlin, A. Bunde (1994). Fractals in Science. Springer. http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php.
6. ^ CMS Collaoration, "Search for Microscopic Black Hole Signatures at the Large Hadron Collider," http://arxiv.org/abs/1012.3375
7. ^ C.M. Song, S. Havlin, H.A. Makse (2005). "Self-similarity of complex networks". Nature 433: 7024. arXiv:cond-mat/0503078v1. Bibcode 2005Natur.433..392S. doi:10.1038/nature03248. http://havlin.biu.ac.il/Publications.php?keyword=Self-similarity+of+complex+networks&year=*&match=all.
8. ^ D. Li, K. Kosmidis, A. Bunde, S. Havlin (2011). "Dimension of spatially embedded networks". Nature Physics. Bibcode 2011NatPh...7..481D. doi:10.1038/nphys1932. http://havlin.biu.ac.il/Publications.php?keyword=Dimension+of+spatially+embedded+networks&year=*&match=all.
9. ^ Prolegomena, § 12

[edit] 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.
• Michio Kaku, (1994) Hyperspace, a Scientific Odyssey Through the 10th Dimension, Oxford University Press.

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