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Copenhagen aims to be carbon-neutral by 2025. Commercial and residential buildings are to reduce electricity consumption by 20 per cent and 10 per cent respectively, and total heat consumption is to fall by 20 per cent by 2025. Renewable energy features such as solar panels are becoming increasingly common in the newest buildings in Copenhagen. District heating will be carbon-neutral by 2025, by waste incineration and biomass. New buildings must now be constructed according to Low Energy Class ratings and in 2020 near net-zero energy buildings. By 2025, 75% of trips should be made on foot, by bike, or by using public transit. The city plans that 20–30% of cars will run on electricity or biofuel by 2025. The investment is estimated at $472 million public funds and $4.78 billion private funds. The city's urban planning authorities continue to take full account of these priorities. Special attention is given both to climate issues and efforts to ensure maximum application of low-energy standards. Priorities include sustainable drainage systems, recycling rainwater, green roofs and efficient waste management solutions. In city planning, streets and squares are to be designed to encourage cycling and walking rather than driving.
Demographics. Copenhagen is the most populous city in Denmark and one of the most populous in the Nordic countries. For statistical purposes, Statistics Denmark considers the City of Copenhagen () to consist of the Municipality of Copenhagen plus three adjacent municipalities: Dragør, Frederiksberg, and Tårnby. Their combined population stands at 763,908 (). The Municipality of Copenhagen is by far the most populous in the country and one of the most populous Nordic municipalities with 644,431 inhabitants (as of 2022). There was a demographic boom in the 1990s and first decades of the 21st century, largely due to immigration to Denmark. According to figures from the first quarter of 2022, 73.7% of the municipality's population was of Danish descent, defined as having at least one parent who was born in Denmark and has Danish citizenship. Much of the remaining 26.3% were of a foreign background, defined as immigrants (20.3%) or descendants of recent immigrants (6%). There are no official statistics on ethnic groups. The adjacent table shows the most common countries of origin of Copenhagen residents. Largest foreign groups are Pakistanis (1.3%), Turks (1.2%), Iraqis (1.1%), Germans (1.0%) and Poles (1.0%).
According to Statistics Denmark, Copenhagen's urban area has a larger population of 1,280,371 (). The urban area consists of the municipalities of Copenhagen and Frederiksberg plus 16 of the 20 municipalities of the former counties Copenhagen and Roskilde, though five of them only partially. Metropolitan Copenhagen has a total of 2,016,285 inhabitants (). The area of Metropolitan Copenhagen is defined by the Finger Plan. Since the opening of the Øresund Bridge in 2000, commuting between Zealand and Scania in Sweden has increased rapidly, leading to a wider, integrated area. Known as the Øresund Region, it has 4.1 million inhabitants—of whom 2.7 million (August 2021) live in the Danish part of the region. Religion. A majority (56.9%) of those living in Copenhagen are members of the Lutheran Church of Denmark which is 0.6% lower than one year earlier according to 2019 figures. The National Cathedral, the Church of Our Lady, is one of the dozens of churches in Copenhagen. There are also several other Christian communities in the city, of which the largest is Roman Catholic.
Foreign migration to Copenhagen, rising over the last three decades, has contributed to increasing religious diversity; the Grand Mosque of Copenhagen, the first in Denmark, opened in 2014. Islam is the second largest religion in Copenhagen, accounting for approximately 10% of the population. While there are no official statistics, a significant portion of the estimated 175,000–200,000 Muslims in the country live in the Copenhagen urban area, with the highest concentration in Nørrebro and the Vestegnen. There are also some 7,000 Jews in Denmark, most of them in the Copenhagen area where there are several synagogues. It has a membership of 1,800 members. There is a long history of Jews in the city, and the first synagogue in Copenhagen was built in 1684. Today, the history of the Jews of Denmark can be explored at the Danish Jewish Museum in Copenhagen. Quality of living. For a number of years, Copenhagen has ranked high in international surveys for its quality of life. Its stable economy together with its education services and level of social safety make it attractive for locals and visitors alike. Although it is one of the world's most expensive cities, it is also one of the most liveable with its public transport, facilities for cyclists and its environmental policies. In elevating Copenhagen to "most liveable city" in 2013, "Monocle" pointed to its open spaces, increasing activity on the streets, city planning in favour of cyclists and pedestrians, and features to encourage inhabitants to enjoy city life with an emphasis on community, culture and cuisine. The city is voted 2024 second most liveable city by Economist Intelligence Unit. Other sources have ranked Copenhagen high for its business environment, accessibility, restaurants and environmental planning. However, Copenhagen ranks only 39th for student friendliness in 2012. Despite a top score for quality of living, its scores were low for employer activity and affordability.
Economy. Copenhagen is the major economic and financial centre of Denmark. The city's economy is based largely on services and commerce. Statistics for 2010 show that the vast majority of the 350,000 workers in Copenhagen are employed in the service sector, especially transport and communications, trade, and finance, while less than 10,000 work in the manufacturing industries. The public sector workforce is around 110,000, including education and healthcare. From 2006 to 2011, the economy grew by 2.5% in Copenhagen, while it fell by some 4% in the rest of Denmark. In 2017, the wider Capital Region of Denmark had a gross domestic product (GDP) of €120 billion, and the 15th largest GDP per capita of regions in the European Union. As of Copenhagen Green Economy Leader Report made by London School of Economics and Political Science – Copenhagen is widely recognised as a leader in the global green economy. The Copenhagen region accounts for almost 40% of Denmark's output and has enjoyed long-term stable growth. At a national level, Danish GDP per capita is ranked among the top 10 countries in the world. At the same time, the city's growth has been delivered while improving environmental performance and transitioning to a low-carbon economy.
Several financial institutions and banks have headquarters in Copenhagen, including Alm. Brand, Danske Bank, Nykredit and Nordea Bank Danmark. The Copenhagen Stock Exchange (CSE) was founded in 1620 and is now owned by Nasdaq, Inc. Copenhagen is also home to a number of international companies including A.P. Møller-Mærsk, Novo Nordisk, Carlsberg and Novozymes. City authorities have encouraged the development of business clusters in several innovative sectors, which include information technology, biotechnology, pharmaceuticals, clean technology and smart city solutions. Life science is a key sector with extensive research and development activities. Medicon Valley is a leading bi-national life sciences cluster in Europe, spanning the Øresund Region. Copenhagen is rich in companies and institutions with a focus on research and development within the field of biotechnology, and the Medicon Valley initiative aims to strengthen this position and to promote cooperation between companies and academia. Many major Danish companies like Novo Nordisk and Lundbeck, both of which are among the 50 largest pharmaceutical and biotech companies in the world, are located in this business cluster.
Shipping is another important sector with Maersk, the world's largest shipping company, having their world headquarters in Copenhagen. The city has an industrial harbour, Copenhagen Port. Following decades of stagnation, it has experienced a resurgence since 1990 following a merger with Malmö harbour. Both ports are operated by Copenhagen Malmö Port (CMP). The central location in the Øresund Region allows the ports to act as a hub for freight that is transported onward to the Baltic countries. CMP annually receives about 8,000 ships and handled some 148,000 TEU in 2012. Copenhagen has some of the highest gross wages in the world. High taxes mean that wages are reduced after mandatory deduction. A "beneficial researcher scheme" with low taxation of foreign specialists has made Denmark an attractive location for foreign labour. It is, however, also among the most expensive cities in Europe. Denmark's Flexicurity model features some of the most flexible hiring and firing legislation in Europe, providing attractive conditions for foreign investment and international companies looking to locate in Copenhagen. In Dansk Industri's 2013 survey of employment factors in the ninety-six municipalities of Denmark, Copenhagen came in first place for educational qualifications and for the development of private companies in recent years, but fell to 86th place in local companies' assessment of the employment climate. The survey revealed considerable dissatisfaction in the level of dialogue companies enjoyed with the municipal authorities.
Tourism. Tourism is a major contributor to Copenhagen's economy, attracting visitors due to the city's harbour, cultural attractions and award-winning restaurants. Since 2009, Copenhagen has been one of the fastest growing metropolitan destinations in Europe. Hotel capacity in the city is growing significantly. From 2009 to 2013, it experienced a 42% growth in international bed nights (total number of nights spent by tourists), tallying a rise of nearly 70% for Chinese visitors. The total number of bed nights in the Capital Region surpassed 9 million in 2013, while international bed nights reached 5 million. In 2010, it is estimated that city break tourism contributed to DKK 2 billion in turnover. However, 2010 was an exceptional year for city break tourism and turnover increased with 29% in that one year. 680,000 cruise passengers visited the port in 2015. In 2019 Copenhagen was ranked first among Lonely Planet's top ten cities to visit. In October 2021, Copenhagen was shortlisted for the European Commission's 2022 European Capital of Smart Tourism award along with Bordeaux, Dublin, Florence, Ljubljana, Palma de Mallorca and Valencia.
Cityscape. The city's appearance today is shaped by the key role it has played as a regional centre for centuries. Copenhagen has a multitude of districts, each with its distinctive character and representing its own period. Other distinctive features of Copenhagen include the abundance of water, its many parks, and the bicycle paths that line most streets. Architecture. The oldest section of Copenhagen's inner city is often referred to as (the medieval city). However, the city's most distinctive district is Frederiksstaden, developed during the reign of Frederick V. It has the Amalienborg Palace at its centre and is dominated by the dome of Frederik's Church (or the Marble Church) and several elegant 18th-century Rococo mansions. The inner city includes Slotsholmen, a little island on which Christiansborg Palace stands and Christianshavn with its canals. Børsen on Slotsholmen and Frederiksborg Palace in Hillerød are prominent examples of the Dutch Renaissance style in Copenhagen. Around the historical city centre lies a band of congenial residential boroughs (Vesterbro, Inner Nørrebro, Inner Østerbro) dating mainly from late 19th century. They were built outside the old ramparts when the city was finally allowed to expand beyond its fortifications.
Sometimes referred to as "the City of Spires", Copenhagen is known for its horizontal skyline, broken only by the spires and towers of its churches and castles. Most characteristic of all is the Baroque spire of the Church of Our Saviour with its narrowing external spiral stairway that visitors can climb to the top. Other important spires are those of Christiansborg Palace, the City Hall and the former Church of St. Nikolaj that now houses a modern art venue. Not quite so high are the Renaissance spires of Rosenborg Castle and the "dragon spire" of Christian IV's former stock exchange, so named because it resembles the intertwined tails of four dragons. Copenhagen is recognised globally as an exemplar of best practice urban planning. Its thriving mixed use city centre is defined by striking contemporary architecture, engaging public spaces and an abundance of human activity. These design outcomes have been deliberately achieved through careful replanning in the second half of the 20th century. Recent years have seen a boom in modern architecture in Copenhagen both for Danish architecture and for works by international architects. For a few hundred years, virtually no foreign architects had worked in Copenhagen, but since the turn of the millennium the city and its immediate surroundings have seen buildings and projects designed by top international architects. British design magazine "Monocle" named Copenhagen the "World's best design city 2008".
Copenhagen's urban development in the first half of the 20th century was heavily influenced by industrialisation. After World War II, Copenhagen Municipality adopted Fordism and repurposed its medieval centre to facilitate private automobile infrastructure in response to innovations in transport, trade and communication. Copenhagen's spatial planning in this time frame was characterised by the separation of land uses: an approach which requires residents to travel by car to access facilities of different uses. The boom in urban development and modern architecture has brought some changes to the city's skyline. A political majority has decided to keep the historical centre free of high-rise buildings, but several areas will see or have already seen massive urban development. Ørestad now has seen most of the recent development. Located near Copenhagen Airport, it currently boasts one of the largest malls in Scandinavia and a variety of office and residential buildings as well as the IT University and a high school.
Parks, gardens and zoo. Copenhagen is a green city with many parks, both large and small. King's Garden (""), the garden of Rosenborg Castle, is the oldest and most frequented of them all. It was Christian IV who first developed its landscaping in 1606. Every year it sees more than 2.5 million visitors and in the summer months it is packed with sunbathers, picnickers and ballplayers. It serves as a sculpture garden with both a permanent display and temporary exhibits during the summer months. Also located in the city centre are the Botanical Gardens noted for their large complex of 19th-century greenhouses donated by Carlsberg founder J. C. Jacobsen. Fælledparken at is the largest park in Copenhagen. It is popular for sports fixtures and hosts several annual events including a free opera concert at the opening of the opera season, other open-air concerts, carnival and Labour Day celebrations, and the Copenhagen Historic Grand Prix, a race for antique cars. A historical green space in the northeastern part of the city is Kastellet, a well-preserved Renaissance citadel that now serves mainly as a park. Another popular park is the Frederiksberg Gardens, a 32-hectare romantic landscape park. It houses a colony of tame grey herons and other waterfowl. The park offers views of the elephants and the elephant house designed by world-famous British architect Norman Foster of the adjacent Copenhagen Zoo. Langelinie, a park and promenade along the inner Øresund coast, is home to one of Copenhagen's most-visited tourist attractions, the Little Mermaid statue.
In Copenhagen, many cemeteries double as parks, though only for the more quiet activities such as sunbathing, reading and meditation. Assistens Cemetery, the burial place of Hans Christian Andersen, is an important green space for the district of Inner Nørrebro and a Copenhagen institution. The lesser known Vestre Kirkegaard is the largest cemetery in Denmark () and offers a maze of dense groves, open lawns, winding paths, hedges, overgrown tombs, monuments, tree-lined avenues, lakes and other garden features. It is official municipal policy in Copenhagen that by 2015 all citizens must be able to reach a park or beach on foot in less than 15 minutes. In line with this policy, several new parks, including the innovative Superkilen in the Nørrebro district, have been completed or are under development in areas lacking green spaces. Landmarks by district. Indre By.
Christianshavn. Christianshavn lies to the southeast of Indre By on the other side of the harbour. The area was developed by Christian IV in the early 17th century. Impressed by the city of Amsterdam, he employed Dutch architects to create canals within its ramparts which are still well preserved today. The canals themselves, branching off the central Christianshavn Canal and lined with house boats and pleasure craft are one of the area's attractions. Another interesting feature is Freetown Christiania, a fairly large area which was initially occupied by squatters during student unrest in 1971. Today it still maintains a measure of autonomy. The inhabitants openly sell drugs on "Pusher Street" as well as their arts and crafts. Other buildings of interest in Christianshavn include the Church of Our Saviour with its spiralling steeple and the magnificent Rococo Christian's Church. Once a warehouse, the North Atlantic House now displays culture from Iceland and Greenland and houses the Noma restaurant, known for its Nordic cuisine.
Vesterbro. Vesterbro, to the southwest of Indre By, begins with the Tivoli Gardens, the city's top tourist attraction with its fairground atmosphere, its Pantomime Theatre, its Concert Hall and its many rides and restaurants. The Carlsberg neighbourhood has some interesting vestiges of the old brewery of the same name including the Elephant Gate and the Ny Carlsberg Brewhouse. The Tycho Brahe Planetarium is located on the edge of Skt. Jørgens Sø, one of the Copenhagen lakes. Halmtorvet, the old hay market behind the Central Station, is an increasingly popular area with its cafés and restaurants. The former cattle market Øksnehallen has been converted into a modern exhibition centre for art and photography. Radisson Blu Royal Hotel, built by Danish architect and designer Arne Jacobsen for the airline Scandinavian Airlines System (SAS) between 1956 and 1960 was once the tallest hotel in Denmark with a height of and the city's only skyscraper until 1969. Completed in 1908, Det Ny Teater (the New Theatre) located in a passage between Vesterbrogade and Gammel Kongevej has become a popular venue for musicals since its reopening in 1994, attracting the largest audiences in the country.
Nørrebro. Nørrebro to the northwest of the city centre has recently developed from a working-class district into a colourful cosmopolitan area with antique shops, non-Danish food stores and restaurants. Much of the activity is centred on Sankt Hans Torv and around Rantzausgade. Copenhagen's historic cemetery, Assistens Kirkegård halfway up Nørrebrogade, is the resting place of many famous figures including Søren Kierkegaard, Niels Bohr, and Hans Christian Andersen but is also used by locals as a park and recreation area. Østerbro. Just north of the city centre, Østerbro is an upper middle-class district with a number of fine mansions, some now serving as embassies. The district stretches from Nørrebro to the waterfront where "The Little Mermaid" statue can be seen from the promenade known as Langelinie. Inspired by Hans Christian Andersen's fairy tale, it was created by Edvard Eriksen and unveiled in 1913. Not far from the Little Mermaid, the old Citadel ("Kastellet") can be seen. Built by Christian IV, it is one of northern Europe's best preserved fortifications. There is also a windmill in the area. The large Gefion Fountain () designed by Anders Bundgaard and completed in 1908 stands close to the southeast corner of Kastellet. Its figures illustrate a Nordic legend.
Frederiksberg. Frederiksberg, a separate municipality within the urban area of Copenhagen, lies to the west of Nørrebro and Indre By and north of Vesterbro. Its landmarks include Copenhagen Zoo founded in 1869 with over 250 species from all over the world and Frederiksberg Palace built as a summer residence by Frederick IV who was inspired by Italian architecture. Now a military academy, it overlooks the extensive landscaped Frederiksberg Gardens with its follies, waterfalls, lakes and decorative buildings. The wide tree-lined avenue of Frederiksberg Allé connecting Vesterbrogade with the Frederiksberg Gardens has long been associated with theatres and entertainment. While a number of the earlier theatres are now closed, the Betty Nansen Theatre and Aveny-T are still active. Amagerbro. Amagerbro (also known as Sønderbro) is the district located immediately south-east of Christianshavn at northernmost Amager. The old city moats and their surrounding parks constitute a clear border between these districts. The main street is Amagerbrogade which after the harbour bridge Langebro, is an extension of H. C. Andersens Boulevard and has a number of various stores and shops as well as restaurants and pubs. Amagerbro was built up during the two first decades of the twentieth century and is the city's southernmost block built area with typically 4–7 floors. Further south follows the Sundbyøster and Sundbyvester districts.
Other districts. Not far from Copenhagen Airport on the Kastrup coast, The Blue Planet completed in March 2013 now houses the national aquarium. With its 53 aquariums, it is the largest facility of its kind in Scandinavia. Grundtvig's Church, located in the northern suburb of Bispebjerg, was designed by P.V. Jensen Klint and completed in 1940. A rare example of Expressionist church architecture, its striking west façade is reminiscent of a church organ. Culture. Apart from being the national capital, Copenhagen also serves as the cultural hub of Denmark and one of the major hubs in wider Scandinavia. Since the late 1990s, it has undergone a transformation from a modest Scandinavian capital into a metropolitan city of international appeal, in the same league as cities such as Barcelona and Amsterdam. This is a result of huge investments in infrastructure and culture as well as the work of successful new Danish architects, designers and chefs. Copenhagen Fashion Week takes place every year in February and August.
Museums. Copenhagen has a wide array of museums of international standing. The National Museum, , is Denmark's largest museum of archaeology and cultural history, comprising the histories of Danish and foreign cultures alike. Denmark's National Gallery () is the national art museum with collections dating from the 12th century to the present. In addition to Danish painters, artists represented in the collections include Rubens, Rembrandt, Picasso, Braque, Léger, Matisse, Emil Nolde, Olafur Eliasson, Elmgreen & Dragset, Superflex, and Jens Haaning. Another important Copenhagen art museum is the Ny Carlsberg Glyptotek founded by second generation Carlsberg philanthropist Carl Jacobsen and built around his personal collections. Its main focus is classical Egyptian, Roman and Greek sculptures and antiquities and a collection of Rodin sculptures, the largest outside France. Besides its sculpture collections, the museum also holds a comprehensive collection of paintings of Impressionist and Post-Impressionist painters such as Monet, Renoir, Cézanne, van Gogh and Toulouse-Lautrec as well as works by the Danish Golden Age painters.
Louisiana is a Museum of Modern Art situated on the coast just north of Copenhagen. It is located in the middle of a sculpture garden on a cliff overlooking Øresund. Its collection of over 3,000 items includes works by Picasso, Giacometti and Dubuffet. The Danish Design Museum is housed in the 18th-century former Frederiks Hospital and displays Danish design as well as international design and crafts. Other museums include: the Thorvaldsens Museum, dedicated to the oeuvre of romantic Danish sculptor Bertel Thorvaldsen who lived and worked in Rome; the Cisternerne museum, an exhibition space for contemporary art, located in former cisterns that come complete with stalactites formed by the changing water levels; and the Ordrupgaard Museum, located just north of Copenhagen, which features 19th-century French and Danish art and is noted for its works by Paul Gauguin. Entertainment and performing arts. The new Copenhagen Concert Hall opened in January 2009. Designed by Jean Nouvel, it has four halls with the main auditorium seating 1,800 people. It serves as the home of the Danish National Symphony Orchestra and along with the Walt Disney Concert Hall in Los Angeles is the most expensive concert hall ever built. Another important venue for classical music is the Tivoli Concert Hall located in the Tivoli Gardens. Designed by Henning Larsen, the Copenhagen Opera House () opened in 2005. It is among the most modern opera houses in the world. The Royal Danish Theatre also stages opera in addition to its drama productions. It is also home to the Royal Danish Ballet. Founded in 1748 along with the theatre, it is one of the oldest ballet troupes in Europe, and is noted for its Bournonville style of ballet.
Copenhagen has a significant jazz scene that has existed for many years. It developed when a number of American jazz musicians such as Ben Webster, Thad Jones, Richard Boone, Ernie Wilkins, Kenny Drew, Ed Thigpen, Bob Rockwell, Dexter Gordon, and others such as rock guitarist Link Wray came to live in Copenhagen during the 1960s. Every year in early July, Copenhagen's streets, squares, parks as well as cafés and concert halls fill up with big and small jazz concerts during the Copenhagen Jazz Festival. One of Europe's top jazz festivals, the annual event features around 900 concerts at 100 venues with over 200,000 guests from Denmark and around the world. The largest venue for popular music in Copenhagen is Vega in the Vesterbro district. It was chosen as "best concert venue in Europe" by international music magazine "Live". The venue has three concert halls: the great hall, Store Vega, accommodates audiences of 1,550, the middle hall, Lille Vega, has space for 500 and Ideal Bar Live has a capacity of 250. Every September since 2006, the Festival of Endless Gratitude (FOEG) has taken place in Copenhagen. This festival focuses on indie counterculture, experimental pop music and left field music combined with visual arts exhibitions.
For free entertainment one can stroll along Strøget, especially between Nytorv and Højbro Plads, which in the late afternoon and evening is a bit like an impromptu three-ring circus with musicians, magicians, jugglers and other street performers. Literature. Most of Denmarks's major publishing houses are based in Copenhagen. These include the book publishers Gyldendal and Akademisk Forlag and newspaper publishers Berlingske and Politiken (the latter also publishing books). Many of the most important contributors to Danish literature such as Hans Christian Andersen (1805–1875) with his fairy tales, the philosopher Søren Kierkegaard (1813–1855) and playwright Ludvig Holberg (1684–1754) spent much of their lives in Copenhagen. Novels set in Copenhagen include "Baby" (1973) by Kirsten Thorup, "The Copenhagen Connection" (1982) by Barbara Mertz, "Number the Stars" (1989) by Lois Lowry, "Miss Smilla's Feeling for Snow" (1992) and "Borderliners" (1993) by Peter Høeg, "Music and Silence" (1999) by Rose Tremain, "The Danish Girl" (2000) by David Ebershoff, and "Sharpe's Prey" (2001) by Bernard Cornwell. Michael Frayn's 1998 play "Copenhagen" about the meeting between the physicists Niels Bohr and Werner Heisenberg in 1941 is also set in the city. On 15–18 August 1973, an oral literature conference took place in Copenhagen as part of the 9th International Congress of Anthropological and Ethnological Sciences.
The Royal Library, belonging to the University of Copenhagen, is the largest library in the Nordic countries with an almost complete collection of all printed Danish books since 1482. Founded in 1648, the Royal Library is located at four sites in the city, the main one being on the Slotsholmen waterfront. Copenhagen's public library network has over 20 outlets, the largest being the Central Library () on Krystalgade in the inner city. Art. Copenhagen has a wide selection of art museums and galleries displaying both historic works and more modern contributions. They include , i.e. the Danish national art gallery, in the Østre Anlæg park, and the adjacent Hirschsprung Collection specialising in the 19th and early 20th century. Kunsthal Charlottenborg in the city centre exhibits national and international contemporary art. Den Frie Udstilling near the Østerport Station exhibits paintings created and selected by contemporary artists themselves rather than by the official authorities. The Arken Museum of Modern Art is located in southwestern Ishøj. Among artists who have painted scenes of Copenhagen are Martinus Rørbye (1803–1848), Christen Købke (1810–1848) and the prolific Paul Gustav Fischer (1860–1934).
A number of notable sculptures can be seen in the city. In addition to "The Little Mermaid" on the waterfront, there are two historic equestrian statues in the city centre: Jacques Saly's "Frederik V on Horseback" (1771) in Amalienborg Square and the statue of Christian V on Kongens Nytorv created by Abraham-César Lamoureux in 1688 who was inspired by the statue of Louis XIII in Paris. Rosenborg Castle Gardens contains several sculptures and monuments including August Saabye's Hans Christian Andersen, Aksel Hansen's Echo, and Vilhelm Bissen's Dowager Queen Caroline Amalie. Copenhagen is believed to have invented the photomarathon photography competition, which has been held in the City each year since 1989. Cuisine. , Copenhagen has 15 Michelin-starred restaurants, the most of any Scandinavian city. The city is increasingly recognized internationally as a gourmet destination. These include Den Røde Cottage, Formel B Restaurant, Grønbech & Churchill, Søllerød Kro, Kadeau, Kiin Kiin (Denmark's first Michelin-starred Asian gourmet restaurant), the French restaurant Kong Hans Kælder, Relæ, Restaurant AOC with two Stars, and Noma (short for "mad", English: Nordic food) as well as Geranium with three. Noma was ranked as the Best Restaurant in the World by "Restaurant" in 2010, 2011, 2012, and again in 2014, sparking interest in the New Nordic Cuisine.
Apart from the selection of upmarket restaurants, Copenhagen offers a great variety of Danish, ethnic and experimental restaurants. It is possible to find modest eateries serving open sandwiches, known as smørrebrød – a traditional, Danish lunch dish; however, most restaurants serve international dishes. Danish pastry can be sampled from any of numerous bakeries found in all parts of the city. The Copenhagen Bakers' Association (Danish: ) dates back to the 1290s and Denmark's oldest confectioner's shop still operating, "Conditori La Glace", was founded in 1870 in Skoubogade by Nicolaus Henningsen, a trained master baker from Flensburg. Copenhagen has long been associated with beer. Carlsberg beer has been brewed at the brewery's premises on the border between the Vesterbro and Valby districts since 1847 and has long been almost synonymous with Danish beer production. However, recent years have seen an explosive growth in the number of microbreweries so that Denmark today has more than 100 breweries, many of which are located in Copenhagen. Some like Nørrebro Bryghus also act as brewpubs where it is also possible to eat on the premises.
Nightlife and festivals. Copenhagen has one of the highest number of restaurants and bars per capita in the world. The nightclubs and bars stay open until 5 or 6 in the morning, some even longer. Denmark has a very liberal alcohol culture and a strong tradition for beer breweries, although binge drinking is frowned upon and the Danish Police take driving under the influence very seriously. Inner city areas such as Istedgade and Enghave Plads in Vesterbro, Sankt Hans Torv in Nørrebro and certain places in Frederiksberg are especially noted for their nightlife. Notable nightclubs include Bakken Kbh, ARCH (previously ZEN), Jolene, The Jane, Chateau Motel, KB3, At Dolores (previously Sunday Club), Rust, Vega Nightclub and Culture Box. Copenhagen has several recurring community festivals, mainly in the summer. Copenhagen Carnival has taken place every year since 1982 during the Whitsun Holiday in Fælledparken and around the city with the participation of 120 bands, 2,000 dancers and 100,000 spectators. Since 2010, the old B&W Shipyard at Refshaleøen in the harbour has been the location for Copenhell, a heavy metal rock music festival. Copenhagen Pride is a LGBT pride festival taking place every year in August. The Pride has a series of different activities all over Copenhagen, but it is at the City Hall Square that most of the celebration takes place. During the Pride the square is renamed Pride Square. Copenhagen Distortion has emerged to be one of the biggest street festivals in Europe with 100,000 people joining to parties in the beginning of June every year.
Amusement parks. Copenhagen has the oldest and third-oldest amusement parks in the world. Dyrehavsbakken, a fair-ground and pleasure-park established in 1583, is located in Klampenborg just north of Copenhagen in a forested area known as Dyrehaven. Created as an amusement park complete with rides, games and restaurants by Christian IV, it is the oldest surviving amusement park in the world. Pierrot (), a nitwit dressed in white with a scarlet grin wearing a boat-like hat while entertaining children, remains one of the park's key attractions. In Danish, Dyrehavsbakken is often abbreviated as . There is no entrance fee to pay and Klampenborg Station on the C-line, is situated nearby. The Tivoli Gardens is an amusement park and pleasure garden located in central Copenhagen between the City Hall Square and the Central Station. It opened in 1843, making it the third-oldest amusement park in the world, the second being Wurstelprater in Vienna. Among its rides are the oldest still operating rollercoaster from 1915 and the oldest ferris wheel still in use, opened in 1943. Tivoli Gardens also serves as a venue for various performing arts and as an active part of the cultural scene in Copenhagen.
Education. Copenhagen has over 94,000 students enrolled in its largest universities and institutions: University of Copenhagen (38,867 students), Copenhagen Business School (20,000 students), Metropolitan University College and University College Capital (10,000 students each), Technical University of Denmark (7,000 students), KEA (c. 4,500 students), IT University of Copenhagen (2,000 students) and the Copenhagen campus of Aalborg University (2,300 students). The University of Copenhagen is Denmark's oldest university founded in 1479. It attracts some 1,500 international and exchange students every year. The Academic Ranking of World Universities placed it 30th in the world in 2016. The Technical University of Denmark is located in Lyngby in the northern outskirts of Copenhagen. In 2013, it was ranked as one of the leading technical universities in Northern Europe. The IT University is Denmark's youngest university, a mono-faculty institution focusing on technical, societal and business aspects of information technology.
The Danish Academy of Fine Arts has provided education in the arts for more than 250 years. It includes the historic School of Visual Arts, and has in later years come to include a School of Architecture, a School of Design and a School of Conservation. Copenhagen Business School (CBS) is an EQUIS-accredited business school located in Frederiksberg. There are also branches of both University College Capital and Metropolitan University College inside and outside Copenhagen. Sport. The city has a variety of sporting teams. The major football teams are the historically successful FC København and Brøndby. FC København plays at Parken in Østerbro. Formed in 1992, it is a merger of two older Copenhagen clubs, B 1903 (from the inner suburb Gentofte) and KB (from Frederiksberg). Brøndby plays at Brøndby Stadion in the inner suburb of Brøndbyvester. BK Frem is based in the southern part of Copenhagen (Sydhavnen, Valby). Other teams of more significant stature are FC Nordsjælland (from suburban Farum), Fremad Amager, B93, AB, Lyngby and Hvidovre IF.
Copenhagen has several handball teams—a sport which is particularly popular in Denmark. Of clubs playing in the "highest" leagues, there are Ajax, Ydun, and HIK (Hellerup). The København Håndbold women's club has recently been established. Copenhagen also has ice hockey teams, of which three play in the top league, Rødovre Mighty Bulls, Herlev Eagles and Hvidovre Ligahockey all inner suburban clubs. Copenhagen Ice Skating Club founded in 1869 is the oldest ice hockey team in Denmark but is no longer in the top league. Rugby union is also played in the Danish capital with teams such as CSR-Nanok, Copenhagen Business School Sport Rugby, Frederiksberg RK, Exiles RUFC and Rugbyklubben Speed. Rugby league is now played in Copenhagen, with the national team playing out of Gentofte Stadion. The Danish Australian Football League, based in Copenhagen is the largest Australian rules football competition outside of the English-speaking world. Copenhagen Marathon, Copenhagen's annual marathon event, was established in 1980.
Round Christiansborg Open Water Swim Race is a open water swimming competition taking place each year in late August. This amateur event is combined with a Danish championship. In 2009 the event included a FINA World Cup competition in the morning. Copenhagen hosted the 2011 UCI Road World Championships in September 2011, taking advantage of its bicycle-friendly infrastructure. It was the first time that Denmark had hosted the event since 1956, when it was also held in Copenhagen. Transport. Airport. The greater Copenhagen area has a very well established transportation infrastructure making it a hub in Northern Europe. Copenhagen Airport, opened in 1925, is Scandinavia's largest airport, located in Kastrup on the island of Amager. It is connected to the city centre by metro and main line railway services. October 2013 was a record month with 2.2 million passengers, and November 2013 figures reveal that the number of passengers is increasing by some 3% annually, about 50% more than the European average. Road, rail and ferry.
Copenhagen has an extensive road network including motorways connecting the city to other parts of Denmark and to Sweden over the Øresund Bridge. The car is still the most popular form of transport within the city itself, representing two-thirds of all distances travelled. This can however lead to serious congestion in rush hour traffic. The Øresund train links Copenhagen with Malmö 24 hours a day, 7 days a week. Copenhagen is also served by a daily ferry connection to Oslo in Norway. In 2012, Copenhagen Harbour handled 372 cruise ships and 840,000 passengers. The Copenhagen S-Train, Copenhagen Metro and the regional train networks are used by about half of the city's passengers, the remainder using bus services. Nørreport Station near the city centre serves passengers travelling by main-line rail, S-train, regional train, metro and bus. Some 750,000 passengers make use of public transport facilities every day. Copenhagen Central Station is the hub of the DSB railway network serving Denmark and international destinations.
The Copenhagen Metro expanded radically with the opening of the City Circle Line (M3) on 29 September 2019. The new line connects all inner boroughs of the city by metro, including the Central Station, and opens up 17 new stations for Copenhageners. On 28 March 2020, the Nordhavn extension of the Harbour Line (M4) opened. Running from Copenhagen Central Station, the new extension is a branch line of M3 Cityring to Østerport. The new metro lines are part of the city's strategy to transform mobility towards sustainable modes of transport such as public transport and cycling as opposed to automobility. Copenhagen is cited by urban planners for its exemplary integration of public transport and urban development. In implementing its Finger Plan, Copenhagen is considered the world's first example of a transit metropolis, and areas around S-Train stations like Ballerup and Brøndby Strand are among the earliest examples of transit-oriented development. Cycling. Copenhagen has been rated as one of the most bicycle-friendly cities in the world since 2015, with bicycles outnumbering its inhabitants. In 2012 some 36% of all working or studying city-dwellers cycled to work, school, or university. With 1.27 million km covered every working day by Copenhagen's cyclists (including both residents and commuters), and 75% of Copenhageners cycling throughout the year. The city's bicycle paths are extensive and well used, boasting of cycle lanes not shared with cars or pedestrians, and sometimes have their own signal systems – giving the cyclists a lead of a couple of seconds to accelerate.
Healthcare. Promoting health is an important issue for Copenhagen's municipal authorities. Central to its sustainability mission is its "Long Live Copenhagen" () scheme in which it has the goal of increasing the life expectancy of citizens, improving quality of life through better standards of health, and encouraging more productive lives and equal opportunities. The city has targets to encourage people to exercise regularly and to reduce the number who smoke and consume alcohol. Copenhagen University Hospital forms a conglomerate of several hospitals in Region Hovedstaden and Region Sjælland, together with the faculty of health sciences at the University of Copenhagen; Rigshospitalet and Bispebjerg Hospital in Copenhagen belong to this group of university hospitals. Rigshospitalet began operating in March 1757 as Frederiks Hospital, and became state-owned in 1903. With 1,120 beds, Rigshospitalet has responsibility for 65,000 inpatients and approximately 420,000 outpatients annually. It seeks to be the number one specialist hospital in the country, with an extensive team of researchers into cancer treatment, surgery and radiotherapy. In addition to its 8,000 personnel, the hospital has training and hosting functions. It benefits from the presence of in-service students of medicine and other healthcare sciences, as well as scientists working under a variety of research grants. The hospital became internationally famous as the location of Lars von Trier's television horror mini-series "The Kingdom". Bispebjerg Hospital was built in 1913, and serves about 400,000 people in the Greater Copenhagen area, with some 3,000 employees. Other large hospitals in the city include Amager Hospital (1997), Herlev Hospital (1976), Hvidovre Hospital (1970), and Gentofte Hospital (1927).
Media. Many Danish media corporations are located in Copenhagen. DR, the major Danish public service broadcasting corporation consolidated its activities in a new headquarters, DR Byen, in 2006 and 2007. Similarly TV2, which is based in Odense, has concentrated its Copenhagen activities in a modern media house in Teglholmen. The two national daily newspapers "Politiken" and "Berlingske" and the two tabloids and "BT" are based in Copenhagen. "Kristeligt Dagblad" is based in Copenhagen and is published six days a week. Other important media corporations include Aller Media which is the largest publisher of weekly and monthly magazines in Scandinavia, the Egmont media group and Gyldendal, the largest Danish publisher of books. Copenhagen has a large film and television industry. Nordisk Film, established in Valby, Copenhagen in 1906 is the oldest continuously operating film production company in the world. In 1992 it merged with the Egmont media group and currently runs the 17-screen Palads Cinema in Copenhagen. Filmbyen (movie city), located in a former military camp in the suburb of Hvidovre, houses several movie companies and studios. Zentropa is a film company, co-owned by Danish director Lars von Trier. He is behind several international movie productions as well and founded the Dogme Movement. is Copenhagen's international feature film festival, established in 2009 as a fusion of the 20-year-old NatFilm Festival and the four-year-old CIFF. The CPH:PIX festival takes place in mid-April. CPH:DOX is Copenhagen's international documentary film festival, every year in November. In addition to a documentary film programme of over 100 films, CPH:DOX includes a wide event programme with dozens of events, concerts, exhibitions and parties all over town. Twin towns – sister cities. Copenhagen is twinned with: Honorary citizens. People awarded the honorary citizenship of Copenhagen are: While honorary citizenship is no longer granted in Copenhagen, three people have been awarded the title of honorary Copenhageners ("æreskøbenhavnere").
Combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an "ad hoc" solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.
Definition. The full scope of combinatorics is not universally agreed upon. According to H.J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with: Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained." One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella. Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting. History. Basic combinatorial concepts and enumerative results appeared throughout the ancient world.The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus, which dates to the 16th century BC. The problem concerns a certain geometric series, and has similarities to Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total. Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 26 − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers. Earlier, in the "Ostomachion", Archimedes (3rd century BCE) may have considered the number of configurations of a tiling puzzle, while combinatorial interests possibly were present in lost works by Apollonius.
In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra () provided formulae for the number of permutations and combinations, and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE. The philosopher and astronomer Rabbi Abraham ibn Ezra () established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321. The arithmetical triangle—a graphical diagram showing relationships among the binomial coefficients—was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics. Graph theory also enjoyed an increase of interest at the same time, especially in connection with the four color problem.
In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject. In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field. Approaches and subfields of combinatorics. Enumerative combinatorics. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.
Analytic combinatorics. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory and has connections with statistical mechanics. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.
Graph theory. Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on "n" vertices with "k" edges) to existing structures (e.g., Hamiltonian cycles) to algebraic representations (e.g., given a graph "G" and two numbers "x" and "y", does the Tutte polynomial "T""G"("x","y") have a combinatorial interpretation?). Although there are very strong connections between graph theory and combinatorics, they are sometimes thought of as separate subjects. While combinatorial methods apply to many graph theory problems, the two disciplines are generally used to seek solutions to different types of problems. Design theory. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics.
Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, group testing and cryptography. Finite geometry. Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory. It should not be confused with discrete geometry (combinatorial geometry). Order theory. Order theory is the study of partially ordered sets, both finite and infinite. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". Various examples of partial orders appear in algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and Boolean algebras.
Matroid theory. Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory. It is now an independent field of study with a number of connections with other parts of combinatorics. Extremal combinatorics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of set systems; this is called extremal set theory. For instance, in an "n"-element set, what is the largest number of "k"-element subsets that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory.
The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on "2n" vertices is a complete bipartite graph "Kn,n". Often it is too hard even to find the extremal answer "f"("n") exactly and one can only give an asymptotic estimate. Ramsey theory is another part of extremal combinatorics. It states that any sufficiently large configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle. Probabilistic combinatorics. In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find) by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as "the" probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time.
Often associated with Paul Erdős, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. The area recently grew to become an independent field of combinatorics. Algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common. Combinatorics on words. Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory, and linguistics. While many applications are new, the classical Chomsky–Schützenberger hierarchy of classes of formal grammars is perhaps the best-known result in the field.
Geometric combinatorics. Geometric combinatorics is related to convex and discrete geometry. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. Combinatorial geometry is a historical name for discrete geometry. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. The study of regular polytopes, Archimedean solids, and kissing numbers is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron, associahedron and Birkhoff polytope. Topological combinatorics. Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.
Arithmetic combinatorics. Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory, and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems. Infinitary combinatorics. Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
Gian-Carlo Rota used the name "continuous combinatorics" to describe geometric probability, since there are many analogies between "counting" and "measure". Related fields. Combinatorial optimization. Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory. Coding theory. Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of information theory. Discrete and computational geometry. Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.
Combinatorics and dynamical systems. Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system. Combinatorics and physics. There are increasing interactions between combinatorics and physics, particularly statistical physics. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic and Tutte polynomials on the other hand.
Calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. Infinitesimal calculus was formulated separately in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus is widely used in science, engineering, biology, and even has applications in social science and other branches of math.
Etymology. In mathematics education, "calculus" is an abbreviation of both infinitesimal calculus and integral calculus, which denotes courses of elementary mathematical analysis. In Latin, the word "calculus" means “small pebble”, (the diminutive of "calx," meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to be the Latin word for "calculation". In this sense, it was used in English at least as early as 1672, several years before the publications of Leibniz and Newton, who wrote their mathematical texts in Latin. In addition to differential calculus and integral calculus, the term is also used for naming specific methods of computation or theories that imply some sort of computation. Examples of this usage include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.
History. Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval Europe and India. Ancient precursors. Egypt. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (), but the formulae are simple instructions, with no indication as to how they were obtained. Greece. Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus () developed the method of exhaustion to prove the formulas for cone and pyramid volumes. During the Hellenistic period, this method was further developed by Archimedes (BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus. In "The Method of Mechanical Theorems" he describes, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines.
China. The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri's principle to find the volume of a sphere. Medieval. Middle East. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (AD) derived a formula for the sum of fourth powers. He determined the equations to calculate the area enclosed by the curve represented by formula_1 (which translates to the integral formula_2 in contemporary notation), for any given non-negative integer value of formula_3.He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. India. Bhāskara II () was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function. In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if formula_4 then formula_5 This can be interpreted as the discovery that cosine is the derivative of sine. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics stated components of calculus. They studied series equivalent to the Maclaurin expansions of , , and more than two hundred years before their introduction in Europe. According to Victor J. Katz they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".
Modern. Johannes Kepler's work "Stereometria Doliorum" (1615) formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse. Significant work was a treatise, the origin being Kepler's methods, written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in "The Method", but this treatise is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.
The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his "Principia Mathematica" (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.
Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics. Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his "Method of Fluxions"), but Leibniz published his "Nova Methodus pro Maximis et Minimis" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions", a term that endured in English schools into the 19th century. The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815.
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. Foundations. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. In early calculus, the use of infinitesimal quantities was thought unrigorous and was fiercely criticized by several authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book "The Analyst" in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today. Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities. The foundations of differential and integral calculus had been laid. In Cauchy's "Cours d'Analyse", we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. In his work, Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can validate nilsquare infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to the complex plane with the development of complex analysis.
In modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus. The reach of calculus has also been greatly extended. Henri Lebesgue invented measure theory, based on earlier developments by Émile Borel, and used it to define integrals of all but the most pathological functions. Laurent Schwartz introduced distributions, which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to the foundation of calculus. Another way is to use Abraham Robinson's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus. There is also smooth infinitesimal analysis, which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations. Based on the ideas of F. W. Lawvere and employing the methods of category theory, smooth infinitesimal analysis views all functions as being continuous and incapable of being expressed in terms of discrete entities. One aspect of this formulation is that the law of excluded middle does not hold. The law of excluded middle is also rejected in constructive mathematics, a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of constructive analysis.
Significance. While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles. The Hungarian polymath John von Neumann wrote of this work, Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, that resolve the paradoxes.
Principles. Limits and infinitesimals. Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols formula_6 and formula_7 were taken to be infinitesimal, and the derivative formula_8 was their ratio. The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the epsilon, delta approach to limits. Limits describe the behavior of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the real number system (as a metric space with the least-upper-bound property). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason, they became the standard approach during the 20th century. However, the infinitesimal concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.
Differential calculus. Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called "differentiation". Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the "derivative function" or just the "derivative" of the original function. In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by differentiating the squaring function turns out to be the doubling function.
In more explicit terms the "doubling function" may be denoted by and the "squaring function" by . The "derivative" now takes the function , defined by the expression "", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function , as will turn out. In Lagrange's notation, the symbol for a derivative is an apostrophe-like mark called a prime. Thus, the derivative of a function called is denoted by , pronounced "f prime" or "f dash". For instance, if is the squaring function, then is its derivative (the doubling function from above). If the input of the function represents time, then the derivative represents change concerning time. For example, if is a function that takes time as input and gives the position of a ball at that time as output, then the derivative of is how the position is changing in time, that is, it is the velocity of the ball. If a function is linear (that is if the graph of the function is a straight line), then the function can be written as , where is the independent variable, is the dependent variable, is the "y"-intercept, and:
This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in divided by the change in varies. Derivatives give an exact meaning to the notion of change in output concerning change in input. To be concrete, let be a function, and fix a point in the domain of . is a point on the graph of the function. If is a number close to zero, then is a number close to . Therefore, is close to . The slope between these two points is This expression is called a "difference quotient". A line through two points on a curve is called a "secant line", so is the slope of the secant line between and . The second line is only an approximation to the behavior of the function at the point because it does not account for what happens between and . It is not possible to discover the behavior at by setting to zero because this would require dividing by zero, which is undefined. The derivative is defined by taking the limit as tends to zero, meaning that it considers the behavior of for all small values of and extracts a consistent value for the case when equals zero:
Geometrically, the derivative is the slope of the tangent line to the graph of at . The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function . Here is a particular example, the derivative of the squaring function at the input 3. Let be the squaring function. The slope of the tangent line to the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the "derivative function" of the squaring function or just the "derivative" of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function is the doubling function. Leibniz notation. A common notation, introduced by Leibniz, for the derivative in the example above is In an approach based on limits, the symbol is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, being the infinitesimally small change in caused by an infinitesimally small change applied to . We can also think of as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
In this usage, the in the denominator is read as "with respect to ". Another example of correct notation could be: Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like and as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative. Integral calculus. "Integral calculus" is the study of the definitions, properties, and applications of two related concepts, the "indefinite integral" and the "definite integral". The process of finding the value of an integral is called "integration". The indefinite integral, also known as the "antiderivative", is the inverse operation to the derivative. is an indefinite integral of when is a derivative of . (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum.
A motivating example is the distance traveled in a given time. If the speed is constant, only multiplication is needed: But if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled. When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, traveling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve. This connection between the area under a curve and the distance traveled can be extended to "any" irregularly shaped region exhibiting a fluctuating velocity over a given period. If represents speed as it varies over time, the distance traveled between the times represented by and is the area of the region between and the -axis, between and .
To approximate that area, an intuitive method would be to divide up the distance between and into several equal segments, the length of each segment represented by the symbol . For each small segment, we can choose one value of the function . Call that value . Then the area of the rectangle with base and height gives the distance (time multiplied by speed ) traveled in that segment. Associated with each segment is the average value of the function above it, . The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for will give more rectangles and in most cases a better approximation, but for an exact answer, we need to take a limit as approaches zero. The symbol of integration is formula_17, an elongated "S" chosen to suggest summation. The definite integral is written as: and is read "the integral from "a" to "b" of "f"-of-"x" with respect to "x"." The Leibniz notation is intended to suggest dividing the area under the curve into an infinite number of rectangles so that their width becomes the infinitesimally small .
The indefinite integral, or antiderivative, is written: Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is a family of functions differing only by a constant. Since the derivative of the function , where is any constant, is , the antiderivative of the latter is given by: The unspecified constant present in the indefinite integral or antiderivative is known as the constant of integration. Fundamental theorem. The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration. The fundamental theorem of calculus states: If a function is continuous on the interval and if is a function whose derivative is on the interval , then
Furthermore, for every in the interval , This realization, made by both Newton and Leibniz, was key to the proliferation of analytic results after their work became known. (The extent to which Newton and Leibniz were influenced by immediate predecessors, and particularly what Leibniz may have learned from the work of Isaac Barrow, is difficult to determine because of the priority dispute between them.) The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulae for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives and are ubiquitous in the sciences. Applications.
Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of objects, and the potential energies due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion, which states that the derivative of an object's momentum concerning time equals the net force upon it. Alternatively, Newton's second law can be expressed by saying that the net force equals the object's mass times its acceleration, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path. Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and in studying radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.
Green's theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter, which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property. In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel to maximize flow. Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a cancerous tumor grows. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.
Communication Communication is commonly defined as the transmission of information. Its precise definition is disputed and there are disagreements about whether unintentional or failed transmissions are included and whether communication not only transmits meaning but also creates it. Models of communication are simplified overviews of its main components and their interactions. Many models include the idea that a source uses a coding system to express information in the form of a message. The message is sent through a channel to a receiver who has to decode it to understand it. The main field of inquiry investigating communication is called communication studies. A common way to classify communication is by whether information is exchanged between humans, members of other species, or non-living entities such as computers. For human communication, a central contrast is between verbal and non-verbal communication. Verbal communication involves the exchange of messages in linguistic form, including spoken and written messages as well as sign language. Non-verbal communication happens without the use of a linguistic system, for example, using body language, touch, and facial expressions. Another distinction is between interpersonal communication, which happens between distinct persons, and intrapersonal communication, which is communication with oneself. Communicative competence is the ability to communicate well and applies to the skills of formulating messages and understanding them.
Non-human forms of communication include animal and plant communication. Researchers in this field often refine their definition of communicative behavior by including the criteria that observable responses are present and that the participants benefit from the exchange. Animal communication is used in areas like courtship and mating, parent–offspring relations, navigation, and self-defense. Communication through chemicals is particularly important for the relatively immobile plants. For example, maple trees release so-called volatile organic compounds into the air to warn other plants of a herbivore attack. Most communication takes place between members of the same species. The reason is that its purpose is usually some form of cooperation, which is not as common between different species. Interspecies communication happens mainly in cases of symbiotic relationships. For instance, many flowers use symmetrical shapes and distinctive colors to signal to insects where nectar is located. Humans engage in interspecies communication when interacting with pets and working animals.
Human communication has a long history and how people exchange information has changed over time. These changes were usually triggered by the development of new communication technologies. Examples are the invention of writing systems, the development of mass printing, the use of radio and television, and the invention of the internet. The technological advances also led to new forms of communication, such as the exchange of data between computers. Definitions. The word "" has its root in the Latin verb , which means or . Communication is usually understood as the transmission of information: a message is conveyed from a sender to a receiver using some medium, such as sound, written signs, bodily movements, or electricity. Sender and receiver are often distinct individuals but it is also possible for an individual to communicate with themselves. In some cases, sender and receiver are not individuals but groups like organizations, social classes, or nations. In a different sense, the term "communication" refers to the message that is being communicated or to the field of inquiry studying communicational phenomena.
The precise characterization of communication is disputed. Many scholars have raised doubts that any single definition can capture the term accurately. These difficulties come from the fact that the term is applied to diverse phenomena in different contexts, often with slightly different meanings. The issue of the right definition affects the research process on many levels. This includes issues like which empirical phenomena are observed, how they are categorized, which hypotheses and laws are formulated as well as how systematic theories based on these steps are articulated. Some definitions are broad and encompass unconscious and non-human behavior. Under a broad definition, many animals communicate within their own species and flowers communicate by signaling the location of nectar to bees through their colors and shapes. Other definitions restrict communication to conscious interactions among human beings. Some approaches focus on the use of symbols and signs while others stress the role of understanding, interaction, power, or transmission of ideas. Various characterizations see the communicator's intent to send a message as a central component. In this view, the transmission of information is not sufficient for communication if it happens unintentionally. A version of this view is given by philosopher Paul Grice, who identifies communication with actions that aim to make the recipient aware of the communicator's intention. One question in this regard is whether only successful transmissions of information should be regarded as communication. For example, distortion may interfere with and change the actual message from what was originally intended. A closely related problem is whether acts of deliberate deception constitute communication.
According to a broad definition by literary critic I. A. Richards, communication happens when one mind acts upon its environment to transmit its own experience to another mind. Another interpretation is given by communication theorists Claude Shannon and Warren Weaver, who characterize communication as a transmission of information brought about by the interaction of several components, such as a source, a message, an encoder, a channel, a decoder, and a receiver. The transmission view is rejected by transactional and constitutive views, which hold that communication is not just about the transmission of information but also about the creation of meaning. Transactional and constitutive perspectives hold that communication shapes the participant's experience by conceptualizing the world and making sense of their environment and themselves. Researchers studying animal and plant communication focus less on meaning-making. Instead, they often define communicative behavior as having other features, such as playing a beneficial role in survival and reproduction, or having an observable response.
Models of communication. Models of communication are conceptual representations of the process of communication. Their goal is to provide a simplified overview of its main components. This makes it easier for researchers to formulate hypotheses, apply communication-related concepts to real-world cases, and test predictions. Due to their simplified presentation, they may lack the conceptual complexity needed for a comprehensive understanding of all the essential aspects of communication. They are usually presented visually in the form of diagrams showing the basic components and their interaction. Models of communication are often categorized based on their intended applications and how they conceptualize communication. Some models are general in the sense that they are intended for all forms of communication. Specialized models aim to describe specific forms, such as models of mass communication.
All the early models, developed in the middle of the 20th century, are linear transmission models. Lasswell's model, for example, is based on five fundamental questions: "Who?", "Says what?", "In which channel?", "To whom?", and "With what effect?". The goal of these questions is to identify the basic components involved in the communicative process: the sender, the message, the channel, the receiver, and the effect. Lasswell's model was initially only conceived as a model of mass communication, but it has been applied to other fields as well. Some communication theorists, like Richard Braddock, have expanded it by including additional questions, like "Under what circumstances?" and "For what purpose?". The Shannon–Weaver model is another influential linear transmission model. It is based on the idea that a source creates a message, which is then translated into a signal by a transmitter. Noise may interfere with and distort the signal. Once the signal reaches the receiver, it is translated back into a message and made available to the destination. For a landline telephone call, the person calling is the source and their telephone is the transmitter. The transmitter translates the message into an electrical signal that travels through the wire, which acts as the channel. The person taking the call is the destination and their telephone is the receiver. The Shannon–Weaver model includes an in-depth discussion of how noise can distort the signal and how successful communication can be achieved despite noise. This can happen by making the message partially redundant so that decoding is possible nonetheless. Other influential linear transmission models include Gerbner's model and Berlo's model.
The earliest interaction model was developed by communication theorist Wilbur Schramm. He states that communication starts when a source has an idea and expresses it in the form of a message. This process is called "encoding" and happens using a code, i.e. a sign system that is able to express the idea, for instance, through visual or auditory signs. The message is sent to a destination, who has to decode and interpret it to understand it. In response, they formulate their own idea, encode it into a message, and send it back as a form of feedback. Another innovation of Schramm's model is that previous experience is necessary to be able to encode and decode messages. For communication to be successful, the fields of experience of source and destination have to overlap. The first transactional model was proposed by communication theorist Dean Barnlund in 1970. He understands communication as "the production of meaning, rather than the production of messages". Its goal is to decrease uncertainty and arrive at a shared understanding. This happens in response to external and internal cues. Decoding is the process of ascribing meaning to them and encoding consists in producing new behavioral cues as a response.
Human. There are many forms of human communication. A central distinction is whether language is used, as in the contrast between verbal and non-verbal communication. A further distinction concerns whether one communicates with others or with oneself, as in the contrast between interpersonal and intrapersonal communication. Forms of human communication are also categorized by their channel or the medium used to transmit messages. The field studying human communication is known as anthroposemiotics. Verbal. Verbal communication is the exchange of messages in linguistic form, i.e., by means of language. In colloquial usage, verbal communication is sometimes restricted to oral communication and may exclude writing and sign language. However, in academic discourse, the term is usually used in a wider sense, encompassing any form of linguistic communication, whether through speech, writing, or gestures. Some of the challenges in distinguishing verbal from non-verbal communication come from the difficulties in defining what exactly "language" means. Language is usually understood as a conventional system of symbols and rules used for communication. Such systems are based on a set of simple units of meaning that can be combined to express more complex ideas. The rules for combining the units into compound expressions are called grammar. Words are combined to form sentences.
One hallmark of human language, in contrast to animal communication, lies in its complexity and expressive power. Human language can be used to refer not just to concrete objects in the here-and-now but also to spatially and temporally distant objects and to abstract ideas. Humans have a natural tendency to acquire their native language in childhood. They are also able to learn other languages later in life as second languages. However, this process is less intuitive and often does not result in the same level of linguistic competence. The academic discipline studying language is called "linguistics". Its subfields include semantics (the study of meaning), morphology (the study of word formation), syntax (the study of sentence structure), pragmatics (the study of language use), and phonetics (the study of basic sounds). A central contrast among languages is between natural and artificial or constructed languages. Natural languages, like English, Spanish, and Japanese, developed naturally and for the most part unplanned in the course of history. Artificial languages, like Esperanto, Quenya, C++, and the language of first-order logic, are purposefully designed from the ground up. Most everyday verbal communication happens using natural languages. Central forms of verbal communication are speech and writing together with their counterparts of listening and reading. Spoken languages use sounds to produce signs and transmit meaning while for writing, the signs are physically inscribed on a surface. Sign languages, like American Sign Language and Nicaraguan Sign Language, are another form of verbal communication. They rely on visual means, mostly by using gestures with hands and arms, to form sentences and convey meaning.
Verbal communication serves various functions. One key function is to exchange information, i.e. an attempt by the speaker to make the audience aware of something, usually of an external event. But language can also be used to express the speaker's feelings and attitudes. A closely related role is to establish and maintain social relations with other people. Verbal communication is also utilized to coordinate one's behavior with others and influence them. In some cases, language is not employed for an external purpose but only for entertainment or personal enjoyment. Verbal communication further helps individuals conceptualize the world around them and themselves. This affects how perceptions of external events are interpreted, how things are categorized, and how ideas are organized and related to each other. Non-verbal. Non-verbal communication is the exchange of information through non-linguistic modes, like facial expressions, gestures, and postures. However, not every form of non-verbal behavior constitutes non-verbal communication. Some theorists, like Judee Burgoon, hold that it depends on the existence of a socially shared coding system that is used to interpret the meaning of non-verbal behavior. Non-verbal communication has many functions. It frequently contains information about emotions, attitudes, personality, interpersonal relations, and private thoughts.
Non-verbal communication often happens unintentionally and unconsciously, like sweating or blushing, but there are also conscious intentional forms, like shaking hands or raising a thumb. It often happens simultaneously with verbal communication and helps optimize the exchange through emphasis and illustration or by adding additional information. Non-verbal cues can clarify the intent behind a verbal message. Using multiple modalities of communication in this way usually makes communication more effective if the messages of each modality are consistent. However, in some cases different modalities can contain conflicting messages. For example, a person may verbally agree with a statement but press their lips together, thereby indicating disagreement non-verbally. There are many forms of non-verbal communication. They include kinesics, proxemics, haptics, paralanguage, chronemics, and physical appearance. Kinesics studies the role of bodily behavior in conveying information. It is commonly referred to as body language, even though it is, strictly speaking, not a language but rather non-verbal communication. It includes many forms, like gestures, postures, walking styles, and dance. Facial expressions, like laughing, smiling, and frowning, all belong to kinesics and are expressive and flexible forms of communication. Oculesics is another subcategory of kinesics in regard to the eyes. It covers questions like how eye contact, gaze, blink rate, and pupil dilation form part of communication. Some kinesic patterns are inborn and involuntary, like blinking, while others are learned and voluntary, like giving a military salute.
Proxemics studies how personal space is used in communication. The distance between the speakers reflects their degree of familiarity and intimacy with each other as well as their social status. Haptics examines how information is conveyed using touching behavior, like handshakes, holding hands, kissing, or slapping. Meanings linked to haptics include care, concern, anger, and violence. For instance, handshaking is often seen as a symbol of equality and fairness, while refusing to shake hands can indicate aggressiveness. Kissing is another form often used to show affection and erotic closeness. Paralanguage, also known as vocalics, encompasses non-verbal elements in speech that convey information. Paralanguage is often used to express the feelings and emotions that the speaker has but does not explicitly stated in the verbal part of the message. It is not concerned with the words used but with how they are expressed. This includes elements like articulation, lip control, rhythm, intensity, pitch, fluency, and loudness. For example, saying something loudly and in a high pitch conveys a different meaning on the non-verbal level than whispering the same words. Paralanguage is mainly concerned with spoken language but also includes aspects of written language, like the use of colors and fonts as well as spatial arrangement in paragraphs and tables. Non-linguistic sounds may also convey information; crying indicates that an infant is distressed, and babbling conveys information about infant health and well-being.
Chronemics concerns the use of time, such as what messages are sent by being on time versus late for a meeting. The physical appearance of the communicator, such as height, weight, hair, skin color, gender, clothing, tattooing, and piercing, also carries information. Appearance is an important factor for first impressions but is more limited as a mode of communication since it is less changeable. Some forms of non-verbal communication happen using such artifacts as drums, smoke, batons, traffic lights, and flags. Non-verbal communication can also happen through visual media like paintings and drawings. They can express what a person or an object looks like and can also convey other ideas and emotions. In some cases, this type of non-verbal communication is used in combination with verbal communication, for example, when diagrams or maps employ labels to include additional linguistic information. Traditionally, most research focused on verbal communication. However, this paradigm began to shift in the 1950s when research interest in non-verbal communication increased and emphasized its influence. For example, many judgments about the nature and behavior of other people are based on non-verbal cues. It is further present in almost every communicative act to some extent and certain parts of it are universally understood. These considerations have prompted some communication theorists, like Ray Birdwhistell, to claim that the majority of ideas and information is conveyed this way. It has also been suggested that human communication is at its core non-verbal and that words can only acquire meaning because of non-verbal communication. The earliest forms of human communication, such as crying and babbling, are non-verbal. Some basic forms of communication happen even before birth between mother and embryo and include information about nutrition and emotions. Non-verbal communication is studied in various fields besides communication studies, like linguistics, semiotics, anthropology, and social psychology.
Interpersonal. Interpersonal communication is communication between distinct people. Its typical form is dyadic communication, i.e. between two people, but it can also refer to communication within groups. It can be planned or unplanned and occurs in many forms, like when greeting someone, during salary negotiations, or when making a phone call. Some communication theorists, like Virginia M. McDermott, understand interpersonal communication as a fuzzy concept that manifests in degrees. In this view, an exchange varies in how interpersonal it is based on several factors. It depends on how many people are present, and whether it happens face-to-face rather than through telephone or email. A further factor concerns the relation between the communicators: group communication and mass communication are less typical forms of interpersonal communication and some theorists treat them as distinct types. Interpersonal communication can be synchronous or asynchronous. For asynchronous communication, the parties take turns in sending and receiving messages. This occurs when exchanging letters or emails. For synchronous communication, both parties send messages at the same time. This happens when one person is talking while the other person sends non-verbal messages in response signaling whether they agree with what is being said. Some communication theorists, like Sarah Trenholm and Arthur Jensen, distinguish between content messages and relational messages. Content messages express the speaker's feelings toward the topic of discussion. Relational messages, on the other hand, demonstrate the speaker's feelings toward their relation with the other participants.
Various theories of the function of interpersonal communication have been proposed. Some focus on how it helps people make sense of their world and create society. Others hold that its primary purpose is to understand why other people act the way they do and to adjust one's behavior accordingly. A closely related approach is to focus on information and see interpersonal communication as an attempt to reduce uncertainty about others and external events. Other explanations understand it in terms of the needs it satisfies. This includes the needs of belonging somewhere, being included, being liked, maintaining relationships, and influencing the behavior of others. On a practical level, interpersonal communication is used to coordinate one's actions with the actions of others to get things done. Research on interpersonal communication includes topics like how people build, maintain, and dissolve relationships through communication. Other questions are why people choose one message rather than another and what effects these messages have on the communicators and their relation. A further topic is how to predict whether two people would like each other.
Intrapersonal. Intrapersonal communication is communication with oneself. In some cases this manifests externally, such as when engaged in a monologue, taking notes, highlighting a passage, and writing a diary or a shopping list. But many forms of intrapersonal communication happen internally in the form of an inner exchange with oneself, as when thinking about something or daydreaming. Closely related to intrapersonal communication is communication that takes place within an organism below the personal level, such as exchange of information between organs or cells. Intrapersonal communication can be triggered by internal and external stimuli. It may happen in the form of articulating a phrase before expressing it externally. Other forms are to make plans for the future and to attempt to process emotions to calm oneself down in stressful situations. It can help regulate one's own mental activity and outward behavior as well as internalize cultural norms and ways of thinking. External forms of intrapersonal communication can aid one's memory. This happens, for example, when making a shopping list. Another use is to unravel difficult problems, as when solving a complex mathematical equation line by line. New knowledge can also be internalized this way, such as when repeating new vocabulary to oneself. Because of these functions, intrapersonal communication can be understood as "an exceptionally powerful and pervasive tool for thinking."
Based on its role in self-regulation, some theorists have suggested that intrapersonal communication is more basic than interpersonal communication. Young children sometimes use egocentric speech while playing in an attempt to direct their own behavior. In this view, interpersonal communication only develops later when the child moves from their early egocentric perspective to a more social perspective. A different explanation holds that interpersonal communication is more basic since it is first used by parents to regulate what their child does. Once the child has learned this, they can apply the same technique to themselves to get more control over their own behavior. Channels. For communication to be successful, the message has to travel from the sender to the receiver. The "channel" is the way this is accomplished. It is not concerned with the meaning of the message but only with the technical means of how the meaning is conveyed. Channels are often understood in terms of the senses used to perceive the message, i.e. hearing, seeing, smelling, touching, and tasting. But in the widest sense, channels encompass any form of transmission, including technological means like books, cables, radio waves, telephones, or television. Naturally transmitted messages usually fade rapidly whereas some messages using artificial channels have a much longer lifespan, as in the case of books or sculptures.
The physical characteristics of a channel have an impact on the code and cues that can be used to express information. For example, typical telephone calls are restricted to the use of verbal language and paralanguage but exclude facial expressions. It is often possible to translate messages from one code into another to make them available to a different channel. An example is writing down a spoken message or expressing it using sign language. The transmission of information can occur through multiple channels at once. For example, face-to-face communication often combines the auditory channel to convey verbal information with the visual channel to transmit non-verbal information using gestures and facial expressions. Employing multiple channels can enhance the effectiveness of communication by helping the receiver better understand the subject matter. The choice of channels often matters since the receiver's ability to understand may vary depending on the chosen channel. For instance, a teacher may decide to present some information orally and other information visually, depending on the content and the student's preferred learning style. This underlines the role of a media-adequate approach.
Communicative competence. Communicative competence is the ability to communicate effectively or to choose the appropriate communicative behavior in a given situation. It concerns what to say, when to say it, and how to say it. It further includes the ability to receive and understand messages. Competence is often contrasted with performance since competence can be present even if it is not exercised, while performance consists in the realization of this competence. However, some theorists reject a stark contrast and hold that performance is the observable part and is used to infer competence in relation to future performances. Two central components of communicative competence are effectiveness and appropriateness. Effectiveness is the degree to which the speaker achieves their desired outcomes or the degree to which preferred alternatives are realized. This means that whether a communicative behavior is effective does not just depend on the actual outcome but also on the speaker's intention, i.e. whether this outcome was what they intended to achieve. Because of this, some theorists additionally require that the speaker be able to give an explanation of why they engaged in one behavior rather than another. Effectiveness is closely related to efficiency, the difference being that effectiveness is about achieving goals while efficiency is about using few resources (such as time, effort, and money) in the process.
Appropriateness means that the communicative behavior meets social standards and expectations. Communication theorist Brian H. Spitzberg defines it as "the perceived legitimacy or acceptability of behavior or enactments in a given context". This means that the speaker is aware of the social and cultural context in order to adapt and express the message in a way that is considered acceptable in the given situation. For example, to bid farewell to their teacher, a student may use the expression "Goodbye, sir" but not the expression "I gotta split, man", which they may use when talking to a peer. To be both effective and appropriate means to achieve one's preferred outcomes in a way that follows social standards and expectations. Some definitions of communicative competence put their main emphasis on either effectiveness or appropriateness while others combine both features. Many additional components of communicative competence have been suggested, such as empathy, control, flexibility, sensitivity, and knowledge. It is often discussed in terms of the individual skills employed in the process, i.e. the specific behavioral components that make up communicative competence. Message production skills include reading and writing. They are correlated with the reception skills of listening and reading. There are both verbal and non-verbal communication skills. For example, verbal communication skills involve the proper understanding of a language, including its phonology, orthography, syntax, lexicon, and semantics.
Many aspects of human life depend on successful communication, from ensuring basic necessities of survival to building and maintaining relationships. Communicative competence is a key factor regarding whether a person is able to reach their goals in social life, like having a successful career and finding a suitable spouse. Because of this, it can have a large impact on the individual's well-being. The lack of communicative competence can cause problems both on the individual and the societal level, including professional, academic, and health problems. Barriers to effective communication can distort the message. They may result in failed communication and cause undesirable effects. This can happen if the message is poorly expressed because it uses terms with which the receiver is not familiar, or because it is not relevant to the receiver's needs, or because it contains too little or too much information. Distraction, selective perception, and lack of attention to feedback may also be responsible. Noise is another negative factor. It concerns influences that interfere with the message on its way to the receiver and distort it. Crackling sounds during a telephone call are one form of noise. Ambiguous expressions can also inhibit effective communication and make it necessary to disambiguate between possible interpretations to discern the sender's intention. These interpretations depend also on the cultural background of the participants. Significant cultural differences constitute an additional obstacle and make it more likely that messages are misinterpreted.
Other species. Besides human communication, there are many other forms of communication found in the animal kingdom and among plants. They are studied in fields like biocommunication and biosemiotics. There are additional obstacles in this area for judging whether communication has taken place between two individuals. Acoustic signals are often easy to notice and analyze for scientists, but it is more difficult to judge whether tactile or chemical changes should be understood as communicative signals rather than as other biological processes. For this reason, researchers often use slightly altered definitions of communication to facilitate their work. A common assumption in this regard comes from evolutionary biology and holds that communication should somehow benefit the communicators in terms of natural selection. The biologists Rumsaïs Blatrix and Veronika Mayer define communication as "the exchange of information between individuals, wherein both the signaller and receiver may expect to benefit from the exchange". According to this view, the sender benefits by influencing the receiver's behavior and the receiver benefits by responding to the signal. These benefits should exist on average but not necessarily in every single case. This way, deceptive signaling can also be understood as a form of communication. One problem with the evolutionary approach is that it is often difficult to assess the impact of such behavior on natural selection. Another common pragmatic constraint is to hold that it is necessary to observe a response by the receiver following the signal when judging whether communication has occurred.
Animals. Animal communication is the process of giving and taking information among animals. The field studying animal communication is called zoosemiotics. There are many parallels to human communication. One is that humans and many animals express sympathy by synchronizing their movements and postures. Nonetheless, there are also significant differences, like the fact that humans also engage in verbal communication, which uses language, while animal communication is restricted to non-verbal (i.e. non-linguistic) communication. Some theorists have tried to distinguish human from animal communication based on the claim that animal communication lacks a referential function and is thus not able to refer to external phenomena. However, various observations seem to contradict this view, such as the warning signals in response to different types of predators used by vervet monkeys, Gunnison's prairie dogs, and red squirrels. A further approach is to draw the distinction based on the complexity of human language, especially its almost limitless ability to combine basic units of meaning into more complex meaning structures. One view states that recursion sets human language apart from all non-human communicative systems. Another difference is that human communication is frequently linked to the conscious intention to send information, which is often not discernable for animal communication. Despite these differences, some theorists use the term "animal language" to refer to certain communicative patterns in animal behavior that have similarities with human language.
Animal communication can take a variety of forms, including visual, auditory, tactile, olfactory, and gustatory communication. Visual communication happens in the form of movements, gestures, facial expressions, and colors. Examples are movements seen during mating rituals, the colors of birds, and the rhythmic light of fireflies. Auditory communication takes place through vocalizations by species like birds, primates, and dogs. Auditory signals are frequently used to alert and warn. Lower-order living systems often have simple response patterns to auditory messages, reacting either by approach or avoidance. More complex response patterns are observed for higher animals, which may use different signals for different types of predators and responses. For example, some primates use one set of signals for airborne predators and another for land predators. Tactile communication occurs through touch, vibration, stroking, rubbing, and pressure. It is especially relevant for parent-young relations, courtship, social greetings, and defense. Olfactory and gustatory communication happen chemically through smells and tastes, respectively.
There are large differences between species concerning what functions communication plays, how much it is realized, and the behavior used to communicate. Common functions include the fields of courtship and mating, parent-offspring relations, social relations, navigation, self-defense, and territoriality. One part of courtship and mating consists in identifying and attracting potential mates. This can happen through various means. Grasshoppers and crickets communicate acoustically by using songs, moths rely on chemical means by releasing pheromones, and fireflies send visual messages by flashing light. For some species, the offspring depends on the parent for its survival. One central function of parent-offspring communication is to recognize each other. In some cases, the parents are also able to guide the offspring's behavior. Social animals, like chimpanzees, bonobos, wolves, and dogs, engage in various forms of communication to express their feelings and build relations. Communication can aid navigation by helping animals move through their environment in a purposeful way, e.g. to locate food, avoid enemies, and follow other animals. In bats, this happens through echolocation, i.e. by sending auditory signals and processing the information from the echoes. Bees are another often-discussed case in this respect since they perform a type of dance to indicate to other bees where flowers are located. In regard to self-defense, communication is used to warn others and to assess whether a costly fight can be avoided. Another function of communication is to mark and claim territories used for food and mating. For example, some male birds claim a hedge or part of a meadow by using songs to keep other males away and attract females.
Two competing theories in the study of animal communication are nature theory and nurture theory. Their conflict concerns to what extent animal communication is programmed into the genes as a form of adaptation rather than learned from previous experience as a form of conditioning. To the degree that it is learned, it usually happens through imprinting, i.e. as a form of learning that only occurs in a certain phase and is then mostly irreversible. Plants, fungi, and bacteria. Plant communication refers to plant processes involving the sending and receiving of information. The field studying plant communication is called phytosemiotics. This field poses additional difficulties for researchers since plants are different from humans and other animals in that they lack a central nervous system and have rigid cell walls. These walls restrict movement and usually prevent plants from sending and receiving signals that depend on rapid movement. However, there are some similarities since plants face many of the same challenges as animals. For example, they need to find resources, avoid predators and pathogens, find mates, and ensure that their offspring survive. Many of the evolutionary responses to these challenges are analogous to those in animals but are implemented using different means. One crucial difference is that chemical communication is much more prominent in the plant kingdom in contrast to the importance of visual and auditory communication for animals.
In plants, the term "behavior" is usually not defined in terms of physical movement, as is the case for animals, but as a biochemical response to a stimulus. This response has to be short relative to the plant's lifespan. Communication is a special form of behavior that involves conveying information from a sender to a receiver. It is distinguished from other types of behavior, like defensive reactions and mere sensing. Like in the field of animal communication, plant communication researchers often require as additional criteria that there is some form of response in the receiver and that the communicative behavior is beneficial to sender and receiver. Biologist Richard Karban distinguishes three steps of plant communication: the emission of a cue by a sender, the perception of the cue by a receiver, and the receiver's response. For plant communication, it is not relevant to what extent the emission of a cue is intentional. However, it should be possible for the receiver to ignore the signal. This criterion can be used to distinguish a response to a signal from a defense mechanism against an unwanted change like intense heat.
Plant communication happens in various forms. It includes communication within plants, i.e. within plant cells and between plant cells, between plants of the same or related species, and between plants and non-plant organisms, especially in the root zone. A prominent form of communication is airborne and happens through volatile organic compounds (VOCs). For example, maple trees release VOCs when they are attacked by a herbivore to warn neighboring plants, which then react accordingly by adjusting their defenses. Another form of plant-to-plant communication happens through mycorrhizal fungi. These fungi form underground networks, colloquially referred to as the Wood-Wide Web, and connect the roots of different plants. The plants use the network to send messages to each other, specifically to warn other plants of a pest attack and to help prepare their defenses. Communication can also be observed for fungi and bacteria. Some fungal species communicate by releasing pheromones into the external environment. For instance, they are used to promote sexual interaction in several aquatic fungal species. One form of communication between bacteria is called quorum sensing. It happens by releasing hormone-like molecules, which other bacteria detect and respond to. This process is used to monitor the environment for other bacteria and to coordinate population-wide responses, for example, by sensing the density of bacteria and regulating gene expression accordingly. Other possible responses include the induction of bioluminescence and the formation of biofilms.