Cantonese is a variety of Chinese spoken in the city of Guangzhou and its surrounding area in Southeastern China. It is the traditional prestige variety and standard form of Yue Chinese, one of the major subgroups of Chinese. In mainland China, it is the lingua franca of the province of Guangdong and neighbouring areas such as Guangxi, it is the official language of Hong Kong and Macau. Cantonese is widely spoken amongst Overseas Chinese in Southeast Asia and throughout the Western world. While the term Cantonese refers to the prestige variety, it is used in a broader sense for the entire Yue subgroup of Chinese, including related but mutually unintelligible languages and dialects such as Taishanese; when Cantonese and the related Yuehai dialects are classified together, there are about 80 million total speakers. Cantonese is viewed as a vital and inseparable part of the cultural identity for its native speakers across large swaths of Southeastern China, Hong Kong and Macau, as well as in overseas communities.
Although Cantonese shares a lot of vocabulary with Mandarin, the two varieties are mutually unintelligible because of differences in pronunciation and lexicon. Sentence structure, in particular the placement of verbs, sometimes differs between the two varieties. A notable difference between Cantonese and Mandarin is; this results in the situation in which a Cantonese and a Mandarin text may look similar but are pronounced differently. In English, the term "Cantonese" can be ambiguous. Cantonese proper is the variety native to the city of Canton, the traditional English name of Guangzhou; this narrow sense may be specified as "Canton language" or "Guangzhou language". However, "Cantonese" may refer to the primary branch of Chinese that contains Cantonese proper as well as Taishanese and Gaoyang. In this article, "Cantonese" is used for Cantonese proper. Speakers called this variety "Canton speech" or "Guangzhou speech", although this term is now used outside Guangzhou. In Guangdong and Guangxi, people call it "provincial capital speech" or "plain speech".
Academically called "Canton prefecture speech". In Hong Kong and Macau, as well as among overseas Chinese communities, the language is referred to as "Guangdong speech" or "Canton Province speech", or as "Chinese". In mainland China, the term "Guangdong speech" is increasingly being used amongst both native and non-native speakers. Given the history of the development of the Yue languages and dialects during the Tang dynasty migrations to the region, in overseas Chinese communities, it is referred to as "Tang speech", given that the Cantonese people refer to themselves as "people of Tang". Due to its status as a prestige dialect among all the dialects of the Yue branch of Chinese varieties, it is called "Standard Cantonese"; the official languages of Hong Kong are English, as defined in the Hong Kong Basic Law. The Chinese language has many different varieties. Given the traditional predominance of Cantonese within Hong Kong, it is the de facto official spoken form of the Chinese language used in the Hong Kong Government and all courts and tribunals.
It is used as the medium of instruction in schools, alongside English. A similar situation exists in neighboring Macau, where Chinese is an official language alongside Portuguese; as in Hong Kong, Cantonese is the predominant spoken variety of Chinese used in everyday life and is thus the official form of Chinese used in the government. The Cantonese spoken in Hong Kong and Macau is mutually intelligible with the Cantonese spoken in the mainland city of Guangzhou, although there exist some minor differences in accent and vocabulary. Cantonese first developed around the port city of Guangzhou in the Pearl River Delta region of southeastern China. Due to the city's long standing as an important cultural center, Cantonese emerged as the prestige dialect of the Yue varieties of Chinese in the Southern Song dynasty and its usage spread around most of what is now the provinces of Guangdong and Guangxi. Despite the cession of Macau to Portugal in 1557 and Hong Kong to Britain in 1842, the ethnic Chinese population of the two territories originated from the 19th and 20th century immigration from Guangzhou and surrounding areas, making Cantonese the predominant Chinese language in the territories.
On the mainland, Cantonese continued to serve as the lingua franca of Guangdong and Guangxi provinces after Mandarin was made the official language of the government by the Qing dynasty in the early 1900s. Cantonese remained a dominant and influential language in southeastern China until the establishment of the People's Republic of China in 1949 and its promotion of Standard Chinese as the sole official language of the nation throughout the last half of the 20th century, although its influence still remains strong within the region. While the Chinese government vehemently discourages the official use of all forms of Chinese except Standard Chinese, Cantonese enjoys a higher standing than other Chinese langua
Serbo-Croatian is a South Slavic language and the primary language of Serbia, Croatia and Herzegovina, Montenegro. It is a pluricentric language with four mutually intelligible standard varieties. South Slavic dialects formed a continuum; the turbulent history of the area due to expansion of the Ottoman Empire, resulted in a patchwork of dialectal and religious differences. Due to population migrations, Shtokavian became the most widespread dialect in the western Balkans, intruding westwards into the area occupied by Chakavian and Kajkavian. Bosniaks and Serbs differ in religion and were often part of different cultural circles, although a large part of the nations have lived side by side under foreign overlords. During that period, the language was referred to under a variety of names, such as "Slavic" in general or "Serbian", "Croatian", ”Bosnian”, "Slavonian" or "Dalmatian" in particular. In a classicizing manner, it was referred to as "Illyrian"; the process of linguistic standardization of Serbo-Croatian was initiated in the mid-19th-century Vienna Literary Agreement by Croatian and Serbian writers and philologists, decades before a Yugoslav state was established.
From the beginning, there were different literary Serbian and Croatian standards, although both were based on the same Shtokavian subdialect, Eastern Herzegovinian. In the 20th century, Serbo-Croatian served as the official language of the Kingdom of Yugoslavia, as one of the official languages of the Socialist Federal Republic of Yugoslavia; the breakup of Yugoslavia affected language attitudes, so that social conceptions of the language separated on ethnic and political lines. Since the breakup of Yugoslavia, Bosnian has been established as an official standard in Bosnia and Herzegovina, there is an ongoing movement to codify a separate Montenegrin standard. Serbo-Croatian thus goes by the names Serbian, Croatian and sometimes Montenegrin and Bunjevac. Like other South Slavic languages, Serbo-Croatian has a simple phonology, with the common five-vowel system and twenty-five consonants, its grammar evolved from Common Slavic, with complex inflection, preserving seven grammatical cases in nouns and adjectives.
Verbs exhibit imperfective or perfective aspect, with a moderately complex tense system. Serbo-Croatian is a pro-drop language with flexible word order, subject–verb–object being the default, it can be written in Serbian Cyrillic or Gaj's Latin alphabet, whose thirty letters mutually map one-to-one, the orthography is phonemic in all standards. Throughout the history of the South Slavs, the vernacular and written languages of the various regions and ethnicities developed and diverged independently. Prior to the 19th century, they were collectively called "Illyric", "Slavic", "Slavonian", "Bosnian", "Dalmatian", "Serbian" or "Croatian". Since the XIX century the term Illyric was used quite often. Although the word Illyrian was used on a few occasions before, the widespread usage of the term began after Ljudevit Gaj and several other prominent linguists met at Ljudevit Vukotinović's house to discuss the issue in 1832; the term Serbo-Croatian was first used by Jacob Grimm in 1824, popularized by the Viennese philologist Jernej Kopitar in the following decades, accepted by Croatian Zagreb grammarians in 1854 and 1859.
At that time and Croat lands were still part of the Ottoman and Austrian Empires. The language was called variously Serbo-Croat, Croato-Serbian and Croatian, Croatian and Serbian, Serbian or Croatian, Croatian or Serbian. Unofficially and Croats called the language "Serbian" or "Croatian" without implying a distinction between the two, again in independent Bosnia and Herzegovina, "Bosnian", "Croatian", "Serbian" were considered to be three names of a single official language. Croatian linguist Dalibor Brozović advocated the term Serbo-Croatian as late as 1988, claiming that in an analogy with Indo-European, Serbo-Croatian does not only name the two components of the same language, but charts the limits of the region in which it is spoken and includes everything between the limits. Today, use of the term "Serbo-Croatian" is controversial due to the prejudice that nation and language must match, it is still used for lack of a succinct alternative, though alternative names have emerged, such as Bosnian/Croatian/Serbian, seen in political contexts such as the International Criminal Tribunal for the former Yugoslavia.
Old Church Slavonic was adopted as the language of the liturgy. This language was adapted to non-liturgical purposes and became known as the Croatian version of Old Slavonic; the two variants of the language and non-liturgical, continued to be a part of the Glagolitic service as late as the middle of the 19th century. The earliest known Croatian Church Slavonic Glagolitic manuscripts are the Glagolita Clozianus and the Vienna Folia from the 11th century; the beginning of written Serbo-Croatian can be traced from the 10th century and on when Serbo-Croatian medieval texts were written in five scripts: Latin, Early Cyrillic, Bosnian Cyrillic, Arebica, the last principally by Bosniak nobility. Serbo-Croatian competed with the more established literary languages of Latin
Classical Chinese known as Literary Chinese, is the language of the classic literature from the end of the Spring and Autumn period through to the end of the Han dynasty, a written form of Old Chinese. Classical Chinese is a traditional style of written Chinese that evolved from the classical language, making it different from any modern spoken form of Chinese. Literary Chinese was used for all formal writing in China until the early 20th century, during various periods, in Japan and Vietnam. Among Chinese speakers, Literary Chinese has been replaced by written vernacular Chinese, a style of writing, similar to modern spoken Mandarin Chinese, while speakers of non-Chinese languages have abandoned Literary Chinese in favor of local vernaculars. Literary Chinese is known as kanbun in Japanese, hanmun in Korean, cổ văn or văn ngôn in Vietnamese. Speaking, Classical Chinese refers to the written language of the classical period of Chinese literature, from the end of the Spring and Autumn period to the end of the Han dynasty, while Literary Chinese is the form of written Chinese used from the end of the Han Dynasty to the early 20th century, when it was replaced by vernacular written Chinese.
It is also referred to as "Classical Chinese", but sinologists distinguish it from the language of the early period. During this period the dialects of China became more and more disparate and thus the Classical written language became less and less representative of the varieties of Chinese. Although authors sought to write in the style of the Classics, the similarity decreased over the centuries due to their imperfect understanding of the older language, the influence of their own speech, the addition of new words; this situation, the use of Literary Chinese throughout the Chinese cultural sphere despite the existence of disparate regional vernaculars, is called diglossia. It can be compared to the position of Classical Arabic relative to the various regional vernaculars in Arab lands, or of Latin in medieval Europe; the Romance languages continued to evolve, influencing Latin texts of the same period, so that by the Middle Ages, Medieval Latin included many usages that would have baffled the Romans.
The coexistence of Classical Chinese and the native languages of Japan and Vietnam can be compared to the use of Latin in nations that natively speak non-Latin-derived Germanic languages or Slavic languages, to the position of Arabic in Persia or the position of the Indic language, Sanskrit, in South India and Southeast Asia. However, the non-phonetic Chinese writing system causes a unique situation where the modern pronunciation of the classical language is far more divergent than in analogous cases, complicating understanding and study of Classical Chinese further compared to other classical languages. Christian missionaries coined the term Wen-li for Literary Chinese. Though composed from Chinese roots, this term was never used in that sense in Chinese, was rejected by non-missionary sinologues. Chinese characters are not alphabetic and only reflect sound changes; the tentative reconstruction of Old Chinese is an endeavor only a few centuries old. As a result, Classical Chinese is not read with a reconstruction of Old Chinese pronunciation.
With the progress of time, every dynasty has modified the official Phonology Dictionary. By the time of the Yuan Dynasty and Ming Dynasty, the Phonology Dictionary was based on early Mandarin, but since the Imperial Examination required the composition of Shi genre, in non-Mandarin speaking parts of China such as Zhejiang and Fujian, pronunciation is either based on everyday speech as in Cantonese. In practice, all varieties of Chinese combine these two extremes. Mandarin and Cantonese, for example have words that are pronounced one way in colloquial usage and another way when used in Classical Chinese or in specialized terms coming from Classical Chinese, though the system is not as extensive as that of Southern Min or Wu. Japanese, Hokkien-Taiwanese, Cantonese or Vietnamese readers of Classical Chinese use systems of pronunciation specific to their own languages. For example, Japanese speakers use On'yomi pronunciation when reading the kanji of words of Chinese origin such as 銀行 or the name for the city of Tōkyō, but use Kun'yomi when the kanji represents a native word such as the reading of 行 in 行く or the reading of both characters in the name for the city of Ōsaka, a system that aids Japanese speakers with Classical Chinese word order.
Since the pronunciation of all modern varieties of Chinese is different from Old Chinese or other forms of historical Chinese, characters that once rhymed in poetry may not rhyme any longer, or vice versa, which may still rhyme in Min or Cantonese. Poetry and other rhyme-based writing thus becomes less coherent than the original reading must have been. However, some modern Chinese varieties have ce
Caribbean Spanish is the general name of the Spanish dialects spoken in the Caribbean region. It resembles the Spanish spoken in the Canary Islands and more distantly the one spoken in western Andalusia. More the term refers to the Spanish language as spoken in the Caribbean islands of Cuba, Puerto Rico, the Dominican Republic as well as in Panama and the Caribbean coast of Colombia. Seseo, where /θ/ and /s/ merge to /s/, as in the rest of Latin America, in the Canary Islands and in southern Spain. Yeismo, where /ʎ/ and /ʝ/ merge to /ʝ/, as in many other Spanish dialects. /s/ is debuccalized to at the end of syllables, as is common in the southern half of Spain, the Canaries and much of Spanish America: los amigos, dos. Syllable-initial /s/ is sporadically debuccalized, although this process is documented only in certain areas, such as parts of Puerto Rico: cinco centavos, la semana pasada. /x/ pronounced, as is common in Andalusia, the Canary islands and various parts of South America. Lenition of /tʃ/ to mucho →, as in part of Andalusia or in Chile.
Word-final /n/ is realized as a velar nasal. It can be elided, with backwards nasalization of the preceding vowel: →. Deletion of intervocalic /ð/ and word final /d/, as in many Spanish dialects: cansado, nada →, perdido, mitad → Syllable final'r' has a variety of realisations: lambdacism /ɾ/→/l/ porque → deletion of /ɾ/ hablar → assimilation to following consonant, causing gemination. Carne →, →. Most notable of Spanish spoken around Havana. /ɹ/ is a common realisation in middle and upper classes in Puerto Rico under the influence of English. Vocalisation of /ɾ/ to /j/ hacer → in the Cibao region of the Dominican Republic. 6. Aspiration /ɾ/→/h/ carne → /r/ is devoiced to /r̥/ in the Dominican Republic and Puerto Rico: cotorra → and realised as a uvular fricative /ʀ/, /χ/ in rural Puerto rican dialects Several neutralizations occur in the syllable coda; the liquids /l/ and /ɾ/ may neutralize to, or as complete regressive assimilation. The deletions and neutralizations show variability in their occurrence with the same speaker in the same utterance, which implies that nondeleted forms exist in the underlying structure.
That is not to say that these dialects are on the path to eliminating coda consonants since such processes have existed for more than four centuries in these dialects. Guitart argues that it is the result of speakers acquiring multiple phonological systems with uneven control, like that of second language learners; as in all Latin American variants of Spanish the third person plural pronoun ustedes has supplanted the pronoun vosotros/vosotras. Voseo is now absent from insular Caribbean Spanish. Contemporary commentators such as the Cuban Esteban Pichardo speak of its survival as late as the 1830s but by the 1870s it appears to have become confined to a small number of speakers from the lowest social strata. In the north east of Venezuela, in the states of Falcon and Zulia, in the north of the Cesar department, in the south of La Guajira department on Colombia's Atlantic coast and the Azurero Peninsula in Panama voseo is still used; the diminutive takes the form after /t/: pato→patico, pregunta→preguntica.
BUT perro→perrito. As a result of the routine elision of word-final, some speakers may use as a plural marker, but this tendency is limited to words with singular forms that end in a stressed vowel: café ‘coffee’ → ‘coffees’, sofá ‘sofa’ → ‘sofas’; the second-person subject pronouns, tú and usted, are used more than in other varieties of Spanish, contrary to the general Spanish tendency to omit them when meaning is clear from the context. Thus, tú estás hablando instead of estás hablando; the tendency is strongest in the island countries and, on the mainland, in Nicaragua, where voseo is predominant. So-called "wh-questions", which in standard Spanish are marked by subject/verb inversion appear without the inversion in Caribbean Spanish: "¿Qué tú quieres?" for standard "¿Qué quieres?". Cuban Spanish Dominican Spanish Puerto Rican Spanish Languages of the Caribbean Boyd-Bowman, Peter, "A sample of Sixteenth Century'Caribbean' Spanish Phonology.", in Milán, William G.. C.: Georgetown University Press, pp. 1–11 Guitart, Jorge M. "Variability and the organization of phonology in Caribbean Spanish dialects", in Martínez-Gil, Fernando.
Conversation is interactive communication between two or more people. The development of conversational skills and etiquette is an important part of socialization; the development of conversational skills in a new language is a frequent focus of language teaching and learning. Conversation analysis is a branch of sociology which studies the structure and organization of human interaction, with a more specific focus on conversational interaction. No accepted definition of conversation exists, beyond the fact that a conversation involves at least two people talking together; the term is defined by what it is not. A ritualized exchange such as a mutual greeting is not a conversation, an interaction that includes a marked status differential is not a conversation. An interaction with a focused topic or purpose is generally not considered a conversation. Summarizing these properties, one authority writes that "Conversation is the kind of speech that happens informally and for the purposes of establishing and maintaining social ties."From a less technical perspective, a writer on etiquette in the early 20th century defined conversation as the polite give and take of subjects thought of by people talking with each other for company.
Conversations follow rules of etiquette because conversations are social interactions, therefore depend on social convention. Specific rules for conversation arise from the cooperative principle. Failure to adhere to these rules causes the conversation to deteriorate or to end. Contributions to a conversation are responses to what has been said. Conversations may be the optimal form of communication, depending on the participants' intended ends. Conversations may be ideal when, for example, each party desires a equal exchange of information, or when the parties desire to build social ties. On the other hand, if permanency or the ability to review such information is important, written communication may be ideal. Or if time-efficient communication is most important, a speech may be preferable. Conversation involves a implied context that lies beneath just the words. Conversation is face-to-face person-to-person at the same time – online with video applications such as Skype, but might include audio-only phone calls.
It would not include internet written communication which tends to be asynchronous and does not fit the'con'='with' in'conversation'. In face to face conversation it has been suggested that 85% of the communication is non-verbal/body language – a smile, a frown, a shrug, tone of voice conveying much added meaning to the mere words. Short forms of written communication such as sms are thus misunderstood, yet the convenience and apparent control makes them popular now that many people seem to prefer to communicate via short text or Facebook post and/or'like' than meeting face to face. Face to face conversation is deemed less important when people have seen all the relevant news about the other person they have shared online. People would never say face to face some things they might write with the apparent impunity of anonymous online posts. To this extent the decreasing popularity of face to face conversation can be seen as a loss to society and civility. Banter is short witty sentences that bounce forth between individuals.
Banter uses clever put-downs and witty insults, misunderstandings, zippy wisecracks, zingers and puns. The idea is each line of banter should "top" the one before it and in short a verbal war of wit without any physical contact. Films that have used banter as a way of structure in conversations are: The Big Sleep His Girl Friday Bringing Up Baby Important factors in delivering a banter is the subtext and the rapport with the person; every line in a banter should be able to evoke both an emotional response and ownership without hurting one's feelings. Following a structure that the involved parties understand is important if the subject and structure is absurd, a certain level of progression should be kept in a manner that it connects with the involved parties. Different methods of story telling could be used in delivering banter, like making an unexpected turn in the flow of structure, taking the conversation towards an expected crude form with evoking questions, self-conscientiousness or layering the existing pattern with multiple anchors...etc.
It is important to quit the bantering with the sensibility of playground rules, both parties shouldn't obsess on topping each other, continuously after a certain point of interest. It is as Shakespeare said "Brevity is the soul of wit." One element of conversation is discussion: sharing opinions on subjects that are thought of during the conversation. In polite society the subject changes before discussion becomes controversial. For example, if theology is being discussed,maybe no one is insisting a particular view be accepted. Many conversations can be divided into four categories according to their major subject content: Subjective ideas, which serve to extend understanding and awareness. Objective facts, which may serve to consolidate a held view. Other people, which may be either critical, competitive, or supportive; this includes gossip. Oneself, which sometimes indicate attention-seeking behavior or can provide relevant information about oneself to participants in the conversation. Few conversations fall i
Classical Arabic is the form of the Arabic language used in Umayyad and Abbasid literary texts from the 7th century AD to the 9th century AD. The orthography of the Qurʾān was not developed for the standardized form of Classical Arabic. Modern Standard Arabic is its direct descendant used today throughout the Arab world in writing and in formal speaking, for example, prepared speeches, some radio broadcasts, non-entertainment content. While the lexis and stylistics of Modern Standard Arabic are different from Classical Arabic, the morphology and syntax have remained unchanged. In the Arab world, little distinction is made between CA and MSA, both are called al-fuṣḥá in Arabic, meaning'the most eloquent'. In the late 6th century AD, a uniform intertribal ‘poetic koiné’ distinct from the spoken vernaculars developed based on the Bedouin dialects of Najd in connection with the Lakhmid court of al-Ḥīra. During the first Islamic century the majority of Arabic poets and Arabic-writing persons spoke a form of Arabic as their mother tongue.
Their texts, although preserved in far manuscripts, contain traces of non-standardized Classical Arabic elements in morphology and syntax. The standardization of Classical Arabic reached completion around the end of the 8th century; the first comprehensive description of the ʿarabiyya "Arabic", Sībawayhi's al-Kitāb, is based first of all upon a corpus of poetic texts, in addition to the Qurʾān and Bedouin informants whom he considered to be reliable speakers of the ʿarabiyya. "Colloquial" Arabic refers to the many regional dialects derived from Arabic spoken daily across the region and learned as a first language, as second language if people speak other languages native to their particular country. By the 8th century, knowledge of Classical Arabic had become an essential prerequisite for rising into the higher classes throughout the Islamic world, as it was the lingua franca across the Middle East, North Africa, Horn of Africa during those times. Various Arabic dialects borrowed words from Classical Arabic, this situation is similar to Romance languages, wherein scores of words were borrowed directly from Classical Latin.
People speak Classical Arabic as a second language if they speak colloquial Arabic dialects as their first language, but as a third language if others speak other languages native to a country as their first language and colloquial Arabic dialects as their second language. But Classical Arabic was spoken with different pronunciations influenced by informal dialects; the differentiation of the pronunciation of informal dialects is the influence from native languages spoken and some presently spoken in the regions, such as Coptic in Egypt, Punic or Phoenician in North Africa, Modern South Arabian and Old South Arabian in Yemen, Aramaic in the Levant. Like Modern Standard Arabic, Classical Arabic had 28 consonant phonemes: Notes: ^1 Sibawayh described the consonant ⟨ط⟩ as voiced, but some modern linguists cast doubt upon this testimony. ^2 Ibn Khaldun described the pronunciation of ⟨ق⟩ as a voiced velar /g/ and that it might have been the old Arabic pronunciation of the letter, he describes that prophet Muhammad may have had the /g/ pronunciation.
^3 Non-emphatic /s/ may have been, shifting forward in the mouth before or with the fronting of the palatals. ^4 As it derives from Proto-Semitic *g, /ɟ/ may have been a palatalized velar: /ɡʲ/. ^5 /l/ is emphatic only in /ʔaɫɫɑːh/, the name of God, except after /i/ or /iː/ when it is unemphatic: bismi l-lāhi /bismillaːhi/. ^6 /ɾˠ/ is pronounced without velarization before /i/:. Notes: might have been an allophone of short /a/ in certain imalah contexts In pre-Classical Arabic, arose out of contraction of certain Old Arabic triphthongs; some Arabs said banē for zēda for zāda. This /eː/ merged with /aː/ in Classical Arabic. A different phenomenon called imāla led to the raising of /a/ and /aː/ adjacent to a sequence iC or Ci, where C was a non-emphatic, non-uvular consonant, e.g. al-kēfirīna < al-kāfirīna might have been an allophone of /a/ and /aː/ after uvular and emphatic consonants The A1 inscription dated to the 3rd or 4th c. AD in the Greek alphabet in a dialect showing affinities to that of the Safaitic inscriptions shows that short final high vowels had been lost in at least some dialects of Old Arabic at that time, obliterating the distinction between nominative and genitive case in the singular, leaving the accusative the only marked case:أوس عوذ بناء كازم الإداميْ أتو من شحاصْ؛ أتو بناءَ الدَّورَ ويرعو بقلَ بكانون ʾAws ʿūḏ Bannāʾ Kāzim ʾal-ʾidāmiyy ʾatawa miś-śiḥāṣ.
Classical Arabic however, shows a far more archaic system identical with that of Proto-Arabic: The definite article spread areally among the Central Semitic languages and it would seem that Proto-Arabic lacked any overt marking of definiteness. Besides dialects with no definite article, the Safaitic inscriptions exhibit about four different article forms, ordered by frequency: h-, ʾ-, ʾl-, hn-; the Old Arabic
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space; when n = 3, the set of all such locations is called three-dimensional Euclidean space. It is represented by the symbol ℝ3; this serves as a three-parameter model of the physical universe. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions, any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space. Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height and length. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates.
Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are labeled x, y, z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane. Four distinct points can either coplanar or determine the entire space. Two distinct lines can either be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes are parallel. Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane. A hyperplane is a subspace of one dimension less than the dimension of the full space; the hyperplanes of a three-dimensional space are the two-dimensional subspaces. In terms of cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations, each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, so, are coplanar. A sphere in 3-space consists of the set of all points in 3-space at a fixed distance r from a central point P.
The solid enclosed by the sphere is called a ball. The volume of the ball is given by V = 4 3 π r 3. Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution; the plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane, perpendicular to the axis, is a circle. Simple examples occur. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex the point of intersection. However, if the generatrix and axis are parallel, the surface of revolution is a circular cylinder.
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0, where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloidThe degenerate quadric surfaces are the empty set, a single point, a single li