In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple hexagon is 720°. A regular hexagon has Schläfli symbol and can be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equilateral and equiangular, it is bicentric, meaning that it is both tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 2 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, making up the dihedral group D6; the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations.
The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons, it is not considered a triambus, although it is equilateral. The maximal diameter, D, is twice the maximal radius or circumradius, R, which equals the side length, t; the minimal diameter or the diameter of the inscribed circle, d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: 1 2 d = r = cos R = 3 2 R = 3 2 t and d = 3 2 D; the area of a regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 2.598 R 2 ≈ 3.464 r 2 ≈ 0.6495 D 2 ≈ 0.866 d 2. For any regular polygon, the area can be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, p = 6 R = 4 r 3, so A = a p 2 = r ⋅ 4 r 3 2 = 2 r 2 3 ≈ 3.464 r 2. The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C PE + PF = PA + PB + PC + PD. The regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, Dih1, 4 cyclic subgroups: Z6, Z3, Z2, Z1; these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a group order. R12 is full symmetry, a1 is no symmetry. P6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles; these two forms have half the symmetry order of the regular hexagon. The
Time in France
Metropolitan France uses Central European Time and Central European Summer Time. Daylight saving time is observed in Metropolitan France from the last Sunday in March to the last Sunday in October. With its overseas territories, France uses 12 different time zones, more than any other country in the world; the overseas territories of France use different timezones. Before 1891, each town and city in Metropolitan France had its own time based on local solar time. In 1891, to avoid complications with railway timetables, time was unified in Metropolitan France and based on the solar time of Paris. In detail, the railway companies used a unified time which lagged behind Paris solar time by 5 minutes, for the benefit of non-punctual travellers. In 1911, Metropolitan France adopted GMT+0 as its official time, used it until 1940. In the summer of 1940, the German military authorities switched the occupied northern part of Metropolitan France to GMT+2, while the non-occupied southern part of Metropolitan France remained at GMT+1.
The Vichy authorities kept GMT+1 during the winter of 1940–1941 and adopted GMT+2 in May 1941 in order to unify the railway timetables between occupied and non-occupied Metropolitan France. In 1942, 1943, 1944 the whole of Metropolitan France thus used GMT+2 during the summer, GMT+1 during the winter. At the Liberation of France in the summer of 1944, Metropolitan France kept GMT+2 as it was the time used by the Allies. In the winter of 1944–1945, Metropolitan France switched to GMT+1, same as in the United Kingdom, switched again to GMT+2 in April 1945 like its British ally. In September 1945, Metropolitan France returned to GMT+1, which the British had done in July 1945. Metropolitan France was scheduled to return to GMT+0 on November 18, 1945, but the French government canceled the decision on November 5, 1945, GMT+1 has since remained the official time of Metropolitan France. In 1976, daylight saving time was reintroduced in Metropolitan France for the first time since WW2 because of the oil crisis, since 1976 Metropolitan France has thus been at GMT+1 during the winter and GMT+2 during the summer.
In 1996, daylight saving time was harmonized throughout the European Union by Directive 2000/84/EC, which moved the end of DST to the last Sunday in October. Since GMT is Metropolitan France's "natural" time zone its use of UTC+01:00 in winter can be seen as a form of daylight saving time in winter, while Central European Summer Time can be seen as a form of "double summer time." List of time zones
The English Channel called the Channel, is the body of water that separates Southern England from northern France and links the southern part of the North Sea to the Atlantic Ocean. It is the busiest shipping area in the world, it is about 560 km long and varies in width from 240 km at its widest to 33.3 km in the Strait of Dover. It is the smallest of the shallow seas around the continental shelf of Europe, covering an area of some 75,000 km2; until the 18th century, the English Channel had no fixed name either in French. It was never defined as a political border, the names were more or less descriptive, it was not considered as the property of a nation. Before the development of the modern nations, British scholars often referred to it as "Gaulish" and French scholars as "British" or "English"; the name "English Channel" has been used since the early 18th century originating from the designation Engelse Kanaal in Dutch sea maps from the 16th century onwards. In modern Dutch, however, it is known as Het Kanaal.
It has been known as the "British Channel" or the "British Sea". It was called Oceanus Britannicus by the 2nd-century geographer Ptolemy; the same name is used on an Italian map of about 1450, which gives the alternative name of canalites Anglie—possibly the first recorded use of the "Channel" designation. The Anglo-Saxon texts call it Sūð-sǣ as opposed to Norð-sǣ; the common word channel was first recorded in Middle English in the 13th century and was borrowed from Old French chanel, variant form of chenel "canal". The French name la Manche has been in use since at least the 17th century; the name is said to refer to the Channel's sleeve shape. Folk etymology has derived it from a Celtic word meaning channel, the source of the name for the Minch in Scotland, but this name was never mentioned before the 17th century, French and British sources of that time are clear about its etymology; the name in Breton means "Breton Sea", its Cornish name means "British Sea". The International Hydrographic Organization defines the limits of the English Channel as follows: The IHO defines the southwestern limit of the North Sea as "a line joining the Walde Lighthouse and Leathercoat Point".
The Walde Lighthouse is 6 km east of Calais, Leathercoat Point is at the north end of St Margaret's Bay, Kent. The Strait of Dover, at the Channel's eastern end, is its narrowest point, while its widest point lies between Lyme Bay and the Gulf of Saint Malo, near its midpoint, it is shallow, with an average depth of about 120 m at its widest part, reducing to a depth of about 45 m between Dover and Calais. Eastwards from there the adjoining North Sea reduces to about 26 m in the Broad Fourteens where it lies over the watershed of the former land bridge between East Anglia and the Low Countries, it reaches a maximum depth of 180 m in the submerged valley of Hurd's Deep, 48 km west-northwest of Guernsey. The eastern region along the French coast between Cherbourg and the mouth of the Seine river at Le Havre is referred to as the Bay of the Seine. There are several major islands in the Channel, the most notable being the Isle of Wight off the English coast, the Channel Islands, British Crown dependencies off the coast of France.
The coastline on the French shore, is indented. The Cotentin Peninsula in France juts out into the Channel, whilst on the English side there is a small parallel strait known as the Solent between the Isle of Wight and the mainland; the Celtic Sea is to the west of the Channel. The Channel acts as a funnel that amplifies the tidal range from less than a metre as observed at sea to more than 6 metres as observed in the Channel Islands, the west coast of the Cotentin Peninsula and the north coast of Brittany; the time difference of about six hours between high water at the eastern and western limits of the Channel is indicative of the tidal range being amplified further by resonance. In the UK Shipping Forecast the Channel is divided into the following areas, from the east: Dover Wight Portland Plymouth The Channel is of geologically recent origin, having been dry land for most of the Pleistocene period. Before the Devensian glaciation and Ireland were part of continental Europe, linked by an unbroken Weald-Artois Anticline, a ridge that acted as a natural dam holding back a large freshwater pro-glacial lake in the Doggerland region, now submerged under the North Sea.
During this period the North Sea and all of the British Isles were covered by ice. The lake was fed by meltwater from the Baltic and from the Caledonian and Scandinavian ice sheets that joined to the north, blocking its exit; the sea level was about 120 m lower. Between 450,000 and 180,000 years ago, at least two catastrophic glacial lake outburst floods breached the Weald–Artois anticline; the first flood would have lasted for several months, releasing as much as one million cubic metres of water per second. The flood started with large but localized waterfalls over the ridge, which excav
The Rhine is one of the major European rivers, which has its sources in Switzerland and flows in an northerly direction through Germany and The Netherlands to the North Sea. The river begins in the Swiss canton of Graubünden in the southeastern Swiss Alps, forms part of the Swiss-Liechtenstein, Swiss-Austrian, Swiss-German and the Franco-German border flows through the German Rhineland and the Netherlands and empties into the North Sea; the largest city on the Rhine is Cologne, with a population of more than 1,050,000 people. It is the second-longest river in Central and Western Europe, at about 1,230 km, with an average discharge of about 2,900 m3/s; the Rhine and the Danube formed most of the northern inland frontier of the Roman Empire and, since those days, the Rhine has been a vital and navigable waterway carrying trade and goods deep inland. Its importance as a waterway in the Holy Roman Empire is supported by the many castles and fortifications built along it. In the modern era, it has become a symbol of German nationalism.
Among the biggest and most important cities on the Rhine are Cologne, Düsseldorf, Rotterdam and Basel. The variants of the name of the Rhine in modern languages are all derived from the Gaulish name Rēnos, adapted in Roman-era geography as Greek Ῥῆνος, Latin Rhenus; the spelling with Rh- in English Rhine as well as in German Rhein and French Rhin is due to the influence of Greek orthography, while the vocalisation -i- is due to the Proto-Germanic adoption of the Gaulish name as *Rīnaz, via Old Frankish giving Old English Rín,Old High German Rīn, early Middle Dutch Rijn. The diphthong in modern German Rhein is a Central German development of the early modern period, the Alemannic name Rī retaining the older vocalism, as does Ripuarian Rhing, while Palatine has diphthongized Rhei, Rhoi. Spanish is with French in adopting the Germanic vocalism Rin-, while Italian and Portuguese retain the Latin Ren-; the Gaulish name Rēnos belongs to a class of river names built from the PIE root *rei- "to move, run" found in other names such as the Reno in Italy.
The grammatical gender of the Celtic name is masculine, the name remains masculine in German and French. The Old English river name was variously inflected as feminine; the length of the Rhine is conventionally measured in "Rhine-kilometers", a scale introduced in 1939 which runs from the Old Rhine Bridge at Constance to Hoek van Holland. The river is shortened from its natural course due to a number of canalisation projects completed in the 19th and 20th century; the "total length of the Rhine", to the inclusion of Lake Constance and the Alpine Rhine is more difficult to measure objectively. Its course is conventionally divided as follows: The Rhine carries its name without distinctive accessories only from the confluence of the Rein Anteriur/Vorderrhein and Rein Posteriur/Hinterrhein next to Reichenau in Tamins. Above this point is the extensive catchment of the headwaters of the Rhine, it belongs exclusively to the Swiss canton of Graubünden, ranging from Saint-Gotthard Massif in the west via one valley lying in Ticino and Italy in the south to the Flüela Pass in the east.
Traditionally, Lake Toma near the Oberalp Pass in the Gotthard region is seen as the source of the Anterior Rhine and the Rhine as a whole. The Posterior Rhine rises in the Rheinwald below the Rheinwaldhorn; the source of the river is considered north of Lai da Tuma/Tomasee on Rein Anteriur/Vorderrhein, although its southern tributary Rein da Medel is longer before its confluence with the Anterior Rhine near Disentis. The Anterior Rhine springs from Lai da Tuma/Tomasee, near the Oberalp Pass and passes the impressive Ruinaulta formed by the largest visible rock slide in the alps, the Flims Rockslide; the Posterior Rhine starts near the Rheinwaldhorn. One of its tributaries, the Reno di Lei, drains the Valle di Lei on politically Italian territory. After three main valleys separated by the two gorges and Viamala, it reaches Reichenau in Tamins; the Anterior Rhine arises from numerous source streams in the upper Surselva and flows in an easterly direction. One source is Lai da Tuma with the Rein da Tuma, indicated as source of the Rhine, flowing through it.
Into it flow tributaries from the south, some longer, some equal in length, such as the Rein da Medel, the Rein da Maighels, the Rein da Curnera. The Cadlimo Valley in the canton of Ticino is drained by the Reno di Medel, which crosses the geomorphologic Alpine main ridge from the south. All streams in the source area are sometimes captured and sent to storage reservoirs for the local hydro-electric power plants; the culminating point of the Anterior Rhine's drainage basin is the Piz Russein of the Tödi massif of the Glarus Alps at 3,613 metres above sea level. It starts with the creek Aua da Russein. In its lower course the Anterior Rhine flows through a gorge named Ruinaulta; the whole stretch of the Anterior Rhine to the Alpine Rhine confluence next to Reichen
The French are an ethnic group and nation who are identified with the country of France. This connection may be ethnic, historical, or cultural; the heritage of the French people is of Celtic and Germanic origin, descending from the ancient and medieval populations of Gauls, Ligures, Iberians, Franks and Norsemen. France has long been a patchwork of local customs and regional differences, while most French people still speak the French language as their mother tongue, languages like Norman, Catalan, Corsican, French Flemish, Lorraine Franconian and Breton remain spoken in their respective regions. Arabic is widely spoken, arguably the largest minority language in France as of the 21st century. Modern French society is a melting pot. From the middle of the 19th century, it experienced a high rate of inward migration consisting of Arab-Berbers, Sub-Saharan Africans and other peoples from Africa, the Middle East and East Asia, the government, defining France as an inclusive nation with universal values, advocated assimilation through which immigrants were expected to adhere to French values and cultural norms.
Nowadays, while the government has let newcomers retain their distinctive cultures since the mid-1980s and requires from them a mere integration, French citizens still equate their nationality with citizenship as does French law. In addition to mainland France, French people and people of French descent can be found internationally, in overseas departments and territories of France such as the French West Indies, in foreign countries with significant French-speaking population groups or not, such as Switzerland, the United States, Argentina, Brazil and Uruguay. To be French, according to the first article of the French Constitution, is to be a citizen of France, regardless of one's origin, race, or religion. According to its principles, France has devoted itself to the destiny of a proposition nation, a generic territory where people are bounded only by the French language and the assumed willingness to live together, as defined by Ernest Renan's "plébiscite de tous les jours" on the willingness to live together, in Renan's 1882 essay "Qu'est-ce qu'une nation?").
The debate concerning the integration of this view with the principles underlying the European Community remains open. A large number of foreigners have traditionally been permitted to live in France and succeeded in doing so. Indeed, the country has long valued its openness and the quality of services available. Application for French citizenship is interpreted as a renunciation of previous state allegiance unless a dual citizenship agreement exists between the two countries; the European treaties have formally permitted movement and European citizens enjoy formal rights to employment in the state sector. Seeing itself as an inclusive nation with universal values, France has always valued and advocated assimilation. However, the success of such assimilation has been called into question. There is increasing dissatisfaction with, within, growing ethno-cultural enclaves; the 2005 French riots in some troubled and impoverished suburbs were an example of such tensions. However they should not be interpreted as ethnic conflicts but as social conflicts born out of socioeconomic problems endangering proper integration.
French people are the descendants of Gauls and Romans, western European Celtic and Italic peoples, as well as Bretons, Aquitanians and Germanic people arriving at the beginning of the Frankish Empire such as the Franks, the Visigoths, the Suebi, the Saxons, the Allemanni and the Burgundians, Germanic groups such as the Vikings, who settled in Normandy and to a lesser extent in Brittany in the 9th century. The name "France" etymologically derives from the territory of the Franks; the Franks were a Germanic tribe. In the pre-Roman era, all of Gaul was inhabited by a variety of peoples who were known collectively as the Gaulish tribes, their ancestors were Celts who came from Central Europe in the 7th century BCE, non-Celtic peoples including the Ligures, Aquitanians in Aquitaine. Some in the northern and eastern areas, may have had Germanic admixture. Gaul was militarily conquered in 58–51 BCE by the Roman legions under the command of General Julius Caesar. Over the next six centuries, the two cultures intermingled, creating a hybridized Gallo-Roman culture.
In the late Roman era, in addition to colonists from elsewhere in the Empire and Gaulish natives, Gallia became home to some in-migrating populations of Germanic and Scythian origin, such as Alans. The Gaulish language is thought to have survived into the 6th century in France, despite considerable Romanizat
France the French Republic, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, from the Rhine to the Atlantic Ocean, it is bordered by Belgium and Germany to the northeast and Italy to the east, Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic and Indian oceans; the country's 18 integral regions span a combined area of 643,801 square kilometres and a total population of 67.3 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Toulouse, Bordeaux and Nice. During the Iron Age, what is now metropolitan France was inhabited by a Celtic people. Rome annexed the area in 51 BC, holding it until the arrival of Germanic Franks in 476, who formed the Kingdom of Francia.
The Treaty of Verdun of 843 partitioned Francia into Middle Francia and West Francia. West Francia which became the Kingdom of France in 987 emerged as a major European power in the Late Middle Ages following its victory in the Hundred Years' War. During the Renaissance, French culture flourished and a global colonial empire was established, which by the 20th century would become the second largest in the world; the 16th century was dominated by religious civil wars between Protestants. France became Europe's dominant cultural and military power in the 17th century under Louis XIV. In the late 18th century, the French Revolution overthrew the absolute monarchy, established one of modern history's earliest republics, saw the drafting of the Declaration of the Rights of Man and of the Citizen, which expresses the nation's ideals to this day. In the 19th century, Napoleon established the First French Empire, his subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a tumultuous succession of governments culminating with the establishment of the French Third Republic in 1870.
France was a major participant in World War I, from which it emerged victorious, was one of the Allies in World War II, but came under occupation by the Axis powers in 1940. Following liberation in 1944, a Fourth Republic was established and dissolved in the course of the Algerian War; the Fifth Republic, led by Charles de Gaulle, remains today. Algeria and nearly all the other colonies became independent in the 1960s and retained close economic and military connections with France. France has long been a global centre of art and philosophy, it hosts the world's fourth-largest number of UNESCO World Heritage Sites and is the leading tourist destination, receiving around 83 million foreign visitors annually. France is a developed country with the world's sixth-largest economy by nominal GDP, tenth-largest by purchasing power parity. In terms of aggregate household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, human development.
France is considered a great power in global affairs, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a leading member state of the European Union and the Eurozone, a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, La Francophonie. Applied to the whole Frankish Empire, the name "France" comes from the Latin "Francia", or "country of the Franks". Modern France is still named today "Francia" in Italian and Spanish, "Frankreich" in German and "Frankrijk" in Dutch, all of which have more or less the same historical meaning. There are various theories as to the origin of the name Frank. Following the precedents of Edward Gibbon and Jacob Grimm, the name of the Franks has been linked with the word frank in English, it has been suggested that the meaning of "free" was adopted because, after the conquest of Gaul, only Franks were free of taxation.
Another theory is that it is derived from the Proto-Germanic word frankon, which translates as javelin or lance as the throwing axe of the Franks was known as a francisca. However, it has been determined that these weapons were named because of their use by the Franks, not the other way around; the oldest traces of human life in what is now France date from 1.8 million years ago. Over the ensuing millennia, Humans were confronted by a harsh and variable climate, marked by several glacial eras. Early hominids led a nomadic hunter-gatherer life. France has a large number of decorated caves from the upper Palaeolithic era, including one of the most famous and best preserved, Lascaux. At the end of the last glacial period, the climate became milder. After strong demographic and agricultural development between the 4th and 3rd millennia, metallurgy appeared at the end of the 3rd millennium working gold and bronze, iron. France has numerous megalithic sites from the Neolithic period, including the exceptiona
Etymology is the study of the history of words. By extension, the term "the etymology" means the origin of the particular word and for place names, there is a specific term, toponymy. For Greek—with a long written history—etymologists make use of texts, texts about the language, to gather knowledge about how words were used during earlier periods and when they entered the language. Etymologists apply the methods of comparative linguistics to reconstruct information about languages that are too old for any direct information to be available. By analyzing related languages with a technique known as the comparative method, linguists can make inferences about their shared parent language and its vocabulary. In this way, word roots have been found that can be traced all the way back to the origin of, for instance, the Indo-European language family. Though etymological research grew from the philological tradition, much current etymological research is done on language families where little or no early documentation is available, such as Uralic and Austronesian.
The word etymology derives from the Greek word ἐτυμολογία, itself from ἔτυμον, meaning "true sense", the suffix -logia, denoting "the study of". In linguistics, the term etymon refers to a word or morpheme from which a word derives. For example, the Latin word candidus, which means "white", is the etymon of English candid. Etymologists apply a number of methods to study the origins of words, some of which are: Philological research. Changes in the form and meaning of the word can be traced with the aid of older texts, if such are available. Making use of dialectological data; the form or meaning of the word might show variations between dialects, which may yield clues about its earlier history. The comparative method. By a systematic comparison of related languages, etymologists may be able to detect which words derive from their common ancestor language and which were instead borrowed from another language; the study of semantic change. Etymologists must make hypotheses about changes in the meaning of particular words.
Such hypotheses are tested against the general knowledge of semantic shifts. For example, the assumption of a particular change of meaning may be substantiated by showing that the same type of change has occurred in other languages as well. Etymological theory recognizes that words originate through a limited number of basic mechanisms, the most important of which are language change, borrowing. While the origin of newly emerged words is more or less transparent, it tends to become obscured through time due to sound change or semantic change. Due to sound change, it is not obvious that the English word set is related to the word sit, it is less obvious that bless is related to blood. Semantic change may occur. For example, the English word bead meant "prayer", it acquired its modern meaning through the practice of counting the recitation of prayers by using beads. English derives from Old English, a West Germanic variety, although its current vocabulary includes words from many languages; the Old English roots may be seen in the similarity of numbers in English and German seven/sieben, eight/acht, nine/neun, ten/zehn.
Pronouns are cognate: I/mine/me and ich/mein/mich. However, language change has eroded many grammatical elements, such as the noun case system, simplified in modern English, certain elements of vocabulary, some of which are borrowed from French. Although many of the words in the English lexicon come from Romance languages, most of the common words used in English are of Germanic origin; when the Normans conquered England in 1066, they brought their Norman language with them. During the Anglo-Norman period, which united insular and continental territories, the ruling class spoke Anglo-Norman, while the peasants spoke the vernacular English of the time. Anglo-Norman was the conduit for the introduction of French into England, aided by the circulation of Langue d'oïl literature from France; this led to many paired words of English origin. For example, beef is related, through borrowing, to modern French bœuf, veal to veau, pork to porc, poultry to poulet. All these words and English, refer to the meat rather than to the animal.
Words that refer to farm animals, on the other hand, tend to be cognates of words in other Germanic languages. For example, swine/Schwein, cow/Kuh, calf/Kalb, sheep/Schaf; the variant usage has been explained by the proposition that it was the Norman rulers who ate meat and the Anglo-Saxons who farmed the animals. This explanation has been disputed. English has proved accommodating to words from many languages. Scientific terminology, for example, relies on words of Latin and Greek origin, but there are a great many non-scientific examples. Spanish has contributed many words in the southwestern United States. Examples include buckaroo, rodeo and states' names such as Colorado and Florida. Albino, lingo and coconut from Portuguese. Modern French has contributed café, naive and many more. Smorgasbord, slalom