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Dimension

Dimension

In common usage, the dimensions (from Latin "measured out") of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. In mathematics, the dimensions of a space are the parameters required to describe a particular object in this space. The dimension of a space is the number of these parameters. For example, locating a city on the Earth requires two parameters: longitude and latitude; the corresponding space has therefore two dimensions and its dimension is two. This space is said to be 2-dimensional (for short 2D). Locating an airplane might require a 3D space by addition of the altitude parameter, or even a 6D space, if one adds the three angles required for defining the orientation of the airplane. The airplane is then considered to have six degrees of freedom. Other dimensions can be supplemented like speed, temperature or even colour or price of the plane. Generalisations of the concept are possible and a number of alternative definitions may be introduced. Units are sometimes associated with each dimension, for instance, meters or feet with altitude or dollars with price. In science fiction, a "dimension" can also refer to an alternate universe or plane of existence. This useage is derived from the fact that to get to the alternate universe/plane of existence requires movement in an dimension beyond the normal 3 space + time.

Physical dimensions

The physical dimensions are the parameters required to answer to the question where and when happened or will happen some event; for instance: When did Napoleon die? — On the 5 May 1821 at Saint Helena (15°56′ S 5°42′ W). They play a fundamental role in our perception of the world around us. According to Immanuel Kant, we actually do not perceive them but they form the frame in which we perceive events; they form the a priori background in which events are perceived.

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 amount. 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 Cartesian coordinate system)

Time

Time is often referred to as the "fourth dimension". It is, in essence, 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 movement seems to occur at a fixed rate and in one direction. The equations used by physics to model reality often do not treat time in the same way that humans perceive it. In particular, 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 most well-known treatment of time as a dimension is Einstein's theory of general relativity, which treats perceived space and time as parts of a four-dimensional manifold. Some scientists consider that the fact that Einstein's theory allows time to move at different rates indicates a 2nd time dimension exists.

Additional dimensions

Theories such as string theory predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26), but that the universe measured along these additional dimensions is subatomic in size. As a result, we perceive only the three spatial dimensions that have macroscopic size.

Units

In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of physical unit that such a quantity is measured against. The dimension of speed, for example, is length divided by time. In the SI system, the dimension is given by the seven exponents of the fundamental quantities. See Dimensional analysis.

Mathematical dimensions

In mathematics, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space Eⁿ. The point E⁰ is 0-dimensional. The line E¹ is 1-dimensional. The plane E² is 2-dimensional. And in general Eⁿ is n-dimensional. A tesseract is an example of a four-dimensional object. In the rest of this section we examine some of the more important mathematical definitions of dimension.

Hamel dimension

For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis. See Hamel dimension for details.

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. The theory of manifolds, in the field of geometric topology, is characterised 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 topological space, the Lebesgue covering dimension is defined to be n if n is the smallest integer for which the following holds: any open cover has a refinement (a second 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. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension is infinite.

Inductive dimension

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.

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. The upper and lower box dimensions are a variant of the same idea.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases 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 two dimensions coincide.

Krull dimension of commutative rings

The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Science fiction

Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader. The word gives a sense of authority to a film, and inspires imagination and awe in the minds of the reader, that one could travel to "another dimension". This concept is derived from the idea that in order to travel to 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.

Anaglyph

This section should be merged into Anaglyph image. To understand how Anaglyph 3D works, you must understand how Sterioscope 3D works. It's basically blending the two images taken eye width apart. The eye thinks it's seeing one normal image by blending together one image we see with both eyes. You blend two slightly different pictures together and look at it. It looks 3D! Anaglyph is just taking a single framed 3D image and making one eye only see one image... the red sees the blue (because the red of the glasses blends in with the red) and the blue sees the red (the blue of the glasses blends in with the blue). This creates a normal stereo graph image without the need of crossing your eyes the whole time you're looking at the image.

3-D film

This section should be merged into 3-D film. In the 1950's, 3D movies were very popular, but since they didn't have color film then they had an entirely different method. They would take two black and white cameras, line them up eye-with apart and take the film from both at the same time. They took two projectors in the projection booth, both aimed at the screen. The left one with the left film with a blue filter over the lense, and the right with the right film and a red filter over the lense, rather than just having one 3D image blended together.

Modern 3-D films

Spy Kids 3D has developed a new craze of 3D, mostly in amusement parks with motion simulators. Spongebob 3D in Paramount's Great America, California has a Polarization 3D movie along with a motion simulator. Shark Boy and Lava Girl is another modern 3D movie, by the same producer of Spy Kids.

More dimensions

- Dimension of an algebraic variety
- Topological dimension
- Isoperimetric dimension
- Poset dimension
- Pointwise dimension
- Lyapunov dimension
- Kaplan-Yorke dimension
- Exterior dimension
- Hurst exponent
- q-dimension; especially:
- Information dimension (corresponding to q=1)
- Correlation dimension (corresponding to q=2)

Latin

Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. It gained great importance as the formal language of the Roman Empire. All Romance languages, those being most notably Spanish, French, Portuguese, Italian, and Romanian, are descended from Latin, and many words based on Latin are found in other modern languages such as English. The Latin alphabet, derived from the Greek, remains the most widely-used alphabet in the world. It is said that 80 percent of scholarly English words are derived from Latin (in a large number of cases by way of French). Moreover, in the Western world, Latin was a lingua franca, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th. Ecclesiastical Latin remains the formal language of the Roman Catholic Church to this day, and thus the official national language of the Vatican. The Church used Latin as its primary liturgical language until the Second Vatican Council in the 1960s. Latin is also still used (drawing heavily on Greek roots) to furnish the names used in the scientific classification of living things. The modern study of Latin, along with Greek, is known as Classics.

Main features

Latin is a synthetic inflectional language: affixes (which usually encode more than one grammatical category) are attached to fixed stems to express gender, number, and case in adjectives, nouns, and pronouns, which is called declension; and person, number, tense, voice, mood, and aspect in verbs, which is called conjugation. There are five declensions (declinationes) of nouns and four conjugations of verbs. There are six noun cases: #nominative (used as the subject of the verb or the predicate nominative), #genitive (used to indicate relation or possession, often represented by the English of or the addition of s to a noun), #dative (used of the indirect object of the verb, often represented by the English to or for), #accusative (used of the direct object of the verb, or object of the preposition in some cases), #ablative (separation, source, cause, or instrument, often represented by the English by, with, from), #vocative (used of the person or thing being addressed). In addition, some nouns have a locative case used to express location (otherwise expressed by the ablative with a preposition such as in), but this survival from Proto-Indo-European is found only in the names of lakes, cities, towns, small islands, and a few other words related to locations, such as "house", "ground", and "countryside". Latin itself, being a very old language, is far closer to Proto-Indo-European than are most modern Western European languages; it has, in fact, about the same relationship with PIE as modern Italian or French has to Latin. There are six general tenses in Latin (technically they are tense/aspect/mood complexes). The indicative mood can be used with all of them. The subjunctive mood, however, has only present, imperfect, perfect, and pluperfect tenses. These tenses in the subjunctive mood do not completely correlate in meaning to the tenses in the indicative. The following examples are of the first conjugation verb "laudare" ("to praise") in the indicative mood and the active voice:
Primary sequence tenses
# present (laudo, "I praise") # imperfect (laudabam, "I was praising") # future (laudabo, "I shall praise," "I will praise")
Secondary sequence tenses
# perfect (laudavi, "I praised", "I have praised") # pluperfect (laudaveram, "I had praised") # future perfect (laudavero, "I shall have praised," "I will have praised") The future perfect tense can also imply a normal future idea (like in "When I will have run...") and so may also sometimes be included in the primary sequence.
Latin and Romance
After the collapse of the Roman Empire, Latin evolved into the various Romance languages. These were for many centuries only spoken languages, Latin still being used for writing. For example, Latin was the official language of Portugal until 1296 when it was replaced by Portuguese. The Romance languages evolved from Vulgar Latin, the spoken language of common usage, which in turn evolved from an older speech which also produced the formal classical standard. Latin and Romance differ (for example) in that Romance had distinctive stress, whereas Latin had distinctive length of vowels. In Italian and Sardo logudorese, there is distinctive length of consonants and stress, in Spanish only distinctive stress, and in French even stress is no longer distinctive. Another major distinction between Romance and Latin is that all Romance languages, excluding Romanian, have lost their case endings in most words except for some pronouns. Romanian retains a direct case (nominative/accusative), an indirect case (dative/genitive), and vocative. In Italy, Latin is still compulsory in secondary schools as Liceo Classico and Liceo Scientifico which are usually attended by people who aim to the highest level of education. In Liceo Classico Ancient Greek is a compulsory subject.
Latin and English
See Latin influence in English for a more complete exposition. English grammar is independent of Latin grammar, though prescriptive grammarians in English have been heavily influenced by Latin. Attempts to make English grammar follow Latin rules — such as the prohibition against the split infinitive — have not worked successfully in regular usage. However, as many as half the words in English were derived from Latin, including many words of Greek origin first adopted by the Romans, not to mention the thousands of French, hundreds of Spanish, Portuguese and Italian words of Latin origin that have also enriched English. During the 16th and on through the 18th century English writers created huge numbers of new words from Latin and Greek roots. These words were dubbed "inkhorn" or "inkpot" words (as if they had spilled from a pot of ink). Many of these words were used once by the author and then forgotten, but some remain. Imbibe, extrapolate, dormant and inebriation are all inkhorn terms carved from Latin words. In fact, the word etymology is derived from the Greek word etymologia, meaning "true sense of the word." Latin was once taught in many of the schools in Britain with academic leanings - perhaps 25% of the total [http://www.channel4.com/history/microsites/T/teachem2/thennow/]. However, the requirement for it was gradually abandoned in the professions such as the law and medicine, and then, from around the late 1960s, for admission to university. After the introduction of the Modern Language GCSE in the 1980s, it was gradually replaced by other languages, although it is now being taught by more schools along with other classical languages.
Latin education
The linguistic element of Latin courses offered in high schools or secondary schools, and in universities, is primarily geared toward an ability to translate Latin texts into modern languages, rather than using it in oral communication. As such, the skill of reading is heavily emphasized, whereas speaking and listening skills are barely touched upon. However, there is a growing movement, sometimes known as the Living Latin movement, whose supporters believe that Latin can, or should, be taught in the same way that modern "living" languages are taught, that is, as a means of both spoken and written communication. One of the most interesting aspects of such an approach is that it assists speculative insight into how many of the ancient authors spoke and incorporated sounds of the language stylistically; without understanding how the language is meant to be heard it is very difficult to identify patterns in Latin poetry. Institutions offering Living Latin instruction include the Vatican and the University of Kentucky. In Britain the Classical Association encourages this approach, and there has been something of a vogue for books describing the adventures of a mouse called Minimus. In the United States there is a thriving competitive organization for high school Latin students, the National Junior Classical League (the second-largest youth organization in the world after the Boy Scouts), backed up by the Senior Classical League for college students. Many would-be international auxiliary languages have been heavily influenced by Latin, and the moderately successful Interlingua considers itself to be the modernized and simplified version of the language (le latino moderne international e simplificate). Latin translations of modern literature such as Paddington Bear, Winnie the Pooh, Harry Potter and the Philosopher's Stone, Le Petit Prince, Max und Moritz, and The Cat in the Hat have also helped boost interest in the language.
See also

About the Latin language

- Latin grammar
- Latin spelling and pronunciation
- Latin declension
- Latin conjugation
- Latin alphabet
- List of Latin words with English derivatives
- Latin verbs with English derivatives
- Latin nouns with English derivatives
- ablative absolute
- Word order in Latin
About the Latin literary heritage

- Latin literature
- Romance languages
- Loeb Classical Library
- List of Latin phrases
- List of Latin proverbs
- Brocard
- List of Latin and Greek words commonly used in systematic names
- List of Latin place names in Europe
- Carmen Possum
Other related topics

- Roman Empire
- Internationalism
References

- Bennett, Charles E. Latin Grammar (Allyn and Bacon, Chicago, 1908)
- N. Vincent: "Latin", in The Romance Languages, M. Harris and N. Vincent, eds., (Oxford Univ. Press. 1990), ISBN 0195208293
- Waquet, Françoise, Latin, or the Empire of a Sign: From the Sixteenth to the Twentieth Centuries (Verso, 2003) ISBN 1859844022; translated from the French by John Howe.
- Wheelock, Frederic. Latin: An Introduction (Collins, 6th ed., 2005) ISBN 0060784237
External links

- [http://www.jambell.com/latin.html Latin Phrases for after dinner conversation (Thanks to Elaine Poole)]
- [http://www.ethnologue.com/show_language.asp?code=lat Ethnologue report for Latin]
- [http://forumromanum.org/literature/index.html Corpus Scriptorum Latinorum] is a comprehensive webography of Latin texts and their translations.
- [http://www.perseus.tufts.edu/ The Perseus Project] has many useful pages for the study of classical languages and literatures, including [http://www.perseus.tufts.edu/cgi-bin/resolveform?lang=Latin an interactive Latin dictionary].
- [http://lysy2.archives.nd.edu/cgi-bin/words.exe words by William whitaker] is a dictionary program online capable of looking up various word forms.
- [http://retiarius.org/ Retiarius.Org] includes a Latin text search engine.
- [http://www.nd.edu/~archives/latgramm.htm Latin-English dictionary and Latin grammar from U of Notre Dame]
- [http://latin-language.co.uk/ Latin language] History of Latin language, Latin texts with English translation and a collection of dictionaries.
- [http://augustinus.eresmas.net/scl/ Societas Circulorum Latinorum] gathers together Latin Circles all over the world.
- [http://www.learnlatin.tk LearnLatin.tk] - Free online course in Latin
- [http://www.latintests.net/ LatinTests.net] - Lets Latin learners test their grammar and vocabulary with self-checking quizzes.
- [http://thelatinlibrary.com/ The Latin Library] contains many Latin etexts
- [http://www.textkit.com/ Textkit] has Latin textbooks and etexts.
- [http://www.websters-online-dictionary.org/definition/Latin-english/ Latin–English Dictionary]: from Webster's Rosetta Edition.
- [http://www.language-reference.com/ Language reference] Cross-foreign-language lexicon powered by its own search engine. All cross combinations between Latin and French, German, Italian, Spanish.
- [http://comp.uark.edu/~mreynold/rhetor.html Rhetor by Gabriel Harvey] was originally published in 1577 and never again reprinted.
- [http://freewebs.com/omniamundamundis omniamundamundis] Latin hypertexts from fourteen ancient Roman authors.
- [http://www.saltspring.com/capewest/pron.htm Pronunciation of Biological Latin, Including Taxonomic Names of Plants and Animals]
- [http://www.yleradio1.fi/nuntii Nuntii Latini (News in Latin)], written and spoken (RealAudio) news in latin. Weekly review of world news in Classical Latin, the only international broadcast of its kind in the world, produced by YLE, the Finnish Broadcasting Company.
- [http://www.tranexp.com:2000/InterTran?url=http%3A%2F%2F&type=text&text=Replace%20Me&from=eng&to=ltt InterTran Latin], Translate from Latin to ENGLISH or vice versa.
- [http://www.latinvulgate.com Latin Vulgate] The Latin and English of the Old & New Testaments in parallel, along with the Complete Sayings of Jesus in parallel Latin and English. Category:Classical languages Category:Ancient languages Category:Fusional languages Category:Languages of Italy Category:Languages of Vatican City als:Latein zh-min-nan:Latin-gí ko:라틴어 ja:ラテン語 simple:Latin language th:ภาษาละติน

Parameter

:For usage in computer science and programming, see parameter (computer science). A parameter is a measurement or value on which something else depends. For example, a parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.

Types of parameter

Mathematical

In mathematics, the difference in meaning between a parameter and an argument of a function is that the parameters are the symbols that are part of the function's definition, while arguments are the symbols that are supplied to the function when it is used. The value or objects assigned to the parameters by the corresponding arguments of a function or system are not reassigned during the function's evaluation. So, parameters are effectively constants during the evaluation or processing of that function or system. The value of arguments can change outside of the function and between function usages. This distinction, the parameter's constancy, is a key part of the meaning of a parameter in any situation, often in usage beyond just mathematics. In some informal situations people regard it as a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters.

Computer science

When the terms formal parameter and actual parameter are used, they generally correspond with the definitions used in computer science. In the definition of a function such as :f(x) = x + 2, x is a formal parameter. When the function is used as in :y = f(3) + 5, 3 is the actual parameter value that is used to solve the equation. These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic. In computing, the parameters passed to a function subroutine are more normally called arguments.

Logic

In logic, the parameters passed to (or operated on by) an open predicate are called parameters by some authors (e.g., Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Parameters locally defined within the predicate are called variables. This extra distinction pays off when defining substitution (without this distinction special provision has to be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables, and when defining substitution have to distinguish between free variables and bound variables.

Analytic geometry

In analytic geometry, curves are often given as the image of some function. The argument of the function is invariably called "the parameter". A circle of radius 1 centered at the origin can be specified in more than one form:
- implicit form :

x^2+y^2=1

- parametric form :

(x,y)=(\cos t,\sin t)

:where t is the parameter. A somewhat more detailed description can be found at parametric equation.

Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form :

F(t)=\int_^f(x;t)\,dx.

In this formula, t is the argument of the function F on the left-hand side, and the parameter that the integral depends on, on the right-hand side. The quantity x is a dummy variable or variable (or parameter) of integration. Now, if we performed the substitution x=g(y), it would be called a change of variable.

Probability theory

In probability theory, one may describe the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters. For example, one talks about "a Poisson distribution with mean value λ", or "a normal distribution with mean μ and variance σ²". The latter formulation and notation leaves some ambiguity whether σ or σ² is the second parameter; the distinction is not always relevant. It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution.

Statistics

In statistics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In classical estimation these parameters are considered "fixed but unknown", but in Bayesian estimation they are random variables with distributions of their own. It is possible to make statistical inferences without assuming a particular parametric family of probability distributions. In that case, one speaks of non-parametric statistics as opposed to the parametric statistics described in the previous paragraph. For example, Spearman is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas Pearson is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship. Statistics are mathematical characteristics of samples which are used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the sample mean (

\overline X

) is an estimate of the mean parameter (μ) of the population from which the sample was drawn.

Measurement

In classical physics and engineering, measurement is the process of estimating or determining the ratio of a magnitude of a quantitative property or relation to a unit of the same type of quantitative property or relation. A process of measurement involves the comparison of physical quantities of objects or phenomena, or the comparison of relations between objects (e.g. angles). A particular measurement is the result of such a process, normally expressed as the multiple of a real number and a unit, where the real number is the ratio obtained from the measurement process. For example, the measurement of the length of an object might be 5 m, which is an estimate of the object's length, a magnitude, relative to a unit of length, the meter. Measurement is not limited to physical quantities and relations but can in principle extend to the quantification of a magnitude of any type, through application of a measurement model such as the Rasch model, and subjecting empirical data derived from comparisons to appropriate testing in order to ascertain whether specific criteria for measurement have been satisfied. In addition, the term measurement is often used in a somewhat looser fashion than defined above, to refer to any process in which numbers are assigned to entities such that the numbers are intended to represent increasing amount, in some sense, without a process that involves the estimation of ratios of magnitudes to a unit. Such examples of measurement range from degrees of uncertainty to consumer confidence to the rate of increase in the fall in the price of a good or service. It is generally proposed that there are four different levels of measurement, and that different levels are applicable to different contexts and types of measurement process. In scientific research, measurement is essential. It includes the process of collecting data which can be used to make claims about learning. Measurement is also used to evaluate the effectiveness of a program or product (known as an evaluand). :: A measurement is a comparison to a standard. -- William Shockley :: By number we understand not so much a multitude of Unities, as the abstracted Ratio of any Quantity to another Quantity of the same kind, which we take for Unity -- Sir Isaac Newton

Units and systems of measurement

:Main articles: Units of measurement and Systems of measurement Because measurement involves the estimation of magnitudes of quantities relative to particular quantities, called units, the specification of units is of fundamental importance to measurement. The definition or specification of precise standards of measurement involves two key features, which are evident in the Système International d'Unités (SI). Specifically, in this system the definition of each of the base units makes reference to specific empirical conditions and, with the exception of the kilogram, also to other quantitative attributes. Each derived SI unit is defined purely in terms of a relationship involving itself and other units; for example, the unit of velocity is 1 m/s. Due to the fact that derived units make reference to base units, the specification of empirical conditions is an implied component of the definition of all units. The measurement of a specific entity or relation results in at least two numbers for the relationship between the entity or relation under study and the referenced unit of measurement, where at least one number estimates the statistical uncertainty in the measurement, also referred to as measurement error. Measuring instruments are used to estimate ratios of magnitudes to units. Prior comparisons underlie the calibration, in terms of standard units, of commonly used instruments constructed to measure physical quantities.

Metrology

Metrology is the study of measurement. In general, a metric is a scale of measurement defined in terms of a standard: i.e. in terms of well-defined unit. The quantification of phenomena through the process of measurement relies on the existence of an explicit or implicit metric, which is the standard to which measurements are referenced. If I say I am 5, I am indicating a measurement without supplying an applicable standard. I may mean I am 5 years old or I am 5 feet high, however the implicit metric is that.

History

:Main article: History of measurement Laws to regulate measurement were originally developed to prevent fraud. However, units of measurement are now generally defined on a scientific basis, and are established by international treaties. In the United States, commercial measurements are regulated by the National Institute of Standards and Technology NIST, a division of the United States Department of Commerce. The history of measurements is a topic within the history of science and technology. The metre (us: meter) was standardized as the unit for length after the French revolution, and has since been adopted throughout most of the world. The United States and the UK are in the process of converting to the SI system. This process is known as metrication.

Difficulties in measurement

Measurement of many quantities is very difficult and prone to large error. Part of the difficulty is due to uncertainty, and part of it is due to the limited time available in which to make the measurement. Examples of things that are very difficult to measure in some respects and for some purposes include social related items such as:
- A person's knowledge (as in testing, see also assessment)
- A person's feelings, emotions, or beliefs.
- A person's senses (qualia). Even for physical quantities gaining accurate measurement can be difficult. It is not possible to be exact, instead, repeated measurements will vary due to various factors affecting the quantity such as temperature, time, electromagnetic fields, and especially measurement method. As an example in the measurement of the speed of light, the quantity is now known to a high degree of precision due to modern methods, but even with those methods there is some variability in the measurement. Statistical techniques are applied to the measurement samples to estimate the speed. In earlier sets of measurements, the variability was greater, and comparing the results shows that the variability and bias in the measurement methods was not properly taken into account. Proof of this is that when various group's measurements are plotted with the estimated speed and error bars showing the expected variability of the estimated speed from the actual number, the error bars from each of the experiments did not all overlap. This means a number of groups incorrectly accounted for the true sources of error and overestimated the accuracy of their methods.

Miscellaneous

Measuring the ratios between physical quantities is an important sub-field of physics. Some important physical quantities include:
- Speed of light
- Planck's constant
- Gravitational constant
- Elementary charge (electric charge of electrons, protons, etc.)
- Fine-structure constant

References

Newton, I. (1728/1967). Universal Arithmetic: Or, a Treatise of Arithmetical Composition and Resolution. In D.T. Whiteside (Ed.), The mathematical Works of Isaac Newton, Vol. 2 (pp. 3-134). New York: Johnson Reprint Corp.

External links

- [http://www.unc.edu/~rowlett/units/index.html A Dictionary of Units of Measurement]
- [http://www.geocities.com/qubestrader/conversion.html Conversion Calculator]
- [http://www.euromet.org/docs/pubs/docs/Metrology_in_short_2nd_edition_may_2004.pdf 'Metrology In Short', 2nd Edition] ja:測定 simple:Measurement

Size

You may be looking for one of the following:
- Dimensions: length, width, height
- Geometry
- Measurement Or the following command-line Unix tool:
- Size (Unix)

Width

:This article is about the concept and measurement of distance. For usage in cricket, see line and length. In general English usage, length (symbols: l, L) is but one particular instance of distance – an object's length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with "distance". Height is vertical distance; width (or breadth) is a lateral distance; an object's width is less than its length. No one speaks of "the length from here to Alpha Centauri", but rather of "the distance from here to Alpha Centauri," but when one speaks of distance more abstractly, one says "A kilometre or a mile, is a unit of length" or "...of distance", and the two statements are synonymous. Likewise, a mountain might be a mile in height. Length is the metric of one dimension of space. The metric of space itself is volume, or (length)³. Length is commonly considered to be one of the fundamental units, meaning that it cannot be defined in terms of other dimensions. However, a set of units can be constructed where units of length can be derived from fundamental physical constants - see Planck units and Planck length. Colloquially length sometimes refers to duration, especially when used in context of music.

Units of length(SI)

The SI unit of Length is the metre (U.S. spelling: meter), from which can be derived:from the regular basis of the foundation of the whole world
- centimetre
- kilometre

Other units of length

The Imperial and US customary units of length

- inch
- foot
- yard
- mile

Units are used in astronomy

- Astronomical unit
- Light year
- Parsec

External links

- [http://www.unitconversion.org/unit_converter/length.html Length Converter: convert between units of length, such as meter, yard, mile, and so on]
- [http://www.unitconversion.org/unit_converter/length-v.html Length Conversion table: convert selected unit to all other units of length]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- Category:Norm ko:길이 ja:長さ

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Space

:This article is about space — the scientific and philosophical concepts. For other uses of space, see space (disambiguation). Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. Perhaps as a result of this considerable discussion, it is difficult to provide an uncontroversial and clear definition of the nature of space, except its physical definition (see below). This article looks at the way space is dealt with variously by physicists, mathematicians and philosophers, and at the relation between space and the mind.

Physics and Space

Space is one of the few fundamental quantities in physics meaning it can't be defined via other quantities because there is nothing more fundamental known at present. Thus, similar to the definition of other fundamental quantities (like time and mass), space is defined via measurement. Currently, the standard space interval, called a standard meter or simply meter, is defined as the distance traveled by light in a vacuum during a time interval of 1/299 792 458 of a second (exact). In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinates. Relativistic physics examines spacetime rather than space; spacetime is modeled as a four-dimensional manifold, and currently, there are theories that can support even eleven-dimensional spaces. Before Einstein's work on relativistic physics, time and space were seen as independent dimensions. Einstein's work unified the two into spacetime. In spacetime, measurements of space and time are held to be relative to velocity.

Measurement

The measurement of physical space has long been important. Geometry, the name given to the branch of mathematics which measures spatial relations, was popularised by the ancient Greeks, although earlier societies had developed measuring systems. The International System of Units, (SI), is now the most common system of units used in the measuring of space, and is almost universally used within science. Geography is the branch of science concerned with identifying and describing the Earth, utilising spatial awareness to try and understand why things exist in specific locations. Cartography is the mapping of spaces to allow better navigation, for visualisation purposes and to act as a locational device. Astronomy is the science involved with the observation, explanation and measuring of objects in outer space.

Astronomy and space

In astronomy, space refers collectively to the relatively empty parts of the universe. Any area outside the atmospheres of any celestial body can be considered 'space'. Although space is certainly spacious, it is now known to be far from empty, and filled with a tenuous plasma. In particular, the boundary between space and Earth's atmosphere is conventionally set at the Karman line.

Mathematics and space

In mathematics, a space is a set, with some particular properties and usually some additional structure. It is not a formally defined concept as such, but a generic name for a number of similar concepts, most of which generalize some abstract properties of the physical concept of space. In particular, a vector space and specifically a Euclidean space can be seen as generalizations of the concept of a Euclidean coordinate system. Important varieties of vector spaces with more imposed structure include Banach space and Hilbert space. Distance measurement is abstracted as the concept of metric space and volume measurement leads to the concept of measure space. As far as the concept of dimension is defined, this need not be 3: it can also be 0 (a point), 1 (a line), 2 (a plane), more than 3, and with some definitions, a non-integer value. Mathematicians often study general structures that hold regardless of the number of dimensions. Kinds of mathematical spaces include:
- Banach space
- Euclidean space
- Hilbert space
- Metric space
- Probability space
- Projective space
- Topological space
- Vector space

The philosophy of space

Space has a range of definitions.
- One view of space is that it is part of the fundamental structure of the universe, a set of dimensions in which objects are separated and located, have size and shape, and through which they can move.
- A contrasting view is that space is part of a fundamental abstract mathematical conceptual framework (together with time and number) within which we compare and quantify the distance between objects, their sizes, their shapes, and their speeds. In this view space does not refer to any kind of entity that is a "container" that objects "move through". These opposing views are relevant also to definitions of time. Space is typically described as having three dimensions, and that three numbers are needed to specify the size of any object and/or its location with respect to another location. Modern physics does not treat space and time as independent dimensions, but treats both as features of spacetime – a conception that challenges intuitive notions of distance and time. An issue of philosophical debate is whether space is an ontological entity itself, or simply a conceptual framework we need to think (and talk) about the world. Another way to frame this is to ask, "Can space itself be measured, or is space part of the measurement system?" The same debate applies also to time, and an important formulation in both areas was given by Immanuel Kant. In his Critique of Pure Reason, Kant described space as an a priori notion that (together with other a priori notions such as time) allows us to comprehend sense experience. With Kant, neither space nor time are conceived as substances, but rather both are elements of a systematic framework we use to structure our experience. Spatial measurements are used to quantify how far apart objects are, and temporal measurements are used to quantify how far apart events occur. Similar philosophical questions concerning space include: Is space absolute or purely relational? Does space have one correct geometry, or is the geometry of space just a convention? Historical positions in these debates have been taken by Isaac Newton (space is absolute), Gottfried Leibniz (space is relational), and Henri Poincaré (spatial geometry is a convention). Two important thought-experiments connected with these questions are: Newton's bucket argument and Poincaré's sphere-world.

The psychology of space

The way in which space is perceived is an area which psychologists first began to study in the middle of the 19th century, and it is now thought by those concerned with such studies to be a distinct branch within psychology. Psychologists analysing the perception of space are concerned with how recognition of an object's physical appearance or its interactions are perceived. Other, more specialised topics studied include amodal perception and object permanence. The perception of surroundings is important due to its necessary relevance to survival, especially with regards to hunting and self preservation. "Veridical perception" is the term used to describe the processing of the information provided by the sensory organs to an extent whereby it allows interaction with the actuality of that perceived. It is worth noting that the way we perceive space may not necessarily be representative of the actuality of space.

Anxiety and space

Space can also cause anxiety in people, with agoraphobia manifesting itself in some people as a fear of open spaces, and claustrophobia being the fear of enclosed spaces. Astrophobia is the fear of celestial space, Kenophobia is the fear of empty spaces and spacephobia is the fear of outer space.

Personal space

The term personal space refers to the amount of space a person likes to maintain between their own person and that of other people.

Use of space

The definition of physical space in relation to ownership, in which space is seen as property, has long been an important issue. Whilst some cultures assert the rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership. Spatial planning is a method of regulating the use of space at land-level, with decisions made at regional, national and international levels. Space can also impact on human and cultural behaviour, being an important factor in architecture, where it will impact on the design of buildings and structures, and on farming. Ownership of space is not restricted to land. Ownership of Airspace and of waters is decided internationally. Public space is a term used to define areas of land which are open to all, whilst private property is that area of land owned by an individual or company, for their own use and pleasure.

Reference

[http://search.eb.com/eb/article?tocId=46639 Space perception]. Encyclopædia B

Dimension

Physical dimensions

Spatial dimensions

Time

Additional dimensions

Units

Mathematical dimensions

Hamel dimension

Manifolds

Lebesgue covering dimension

Inductive dimension

Hausdorff dimension

Hilbert spaces

Krull dimension of commutative rings

Science fiction

Anaglyph

3-D film

Modern 3-D films

More dimensions

See also

Degrees of freedom

Other

Further reading

Latin

Main features

Primary sequence tenses

Secondary sequence tenses

Latin and Romance

Latin and English

Latin education

See also

About the Latin language

About the Latin literary heritage

Other related topics

References

External links

Parameter

Types of parameter

Mathematical

Computer science

Logic

Analytic geometry

Mathematical analysis

Probability theory

Statistics

See also

Measurement

Units and systems of measurement

Metrology

History

Difficulties in measurement

See also

Miscellaneous

References

External links

Size

Width

Units of length(SI)

Other units of length

The Imperial and US customary units of length

Units are used in astronomy

See also

External links

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians