In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with internal angles of the same size, it has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a truncated hexagon, t, or a twice-truncated triangle, tt; the internal angle at each vertex of a regular dodecagon is 150°. The area of a regular dodecagon of side length a is given by: A = 3 cot a 2 = 3 a 2 ≃ 11.19615242 a 2 And in terms of the apothem r, the area is: A = 12 tan r 2 = 12 r 2 ≃ 3.2153903 r 2 In terms of the circumradius R, the area is: A = 6 sin R 2 = 3 R 2 The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area is: A = 3 a S This can be verified with the trigonometric relationship: S = a The perimeter of a regular dodecagon in terms of circumradius is: p = 24 R tan = 12 R 2 − 3 ≃ 6.21165708246 R The perimeter in terms of apothem is: p = 24 r tan = 24 r ≃ 6.43078061835 r This coefficient is double the coefficient found in the apothem equation for area.
As 12 = 22 × 3, regular dodecagon is constructible using compass-and-straightedge construction: Coxeter states that every zonogon can be dissected into m/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular dodecagon, m=6, it can be divided into 15: 3 squares, 6 wide 30° rhombs and 6 narrow 15° rhombs; this decomposition is based with 15 of 240 faces. The sequence OEIS sequence A006245 defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, they are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles. The regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup cyclic symmetries; each subgroup symmetry allows one or more degrees of freedom for irregular forms.
Only the g12 subgroup has no degrees of freedom. A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways: Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration: A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane; the interior of such an dodecagon is not defined. A skew zig-zag. A regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D5d, order
Cranbrook is a small town in the civil parish of Cranbrook and Sissinghurst, in the Weald of Kent in South East England. It lies half-way between Maidstone and Hastings, about 38 miles southeast of central London; the smaller settlements of Sissinghurst, Colliers Green and Hartley lie within the civil parish. The population of the parish was 6,717 in 2011; the place name Cranbrook derives from Old English cran broc, meaning Crane Marsh, marshy ground frequented by cranes. Spelling of the place name has evolved over the centuries from Cranebroca. By 1610 the name had become Cranbrooke. Edward III brought over Flemish weavers to develop the Wealden cloth industry using wool from Romney Marsh. Iron-making was carried on at Bedgebury on the River Teise, an industry which dates back to Roman times; the tributaries of the River Beult around Cranbrook powered 17 watermills at one time. In 1290 the town received a charter from Archbishop Peckham, allowing it to hold a market in the High Street. Baker's Cross on the eastern edge of the town is linked to John Baker, Chancellor of the Exchequer under Queen Mary, a Catholic.
Legend holds that he was riding on his way to Cranbrook in order to have two local Protestants executed, when he turned back after the news reached him that Queen Mary was dead. Different versions of the legend have it that he heard the parish church bells ringing, or that he was met by a messenger; the place where this happened was, in the words of biographer and historian Arthur Irwin Dasent, "at a place where three roads meet, known to this day as Baker's Cross". Popular legend has it that Baker was killed at Baker's Cross; the town developed around the "King's High Road" until the Second World War. Following the war, additional housing was built adjacent to the historic centre - the Wheatfield Estate to the north and the Frythe Estate to the south. In the 1970s a Conservation Area was designated in the town centre. In 1974 Cranbrook Rural District was merged into the Borough of Tunbridge Wells. In 2010 Francis Rook of the Liberal Democrats won one of the three council seats in the Benenden and Cranbrook ward from the Conservatives to become one of only 6 non-Conservative councillors out of 48 in the borough.
The name of the parish council was changed from Cranbrook Parish Council to Cranbrook and Sissinghurst Parish Council in 2009. The parish council is based in the Old Fire Station on Stone Street. Located on the Maidstone to Hastings road, it is five miles north of Hawkhurst. Baker's Cross is on the eastern outskirts of the town. Cranbrook is on the Hastings Beds, alternating sands and clays which are more resistant to erosion than the surrounding clays and so form the hills of the High Weald; the geology of the area has played a major role in the town's development, deposits of iron ore and fuller's earth were important in the iron industry and cloth industry respectively. At the 2011 census, Cranbrook had 6,717 residents; the Kent Structure Plan calls it the smallest town in Kent, although Fordwich has a town council and just 381 residents. Since the decline of the cloth trade, agriculture became the mainstay of the economy; the first bank was opened in Cranbrook in 1803 by Samuel Waddington. It closed in 1805.
In 1804, the Cranbrook Bank was opened. It changed its name to the Weald of Kent Bank in 1812 and to Bishop & Co's Bank in 1813 before being declared bankrupt in October 1814; the Tooth family of Great Swifts, near Cranbrook, established a brewery at Baker's Cross. A large part of their trade was the export of beer to Australia. Subsequently, John Tooth emigrated to Australia in the early 1830s, traded for a time as a general merchant, in 1835, with his brother-in-law, John Newnham, opened a brewery in Sydney, he named the brewery Kent Brewery, which continued to 1985. Meanwhile, the brewery at Cranbrook had been sold to one William Barling Sharpe, whose daughter had married the local estate agent, William Winch; the brewery Sharpe & Winch was established in Baker's Cross at some point prior to 1846 by William Barling Sharpe. The brewery assumed the name Sharpe & Winch in 1892, was purchased and taken over by Frederick Leney & Sons Ltd, a Wateringbury company, in 1927; the brewery were responsible for the mock-Tutor extension to the 18th century Baker's Cross House.
During the 19th century, a group of artists known as the "Cranbrook Colony" were located here. The Colony artists tended to paint scenes of domestic life in rural Kent – cooking and washing, children playing, other family activities. Queen's Hall Theatre, part of Cranbrook School, sponsors many theatre groups, including the Cambridge Footlights and Cranbrook Opera and Dramatic Society; the Showtimers pantomime group produces an annual show. Cranbrook Town Band, founded in the 1920s, is a British-style brass band, which performs regular concerts in the Queen's Hall, St Dunstan's Church and around Kent. There have been many plans to create a community hub, starting with a proposal to convert the old council offices; the focus switched to a £2m building planned on Wilkes Field, next to the Co-op carpark. As of 2013 plans included small community rooms and three large day rooms which could convert into a hall for 300 people, along with a day care centre, council offices, public toilets and
Amsterdam is the capital city and most populous municipality of the Netherlands. Its status as the capital is mandated by the Constitution of the Netherlands, although it is not the seat of the government, The Hague. Amsterdam has a population of 854,047 within the city proper, 1,357,675 in the urban area and 2,410,960 in the metropolitan area; the city is located in the province of North Holland in the west of the country but is not its capital, Haarlem. The Amsterdam metropolitan area comprises much of the northern part of the Randstad, one of the larger conurbations in Europe, which has a population of 8.1 million. Amsterdam's name derives from Amstelredamme, indicative of the city's origin around a dam in the river Amstel. Originating as a small fishing village in the late 12th century, Amsterdam became one of the most important ports in the world during the Dutch Golden Age, as a result of its innovative developments in trade. During that time, the city was the leading centre for trade. In the 19th and 20th centuries the city expanded, many new neighbourhoods and suburbs were planned and built.
The 17th-century canals of Amsterdam and the 19–20th century Defence Line of Amsterdam are on the UNESCO World Heritage List. Since the annexation of the municipality of Sloten in 1921 by the municipality of Amsterdam, the oldest historic part of the city lies in Sloten, dating to the 9th century; as the commercial capital of the Netherlands and one of the top financial centres in Europe, Amsterdam is considered an alpha- world city by the Globalization and World Cities study group. The city is the cultural capital of the Netherlands. Many large Dutch institutions have their headquarters there, including Philips, AkzoNobel, TomTom and ING. Many of the world's largest companies are based in Amsterdam or established their European headquarters in the city, such as leading technology companies Uber and Tesla. In 2012, Amsterdam was ranked the second best city to live in by the Economist Intelligence Unit and 12th globally on quality of living for environment and infrastructure by Mercer; the city was ranked 4th place globally as top tech hub in the Savills Tech Cities 2019 report, 3rd in innovation by Australian innovation agency 2thinknow in their Innovation Cities Index 2009.
The Port of Amsterdam to this day remains the second in the country, the fifth largest seaport in Europe. Famous Amsterdam residents include the diarist Anne Frank, artists Rembrandt van Rijn and Vincent van Gogh, philosopher Baruch Spinoza; the Amsterdam Stock Exchange, the oldest stock exchange in the world, is located in the city centre. Amsterdam's main attractions include its historic canals, the Rijksmuseum, the Van Gogh Museum, the Stedelijk Museum, Hermitage Amsterdam, the Anne Frank House, the Scheepvaartmuseum, the Amsterdam Museum, the Heineken Experience, the Royal Palace of Amsterdam, Natura Artis Magistra, Hortus Botanicus Amsterdam, NEMO, the red-light district and many cannabis coffee shops, they draw more than 5 million international visitors annually. The city is well known for its nightlife and festival activity, it is one of the world's most multicultural cities, with at least 177 nationalities represented. After the floods of 1170 and 1173, locals near the river Amstel built a bridge over the river and a dam across it, giving its name to the village: "Aemstelredamme".
The earliest recorded use of that name is in a document dated 27 October 1275, which exempted inhabitants of the village from paying bridge tolls to Count Floris V. This allowed the inhabitants of the village of Aemstelredamme to travel through the County of Holland, paying no tolls at bridges and dams; the certificate describes the inhabitants. By 1327, the name had developed into Aemsterdam. Amsterdam is much younger than Dutch cities such as Nijmegen and Utrecht. In October 2008, historical geographer Chris de Bont suggested that the land around Amsterdam was being reclaimed as early as the late 10th century; this does not mean that there was a settlement since reclamation of land may not have been for farming—it may have been for peat, for use as fuel. Amsterdam was granted city rights in either 1300 or 1306. From the 14th century on, Amsterdam flourished from trade with the Hanseatic League. In 1345, an alleged Eucharistic miracle in the Kalverstraat rendered the city an important place of pilgrimage until the adoption of the Protestant faith.
The Miracle devotion was kept alive. In the 19th century after the jubilee of 1845, the devotion was revitalized and became an important national point of reference for Dutch Catholics; the Stille Omgang—a silent walk or procession in civil attire—is the expression of the pilgrimage within the Protestant Netherlands since the late 19th century. In the heyday of the Silent Walk, up to 90,000 pilgrims came to Amsterdam. In the 21st century this has reduced to about 5000. In the 16th century, the Dutch rebelled against Philip II of his successors; the main reasons for the uprising were the imposition of new taxes, the tenth penny, the religious persecution of Protestants by the newly introduced Inquisition. The revolt escalated into the Eighty Years' War, which led to Dutch independence. Pushed by Dutch Revolt leader William the Silent, the Dutch Republic became known for its relative religious tolerance. Jews from the Iberian Peninsula, Huguenots from France, prosperous merchants and printers from Flanders, economic and religious refugees
Kent is a county in South East England and one of the home counties. It borders Surrey to the west and East Sussex to the south-west; the county shares borders with Essex along the estuary of the River Thames, with the French department of Pas-de-Calais through the Channel Tunnel. The county town is Maidstone. Canterbury Cathedral in Kent has been the seat of the Archbishop of Canterbury, leader of the Church of England, since the Reformation. Prior to that it was built by Catholics, dating back to the conversion of England to Catholicism by Saint Augustine that began in the 6th century. Before the English Reformation the cathedral was part of a Benedictine monastic community known as Christ Church, Canterbury, as well as being the seat of the Catholic Archbishop of Canterbury; the last Catholic Archbishop of Canterbury was Reginald Pole. Rochester Cathedral is in Kent, in Medway, it is the second-oldest cathedral in England, with Canterbury Cathedral being the oldest. Between London and the Strait of Dover, which separates it from mainland Europe, Kent has seen both diplomacy and conflict, ranging from the Leeds Castle peace talks of 1978 and 2004 to the Battle of Britain in World War II.
England relied on the county's ports to provide warships through much of its history. France can be seen in fine weather from Folkestone and the White Cliffs of Dover. Hills in the form of the North Downs and the Greensand Ridge span the length of the county and in the series of valleys in between and to the south are most of the county's 26 castles; because of its relative abundance of fruit-growing and hop gardens, Kent is known as "The Garden of England". Kent's economy is diversified. In northwest Kent industries include extraction of aggregate building materials and scientific research. Coal mining has played its part in Kent's industrial heritage. Large parts of Kent are within the London commuter belt and its strong transport connections to the capital and the nearby continent makes Kent a high-income county. Twenty-eight per cent of the county forms part of two Areas of Outstanding Natural Beauty: the North Downs and The High Weald; the name Kent is believed to be of British Celtic origin and was known in Old English as Cent, Cent lond, Centrice.
In Latin sources Kent is mentioned as Canticum. The meaning is explained by some researchers as "coastal district," or "corner-land, land on the edge". If so, the name could be etymologically related to the placename Cantabria a Celtiberian-speaking coastal region in pre-Roman Iberia, today a province of Spain; the area has been occupied since the Palaeolithic era, as attested by finds from the quarries at Swanscombe. The Medway megaliths were built during the Neolithic era. There is a rich sequence of Bronze Age, Iron Age, Roman era occupation, as indicated by finds and features such as the Ringlemere gold cup and the Roman villas of the Darent valley; the modern name of Kent is derived from the Brythonic word kantos meaning "rim" or "border", or from a homonymous word kanto "horn, hook". This describes the eastern part of the current county area as coastal district. Julius Caesar had described the area as um, or home of the Cantiaci in 51 BC; the extreme west of the modern county was by the time of Roman Britain occupied by Iron Age tribes, known as the Regnenses.
Caesar wrote that the people of Kent are'by far the most civilised inhabitants of Britain'. East Kent became a kingdom of the Jutes during the 5th century and was known as Cantia from about 730 and recorded as Cent in 835; the early medieval inhabitants of the county were known as the Kent people. These people regarded the city of Canterbury as their capital. In 597, Pope Gregory I appointed the religious missionary as the first Archbishop of Canterbury. In the previous year, Augustine converted the pagan King Æthelberht of Kent to Christianity; the Diocese of Canterbury became England's first Episcopal See with first cathedral and has since remained England's centre of Christianity. The second designated English cathedral was in Kent at Rochester Cathedral. In the 11th century, the people of Kent adopted the motto Invicta, meaning "undefeated" or "unconquered"; this naming followed the invasion of Britain by William of Normandy. The Kent people's continued resistance against the Normans led to Kent's designation as a semi-autonomous county palatine in 1067.
Under the nominal rule of William's half-brother Odo of Bayeux, the county was granted similar powers to those granted in the areas bordering Wales and Scotland. Kent was traditionally partitioned into East and West Kent, into lathes and hundreds; the traditional border of East and West Kent was the Medway. Men and women from east of the Medway are Men of Kent, those from the west are Kentishmen or Kentish Maids. During the medieval and early modern period, Kent played a major role in several of England's most notable rebellions, including the Peasants' Revolt of 1381, led by Wat Tyler,Jack Cade's Kent rebellion of 1450, Wyatt's Rebellion of 1554 against Queen Mary I; the Royal Navy first used the River Medway in 1547. By the reign of Elizabeth I a small dockyard had been established at Chatham. By 1618, storehouses, a ropewalk, a drydock, houses for officials had
A tower mill is a type of vertical windmill consisting of a brick or stone tower, on which sits a wooden'cap' or roof, which can rotate to bring the sails into the wind. This rotating cap on a firm masonry base gave tower mills great advantages over earlier post mills, as they could stand much higher, bear larger sails, thus afford greater reach into the wind. Windmills in general had been known to civilization for centuries, but the tower mill represented an improvement on traditional western-style windmills; the tower mill was an important source of power for Europe for nearly 600 years from 1300–1900, contributing to 25 percent of the industrial power of all wind machines before the advent of the steam engine and coal power. It represented a modification or a demonstration of improving and adapting technology, known by humans for ages. Although these types of mills were effective, some argue that, owing to their complexity, they would have been built by the most wealthy individuals; the tower mill originated in written history in the late 13th century in western Europe.
Other early examples come from Buckinghamshire. Other sources pin its earliest inception back in 1180 in the form of an illustration on a Norman deed, showing this new western-style windmill; the Netherlands has six mills recorded before the year 1407. One of the earliest tower mills in Britain was Chesterton Mill, which has a hollowed conical base with arches; the large part of its development continued through the Late Middle Ages, towards the end of the 15th century tower mills began appearing across Europe in greater numbers. The origins of the tower mill can be found in a growing economy of Europe, which needed a more reliable and efficient form of power one that could be used away from a river bank; the spread of tower mills came with a growing economy that called for larger and more stable sources of power. Post mills dominated the scene in Europe until the 19th century when tower mills began to replace them in such places as Billingford Mill in Norfolk, Upper Hellesden Mill in Norwich, Stretham Mill in Cambridgeshire.
The tower mill was seen as a cultural object, being painted and designed with aesthetic appeal in mind. Styles of the mills reflected on local tradition and weather conditions, for example mills built on the western coast of Britain were built of stone to withstand the stronger winds, those built in the east were of brick. In England around 12 eight-sailers, more than 50 six- and 50 five-sailers were built in the late 18th century and 19th century, half of them in Lincolnshire. Of the eight sailed mills only Pocklington's Mill in Heckington survived in functional state. A few of the other ones exist as four-sailed mills, as residences, as ruins, or have been dismantled. In Lincolnshire some of the six-sailed and five-sailed slender tower mills with their white onion-shaped cap and a huge fantail are still there and working today. Other former five- and six-sailed Lincolnshire and Yorkshire tower mills now without sails and without cap are LeTall's Mill in Lincoln, Holgate Windmill in Holgate, Black, Cliff, or Whiting's Mill in Hessle and Barton-upon-Humber Tower mill, Brunswick Mill in Long Sutton, Metheringham Windmill, Penny Hill Windmill in Holbeach, Wragby Mill, Wellingore Tower Mill.
Another fine six-sailer can be found in Derbyshire – England's only sandstone towered windmill at Heage of 1791. The advantage of the tower mill over the earlier post mill is that it is not necessary to turn the whole mill with all its machinery into the wind. However, select tower mills found around Holland were constructed on a wooden frame so as to rotate the entire foundation of the mill along with the cap; these towers were constructed out of wood rather than masonry as well. A movable head which could pivot to react to the changing wind patterns was the most important aspect of the tower mill; this ability gave the advantage of a larger and more stable frame that could deal with harsh weather. Only moving a cap was much easier than moving an entire structure. In the earliest tower mills the cap was turned into the wind with a long tail-pole which stretched down to the ground at the back of the mill. An endless chain was used which drove the cap through gearing. In 1745 an English engineer, Edmund Lee, invented the windmill fantail – a little windmill mounted at right angles to the sails, at the rear of the mill, which turned the cap automatically to bring it into the wind.
Like other windmills tower mills have four blades. To increase windmill efficiency millwrights experimented with different methods: automated patent-sails instead of cloth spread type sails didn't need the sail cross to be stopped to spread or remove the cloth sails because they altered the surface from inside the mill by means of a controlling gear. More than four blades to increase the sail surface. Therefore, engineer John Smeaton invented the cast-iron Lincolnshire cross to make sail-crosses with five and eight blades possible; the cross was named after Lincolnshire where it was most used. There are several components to the tower mil
Massachusetts the Commonwealth of Massachusetts, is the most populous state in the New England region of the northeastern United States. It borders on the Atlantic Ocean to the east, the states of Connecticut and Rhode Island to the south, New Hampshire and Vermont to the north, New York to the west; the state is named after the Massachusett tribe, which once inhabited the east side of the area, is one of the original thirteen states. The capital of Massachusetts is Boston, the most populous city in New England. Over 80% of Massachusetts's population lives in the Greater Boston metropolitan area, a region influential upon American history and industry. Dependent on agriculture and trade, Massachusetts was transformed into a manufacturing center during the Industrial Revolution. During the 20th century, Massachusetts's economy shifted from manufacturing to services. Modern Massachusetts is a global leader in biotechnology, higher education and maritime trade. Plymouth was the site of the second colony in New England after Popham Colony in 1607 in what is now Maine.
Plymouth was founded in 1620 by passengers of the Mayflower. In 1692, the town of Salem and surrounding areas experienced one of America's most infamous cases of mass hysteria, the Salem witch trials. In 1777, General Henry Knox founded the Springfield Armory, which during the Industrial Revolution catalyzed numerous important technological advances, including interchangeable parts. In 1786, Shays' Rebellion, a populist revolt led by disaffected American Revolutionary War veterans, influenced the United States Constitutional Convention. In the 18th century, the Protestant First Great Awakening, which swept the Atlantic World, originated from the pulpit of Northampton preacher Jonathan Edwards. In the late 18th century, Boston became known as the "Cradle of Liberty" for the agitation there that led to the American Revolution; the entire Commonwealth of Massachusetts has played a powerful commercial and cultural role in the history of the United States. Before the American Civil War, Massachusetts was a center for the abolitionist and transcendentalist movements.
In the late 19th century, the sports of basketball and volleyball were invented in the western Massachusetts cities of Springfield and Holyoke, respectively. In 2004, Massachusetts became the first U. S. state to recognize same-sex marriage as a result of the decision in Goodridge v. Department of Public Health by the Massachusetts Supreme Judicial Court. Many prominent American political dynasties have hailed from the state, including the Adams and Kennedy families. Harvard University in Cambridge is the oldest institution of higher learning in the United States, with the largest financial endowment of any university, Harvard Law School has educated a contemporaneous majority of Justices of the Supreme Court of the United States. Kendall Square in Cambridge has been called "the most innovative square mile on the planet", in reference to the high concentration of entrepreneurial start-ups and quality of innovation which have emerged in the vicinity of the square since 2010. Both Harvard University and the Massachusetts Institute of Technology in Cambridge, have been ranked among the most regarded academic institutions in the world.
Massachusetts' public-school students place among the top tier in the world in academic performance, the state has been ranked as one of the top states in the United States for citizens to live in, as well as one of the most expensive. The Massachusetts Bay Colony was named after the indigenous population, the Massachusett derived from a Wôpanâak word muswach8sut, segmented as mus "big" + wach8 "mountain" + -s "diminutive" + -ut "locative", it has been translated as "near the great hill", "by the blue hills", "at the little big hill", or "at the range of hills", referring to the Blue Hills, or in particular the Great Blue Hill, located on the boundary of Milton and Canton. Alternatively, Massachusett has been represented as Moswetuset—from the name of the Moswetuset Hummock in Quincy, where Plymouth Colony commander Myles Standish, hired English military officer, Squanto, part of the now disappeared Patuxet band of the Wampanoag peoples, met Chief Chickatawbut in 1621; the official name of the state is the "Commonwealth of Massachusetts".
While this designation is part of the state's official name, it has no practical implications. Massachusetts has powers within the United States as other states, it may have been chosen by John Adams for the second draft of the Massachusetts Constitution because unlike the word "state", "commonwealth" at the time had the connotation of a republic, in contrast to the monarchy the former American colonies were fighting against. Massachusetts was inhabited by tribes of the Algonquian language family such as the Wampanoag, Nipmuc, Pocomtuc and Massachusett. While cultivation of crops like squash and corn supplemented their diets, these tribes were dependent on hunting and fishing for most of their food. Villages consisted of lodges called wigwams as well as longhouses, tribes were led by male or female elders known as sachems. In the early 1600s, after contact had been made with Europeans, large numbers of the indigenous peoples in the northeast of what is now the United States were killed by virgin soil epidemics such as smallpox, measles and leptospirosis.
Between 1617 and 1619, smallpox killed ap
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