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has at least one jurisdiction while others may have several more, and each jurisdiction is usually composed of between 30 and 100 churches. Each jurisdiction is then broken down into several districts, which are smaller groups of churches (either grouped by geographical situation or by similar affiliations) which are each under the authority of District Superintendents who answer to the authority of their jurisdictional/state bishop. There are currently over 170 jurisdictions in the United States, and over 30 jurisdictions in other countries. The bishops of each jurisdiction, according to the COGIC Manual, are considered to be the modern day equivalent in the church of the early apostles and overseers of the New Testament church, and as the highest ranking clergymen in the COGIC, they are tasked with the responsibilities of being the head overseers of all religious, civil, and economic ministries and protocol for the church denomination. They also have the authority to appoint and ordain local pastors, elders, ministers, and reverends within the denomination.
They also have the authority to appoint and ordain local pastors, elders, ministers, and reverends within the denomination. The bishops of the COGIC denomination are all collectively called "The Board of Bishops". From the Board of Bishops, and the General Assembly of the COGIC, the body of the church composed of clergy and lay delegates that are responsible for making and enforcing the bylaws of the denomination, every four years, twelve bishops from the COGIC are elected as "The General Board" of the church, who work alongside the delegates of the General Assembly and Board of Bishops to provide administration over the denomination as the church's head executive leaders. One of twelve bishops of the General Board is also elected the "presiding bishop" of the church, and two others are appointed by the presiding bishop himself, as his first and second assistant presiding bishops.
Bishops in the Church of God in Christ usually wear black clergy suits which consist of a black suit blazer, black pants, a purple or scarlet clergy shirt and a white clerical collar, which is usually referred to as "Class B Civic attire". Bishops in COGIC also typically wear the Anglican Choir Dress style vestments of a long purple or scarlet chimere, cuffs, and tippet worn over a long white rochet, and a gold pectoral cross worn around the neck with the tippet. This is usually referred to as "Class A Ceremonial attire". The bishops of COGIC alternate between Class A Ceremonial attire and Class B Civic attire depending on the protocol of the religious services and other events they have to attend. Church of God (Cleveland, Tennessee). In the polity of the Church of God (Cleveland, Tennessee), the international leader is the presiding bishop, and the members of the executive committee are executive bishops. Collectively, they supervise and appoint national and state leaders across the world. Leaders of individual states and regions are administrative bishops, who have jurisdiction over local churches in their respective states and are vested with appointment authority for local pastorates. All ministers are credentialed at one of three levels of licensure, the most senior of which is the rank of ordained bishop. To be eligible to serve in state, national, or international positions of authority, a minister must hold the rank of ordained bishop.
Pentecostal Church of God. In 2002, the general convention of the Pentecostal Church of God came to a consensus to change the title of their overseer from general superintendent to bishop. The change was brought on because internationally, the term "bishop" is more commonly related to religious leaders than the previous title. The title "bishop" is used for both the general (international leader) and the district (state) leaders. The title is sometimes used in conjunction with the previous, thus becoming general (district) superintendent/bishop. Seventh-day Adventists. According to the Seventh-day Adventist understanding of the doctrine of the church: "The "elders" (Greek, ) or "bishops" () were the most important officers of the church. The term elder means older one, implying dignity and respect. His position was similar to that of the one who had supervision of the synagogue. The term bishop means "overseer". Paul used these terms interchangeably, equating elders with overseers or bishops (Acts 20:17,; Titus 1:5, 7).
"Those who held this position supervised the newly formed churches. Elder referred to the status or rank of the office, while bishop denoted the duty or responsibility of the office—"overseer". Since the apostles also called themselves elders (1 Peter 5:1; 2 John 1; 3 John 1), it is apparent that there were both local elders and itinerant elders, or elders at large. But both kinds of elder functioned as shepherds of the congregations." The above understanding is part of the basis of Adventist organizational structure. The world wide Seventh-day Adventist church is organized into local districts, conferences or missions, union conferences or union missions, divisions, and finally at the top is the general conference. At each level (with exception to the local districts), there is an elder who is elected president and a group of elders who serve on the executive committee with the elected president. Those who have been elected president would in effect be the "bishop" while never actually carrying the title or ordained as such because the term is usually associated with the episcopal style of church governance most often found in Catholic, Anglican, Methodist and some Pentecostal/Charismatic circles.
Others. Some Baptists also have begun taking on the title of "bishop". In some smaller Protestant denominations and independent churches, the term "bishop" is used in the same way as "pastor", to refer to the leader of the local congregation, and may be male or female. This usage is especially common in African-American churches in the US. In the Church of Scotland, which has a Presbyterian church structure, the word "bishop" refers to an ordained person, usually a normal parish minister, who has temporary oversight of a trainee minister. In the Presbyterian Church (USA), the term bishop is an expressive name for a Minister of Word and Sacrament who serves a congregation and exercises "the oversight of the flock of Christ." The term is traceable to the 1789 Form of Government of the PC (USA) and the Presbyterian understanding of the pastoral office. While not considered orthodox Christian, the Ecclesia Gnostica Catholica uses roles and titles derived from Christianity for its clerical hierarchy, including bishops who have much the same authority and responsibilities as in Catholicism.
The Salvation Army does not have bishops but has appointed leaders of geographical areas, known as Divisional Commanders. Larger geographical areas, called Territories, are led by a Territorial Commander, who is the highest-ranking officer in that Territory. Jehovah's Witnesses do not use the title 'Bishop' within their organizational structure, but appoint elders to be overseers (to fulfill the role of oversight) within their congregations. The Batak Christian Protestant Church of Indonesia, the most prominent Protestant denomination in Indonesia, uses the term "Ephorus" instead of "bishop". In the Vietnamese syncretist religion of Caodaism, bishops () comprise the fifth of nine hierarchical levels, and are responsible for spiritual and temporal education as well as record-keeping and ceremonies in their parishes. At any one time there are seventy-two bishops. Their authority is described in Section I of the text (revealed through seances in December 1926). Caodai bishops wear robes and headgear of embroidered silk depicting the Divine Eye and the Eight Trigrams. (The color varies according to branch.) This is the full ceremonial dress; the simple version consists of a seven-layered turban.
Dress and insignia in Christianity. Traditionally, a number of items are associated with the office of a bishop, most notably the mitre and the crosier. Other vestments and insignia vary between Eastern and Western Christianity. In the Latin Rite of the Catholic Church, the choir dress of a bishop includes the purple cassock with amaranth trim, rochet, purple zucchetto (skull cap), purple biretta, and pectoral cross. The cappa magna may be worn, but only within the bishop's own diocese and on especially solemn occasions. The mitre, zucchetto, and stole are generally worn by bishops when presiding over liturgical functions. For liturgical functions other than the Mass the bishop typically wears the cope. Within his own diocese and when celebrating solemnly elsewhere with the consent of the local ordinary, he also uses the crosier. When celebrating Mass, a bishop, like a priest, wears the chasuble. The Caeremoniale Episcoporum recommends, but does not impose, that in solemn celebrations a bishop should also wear a dalmatic, which can always be white, beneath the chasuble, especially when administering the sacrament of holy orders, blessing an abbot or abbess, and dedicating a church or an altar. The Caeremoniale Episcoporum no longer makes mention of episcopal gloves, episcopal sandals, liturgical stockings (also known as buskins), or the accoutrements that it once prescribed for the bishop's horse. The coat of arms of a Latin Church Catholic bishop usually displays a galero with a cross and crosier behind the escutcheon; the specifics differ by location and ecclesiastical rank (see Ecclesiastical heraldry).
Anglican bishops generally make use of the mitre, crosier, ecclesiastical ring, purple cassock, purple zucchetto, and pectoral cross. However, the traditional choir dress of Anglican bishops retains its late mediaeval form, and looks quite different from that of their Catholic counterparts; it consists of a long rochet which is worn with a chimere. In the Eastern Churches (Eastern Orthodox, Eastern Rite Catholic) a bishop will wear the mandyas, panagia (and perhaps an enkolpion), sakkos, omophorion and an Eastern-style mitre. Eastern bishops do not normally wear an episcopal ring; the faithful kiss (or, alternatively, touch their forehead to) the bishop's hand. To seal official documents, he will usually use an inked stamp. An Eastern bishop's coat of arms will normally display an Eastern-style mitre, cross, eastern style crosier and a red and white (or red and gold) mantle. The arms of Oriental Orthodox bishops will display the episcopal insignia (mitre or turban) specific to their own liturgical traditions. Variations occur based upon jurisdiction and national customs.
Cathedra. In Catholic, Eastern Orthodox, Oriental Orthodox, Lutheran and Anglican cathedrals there is a special chair set aside for the exclusive use of the bishop. This is the bishop's "cathedra" and is often called the throne. In some Christian denominations, for example, the Anglican Communion, parish churches may maintain a chair for the use of the bishop when he visits; this is to signify the parish's union with the bishop. The term's use in non-Christian religions. Buddhism. The leader of the Buddhist Churches of America (BCA) is their bishop, The Japanese title for the bishop of the BCA is , although the English title is favored over the Japanese. When it comes to many other Buddhist terms, the BCA chose to keep them in their original language (terms such as and ), but with some words (including ), they changed/translated these terms into English words. Between 1899 and 1944, the BCA held the name Buddhist Mission of North America. The leader of the Buddhist Mission of North America was called (superintendent/director) between 1899 and 1918. In 1918 the was promoted to bishop (). However, according to George J. Tanabe, the title "bishop" was in practice already used by Hawaiian Shin Buddhists (in Honpa Hongwanji Mission of Hawaii) even when the official title was "kantoku".
Bishops are also present in other Japanese Buddhist organizations. Higashi Hongan-ji's North American District, Honpa Honganji Mission of Hawaii, Jodo Shinshu Buddhist Temples of Canada, a Jodo Shu temple in Los Angeles, the Shingon temple Koyasan Buddhist Temple, Sōtō Mission in Hawai‘i (a Soto Zen Buddhist institution), and the Sōtō Zen Buddhist Community of South America () all have or have had leaders with the title bishop. As for the Sōtō Zen Buddhist Community of South America, the Japanese title is , but the leader is in practice referred to as "bishop". Tenrikyo. Tenrikyo is a Japanese New Religion with influences from both Shinto and Buddhism. The leader of the Tenrikyo North American Mission has the title of bishop.
Bertrand Andrieu Bertrand Andrieu (24 November 1761 – 6 December 1822) was a French engraver of medals. He was born in Bordeaux. In France, he was considered as the restorer of the art, which had declined after the time of Louis XIV. During the last twenty years of his life, the French government commissioned him to undertake every major work of importance.
Bordeaux Bordeaux (, ; ; Gascon ; ) is a city on the river Garonne in the Gironde department, southwestern France. A port city, it is the capital of the Nouvelle-Aquitaine region, as well as the prefecture of the Gironde department. Its inhabitants are called ""Bordelais" (masculine) or ""Bordelaises" (feminine). The term "Bordelais" may also refer to the city and its surrounding region. The city of Bordeaux proper had a population of 259,809 in 2020 within its small municipal territory of , but together with its suburbs and exurbs the Bordeaux metropolitan area had a population of 1,376,375 that same year (Jan. 2020 census), the sixth-most populated in France after Paris, Lyon, Marseille, Lille, and Toulouse. Bordeaux and 27 suburban municipalities form the Bordeaux Metropolis, an indirectly elected metropolitan authority now in charge of wider metropolitan issues. The Bordeaux Metropolis, with a population of 819,604 at the January 2020 census, is the fifth most populated metropolitan council in France after those of Paris, Marseille, Lyon and Lille.
Bordeaux is a world capital of wine: many châteaux and vineyards stand on the hillsides of the Gironde, and the city is home to the world's main wine fair, Vinexpo. Bordeaux is also one of the centers of gastronomy and business tourism for the organization of international congresses. It is a central and strategic hub for the aeronautics, military and space sector, home to major companies such as Dassault Aviation, ArianeGroup, Safran and Thales. The link with aviation dates back to 1910, the year the first airplane flew over the city. A crossroads of knowledge through university research, it is home to one of the only two megajoule lasers in the world, as well as a university population of more than 130,000 students within the Bordeaux Metropolis. Bordeaux is an international tourist destination for its architectural and cultural heritage with more than 362 historic , making it, after Paris, the city with the most listed or registered monuments in France. The "Pearl of Aquitaine" has been voted European Destination of the year in a 2015 online poll. The metropolis has also received awards and rankings by international organizations such as in 1957, Bordeaux was awarded the Europe Prize for its efforts in transmitting the European ideal. In June 2007, the Port of the Moon in historic Bordeaux was inscribed on the UNESCO World Heritage List, for its outstanding architecture and urban ensemble and in recognition of Bordeaux's international importance over the last 2000 years. Bordeaux is also ranked as a Sufficiency city by the Globalization and World Cities Research Network.
History. 5th century BC to 11th century AD. Around 300 BC, the region was the settlement of a Celtic tribe, the Bituriges Vivisci, who named the town Burdigala, probably of Aquitanian origin. In 107 BC, the Battle of Burdigala was fought by the Romans who were defending the Allobroges, a Gallic tribe allied to Rome, and the Tigurini led by Divico. The Romans were defeated and their commander, the consul Lucius Cassius Longinus, was killed in battle. The city came under Roman rule around 60 BC, and it became an important commercial centre for tin and lead. During this period were built the amphitheatre and the monument "Les Piliers de Tutelle". In 276 AD, it was sacked by the Vandals. The Vandals attacked again in 409, followed by the Visigoths in 414, and the Franks in 498, and afterwards the city fell into a period of relative obscurity. In the late 6th century AD the city re-emerged as the seat of a county and an archdiocese within the Merovingian kingdom of the Franks, but royal Frankish power was never strong. The city started to play a regional role as a major urban center on the fringes of the newly founded Frankish Duchy of Vasconia. Around 585 Gallactorius was made Count of Bordeaux and fought the Basques.
In 732, the city was plundered by the troops of Abd er Rahman who stormed the fortifications and overwhelmed the Aquitanian garrison. Duke Eudes mustered a force to engage the Umayyads, eventually engaging them in the Battle of the River Garonne somewhere near the river Dordogne. The battle had a high death toll, and although Eudes was defeated he had enough troops to engage in the Battle of Poitiers and so retain his grip on Aquitaine. In 737, following his father Eudes's death, the Aquitanian duke Hunald led a rebellion to which Charles responded by launching an expedition that captured Bordeaux. However, it was not retained for long, during the following year the Frankish commander clashed in battle with the Aquitanians but then left to take on hostile Burgundian authorities and magnates. In 745 Aquitaine faced another expedition where Charles's sons Pepin and Carloman challenged Hunald's power and defeated him. Hunald's son Waifer replaced him and confirmed Bordeaux as the capital city (along with Bourges in the north).
During the last stage of the war against Aquitaine (760–768), it was one of Waifer's last important strongholds to fall to the troops of King Pepin the Short. Charlemagne built the fortress of Fronsac ("Frontiacus", "Franciacus") near Bordeaux on a hill across the border with the Basques ("Wascones"), where Basque commanders came and pledged their loyalty (769). In 778, Seguin (or Sihimin) was appointed count of Bordeaux, probably undermining the power of the Duke Lupo, and possibly leading to the Battle of Roncevaux Pass. In 814, Seguin was made Duke of Vasconia, but was deposed in 816 for failing to suppress a Basque rebellion. Under the Carolingians, sometimes the Counts of Bordeaux held the title concomitantly with that of Duke of Vasconia. They were to keep the Basques in check and defend the mouth of the Garonne from the Vikings when they appeared in c. 844. In Autumn 845, the Vikings were raiding Bordeaux and Saintes, count Seguin II marched on them but was captured and executed. Although the port of Bordeaux was a buzzing trade center, the stability and success of the city was threatened by Viking and Norman incursions and political instability. The restoration of the Ramnulfid Dukes of Aquitaine under William IV and his successors (known as the House of Poitiers) brought continuity of government. 12th century to 15th century, the English era.
From the 12th to the 15th century, Bordeaux flourished once more following the marriage of Eléonore, Duchess of Aquitaine and the last of the House of Poitiers, to Henry II Plantagenêt, Count of Anjou and the grandson of Henry I of England, who succeeded to the English crown months after their wedding, bringing into being the vast Angevin Empire, which stretched from the Pyrenees to Ireland. After granting a tax-free trade status with England, Henry was adored by the locals as they could be even more profitable in the wine trade, their main source of income, and the city benefited from imports of cloth and wheat. The belfry (Grosse Cloche) and city cathedral St-André were built, the latter in 1227, incorporating the artisan quarter of Saint-Paul. Under the terms of the Treaty of Brétigny it became briefly the capital of an independent state (1362–1372) under Edward, the Black Prince, but after the Battle of Castillon (1453) it was annexed by France. 15th century to 17th century. In 1462, Bordeaux created a local parliament.
Bordeaux adhered to the Fronde, being effectively annexed to the Kingdom of France only in 1653, when the army of Louis XIV entered the city. 18th century, the golden era. The 18th century saw another golden age of Bordeaux. The Port of the Moon supplied the majority of Europe with coffee, cocoa, sugar, cotton and indigo, becoming France's busiest port and the second busiest port in the world after London. Many downtown buildings (about 5,000), including those on the quays, are from this period. Bordeaux was also a major trading centre for slaves. In total, the Bordeaux shipowners deported 150,000 Africans in some 500 expeditions. French Revolution: political disruption and loss of the most profitable colony. At the beginning of the French Revolution (1789), many local revolutionaries were members of the Girondists. This Party represented the provincial bourgeoisie, favorable towards abolishing aristocracy privileges, but opposed to the Revolution's social dimension. The Gironde valley's economic value and significance was satiated by the city's commercial power which was in dire contrast to the emerging widespread poverty affecting its inhabitants. Trade and commerce were the driving factors in the region's economic prosperity, still this resulted in a significant number of locals struggling to survive on a daily basis due to lack of food and resources. This socioeconomic disparity served as fertile ground for discontent, sparking frequent episodes of mass unrest well before the tumultuous events of 1783. [1]
In 1793, the Montagnards led by Robespierre and Marat came to power. Fearing a bourgeois misappropriation of the Revolution, they executed a great number of Girondists. During the purge, the local Montagnard Section renamed the city of Bordeaux "Commune-Franklin" (Franklin-municipality) in homage to Benjamin Franklin. At the same time, in 1791, a slave revolt broke out at Saint-Domingue (current Haiti), the most profitable of the French colonies. In the lively era of the 18th century, Bordeaux emerged as a center of economic activity, particularly known at first for its successful wine trade. The city's placement along the Gironde River was very strategic, helping to facilitate the transportation of produce to markets both internationally and domestically, which led to an increase in exports and Bordeaux's economic prosperity. There was a significant transformation to the economic landscape of Bordeaux in 1785, which was spurred by the attraction of large profits, traders and merchants in Bordeaux began to turn their attention to the slave trade. This was a very important moment in the city's economic history seeing as it diversified its commercial expansion, at a serious moral cost. This introduced a new layer of difficulty to Bordeaux's economic activities. Even though it brought along significant wealth to certain segments of society, it complicated the socio-economic inconsistencies within the region. The entry into the slave trade brought even more tension within Bordeaux society. The trade exacerbated the divide between an elite with growing wealth and those living in poverty. This economic divide laid out the foundation for the mass unrest that would break out in the French Revolution. [2]
Three years later, the Montagnard Convention abolished slavery. In 1802, Napoleon revoked the manumission law but lost the war against the army of former slaves. In 1804, Haiti became independent. The loss of this "Pearl" of the West Indies generated the collapse of Bordeaux's port economy, which was dependent on the colonial trade and trade in slaves. Towards the end of the Peninsular War of 1814, the Duke of Wellington sent William Beresford with two divisions and seized Bordeaux, encountering little resistance. Bordeaux was largely anti-Bonapartist and the majority supported the Bourbons. The British troops were treated as liberators. Distinguished historian of the French revolution Suzanne Desan explains that "examining intricate local dynamics" is essential to studying the Revolution by region. [3] 19th century, rebirth of the economy. From the Bourbon Restoration, the economy of Bordeaux was rebuilt by traders and shipowners. They engaged to construct the first bridge of Bordeaux, and customs warehouses. The shipping traffic grew through the new African colonies.
Georges-Eugène Haussmann, a longtime prefect of Bordeaux, used Bordeaux's 18th-century large-scale rebuilding as a model when he was asked by Emperor Napoleon III to transform the quasi-medieval Paris into a "modern" capital that would make France proud. Victor Hugo found the town so beautiful he said: "Take Versailles, add Antwerp, and you have Bordeaux". In 1870, at the beginning of the Franco-Prussian war against Prussia, the French government temporarily relocated to Bordeaux from Paris. That recurred during World War I and again very briefly during World War II, when it became clear that Paris would fall into German hands. 20th century. During World War II, Bordeaux fell under German occupation. In May and June 1940, Bordeaux was the site of the life-saving actions of the Portuguese consul-general, Aristides de Sousa Mendes, who illegally granted thousands of Portuguese visas, which were needed to pass the Spanish border, to refugees fleeing the German occupation. From 1941 to 1943, the Italian Royal Navy established BETASOM, a submarine base at Bordeaux. Italian submarines participated in the Battle of the Atlantic from that base, which was also a major base for German U-boats as headquarters of 12th U-boat Flotilla. The massive, reinforced concrete U-boat pens have proved impractical to demolish and are now partly used as a cultural center for exhibitions. 21st century, listed as World heritage.
In 2007, 40% of the city surface area, located around the Port of the Moon, was listed as World Heritage Site. UNESCO inscribed Bordeaux as "an inhabited historic city, an outstanding urban and architectural ensemble, created in the age of the Enlightenment, whose values continued up to the first half of the 20th century, with more protected buildings than any other French city except Paris". Geography. Bordeaux is located close to the European Atlantic coast, in the southwest of France and in the north of the Aquitaine region. It is around southwest of Paris. The city is built on a bend of the river Garonne, and is divided into two parts: the right bank to the east and left bank in the west. Historically the left bank is more developed because when flowing outside the bend, the water makes a furrow of the required depth to allow the passing of merchant ships, which used to offload on this side of the river. But, today, the right bank is developing, including new urban projects. In Bordeaux, the Garonne River is accessible to ocean liners through the Gironde estuary. The right bank of the Garonne is a low-lying, often marshy plain.
Climate. Bordeaux's climate can be classified as oceanic (Köppen climate classification "Cfb"), bordering on a humid subtropical climate ("Cfa"). However, the Trewartha climate classification system classifies the city as solely humid subtropical, due to a recent rise in temperatures related – to some degree or another – to climate change and the city's urban heat island. The city enjoys cool to mild, wet winters, due to its relatively southerly latitude, and the prevalence of mild, westerly winds from the Atlantic. Its summers are warm and somewhat drier, although wet enough to avoid a Mediterranean classification. Frosts occur annually, but snowfall is quite infrequent, occurring for no more than 3–4 days a year. The summer of 2003 set a record with an average temperature of , while February 1956 was the coldest month on record with an average temperature of −2.00 °C at Bordeaux Mérignac-Airport. Economy. Bordeaux is a major centre for business in France as it has the sixth largest metropolitan population in France. It serves as a major regional center for trade, administration, services and industry.
Wine. The vine was introduced to the Bordeaux region by the Romans, probably in the mid-first century, to provide wine for local consumption, and wine production has been continuous in the region since. Bordeaux wine growing area has about of vineyards, 57 appellations, 10,000 wine-producing estates (châteaux) and 13,000 grape growers. With an annual production of approximately 960 million bottles, the Bordeaux area produces large quantities of everyday wine as well as some of the most expensive wines in the world. Included among the latter are the area's five "premier cru" (First Growth) red wines (four from Médoc and one, Château Haut-Brion, from Graves), established by the Bordeaux Wine Official Classification of 1855: Both red and white wines are made in the Bordeaux region. Red Bordeaux wine is called claret in the United Kingdom. Red wines are generally made from a blend of grapes, and may be made from Cabernet Sauvignon, Merlot, Cabernet Franc, Petit verdot, Malbec, and, less commonly in recent years, Carménère.
White Bordeaux is made from Sauvignon blanc, Sémillon, and Muscadelle. Sauternes is a sub-region of Graves known for its intensely sweet, white, dessert wines such as Château d'Yquem. Because of a wine glut (wine lake) in the generic production, the price squeeze induced by an increasingly strong international competition, and vine pull schemes, the number of growers has recently dropped from 14,000 and the area under vine has also decreased significantly. In the meantime, the global demand for first growths and the most famous labels markedly increased and their prices skyrocketed. The Cité du Vin, a museum as well as a place of exhibitions, shows, movie projections and academic seminars on the theme of wine opened its doors in June 2016. Others. The Laser Mégajoule will be one of the most powerful lasers in the world, allowing fundamental research and the development of the laser and plasma technologies. Some 15,000 people work for the aeronautic industry in Bordeaux. The city has some of the biggest companies including Dassault, EADS Sogerma, Snecma, Thales, SNPE, and others. The Dassault Falcon private jets are built there as well as the military aircraft Rafale and Mirage 2000, the Airbus A380 cockpit, the boosters of Ariane 5, and the M51 SLBM missile.
Tourism, especially wine tourism, is a major industry. Globelink.co.uk mentioned Bordeaux as the best tourist destination in Europe in 2015. Gourmet Touring is a tourism company operating in the Bordeaux wine region. Access to the port from the Atlantic is via the Gironde estuary. Almost nine million tonnes of goods arrive and leave each year. Major companies. This list includes indigenous Bordeaux-based companies and companies that have major presence in Bordeaux, but are not necessarily headquartered there. Population. In January 2020, there were 259,809 inhabitants in the city proper (commune) of Bordeaux. The commune (including Caudéran which was annexed by Bordeaux in 1965) had its largest population of 284,494 at the 1954 census. The majority of the population is French, but there are sizable groups of Italians, Spaniards, Portuguese, Turks, Germans. The built-up area has grown for more than a century beyond the municipal borders of Bordeaux due to the small size of the commune () and urban sprawl. By January 2020 there were 1,376,375 people living in the overall metropolitan area ("aire d'attraction") of Bordeaux, only a fifth of whom lived in the city proper.
Politics. Municipal administration. The Mayor of the city is the environmentalist Pierre Hurmic. Bordeaux is the capital of five cantons and the Prefecture of the Gironde and Aquitaine. The town is divided into three districts, the first three of Gironde. The headquarters of Urban Community of Bordeaux Mériadeck is located in the neighbourhood and the city is at the head of the Chamber of Commerce and Industry that bears his name. The number of inhabitants of Bordeaux is greater than 250,000 and less than 299,999 so the number of municipal councilors is 65. They are divided according to the following composition: Mayors of Bordeaux. Since the Liberation (1944), there have been six mayors of Bordeaux: Elections. Presidential elections of 2007. At the 2007 presidential election, the Bordelais gave 31.37% of their votes to Ségolène Royal of the Socialist Party against 30.84% to Nicolas Sarkozy, president of the UMP. Then came François Bayrou with 22.01%, followed by Jean-Marie Le Pen who recorded 5.42%. None of the other candidates exceeded the 5% mark. Nationally, Nicolas Sarkozy led with 31.18%, then Ségolène Royal with 25.87%, followed by François Bayrou with 18.57%. After these came Jean-Marie Le Pen with 10.44%, none of the other candidates exceeded the 5% mark. In the second round, the city of Bordeaux gave Ségolène Royal 52.44% against 47.56% for Nicolas Sarkozy, the latter being elected President of the Republic with 53.06% against 46.94% for Ségolène Royal. The abstention rates for Bordeaux were 14.52% in the first round and 15.90% in the second round.
Parliamentary elections of 2007. In the parliamentary elections of 2007, the left won eight constituencies against only three for the right. After the partial 2008 elections, the eighth district of Gironde switched to the left, bringing the count to nine. In Bordeaux, the left was for the first time in its history the majority as it held two of three constituencies following the elections. In the first division of the Gironde, the outgoing UMP MP Chantal Bourragué was well ahead with 44.81% against 25.39% for the Socialist candidate Béatrice Desaigues. In the second round, it was Chantal Bourragué who was re-elected with 54.45% against 45.55% for his socialist opponent. In the second district of Gironde the UMP mayor and all new Minister of Ecology, Energy, Sustainable Development and the Sea Alain Juppé confronted the General Counsel PS Michèle Delaunay. In the first round, Alain Juppé was well ahead with 43.73% against 31.36% for Michèle Delaunay. In the second round, it was finally Michèle Delaunay who won the election with 50.93% of the votes against 49.07% for Alain Juppé, the margin being only 670 votes. The defeat of the so-called constituency "Mayor" showed that Bordeaux was rocking increasingly left. Finally, in the third constituency of the Gironde, Noël Mamère was well ahead with 39.82% against 28.42% for the UMP candidate Elizabeth Vine. In the second round, Noël Mamère was re-elected with 62.82% against 37.18% for his right-wing rival.
Municipal elections of 2008. In 2008 municipal elections saw the clash between mayor of Bordeaux, Alain Juppé and the President of the Regional Council of Aquitaine Socialist Alain Rousset. The PS had put up a Socialist heavyweight in the Gironde and had put great hopes in this election after the victory of Ségolène Royal and Michèle Delaunay in 2007. However, after a rather exciting campaign it was Alain Juppé who was widely elected in the first round with 56.62 percent, far ahead of Alain Rousset who garnered 34.14 percent of the vote. At present, of the eight cantons that has Bordeaux, five are held by the PS and three by the UMP, the left eating a little each time into the right's numbers. European elections of 2009. In the European elections of 2009, Bordeaux voters largely voted for the UMP candidate Dominique Baudis, who won 31.54% against 15.00% for PS candidate Kader Arif. The candidate of Europe Ecology José Bové came second with 22.34%. None of the other candidates reached the 10% mark. The 2009 European elections were like the previous ones in eight constituencies. Bordeaux is located in the district "Southwest", here are the results:
UMP candidate Dominique Baudis: 26.89%. His party gained four seats. PS candidate Kader Arif: 17.79%, gaining two seats in the European Parliament. Europe Ecology candidate Bove: 15.83%, obtaining two seats. MoDem candidate Robert Rochefort: 8.61%, winning a seat. Left Front candidate Jean-Luc Mélenchon: 8.16%, gaining the last seat. At regional elections in 2010, the Socialist incumbent president Alain Rousset won the first round by totaling 35.19% in Bordeaux, but this score was lower than the plan for Gironde and Aquitaine. Xavier Darcos, Minister of Labour followed with 28.40% of the votes, scoring above the regional and departmental average. Then came Monique De Marco, Green candidate with 13.40%, followed by the member of Pyrenees-Atlantiques and candidate of the MoDem Jean Lassalle who registered a low 6.78% while qualifying to the second round on the whole Aquitaine, closely followed by Jacques Colombier, candidate of the National Front, who gained 6.48%. Finally the candidate of the Left Front Gérard Boulanger with 5.64%, no other candidate above the 5% mark. In the second round, Alain Rousset had a tidal wave win as national totals rose to 55.83%. If Xavier Darcos largely lost the election, he nevertheless achieved a score above the regional and departmental average obtaining 33.40%. Jean Lassalle, who qualified for the second round, passed the 10% mark by totaling 10.77%. The ballot was marked by abstention amounting to 55.51% in the first round and 53.59% in the second round. 2017 elections.
Bordeaux voted for Emmanuel Macron in the presidential election. In the 2017 parliamentary election, La République En Marche! won most of the constituencies in Bordeaux. 2019 European elections. Bordeaux voted in the 2019 European Parliament election in France. Municipal elections of 2020. After 73 years of right-of-centre rule, the ecologist Pierre Hurmic (EELV) came in ahead of Nicolas Florian (LR/LaREM). Parliamentary representation. The city area is represented by the following constituencies: Gironde's 1st, Gironde's 2nd, Gironde's 3rd, Gironde's 4th, Gironde's 5th, Gironde's 6th, Gironde's 7th. Education. University. During Antiquity, a first university had been created by the Romans in 286. The city was an important administrative centre and the new university had to train administrators. Only rhetoric and grammar were taught. Ausonius and Sulpicius Severus were two of the teachers. In 1441, when Bordeaux was an English town, the Pope Eugene IV created a university by demand of the archbishop Pey Berland. In 1793, during the French Revolution, the National Convention abolished the university, and replace them with the École centrale in 1796. In Bordeaux, this one was located in the former buildings of the college of Guyenne. In 1808, the university reappeared with Napoleon. Bordeaux accommodates approximately 70,000 students on one of the largest campuses of Europe (235 ha).
Schools. Bordeaux has numerous public and private schools offering undergraduate and postgraduate programs. Engineering schools: Business and management schools: Other: Weekend education. The , a part-time Japanese supplementary school, is held in the "Salle de L'Athénée Municipal" in Bordeaux. Attractions and tourism. In October 2021, Bordeaux was shortlisted for the European Commission's 2022 European Capital of Smart Tourism award along with Copenhagen, Dublin, Florence, Ljubljana, Palma de Mallorca, and Valencia. Heritage and architecture. Bordeaux is classified "City of Art and History". The city is home to 362 "monuments historiques" (national heritage sites), with some buildings dating back to Roman times. Bordeaux, Port of the Moon, has been inscribed on UNESCO World Heritage List as "an outstanding urban and architectural ensemble". Bordeaux is home to one of Europe's biggest 18th-century architectural urban areas, making it a sought-after destination for tourists and cinema production crews. It stands out as one of the first French cities, after Nancy, to have entered an era of urbanism and metropolitan big scale projects, with the team Gabriel father and son, architects for King Louis XV, under the supervision of two intendants (Governors), first Nicolas-François Dupré de Saint-Maur then the Marquis de Tourny.
Saint-André Cathedral, Saint-Michel Basilica and Saint-Seurin Basilica are part of the World Heritage Sites of the Routes of Santiago de Compostela in France. The organ in Saint-Louis-des-Chartrons is registered on the French monuments historiques. Notable historic buildings include: Contemporary buildings in contemporary architectural style include: Slavery memorials. Slavery was part of a growing drive for the city. During the 18th and 19th centuries, Bordeaux was an important slave port, which saw some 500 slave expeditions that cause the deportation of 150,000 Africans by Bordeaux shipowners. Secondly, even though the "Triangular trade" represented only 5% of Bordeaux's wealth, the city's direct trade with the Caribbean, that accounted for the other 95%, concerns the colonial stuffs made by the slave (sugar, coffee, cocoa). And thirdly, in that same period, a major migratory movement by Aquitanians took place to the Caribbean colonies, with Saint-Domingue (now Haiti) being the most popular destination. 40% of the white population of the island came from Aquitaine. They prospered with plantations incomes, until the first slave revolts which concluded in 1848 in the final abolition of slavery in France.
A statue of Modeste Testas, an Ethiopian woman who was enslaved by the Bordeaux-based Testas brothers was unveiled in 2019. She was trafficked by them from West Africa, to Philadelphia (where one of the brothers coerced her to have two children by him) and was ultimately freed and lived in Haiti. The bronze sculpture was created by the Haitian artists Woodly Caymitte. A number of traces and memorial sites are visible in the city. Moreover, in May 2009, the Museum of Aquitaine opened the spaces dedicated to "Bordeaux in the 18th century, trans-Atlantic trading and slavery". This work, richly illustrated with original documents, contributes to disseminate the state of knowledge on this question, presenting above all the facts and their chronology. The region of Bordeaux was also the land of several prominent abolitionists, as Montesquieu, Laffon de Ladébat and Elisée Reclus. Others were members of the Society of the Friends of the Blacks as the revolutionaries Boyer-Fonfrède, Gensonné, Guadet and Ducos. Pont Jacques Chaban-Delmas.
Europe's longest-span vertical-lift bridge, the Pont Jacques Chaban-Delmas, was opened in 2013 in Bordeaux, spanning the River Garonne. The central lift span is , weighs 4,600 tons and can be lifted vertically up to to let tall ships pass underneath. The €160 million bridge was inaugurated by President François Hollande and Mayor Alain Juppé on 16 March 2013. The bridge was named after the late Jacques Chaban-Delmas, who was a former Prime Minister and Mayor of Bordeaux. Shopping. Bordeaux has many shopping options. In the heart of Bordeaux is Rue Sainte-Catherine. This pedestrianised street has of shops, restaurants and cafés; it is also one of the longest shopping streets in Europe. Rue Sainte-Catherine starts at Place de la Victoire and ends at Place de la Comédie by the Grand Théâtre. The shops become progressively more upmarket as one moves towards Place de la Comédie and the nearby Cours de l'Intendance is where there are the more exclusive shops and boutiques. Culture. Bordeaux is the first city in France to have created, in the 1980s, an architecture exhibition and research centre, "Arc en rêve".
The city has a large number of cinemas, theatres, and is the home of the Opéra national de Bordeaux. There are many music venues of varying capacity. The city also offers several festivals throughout the year. The Bordeaux International Festival of Women in Cinema (Festival international du cinéma au féminin de Bordeaux) took place in Bordeaux from 2002 until 2005. The Festival international du film indépendant de Bordeaux (Fifib or FIFIB), or Bordeaux International Independent Film Festival, was established in 2012. Transport. Road. Bordeaux is an important road and motorway junction. The city is connected to Paris by the A10 motorway, with Lyon by the A89, with Toulouse by the A62, and with Spain by the A63. There is a ring road called the "Rocade" which is often very busy. Another ring road is under consideration. Bordeaux has five road bridges that cross the Garonne, the Pont de pierre built in the 1820s and three modern bridges built after 1960: the Pont Saint Jean, just south of the Pont de pierre (both located downtown), the Pont d'Aquitaine, a suspension bridge downstream from downtown, and the Pont François Mitterrand, located upstream of downtown. These two bridges are part of the ring-road around Bordeaux. A fifth bridge, the Pont Jacques-Chaban-Delmas, was constructed in 2009–2012 and opened to traffic in March 2013. Located halfway between the Pont de pierre and the Pont d'Aquitaine and serving downtown rather than highway traffic, it is a vertical-lift bridge with a height in closed position comparable to that of Pont de pierre, and to the Pont d'Aquitaine when open. All five road bridges, including the two highway bridges, are open to cyclists and pedestrians as well. Another bridge, the Pont Jean-Jacques Bosc, is to be built in 2018.
Lacking any steep hills, Bordeaux is relatively friendly to cyclists. Cycle paths (separate from the roadways) exist on the highway bridges, along the riverfront, on the university campuses, and incidentally elsewhere in the city. Cycle lanes and bus lanes that explicitly allow cyclists exist on many of the city's boulevards. A paid bicycle-sharing system with automated stations was established in 2010. Rail. The main railway station, Gare de Bordeaux Saint-Jean, near the center of the city, has 12 million passengers a year. It is served by the French national (SNCF) railway's high speed train, the TGV, that gets to Paris in two hours, with connections to major European centers such as Lille, Brussels, Amsterdam, Cologne, Geneva and London. The TGV also serves Toulouse and Irun (Spain) from Bordeaux. A regular train service is provided to Nantes, Nice, Marseille and Lyon. The Gare Saint-Jean is the major hub for regional trains (TER) operated by the SNCF to Arcachon, Limoges, Agen, Périgueux, Langon, Pau, Le Médoc, Angoulême and Bayonne.
Historically the train line used to terminate at a station on the right bank of the river Garonne near the Pont de Pierre, and passengers crossed the bridge to get into the city. Subsequently, a double-track steel railway bridge was constructed in the 1850s, by Gustave Eiffel, to bring trains across the river direct into Gare de Bordeaux Saint-Jean. The old station was later converted and in 2010 comprised a cinema and restaurants. The two-track Eiffel bridge with a speed limit of became a bottleneck and a new bridge was built, opening in 2009. The new bridge has four tracks and allows trains to pass at . During the planning there was much lobbying by the Eiffel family and other supporters to preserve the old bridge as a footbridge across the Garonne, with possibly a museum to document the history of the bridge and Gustave Eiffel's contribution. The decision was taken to save the bridge, but by early 2010 no plans had been announced as to its future use. The bridge remains intact, but unused and without any means of access.
The LGV Sud Europe Atlantique became fully operational in July 2017, shortening the journey time from Bordeaux city to Paris to 2hrs 4mins. Air. Bordeaux is served by Bordeaux–Mérignac Airport, located from the city centre in the suburban city of Mérignac. Trams, buses and boats. Bordeaux has an important public transport system called Transports Bordeaux Métropole (TBM). This company is run by the Keolis group. The network consists of: This network is operated from 5 am to 2 am. There had been several plans for a subway network to be set up, but they stalled for both geological and financial reasons. Work on the Tramway de Bordeaux system was started in the autumn of 2000, and services started in December 2003 connecting Bordeaux with its suburban areas. The tram system uses Alstom APS a form of ground-level power supply technology developed by French company Alstom and designed to preserve the aesthetic environment by eliminating overhead cables in the historic city. Conventional overhead cables are used outside the city. The system was controversial for its considerable cost of installation, maintenance and also for the numerous initial technical problems that paralysed the network. Many streets and squares along the tramway route became pedestrian areas, with limited access for cars.
The Bordeaux Tramway system reached the Mérignac airport on April 29th 2023 with the opening of a 5-km extension of Line A. Taxis. There are more than 400 taxicabs in Bordeaux. Public transportation statistics. The average amount of time people spend commuting with public transit in Bordeaux, for example to and from work, on a weekday is 51 min. 12.% of public transit riders, ride for more than 2 hours every day. The average amount of time people wait at a stop or station for public transit is 13 min, while 15.5% of riders wait for over 20 minutes on average every day. The average distance people usually ride in a single trip with public transit is , while 8% travel for over in a single direction. Sport. The 41,458-capacity Nouveau Stade de Bordeaux is the largest stadium in Bordeaux. The stadium was opened in 2015 and replaced the Stade Chaban-Delmas, which was a venue for the FIFA World Cup in 1938 and 1998, as well as the 2007 Rugby World Cup. In the 1938 FIFA World Cup, it hosted a violent quarter-final known as the Battle of Bordeaux. The ground was formerly known as the "Stade du Parc Lescure" until 2001, when it was renamed in honour of the city's long-time mayor, Jacques Chaban-Delmas.
There are two major sport teams in Bordeaux, Girondins de Bordeaux is the football team who, following administrative relegation, currently play in Championnat National 2, the fourth tier of French football. They are one of the most successful clubs in France, with six Division 1/Ligue 1 titles. Union Bordeaux Bègles is a rugby team in the Top 14 in the Ligue Nationale de Rugby. Skateboarding, rollerblading, and BMX biking are activities enjoyed by many young inhabitants of the city. Bordeaux is home to a quay which runs along the Garonne river. On the quay there is a skate-park divided into three sections. One section is for Vert tricks, one for street style tricks, and one for little action sports athletes with easier features and softer materials. The skate-park is very well maintained by the municipality. Other sports clubs include top flight ice hockey team Boxers de Bordeaux and third-tier basketball team JSA Bordeaux Basket Bordeaux is also the home to one of the strongest cricket teams in France and are champions of the South West League.
There is a wooden velodrome, Vélodrome du Lac, in Bordeaux which hosts international cycling competition in the form of UCI Track Cycling World Cup events. The 2015 Trophee Eric Bompard was in Bordeaux. But the Free Skate was cancelled in all of the divisions due to the Paris and aftermath. The Short Program occurred hours before the bombing. French skaters Chafik Besseghier (68.36) in tenth place, Romain Ponsart (62.86) in 11th. Mae-Berenice-Meite (46.82) in 11th and Laurine Lecavelier (46.53) in 12th. Vanessa James/Morgan Cipres (65.75) in second. Between 1951 and 1955, an annual Formula 1 motor race was held on a 2.5-kilometre circuit which looped around the Esplanade des Quinconces and along the waterfront, attracting drivers such as Juan Manuel Fangio, Stirling Moss, Jean Behra and Maurice Trintignant. International relationships. Twin towns – sister cities. Bordeaux is twinned with:
Puzzle Bobble internationally known as Bust-A-Move, is a 1994 tile-matching puzzle arcade game developed and published by Taito. It is based on the 1986 arcade game "Bubble Bobble", featuring characters and themes from that game. Its characteristically cute Japanese animation and music, along with its play mechanics and level designs, made it successful as an arcade title and spawned several sequels and ports to home gaming systems. Gameplay. At the start of each round, the rectangular playing arena contains a prearranged pattern of colored "bubbles". At the bottom of the screen, the player controls a device called a "pointer", which aims and fires bubbles up the screen. The color of bubbles fired is randomly generated and chosen from the colors of bubbles still left on the screen. The objective of the game is to clear all the bubbles from the arena without any bubble crossing the bottom line. Bubbles will fire automatically if the player remains idle. After clearing the arena, the next round begins with a new pattern of bubbles to clear. The arcade version of the game consists of 30 levels. The fired bubbles travel in straight lines (possibly bouncing off the sidewalls of the arena), stopping when they touch other bubbles or reach the top of the arena. If a bubble touches identically colored bubbles, forming a group of three or more, those bubbles—as well as any bubbles hanging from them—are removed from the field of play, and points are awarded. After every few shots, the "ceiling" of the playing arena drops downwards slightly, along with all the bubbles stuck to it. The number of shots between each drop of the ceiling is influenced by the number of bubble colors remaining. The closer the bubbles get to the bottom of the screen, the faster the music plays and if they cross the line at the bottom then the game is over.
Release. Two different versions of the original game were released. "Puzzle Bobble" was originally released in Japan only in June 1994 by Taito, running on Taito B System hardware (with the preliminary title "Bubble Buster"). Then, six months later in December, the international Neo Geo version of "Puzzle Bobble" was released. It was almost identical aside from being in stereo and having some different sound effects and translated text. Reception. In Japan, "Game Machine" listed the Neo Geo version of "Puzzle Bobble" on their February 15, 1995 issue as being the second most-popular arcade game at the time. It went on to become Japan's second highest-grossing arcade printed circuit board (PCB) software of 1995, below "Virtua Fighter 2". In North America, "RePlay" reported the Neo Geo version of "Puzzle Bobble" to be the fourth most-popular arcade game in February 1995. Reviewing the Super NES version, Mike Weigand of "Electronic Gaming Monthly" called it "a thoroughly enjoyable and incredibly addicting puzzle game". He considered the two player mode the highlight, but also said that the one player mode provides a solid challenge. "GamePro" gave it a generally negative review, saying it starts out fun but that ultimately lacks intricacy and longevity. They elaborated that in one player mode all the levels feel the same, and that two player matches are over too quickly to build up any excitement. They also criticized the lack of any 3D effects in the graphics. "Next Generation" reviewed the SNES version of the game and called it "addictive as hell".
A reviewer for "Next Generation", while questioning the continued viability of the action puzzle genre, admitted that the game is "very simple and "very" addictive". He remarked that though the 3DO version makes no significant additions, none are called for by a game with such simple enjoyment. "GamePro"s brief review of the 3DO version commented that the game's controls are responsive, and they also praised visuals and music. "Edge" magazine ranked the game 73rd on their 100 Best Video Games in 2007. "IGN" rated the SNES version 54th in its Top 100 SNES Games. Legacy. The simplicity of the concept has led to many clones, both commercial and otherwise. 1996's "Snood" replaced the bubbles with small creatures and has been successful in its own right. "Worms Blast" was Team 17's take on the concept. On September 24, 2000, British game publisher Empire Interactive released a similar game, "Spin Jam", for the original PlayStation console. Mobile clones include "Bubble Witch Saga" and "Bubble Shooter". "Frozen Bubble" is a free software clone. For "Bubble Bobble"s 35th anniversary, Taito launched "Puzzle Bobble VR: Vacation Odyssey" on the Oculus Quest and Oculus Quest 2, later coming to PlayStation 4 and PlayStation 5 as "Puzzle Bobble 3D: Vacation Odyssey" in 2021." "Puzzle Bobble Everybubble!". "Puzzle Bobble Everybubble!" was released on May 23, 2023, for Nintendo Switch. The game also comes with an extra mode called ""Puzzle Bobble" vs. "Space Invaders"", where up to four players can work together to erase bubble-encased invaders before they reach the player while only being able to aim straight up.
Bone A bone is a rigid organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red and white blood cells, store minerals, provide structure and support for the body, and enable mobility. Bones come in a variety of shapes and sizes and have complex internal and external structures. They are lightweight yet strong and hard and serve multiple functions. Bone tissue (osseous tissue), which is also called bone in the uncountable sense of that word, is hard tissue, a type of specialised connective tissue. It has a honeycomb-like matrix internally, which helps to give the bone rigidity. Bone tissue is made up of different types of bone cells. Osteoblasts and osteocytes are involved in the formation and mineralisation of bone; osteoclasts are involved in the resorption of bone tissue. Modified (flattened) osteoblasts become the lining cells that form a protective layer on the bone surface. The mineralised matrix of bone tissue has an organic component of mainly collagen called "ossein" and an inorganic component of bone mineral made up of various salts. Bone tissue is mineralized tissue of two types, cortical bone and cancellous bone. Other types of tissue found in bones include bone marrow, endosteum, periosteum, nerves, blood vessels, and cartilage.
In the human body at birth, approximately 300 bones are present. Many of these fuse together during development, leaving a total of 206 separate bones in the adult, not counting numerous small sesamoid bones. The largest bone in the body is the femur or thigh-bone, and the smallest is the "stapes" in the middle ear. The Ancient Greek word for bone is ὀστέον ("osteon"), hence the many terms that use it as a prefix—such as osteopathy. In anatomical terminology, including the "Terminologia Anatomica" international standard, the word for a bone is "os" (for example, "os breve", "os longum", "os sesamoideum"). Structure. Bone is not uniformly solid, but consists of a flexible matrix (about 30%) and bound minerals (about 70%), which are intricately woven and continuously remodeled by a group of specialized bone cells. Their unique composition and design allows bones to be relatively hard and strong, while remaining lightweight. Bone matrix is 90 to 95% composed of elastic collagen fibers, also known as ossein, and the remainder is ground substance. The elasticity of collagen improves fracture resistance. The matrix is hardened by the binding of inorganic mineral salt, calcium phosphate, in a chemical arrangement known as bone mineral, a form of calcium apatite. It is the mineralization that gives bones rigidity.
Bone is actively constructed and remodeled throughout life by specialized bone cells known as osteoblasts and osteoclasts. Within any single bone, the tissue is woven into two main patterns: cortical and cancellous bone, each with a distinct appearance and characteristics. Cortex. The hard outer layer of bones is composed of cortical bone, which is also called compact bone as it is much denser than cancellous bone. It forms the hard exterior (cortex) of bones. The cortical bone gives bone its smooth, white, and solid appearance, and accounts for 80% of the total bone mass of an adult human skeleton. It facilitates bone's main functions—to support the whole body, to protect organs, to provide levers for movement, and to store and release chemical elements, mainly calcium. It consists of multiple microscopic columns, each called an osteon or Haversian system. Each column is multiple layers of osteoblasts and osteocytes around a central canal called the osteonic canal. Volkmann's canals at right angles connect the osteons together. The columns are metabolically active, and as bone is reabsorbed and created the nature and location of the cells within the osteon will change. Cortical bone is covered by a periosteum on its outer surface, and an endosteum on its inner surface. The endosteum is the boundary between the cortical bone and the cancellous bone. The primary anatomical and functional unit of cortical bone is the osteon.
Trabeculae. Cancellous bone or spongy bone, also known as trabecular bone, is the internal tissue of the skeletal bone and is an open cell porous network that follows the material properties of biofoams. Cancellous bone has a higher surface-area-to-volume ratio than cortical bone and it is less dense. This makes it weaker and more flexible. The greater surface area also makes it suitable for metabolic activities such as the exchange of calcium ions. Cancellous bone is typically found at the ends of long bones, near joints, and in the interior of vertebrae. Cancellous bone is highly vascular and often contains red bone marrow where hematopoiesis, the production of blood cells, occurs. The primary anatomical and functional unit of cancellous bone is the trabecula. The trabeculae are aligned towards the mechanical load distribution that a bone experiences within long bones such as the femur. As far as short bones are concerned, trabecular alignment has been studied in the vertebral pedicle. Thin formations of osteoblasts covered in endosteum create an irregular network of spaces, known as trabeculae. Within these spaces are bone marrow and hematopoietic stem cells that give rise to platelets, red blood cells and white blood cells. Trabecular marrow is composed of a network of rod- and plate-like elements that make the overall organ lighter and allow room for blood vessels and marrow. Trabecular bone accounts for the remaining 20% of total bone mass but has nearly ten times the surface area of compact bone.
The words "cancellous" and "trabecular" refer to the tiny lattice-shaped units (trabeculae) that form the tissue. It was first illustrated accurately in the engravings of Crisóstomo Martinez. Marrow. Bone marrow, also known as myeloid tissue in red bone marrow, can be found in almost any bone that holds cancellous tissue. In newborns, all such bones are filled exclusively with red marrow or hematopoietic marrow, but as the child ages the hematopoietic fraction decreases in quantity and the fatty/ yellow fraction called marrow adipose tissue (MAT) increases in quantity. In adults, red marrow is mostly found in the bone marrow of the femur, the ribs, the vertebrae and pelvic bones. Vascular supply. Bone receives about 10% of cardiac output. Blood enters the endosteum, flows through the marrow, and exits through small vessels in the cortex. In humans, blood oxygen tension in bone marrow is about 6.6%, compared to about 12% in arterial blood, and 5% in venous and capillary blood. Cells. Bone is metabolically active tissue composed of several types of cells. These cells include osteoblasts, which are involved in the creation and mineralization of bone tissue, osteocytes, and osteoclasts, which are involved in the reabsorption of bone tissue. Osteoblasts and osteocytes are derived from osteoprogenitor cells, but osteoclasts are derived from the same cells that differentiate to form macrophages and monocytes. Within the marrow of the bone there are also hematopoietic stem cells. These cells give rise to other cells, including white blood cells, red blood cells, and platelets.
Osteoblast. Osteoblasts are mononucleate bone-forming cells. They are located on the surface of osteon seams and make a protein mixture known as osteoid, which mineralizes to become bone. The osteoid seam is a narrow region of a newly formed organic matrix, not yet mineralized, located on the surface of a bone. Osteoid is primarily composed of Type I collagen. Osteoblasts also manufacture hormones, such as prostaglandins, to act on the bone itself. The osteoblast creates and repairs new bone by actually building around itself. First, the osteoblast puts up collagen fibers. These collagen fibers are used as a framework for the osteoblasts' work. The osteoblast then deposits calcium phosphate which is hardened by hydroxide and bicarbonate ions. The brand-new bone created by the osteoblast is called osteoid. Once the osteoblast is finished working it is actually trapped inside the bone once it hardens. When the osteoblast becomes trapped, it becomes known as an osteocyte. Other osteoblasts remain on the top of the new bone and are used to protect the underlying bone, these become known as bone lining cells.
Osteocyte. Osteocytes are cells of mesenchymal origin and originate from osteoblasts that have migrated into and become trapped and surrounded by a bone matrix that they themselves produced. The spaces the cell body of osteocytes occupy within the mineralized collagen type I matrix are known as lacunae, while the osteocyte cell processes occupy channels called canaliculi. The many processes of osteocytes reach out to meet osteoblasts, osteoclasts, bone lining cells, and other osteocytes probably for the purposes of communication. Osteocytes remain in contact with other osteocytes in the bone through gap junctions—coupled cell processes which pass through the canalicular channels. Osteoclast. Osteoclasts are very large multinucleate cells that are responsible for the breakdown of bones by the process of bone resorption. New bone is then formed by the osteoblasts. Bone is constantly remodeled by the resorption of osteoclasts and created by osteoblasts. Osteoclasts are large cells with multiple nuclei located on bone surfaces in what are called "Howship's lacunae" (or "resorption pits"). These lacunae are the result of surrounding bone tissue that has been reabsorbed. Because the osteoclasts are derived from a monocyte stem-cell lineage, they are equipped with phagocytic-like mechanisms similar to circulating macrophages. Osteoclasts mature and/or migrate to discrete bone surfaces. Upon arrival, active enzymes, such as tartrate-resistant acid phosphatase, are secreted against the mineral substrate. The reabsorption of bone by osteoclasts also plays a role in calcium homeostasis.
Composition. Bones consist of living cells (osteoblasts and osteocytes) embedded in a mineralized organic matrix. The primary inorganic component of human bone is hydroxyapatite, the dominant bone mineral, having the nominal composition of Ca10(PO4)6(OH)2. The organic components of this matrix consist mainly of type I collagen—"organic" referring to materials produced as a result of the human body—and inorganic components, which alongside the dominant hydroxyapatite phase, include other compounds of calcium and phosphate including salts. Approximately 30% of the acellular component of bone consists of organic matter, while roughly 70% by mass is attributed to the inorganic phase. The collagen fibers give bone its tensile strength, and the interspersed crystals of hydroxyapatite give bone its compressive strength. These effects are synergistic. The exact composition of the matrix may be subject to change over time due to nutrition and biomineralization, with the ratio of calcium to phosphate varying between 1.3 and 2.0 (per weight), and trace minerals such as magnesium, sodium, potassium and carbonate also be found.
Type I collagen composes 90–95% of the organic matrix, with the remainder of the matrix being a homogenous liquid called ground substance consisting of proteoglycans such as hyaluronic acid and chondroitin sulfate, as well as non-collagenous proteins such as osteocalcin, osteopontin or bone sialoprotein. Collagen consists of strands of repeating units, which give bone tensile strength, and are arranged in an overlapping fashion that prevents shear stress. The function of ground substance is not fully known. Two types of bone can be identified microscopically according to the arrangement of collagen: woven and lamellar. Woven bone is produced when osteoblasts produce osteoid rapidly, which occurs initially in all fetal bones, but is later replaced by more resilient lamellar bone. In adults, woven bone is created after fractures or in Paget's disease. Woven bone is weaker, with a smaller number of randomly oriented collagen fibers, but forms quickly; it is for this appearance of the fibrous matrix that the bone is termed "woven". It is soon replaced by lamellar bone, which is highly organized in concentric sheets with a much lower proportion of osteocytes to surrounding tissue. Lamellar bone, which makes its first appearance in humans in the fetus during the third trimester, is stronger and filled with many collagen fibers parallel to other fibers in the same layer (these parallel columns are called osteons). In cross-section, the fibers run in opposite directions in alternating layers, much like in plywood, assisting in the bone's ability to resist torsion forces. After a fracture, woven bone forms initially and is gradually replaced by lamellar bone during a process known as "bony substitution". Compared to woven bone, lamellar bone formation takes place more slowly. The orderly deposition of collagen fibers restricts the formation of osteoid to about 1 to 2 μm per day. Lamellar bone also requires a relatively flat surface to lay the collagen fibers in parallel or concentric layers.
Deposition. The extracellular matrix of bone is laid down by osteoblasts, which secrete both collagen and ground substance. These cells synthesise collagen alpha polypetpide chains and then secrete collagen molecules. The collagen molecules associate with their neighbors and crosslink via lysyl oxidase to form collagen fibrils. At this stage, they are not yet mineralized, and this zone of unmineralized collagen fibrils is called "osteoid". Around and inside collagen fibrils calcium and phosphate eventually precipitate within days to weeks becoming then fully mineralized bone with an overall carbonate substituted hydroxyapatite inorganic phase. In order to mineralise the bone, the osteoblasts secrete alkaline phosphatase, some of which is carried by vesicles. This cleaves the inhibitory pyrophosphate and simultaneously generates free phosphate ions for mineralization, acting as the foci for calcium and phosphate deposition. Vesicles may initiate some of the early mineralization events by rupturing and acting as a centre for crystals to grow on. Bone mineral may be formed from globular and plate structures, and via initially amorphous phases.
Types. Five types of bones are found in the human body: long, short, flat, irregular, and sesamoid. Terminology. In the study of anatomy, anatomists use a number of anatomical terms to describe the appearance, shape and function of bones. Other anatomical terms are also used to describe the location of bones. Like other anatomical terms, many of these derive from Latin and Greek. Some anatomists still use Latin to refer to bones. The term "osseous", and the prefix "osteo-", referring to things related to bone, are still used commonly today. Some examples of terms used to describe bones include the term "foramen" to describe a hole through which something passes, and a "canal" or "meatus" to describe a tunnel-like structure. A protrusion from a bone can be called a number of terms, including a "condyle", "crest", "spine", "eminence", "tubercle" or "tuberosity", depending on the protrusion's shape and location. In general, long bones are said to have a "head", "neck", and "body". When two bones join, they are said to "articulate". If the two bones have a fibrous connection and are relatively immobile, then the joint is called a "suture".
Development. The formation of bone is called ossification. During the fetal stage of development this occurs by two processes: intramembranous ossification and endochondral ossification. Intramembranous ossification involves the formation of bone from connective tissue whereas endochondral ossification involves the formation of bone from cartilage. Intramembranous ossification mainly occurs during formation of the flat bones of the skull but also the mandible, maxilla, and clavicles; the bone is formed from connective tissue such as mesenchyme tissue rather than from cartilage. The process includes: the development of the ossification center, calcification, trabeculae formation and the development of the periosteum. Endochondral ossification occurs in long bones and most other bones in the body; it involves the development of bone from cartilage. This process includes the development of a cartilage model, its growth and development, development of the primary and secondary ossification centers, and the formation of articular cartilage and the epiphyseal plates.
Endochondral ossification begins with points in the cartilage called "primary ossification centers". They mostly appear during fetal development, though a few short bones begin their primary ossification after birth. They are responsible for the formation of the diaphyses of long bones, short bones and certain parts of irregular bones. Secondary ossification occurs after birth and forms the epiphyses of long bones and the extremities of irregular and flat bones. The diaphysis and both epiphyses of a long bone are separated by a growing zone of cartilage (the epiphyseal plate). At skeletal maturity (18 to 25 years of age), all of the cartilage is replaced by bone, fusing the diaphysis and both epiphyses together (epiphyseal closure). In the upper limbs, only the diaphyses of the long bones and scapula are ossified. The epiphyses, carpal bones, coracoid process, medial border of the scapula, and acromion are still cartilaginous. The following steps are followed in the conversion of cartilage to bone:
Functions. Bones have a variety of functions: Mechanical. Bones serve a variety of mechanical functions. Together the bones in the body form the skeleton. They provide a frame to keep the body supported, and an attachment point for skeletal muscles, tendons, ligaments and joints, which function together to generate and transfer forces so that individual body parts or the whole body can be manipulated in three-dimensional space (the interaction between bone and muscle is studied in biomechanics). Bones protect internal organs, such as the skull protecting the brain or the ribs protecting the heart and lungs. Because of the way that bone is formed, bone has a high compressive strength of about , poor tensile strength of 104–121 MPa, and a very low shear stress strength (51.6 MPa). This means that bone resists pushing (compressional) stress well, resist pulling (tensional) stress less well, but only poorly resists shear stress (such as due to torsional loads). While bone is essentially brittle, bone does have a significant degree of elasticity, contributed chiefly by collagen.
Mechanically, bones also have a special role in hearing. The ossicles are three small bones in the middle ear which are involved in sound transduction. Synthetic. The cancellous part of bones contain bone marrow. Bone marrow produces blood cells in a process called hematopoiesis. Blood cells that are created in bone marrow include red blood cells, platelets and white blood cells. Progenitor cells such as the hematopoietic stem cell divide in a process called mitosis to produce precursor cells. These include precursors which eventually give rise to white blood cells, and erythroblasts which give rise to red blood cells. Unlike red and white blood cells, created by mitosis, platelets are shed from very large cells called megakaryocytes. This process of progressive differentiation occurs within the bone marrow. After the cells are matured, they enter the circulation. Every day, over 2.5 billion red blood cells and platelets, and 50–100 billion granulocytes are produced in this way. As well as creating cells, bone marrow is also one of the major sites where defective or aged red blood cells are destroyed.
Metabolic. Determined by the species, age, and the type of bone, bone cells make up to 15 percent of the bone. Growth factor storage—mineralized bone matrix stores important growth factors such as insulin-like growth factors, transforming growth factor, bone morphogenetic proteins and others. Calcium. Strong bones during our youth is essential for preventing osteoporosis and bone fragility as we age. The importance of insuring factors that could influence increases in BMD while lowering our risks for further bone degradation is necessary during our childhood as these factors lead to a supportive and healthy lifestyle/bone health. Up till the age of 30, the bone stores that we have will ultimately start to decrease as we surpass this age. Influencing factors that can help us have larger stores and higher amounts of BMD will allow us to see less harmful results as we reach older adulthood. The issue of having fragile bones during our childhood leads to an increase in certain disorders and conditions such as juvenile osteoporosis, though it is less common to see, the necessity for a healthy routine especially when it comes to bone development is essential in our youth. Children that naturally have lower bone mineral density have a lower quality of life and therefore lead a life that is less fulfilling and uncomfortable. Factors such as increases in Calcium intake has been shown to increase BMD stores. Studies have shown that increasing calcium stores whether that be through supplementation or intake via foods and beverages such as leafy greens and milk have pushed the notion that prepuberty or even early pubertal children will see increases in BMD with the addition of increase Calcium intake.
Another research study goes on to show that long-term calcium intake has been proven to significantly contribute to overall BMD in children without certain conditions or disorders. This data shows that ensuring adequate calcium intake in children reinforces the structure and rate at which bones will begin to densify. Further detailing how structuring a strong nutritional plan with adequate amounts of Calcium sources can lead to strong bones but also can be a worth-while strategy into preventing further damage or degradation of bone stores as we age. The connection between Calcium intake & BMD and its effects on youth as a whole is a very world-wide issue and has been shown to affect different ethnicities in a variety of differing ways. In a recent study, there was a strong correlation between calcium intake and BMD across a variety of diverse populations of children and adolescence ultimately coming to the conclusion that fundamentally, achieving optimal bone health is necessary for providing our youth with the ability to undergo hormonal changes as well. They found in a study of over 10,000 children ages 8–19 that in females, African Americans, and the 12-15 adolescent groups that at 2.6-2.8g/kg of body weight, they began to see a decrease in BMD. They elaborate on this by determining that this is strongly influenced by a lower baseline in calcium intake throughout puberty. Genetic factors have also been shown to influence lower acceptance of calcium stores.
Ultimately, the window that youth have for accruing and building resilient bones is very minimal. Being able to consistently meet calcium needs while also engaging in weight-bearing exercise is essential for building a strong initial bone foundation at which to build upon. Being able to reach our daily value of 1300 mg for ages 9–18 is becoming more and more necessary and as we progress in health, the chance that osteoporosis and other factors such as bone fragility or potential for stunted growth can be greatly reduced through these resources, ultimately leading to a more fulfilling and healthier lifestyle. Remodeling. Bone is constantly being created and replaced in a process known as remodeling. This ongoing turnover of bone is a process of resorption followed by replacement of bone with little change in shape. This is accomplished through osteoblasts and osteoclasts. Cells are stimulated by a variety of signals, and together referred to as a remodeling unit. Approximately 10% of the skeletal mass of an adult is remodelled each year. The purpose of remodeling is to regulate calcium homeostasis, repair microdamaged bones from everyday stress, and to shape the skeleton during growth. Repeated stress, such as weight-bearing exercise or bone healing, results in the bone thickening at the points of maximum stress (Wolff's law). It has been hypothesized that this is a result of bone's piezoelectric properties, which cause bone to generate small electrical potentials under stress.
The action of osteoblasts and osteoclasts are controlled by a number of chemical enzymes that either promote or inhibit the activity of the bone remodeling cells, controlling the rate at which bone is made, destroyed, or changed in shape. The cells also use paracrine signalling to control the activity of each other. For example, the rate at which osteoclasts resorb bone is inhibited by calcitonin and osteoprotegerin. Calcitonin is produced by parafollicular cells in the thyroid gland, and can bind to receptors on osteoclasts to directly inhibit osteoclast activity. Osteoprotegerin is secreted by osteoblasts and is able to bind RANK-L, inhibiting osteoclast stimulation. Osteoblasts can also be stimulated to increase bone mass through increased secretion of osteoid and by inhibiting the ability of osteoclasts to break down osseous tissue. Increased secretion of osteoid is stimulated by the secretion of growth hormone by the pituitary, thyroid hormone and the sex hormones (estrogens and androgens). These hormones also promote increased secretion of osteoprotegerin. Osteoblasts can also be induced to secrete a number of cytokines that promote reabsorption of bone by stimulating osteoclast activity and differentiation from progenitor cells. Vitamin D, parathyroid hormone and stimulation from osteocytes induce osteoblasts to increase secretion of RANK-ligand and interleukin 6, which cytokines then stimulate increased reabsorption of bone by osteoclasts. These same compounds also increase secretion of macrophage colony-stimulating factor by osteoblasts, which promotes the differentiation of progenitor cells into osteoclasts, and decrease secretion of osteoprotegerin.
Volume. Bone volume is determined by the rates of bone formation and bone resorption. Certain growth factors may work to locally alter bone formation by increasing osteoblast activity. Numerous bone-derived growth factors have been isolated and classified via bone cultures. These factors include insulin-like growth factors I and II, transforming growth factor-beta, fibroblast growth factor, platelet-derived growth factor, and bone morphogenetic proteins. Evidence suggests that bone cells produce growth factors for extracellular storage in the bone matrix. The release of these growth factors from the bone matrix could cause the proliferation of osteoblast precursors. Essentially, bone growth factors may act as potential determinants of local bone formation. Cancellous bone volume in postmenopausal osteoporosis may be determined by the relationship between the total bone forming surface and the percent of surface resorption. Clinical significance. A number of diseases can affect bone, including arthritis, fractures, infections, osteoporosis and tumors. Conditions relating to bone can be managed by a variety of doctors, including rheumatologists for joints, and orthopedic surgeons, who may conduct surgery to fix broken bones. Other doctors, such as rehabilitation specialists may be involved in recovery, radiologists in interpreting the findings on imaging, and pathologists in investigating the cause of the disease, and family doctors may play a role in preventing complications of bone disease such as osteoporosis.
When a doctor sees a patient, a history and exam will be taken. Bones are then often imaged, called radiography. This might include ultrasound X-ray, CT scan, MRI scan and other imaging such as a Bone scan, which may be used to investigate cancer. Other tests such as a blood test for autoimmune markers may be taken, or a synovial fluid aspirate may be taken. Fractures. In normal bone, fractures occur when there is significant force applied or repetitive trauma over a long time. Fractures can also occur when a bone is weakened, such as with osteoporosis, or when there is a structural problem, such as when the bone remodels excessively (such as Paget's disease) or is the site of the growth of cancer. Common fractures include wrist fractures and hip fractures, associated with osteoporosis, vertebral fractures associated with high-energy trauma and cancer, and fractures of long-bones. Not all fractures are painful. When serious, depending on the fractures type and location, complications may include flail chest, compartment syndromes or fat embolism.
Compound fractures involve the bone's penetration through the skin. Some complex fractures can be treated by the use of bone grafting procedures that replace missing bone portions. Fractures and their underlying causes can be investigated by X-rays, CT scans and MRIs. Fractures are described by their location and shape, and several classification systems exist, depending on the location of the fracture. A common long bone fracture in children is a Salter–Harris fracture. When fractures are managed, pain relief is often given, and the fractured area is often immobilised. This is to promote bone healing. In addition, surgical measures such as internal fixation may be used. Because of the immobilisation, people with fractures are often advised to undergo rehabilitation. Tumors. Tumor that can affect bone in several ways. Examples of benign bone tumors include osteoma, osteoid osteoma, osteochondroma, osteoblastoma, enchondroma, giant-cell tumor of bone, and aneurysmal bone cyst. Cancer. Cancer can arise in bone tissue, and bones are also a common site for other cancers to spread (metastasise) to. Cancers that arise in bone are called "primary" cancers, although such cancers are rare. Metastases within bone are "secondary" cancers, with the most common being breast cancer, lung cancer, prostate cancer, thyroid cancer, and kidney cancer. Secondary cancers that affect bone can either destroy bone (called a "lytic" cancer) or create bone (a "sclerotic" cancer). Cancers of the bone marrow inside the bone can also affect bone tissue, examples including leukemia and multiple myeloma. Bone may also be affected by cancers in other parts of the body. Cancers in other parts of the body may release parathyroid hormone or parathyroid hormone-related peptide. This increases bone reabsorption, and can lead to bone fractures.
Bone tissue that is destroyed or altered as a result of cancers is distorted, weakened, and more prone to fracture. This may lead to compression of the spinal cord, destruction of the marrow resulting in bruising, bleeding and immunosuppression, and is one cause of bone pain. If the cancer is metastatic, then there might be other symptoms depending on the site of the original cancer. Some bone cancers can also be felt. Cancers of the bone are managed according to their type, their stage, prognosis, and what symptoms they cause. Many primary cancers of bone are treated with radiotherapy. Cancers of bone marrow may be treated with chemotherapy, and other forms of targeted therapy such as immunotherapy may be used. Palliative care, which focuses on maximising a person's quality of life, may play a role in management, particularly if the likelihood of survival within five years is poor. Diabetes. Type 1 diabetes is an autoimmune disease in which the body attacks the insulin-producing pancreas cells causing the body to not make enough insulin. In contrast type 2 diabetes in which the body creates enough Insulin, but becomes resistant to it over time.
Children makeup approximately 85% of Type 1 Diabetes cases and in America there was an average 22% rise in cases over the first 24 months of the COVID-19 Pandemic. With the increase of developing some form of diabetes across all ranges continually growing the health impacts on bone development and bone health in these populations are still being researched. Most evidence suggests that diabetes, either Type 1 and Type 2, inhibits osteoblastic activity and causes both lower BMD and BMC in both adults and children. The weakening of these developmental aspects is thought to lead to an increased risk of developing many diseases such as osteoarthritis, osteoporosis, osteopenia and fractures. Development of any of these diseases is thought to be correlated with a decrease in ability to perform in athletic environments and activities of daily living. Focusing on therapies that target molecules like osteocalcin or AGEs could provide new ways to improve bone health and help manage the complications of diabetes more effectively.
Osteoporosis. Osteoporosis is a disease of bone where there is reduced bone mineral density, increasing the likelihood of fractures. Osteoporosis is defined in women by the World Health Organization as a bone mineral density of 2.5 standard deviations below peak bone mass, relative to the age and sex-matched average. This density is measured using dual energy X-ray absorptiometry (DEXA), with the term "established osteoporosis" including the presence of a fragility fracture. Osteoporosis is most common in women after menopause, when it is called "postmenopausal osteoporosis", but may develop in men and premenopausal women in the presence of particular hormonal disorders and other chronic diseases or as a result of smoking and medications, specifically glucocorticoids. Osteoporosis usually has no symptoms until a fracture occurs. For this reason, DEXA scans are often done in people with one or more risk factors, who have developed osteoporosis and are at risk of fracture. One of the most important risk factors for osteoporosis is advanced age. Accumulation of oxidative DNA damage in osteoblastic and osteoclastic cells appears to be a key factor in age-related osteoporosis.
Osteoporosis treatment includes advice to stop smoking, decrease alcohol consumption, exercise regularly, and have a healthy diet. Calcium and trace mineral supplements may also be advised, as may Vitamin D. When medication is used, it may include bisphosphonates, Strontium ranelate, and hormone replacement therapy. Osteopathic medicine. Osteopathic medicine is a school of medical thought that links the musculoskeletal system to overall health. , over 77,000 physicians in the United States are trained in osteopathic medical schools. Bone health. Bone health is vastly important all throughout life due to a number of reasons, some of those being, without strong healthy bones we are more at risk for different chronic diseases, and fractures as well as day-to-day function being more difficult with poor bone health. Developing strong bones as a child is one of the most important steps to having healthy bones all throughout life because this is when a strong foundation is built, which will make it much easier to maintain musculoskeletal health in later years.
Developing strong bones as a child is one of the most important steps to having healthy bones all throughout life because this is when a strong foundation is built, which will make it much easier to maintain musculoskeletal health in later years. Adolescence offers a window to really develop bones in either a positive or negative way. It is estimated that diet and exercise during these years can impact peak bone mass as an adult nearly 20-40%. One study done on children with developmental coordination disorder found an increase in bone mass up to 4% and 5% in the cortical areas of the tibia alone from a 13-week training period, which is truly significant when considering how participants only participated in the multimodal workouts twice per week, and it would be reasonable to expect these increases to be greater if workouts were more frequent, especially in youth without developmental coordination disorder.
Osteology. The study of bones and teeth is referred to as osteology. It is frequently used in anthropology, archeology and forensic science for a variety of tasks. This can include determining the nutritional, health, age or injury status of the individual the bones were taken from. Preparing fleshed bones for these types of studies can involve the process of maceration. Typically anthropologists and archeologists study bone tools made by "Homo sapiens" and "Homo neanderthalensis". Bones can serve a number of uses such as projectile points or artistic pigments, and can also be made from external bones such as antlers. Other animals. Bird skeletons are very lightweight. Their bones are smaller and thinner, to aid flight. Among mammals, bats come closest to birds in terms of bone density, suggesting that small dense bones are a flight adaptation. Many bird bones have little marrow due to them being hollow. A bird's beak is primarily made of bone as projections of the mandibles which are covered in keratin. Some bones, primarily formed separately in subcutaneous tissues, include headgears (such as bony core of horns, antlers, ossicones), osteoderm, and os penis/os clitoris. A deer's antlers are composed of bone which is an unusual example of bone being outside the skin of the animal once the velvet is shed.
The extinct predatory fish "Dunkleosteus" had sharp edges of hard exposed bone along its jaws. The proportion of cortical bone that is 80% in the human skeleton may be much lower in other animals, especially in marine mammals and marine turtles, or in various Mesozoic marine reptiles, such as ichthyosaurs, among others. This proportion can vary quickly in evolution; it often increases in early stages of returns to an aquatic lifestyle, as seen in early whales and pinnipeds, among others. It subsequently decreases in pelagic taxa, which typically acquire spongy bone, but aquatic taxa that live in shallow water can retain very thick, pachyostotic, osteosclerotic, or pachyosteosclerotic bones, especially if they move slowly, like sea cows. In some cases, even marine taxa that had acquired spongy bone can revert to thicker, compact bones if they become adapted to live in shallow water, or in hypersaline (denser) water. Many animals, particularly herbivores, practice osteophagy—the eating of bones. This is presumably carried out in order to replenish lacking phosphate.
Many bone diseases that affect humans also affect other vertebrates—an example of one disorder is skeletal fluorosis. Society and culture. Bones from slaughtered animals have a number of uses. In prehistoric times, they have been used for making bone tools. They have further been used in bone carving, already important in prehistoric art, and also in modern time as crafting materials for buttons, beads, handles, bobbins, calculation aids, head nuts, dice, poker chips, pick-up sticks, arrows, scrimshaw, ornaments, etc. Bone glue can be made by prolonged boiling of ground or cracked bones, followed by filtering and evaporation to thicken the resulting fluid. Historically once important, bone glue and other animal glues today have only a few specialized uses, such as in antiques restoration. Essentially the same process, with further refinement, thickening and drying, is used to make gelatin. Broth is made by simmering several ingredients for a long time, traditionally including bones. Bone char, a porous, black, granular material primarily used for filtration and also as a black pigment, is produced by charring mammal bones.
Oracle bone script was a writing system used in ancient China based on inscriptions in bones. Its name originates from oracle bones, which were mainly ox clavicle. The Ancient Chinese (mainly in the Shang dynasty), would write their questions on the oracle bone, and burn the bone, and where the bone cracked would be the answer for the questions. To point the bone at someone is considered bad luck in some cultures, such as Australian aborigines, such as by the Kurdaitcha. The wishbones of fowl have been used for divination, and are still customarily used in a tradition to determine which one of two people pulling on either prong of the bone may make a wish. Various cultures throughout history have adopted the custom of shaping an infant's head by the practice of artificial cranial deformation. A widely practised custom in China was that of foot binding to limit the normal growth of the foot.
Bretwalda Bretwalda (also brytenwalda and bretenanwealda, sometimes capitalised) is an Old English word. The first record comes from the late 9th-century "Anglo-Saxon Chronicle". It is given to some of the rulers of Anglo-Saxon kingdoms from the 5th century onwards who had achieved overlordship of some or all of the other Anglo-Saxon kingdoms. It is unclear whether the word dates back to the 5th century and was used by the kings themselves or whether it is a later, 9th-century, invention. The term "bretwalda" also appears in a 10th-century charter of Æthelstan. The literal meaning of the word is disputed and may translate to either 'wide-ruler' or 'Britain-ruler'. The rulers of Mercia were generally the most powerful of the Anglo-Saxon kings from the mid 7th century to the early 9th century but are not accorded the title of "bretwalda" by the "Chronicle", which had an anti-Mercian bias. The "Annals of Wales" continued to recognise the kings of Northumbria as "Kings of the Saxons" until the death of Osred I of Northumbria in 716.
Etymology. The first syllable of the term "bretwalda" may be related to "Briton" or "Britain". The second element is taken to mean 'ruler' or 'sovereign'. Thus, one interpretation might be 'sovereign of Britain'. Otherwise, the word may be a compound containing the Old English adjective "brytten" ('broad', from the verb "breotan" meaning 'to break' or 'to disperse'), an element also found in the terms "bryten rice" ('kingdom'), "bryten-grund" ('the wide expanse of the earth') and "bryten cyning" ('king whose authority was widely extended'). Though the origin is ambiguous, the draughtsman of the charter issued by Æthelstan used the term in a way that can only mean 'wide-ruler'. The latter etymology was first suggested by John Mitchell Kemble who alluded that "of six manuscripts in which this passage occurs, one only reads "Bretwalda": of the remaining five, four have "Bryten-walda" or "-wealda", and one "Breten-anweald", which is precisely synonymous with Brytenwealda"; that Æthelstan was called "brytenwealda ealles ðyses ealondes", which Kemble translates as 'ruler of all these islands'; and that "bryten-" is a common prefix to words meaning 'wide or general dispersion' and that the similarity to the word "bretwealh" ('Briton') is "merely accidental".
Contemporary use. The first recorded use of the term "Bretwalda" comes from a West Saxon chronicle of the late 9th century that applied the term to Ecgberht, who ruled Wessex from 802 to 839. The chronicler also wrote down the names of seven kings that Bede listed in his "Historia ecclesiastica gentis Anglorum" in 731. All subsequent manuscripts of the "Chronicle" use the term "Brytenwalda", which may have represented the original term or derived from a common error. There is no evidence that the term was a title that had any practical use, with implications of formal rights, powers and office, or even that it had any existence before the 9th-century. Bede wrote in Latin and never used the term and his list of kings holding "imperium" should be treated with caution, not least in that he overlooks kings such as Penda of Mercia, who clearly held some kind of dominance during his reign. Similarly, in his list of bretwaldas, the West Saxon chronicler ignored such Mercian kings as Offa. The use of the term "Bretwalda" was the attempt by a West Saxon chronicler to make some claim of West Saxon kings to the whole of Great Britain. The concept of the overlordship of the whole of Britain was at least recognised in the period, whatever was meant by the term. Quite possibly it was a survival of a Roman concept of "Britain": it is significant that, while the hyperbolic inscriptions on coins and titles in charters often included the title "rex Britanniae", when England was unified the title used was "rex Angulsaxonum", ('king of the Anglo-Saxons'.)
Modern interpretation by historians. For some time, the existence of the word "bretwalda" in the "Anglo-Saxon Chronicle", which was based in part on the list given by Bede in his "Historia Ecclesiastica", led historians to think that there was perhaps a "title" held by Anglo-Saxon overlords. This was particularly attractive as it would lay the foundations for the establishment of an English monarchy. The 20th-century historian Frank Stenton said of the Anglo-Saxon chronicler that "his inaccuracy is more than compensated by his preservation of the English title applied to these outstanding kings". He argued that the term "bretwalda" "falls into line with the other evidence which points to the Germanic origin of the earliest English institutions". Over the later 20th century, this assumption was increasingly challenged. Patrick Wormald interpreted it as "less an objectively realized office than a subjectively perceived status" and emphasised the partiality of its usage in favour of Southumbrian rulers. In 1991, Steven Fanning argued that "it is unlikely that the term ever existed as a title or was in common usage in Anglo-Saxon England". The fact that Bede never mentioned a special title for the kings in his list implies that he was unaware of one. In 1995, Simon Keynes observed that "if Bede's concept of the Southumbrian overlord, and the chronicler's concept of the 'Bretwalda', are to be regarded as artificial constructs, which have no validity outside the context of the literary works in which they appear, we are released from the assumptions about political development which they seem to involve... we might ask whether kings in the eighth and ninth centuries were quite so obsessed with the establishment of a pan-Southumbrian state".
Modern interpretations view the concept of "bretwalda" overlordship as complex and an important indicator of how a 9th-century chronicler interpreted history and attempted to insert the increasingly powerful Saxon kings into that history. Overlordship. A complex array of dominance and subservience existed during the Anglo-Saxon period. A king who used charters to grant land in another kingdom indicated such a relationship. If the other kingdom were fairly large, as when the Mercians dominated the East Anglians, the relationship would have been more equal than in the case of the Mercian dominance of the Hwicce, which was a comparatively small kingdom. Mercia was arguably the most powerful Anglo-Saxon kingdom for much of the late 7th though 8th centuries, though Mercian kings are missing from the two main "lists". For Bede, Mercia was a traditional enemy of his native Northumbria and he regarded powerful kings such as the pagan Penda as standing in the way of the Christian conversion of the Anglo-Saxons. Bede omits them from his list, even though it is evident that Penda held a considerable degree of power. Similarly powerful Mercia kings such as Offa are missed out of the West Saxon "Anglo-Saxon Chronicle", which sought to demonstrate the legitimacy of their kings to rule over other Anglo-Saxon peoples.
Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function formula_1 mapping a nonempty compact convex set to itself, there is a point formula_2 such that formula_3. The simplest forms of Brouwer's theorem are for continuous functions formula_1 from a closed interval formula_5 in the real numbers to itself or from a closed disk formula_6 to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset formula_7 of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.
The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the -dimensional closed ball was first proved in 1910 by Jacques Hadamard and the general case for continuous mappings by Brouwer in 1911. Statement. The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: This can be generalized to an arbitrary finite dimension: A slightly more general version is as follows: An even more general form is better known under a different name: Importance of the pre-conditions. The theorem holds only for functions that are "endomorphisms" (functions that have the same set as the domain and codomain) and for nonempty sets that are "compact" (thus, in particular, bounded and closed) and "convex" (or homeomorphic to convex). The following examples show why the pre-conditions are important.
The function "f" as an endomorphism. Consider the function with domain [-1,1]. The range of the function is [0,2]. Thus, f is not an endomorphism. Boundedness. Consider the function which is a continuous function from formula_10 to itself. As it shifts every point to the right, it cannot have a fixed point. The space formula_10 is convex and closed, but not bounded. Closedness. Consider the function which is a continuous function from the open interval formula_13 to itself. Since the point formula_14 is not part of the interval, there is no point in the domain such that formula_15. The set formula_13 is convex and bounded, but not closed. On the other hand, the function formula_1 does have a fixed point in the "closed" interval formula_18, namely formula_14. The closed interval formula_18 is compact, the open interval formula_13 is not. Convexity. Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball formula_22. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).
The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function formula_23, which is a continuous function from the unit circle to itself. Since "-x≠x" holds for any point of the unit circle, "f" has no fixed point. The analogous example works for the "n"-dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex) . The function "f" have a fixed point for the unit disc, since it takes the origin to itself. A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the Lefschetz fixed-point theorem. Notes. The continuous function in this theorem is not required to be bijective or surjective. Illustrations. The theorem has several "real world" illustrations. Here are some examples. Intuitive approach. Explanations attributed to Brouwer. The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee.
If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving. The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears. Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet." Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness. One-dimensional case.
In one dimension, the result is intuitive and easy to prove. The continuous function "f" is defined on a closed interval ["a", "b"] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval ["a", "b"] which maps "x" to "x" (light green). Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function "g" which maps "x" to "f"("x") − "x". It is ≥ 0 on "a" and ≤ 0 on "b". By the intermediate value theorem, "g" has a zero in ["a", "b"]; this zero is a fixed point. Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."
History. The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case "n" = 3 first was proved by Piers Bohl in 1904 (published in "Journal für die reine und angewandte Mathematik"). It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910, and Brouwer found a different proof in the same year. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to approximate fixed points guaranteed by Brouwer's theorem are now known. Before discovery. At the end of the 19th century, the old problem of the stability of the solar system returned into the focus of the mathematical community. Its solution required new methods. As noted by Henri Poincaré, who worked on the three-body problem, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."
He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision". He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant flow? Poincaré discovered that the answer can be found in what we now call the topological properties in the area containing the trajectory. If this area is compact, i.e. both closed and bounded, then the trajectory either becomes stationary, or it approaches a limit cycle. Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval "t". If the area is a circular band, or if it is not closed, then this is not necessarily the case.
To understand differential equations better, a new branch of mathematics was born. Poincaré called it "analysis situs". The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing". In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem, although the connection with the subject of this article was not yet apparent. A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the fundamental group or sometimes the Poincaré group. This method can be used for a very compact proof of the theorem under discussion. Poincaré's method was analogous to that of Émile Picard, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem. Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after Banach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a contraction.
First proofs. At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations. In 1904 he proved the three-dimensional case of our theorem, but his publication was not noticed. It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially mathematical logic and topology. His initial interest lay in an attempt to solve Hilbert's fifth problem. In 1909, during a voyage to Paris, he met Henri Poincaré, Jacques Hadamard, and Émile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the hairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point. These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem. The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as homotopy, the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."
Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension, as well as other key theorems such as the invariance of dimension. In the context of this work, Brouwer also generalized the Jordan curve theorem to arbitrary dimension and established the properties connected with the degree of a continuous mapping. This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became algebraic topology. Reception. The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.
Other areas are also touched. In game theory, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white. In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria (Hotelling's law), financial equilibria and incomplete markets. Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are not constructive, and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of constructivity. He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand against set theory. Brouwer disavowed his original proof of the fixed-point theorem. Proof outlines. A proof using degree. Brouwer's original 1911 proof relied on the notion of the degree of a continuous mapping, stemming from ideas in differential topology. Several modern accounts of the proof can be found in the literature, notably .
Let formula_24 denote the closed unit ball in formula_25 centered at the origin. Suppose for simplicity that formula_26 is continuously differentiable. A regular value of formula_1 is a point formula_28 such that the Jacobian of formula_1 is non-singular at every point of the preimage of formula_30. In particular, by the inverse function theorem, every point of the preimage of formula_1 lies in formula_32 (the interior of formula_33). The degree of formula_1 at a regular value formula_28 is defined as the sum of the signs of the Jacobian determinant of formula_1 over the preimages of formula_30 under formula_1: The degree is, roughly speaking, the number of "sheets" of the preimage "f" lying over a small open set around "p", with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of winding number to higher dimensions. The degree satisfies the property of "homotopy invariance": let formula_1 and formula_41 be two continuously differentiable functions, and formula_42 for formula_43. Suppose that the point formula_30 is a regular value of formula_45 for all "t". Then formula_46.
If there is no fixed point of the boundary of formula_33, then the function is well-defined, and formula_49 defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so formula_41 also has degree one at the origin. As a consequence, the preimage formula_51 is not empty. The elements of formula_51 are precisely the fixed points of the original function "f". This requires some work to make fully general. The definition of degree must be extended to singular values of "f", and then to continuous functions. The more modern advent of homology theory simplifies the construction of the degree, and so has become a standard proof in the literature. A proof using the hairy ball theorem. The hairy ball theorem states that on the unit sphere in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field on . (The tangency condition means that = 0 for every unit vector .) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in .
In fact, suppose first that is "continuously differentiable". By scaling, it can be assumed that is a continuously differentiable unit tangent vector on . It can be extended radially to a small spherical shell of . For sufficiently small, a routine computation shows that the mapping () = + is a contraction mapping on and that the volume of its image is a polynomial in . On the other hand, as a contraction mapping, must restrict to a homeomorphism of onto (1 + ) and onto (1 + ) . This gives a contradiction, because, if the dimension of the Euclidean space is odd, (1 + )/2 is not a polynomial. If is only a "continuous" unit tangent vector on , by the Weierstrass approximation theorem, it can be uniformly approximated by a polynomial map of into Euclidean space. The orthogonal projection on to the tangent space is given by () = () - () ⋅ . Thus is polynomial and nowhere vanishing on ; by construction /\|\|\|\| is a smooth unit tangent vector field on , a contradiction. The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that is even. If there were a fixed-point-free continuous self-mapping of the closed unit ball of the -dimensional Euclidean space , set
Since has no fixed points, it follows that, for in the interior of , the vector () is non-zero; and for in , the scalar product ⋅ () = 1 – ⋅ () is strictly positive. From the original -dimensional space Euclidean space , construct a new auxiliary ()-dimensional space = x R, with coordinates = (, ). Set By construction is a continuous vector field on the unit sphere of , satisfying the tangency condition ⋅ () = 0. Moreover, () is nowhere vanishing (because, if has norm 1, then ⋅ () is non-zero; while if has norm strictly less than 1, then and () are both non-zero). This contradiction proves the fixed point theorem when is even. For odd, one can apply the fixed point theorem to the closed unit ball in dimensions and the mapping (,) = ((),0). The advantage of this proof is that it uses only elementary techniques; more general results like the Borsuk-Ulam theorem require tools from algebraic topology. A proof using homology or cohomology. The proof uses the observation that the boundary of the "n"-disk "D""n" is "S""n"−1, the ("n" − 1)-sphere.
Suppose, for contradiction, that a continuous function has "no" fixed point. This means that, for every point x in "D""n", the points "x" and "f"("x") are distinct. Because they are distinct, for every point x in "D""n", we can construct a unique ray from "f"("x") to "x" and follow the ray until it intersects the boundary "S""n"−1 (see illustration). By calling this intersection point "F"("x"), we define a function "F" : "D""n" → "S""n"−1 sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever "x" itself is on the boundary, then the intersection point "F"("x") must be "x". Consequently, "F" is a special type of continuous function known as a retraction: every point of the codomain (in this case "S""n"−1) is a fixed point of "F". Intuitively it seems unlikely that there could be a retraction of "D""n" onto "S""n"−1, and in the case "n" = 1, the impossibility is more basic, because "S"0 (i.e., the endpoints of the closed interval "D"1) is not even connected. The case "n" = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce a surjective group homomorphism from the fundamental group of "D"2 to that of "S"1, but the latter group is isomorphic to Z while the first group is trivial, so this is impossible. The case "n" = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields.
For "n" > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: the homology "H""n"−1("D""n") is trivial, while "H""n"−1("S""n"−1) is infinite cyclic. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group. The impossibility of a retraction can also be shown using the de Rham cohomology of open subsets of Euclidean space "E""n". For "n" ≥ 2, the de Rham cohomology of "U" = "E""n" – (0) is one-dimensional in degree 0 and "n" – 1, and vanishes otherwise. If a retraction existed, then "U" would have to be contractible and its de Rham cohomology in degree "n" – 1 would have to vanish, a contradiction. A proof using Stokes' theorem. As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retraction from the ball onto its boundary ∂. In that case it can be assumed that is smooth, since it can be approximated using the Weierstrass approximation theorem or by convolving with non-negative smooth bump functions of sufficiently small support and integral one (i.e. mollifying). If is a volume form on the boundary then by Stokes' theorem,

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