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Prince-Bishopric of Brandenburg The Prince-Bishopric of Brandenburg () was an ecclesiastical principality of the Holy Roman Empire from the 12th century until it was secularized during the second half of the 16th century. It should not be confused with the larger Diocese of Brandenburg () established by King Otto I of Germany in 948, in the territory of the "Marca Geronis" (Saxon Eastern March) east of the Elbe river. The diocese, over which the prince-bishop exercised only spiritual authority, was a suffragan diocese of the Archdiocese of Magdeburg, its seat was Brandenburg an der Havel. History. The Prince-Bishopric of Brandenburg was an imperial estate of the Holy Roman Empire for some time, probably starting about 1161/1165. However, the Brandenburg bishops never managed to gain control over a significant territory, being overshadowed by the Margraviate of Brandenburg, which was originally seated in the same city. Chapter and cathedral, surrounded by further ecclesiastical institutions, were located on the "Dominsel" (Cathedral Island), which formed a prince-episcopal cathedral immunity district ("Domfreiheit"), distinct from the city of Brandenburg. Only in 1929 the - meanwhile former - immunity district was incorporated into the city itself.
As rulers of imperial immediacy, regnant in a, however, dispersed territory partitioned into the four bailiwicks () of Brandenburg/Havel, Ketzin, Teltow and Ziesar. The prince-bishops from the early 14th century onwards resided in their fortress in Ziesar on the road to Magdeburg. The last actual bishop was Matthias von Jagow (d. 1544), who took the side of the Protestant Reformation, married, and in every way furthered the undertakings of the Hohenzollern elector Joachim II. There were two more nominal bishops, but on the petition of the latter of these, the electoral prince John George of Brandenburg appointed in 1560, the secularisation of the bishopric was undertaken and finally accomplished in 1571, in spite of legal proceedings to reassert the imperial immediacy of the prince-bishopric within the Empire and so to likewise preserve the diocese, which dragged on into the 17th century. Prince-bishops. Lutherans. Secularized and merged into Brandenburg.
BASE jumping BASE jumping () is the recreational sport of jumping from fixed objects, using a parachute to descend to the ground. BASE is an acronym that stands for four categories of fixed objects from which one can jump: buildings, antennas (referring to radio masts), spans (bridges) and earth (cliffs). Participants jump from a fixed object such as a cliff and after an optional freefall delay deploy a parachute to slow their descent and land. A popular form of BASE jumping is wingsuit BASE jumping. In contrast to other forms of parachuting, such as skydiving from airplanes, BASE jumps are performed from fixed objects that are generally at much lower altitudes, and BASE jumpers only carry one parachute. BASE jumping is significantly more hazardous than other forms of parachuting and is widely considered to be one of the most dangerous extreme sports. History. Precursors. Fausto Veranzio is widely believed to have been the first person to build and test a parachute, by jumping from St Mark's Campanile in Venice in 1617 when he was more than 65 years old. However these and other sporadic incidents were one-time experiments, not the actual systematic pursuit of a new form of parachuting.
Birth of B.A.S.E. jumping. There are precursors to the sport dating back hundreds of years. In 1966 Michael Pelkey and Brian Schubert jumped from El Capitan in Yosemite National Park. The acronym B.A.S.E. (now more commonly BASE) was later coined by filmmaker Carl Boenish, his wife Jean Boenish, Phil Smith and Phil Mayfield. Carl Boenish was an important catalyst behind modern BASE jumping and in 1978 he filmed jumps from El Capitan made using ram-air parachutes and the freefall tracking technique. While BASE jumps had been made prior to that time, the El Capitan activity was the effective birth of what is now called BASE jumping. After 1978 the filmed jumps from El Capitan were repeated, not as an actual publicity exercise or as a movie stunt but as a true recreational activity. It was this that popularized BASE jumping more widely among parachutists. Carl Boenish continued to publish films and informational magazines on BASE jumping until his death in 1984 after a BASE jump off the Troll Wall. By this time the concept had spread among skydivers worldwide, with hundreds of participants making fixed-object jumps.
During the early eighties nearly all BASE jumps were made using standard skydiving equipment, including two parachutes (main and reserve), and deployment components. Later on, specialized equipment and techniques were developed specifically for the unique needs of BASE jumping. Nowadays, recognizing the sport's growing appeal and the potential for high-impact marketing, companies such as Red Bull have stepped in to sponsor athletes, further elevating the sport's profile. BASE numbers. BASE numbers are awarded to those who have made at least one jump from each of the four categories (buildings, antennae, spans and earth). When Phil Smith and Phil Mayfield jumped together from a Houston skyscraper on 18 January 1981, they became the first to attain the exclusive BASE numbers (BASE #1 and #2, respectively), having already jumped from an antenna, spans, and earthen objects. Jean and Carl Boenish qualified for BASE numbers 3 and 4 soon after. A separate "award" was soon enacted for Night BASE jumping when Mayfield completed each category at night, becoming Night BASE #1, with Smith qualifying a few weeks later.
Upon completing a jump from all of the four object categories, a jumper may choose to apply for a "BASE number", awarded sequentially. The 1000th application for a BASE number was filed in March 2005 and BASE #1000 was awarded to Matt "Harley" Moilanen of Grand Rapids, Michigan. , over 2,000 BASE numbers have been issued. Equipment. In the early days of BASE jumping, people used modified skydiving gear, such as by removing the deployment bag and slider, stowing the lines in a tail pocket, and fitting a large pilot chute. However, modified skydiving gear is then prone to kinds of malfunction that are rare in normal skydiving (such as "line-overs" and broken lines). Modern purpose-built BASE jumping equipment is considered to be much safer and more reliable. Parachute. The biggest difference in gear is that skydivers jump with both a main and a reserve parachute, while BASE jumpers carry only one parachute. BASE jumping parachutes are larger than skydiving parachutes and are typically flown with a wing loading of around . Vents are one element that make a parachute suitable for BASE jumping. BASE jumpers often use extra large pilot chutes to compensate for lower airspeed parachute deployments. On jumps from lower altitudes, the slider is removed for faster parachute opening.
Harness and container. BASE jumpers use a single-parachute harness and container system. Since there is only a single parachute, BASE jumping containers are mechanically much simpler than skydiving containers. This simplicity contributes to the safety and reliability of BASE jumping gear by eliminating many malfunctions that can occur with more complicated skydiving equipment. Since there is no reserve parachute, there is little need to cut-away their parachute, and many BASE harnesses do not contain a 3-ring release system. A modern ultralight BASE system including parachute, container, and harness can weigh as little as . Clothing. When jumping from high mountains, BASE jumpers will often use special clothing to improve control and flight characteristics in the air. Wingsuit flying has become a popular form of BASE jumping in recent years, that allows jumpers to glide over long horizontal distances. Tracking suits inflate like wingsuits to give additional lift to jumpers, but maintain separation of arms and legs to allow for greater mobility and safety.
Technique. BASE jumps can be broadly classified into low jumps and high jumps. The primary distinguishing characteristic of low BASE jumps versus high BASE jumps is the use of a slider reefing device to control the opening speed of the parachute, and whether the jumper falls long enough to reach terminal velocity. Low BASE jumps. Low BASE jumps are those where the jumper does not reach terminal velocity. Sometimes referred to as "slider down" jumps because they are typically performed without a slider reefing device on the parachute. The lack of a slider enables the parachute to open more quickly. Other techniques for low BASE jumps include the use of a static line, direct bag, or P.C.A. (pilot chute assist). These devices form an attachment between the parachute and the jump platform, which stretches out the parachute and suspension lines as the jumper falls, before separating and allowing the parachute to inflate. This enables the very lowest jumps—below to be made. It is common in the UK to jump from around the mark, due to the number of low cliffs at this height. Base jumpers have been known to jump from objects as low as , which leaves little to no canopy time and requires an immediate flare to land safely.
High BASE jumps. Many BASE jumpers are motivated to make jumps from higher objects involving free fall. High BASE jumps are those which are high enough for the jumper to reach terminal velocity. High BASE jumps are often called "slider up" jumps due to the use of a slider reefing device. High BASE jumps present different hazards than low BASE jumps. With greater height and airspeed, jumpers can fly away from the cliff during freefall, allowing them to deploy their parachute far away from the cliff they jumped from and significantly reduce the chance of object striking. However, high BASE jumps also present new hazards such as complications resulting from the use of a wingsuit. Tandem BASE jumps. Tandem BASE jumping is when a skilled pilot jumps with a passenger attached to their front. It is similar to skydiving and is offered in the US and many other countries. Tandem BASE is becoming a more accessible and legal form of BASE jumping. Records. On May 23, 2006, Australians Glenn Singleman and Heather Swan made a BASE jump from an altitude of off Meru Peak in Northern India. They jumped in wingsuits.
On May 5, 2013, Russian Valery Rozov jumped off Changtse (the northern peak of the Mount Everest massif) from a height of . Using a specially-developed wingsuit, he glided down to the Rongbuk glacier more than 1,000 meters below, setting a new world record for highest altitude base jump. He had previously jumped off mountains in Asia, Antarctica and South America in 2004, 2007, 2008, 2010 and 2012. On October 5, 2016, Rozov broke his own record for highest altitude BASE jump when he leapt from a height of from Cho Oyu, the sixth-highest mountain in the world, landing on a glacier approximately two minutes later at an altitude of around . He later died while attempting another high-altitude BASE jump in Nepal in 2017. Competitions. BASE competitions have been held since the early 1980s, with accurate landings or free-fall aerobatics used as the judging criteria. Recent years have seen a formal competition held at the high Petronas Towers in Kuala Lumpur, Malaysia, judged on landing accuracy. In 2012 the World Wingsuit League held their first wingsuit BASE jumping competition in China.
Comparison with skydiving. BASE jumps are typically performed from much lower altitudes than in skydiving. Skydivers are required to deploy their main parachute above altitude. BASE jumps are frequently made from less than . A BASE jump from a object is only about 5.6 seconds from the ground if the jumper remains in free fall. Standard skydiving parachute systems are not designed for this situation, so BASE jumpers use specially designed harnesses and parachute systems. Many BASE jumps, particularly in the UK are made from around due to the number of low cliffs at this height. Jumpers will use a static line method to ensure their canopy is extracted as they jump, as at this height, it is too low to freefall. BASE jumps generally entail slower airspeeds than typical skydives (due to the limited altitude), a BASE jumper does not always reach terminal velocity. Skydivers use the airflow to stabilize their position. BASE jumpers, falling at lower speeds, have less aerodynamic control. The attitude of the body at the moment of jumping determines the stability of flight in the first few seconds, before sufficient airspeed has built up to enable aerodynamic stability. On low BASE jumps, parachute deployment takes place during this early phase of flight. If the parachute is deployed while the jumper is unstable, there is a high risk of entanglement or malfunction. The jumper may also not be facing the right direction. Such an off-heading opening is not as problematic in skydiving, but an off-heading opening that results in object strike has caused many serious injuries and deaths in BASE jumping.
BASE jumps are more hazardous than skydives primarily due to proximity to the object serving as the jump platform. BASE jumping frequently occurs in mountainous terrain, often having much smaller areas in which to land in comparison to a typical skydiving dropzone. BASE jumping is significantly more dangerous than similar sports such as skydiving from aircraft. Legality. BASE jumping is generally not illegal in most places. However, in some cases such as building and antenna jumps, jumping is often done covertly without the permission of owners, which can lead to charges such as trespassing. In some jurisdictions it may be permissible to use land until specifically told not to. The Perrine Bridge in Twin Falls, Idaho, is an example of a man-made structure in the United States where BASE jumping is allowed year-round without a permit. In U.S. National Parks, BASE jumping is allowed pursuant to the terms of a special use permit. There is no record of the U.S. National Parks granting a permit for BASE jumping, with the sole exception of a permit granted for the annual Bridge Day event in New River Gorge National Park. Other U.S. public land, including land controlled by the Bureau of Land Management, does not ban BASE jumping, and there are numerous jumpable objects on BLM land.
The legal position is different at other sites and in other countries. For example, in Norway's Lysefjord (from the mountain Kjerag), BASE jumpers are made welcome. Many sites in the European Alps, near Chamonix and on the Eiger, are also open to jumpers. Some other Norwegian places, like the Troll Wall, are banned because of dangerous rescue missions in the past. In Austria, jumping from mountain cliffs is generally allowed, whereas the use of bridges (such as the Europabruecke near Innsbruck, Tirol) or dams is generally prohibited. Australia has some of the toughest stances on BASE jumping: it specifically bans BASE jumping from certain objects, such as the Sydney Harbour Bridge. Safety. Recent statistics reflect improved safety of the activity. The 2024 Bridge Day Event which saw the participation of 325 BASE jumpers completing 755 BASE jumps resulted in zero jumper injuries, in contrast to two non-jumper related medical transports from the same event. And in the year 2023 the United States saw zero BASE fatalities. A study of BASE jumping fatalities estimated that the overall annual fatality risk in 2002 was one fatality per 60 participants. A study of 20,850 BASE jumps from the Kjerag Massif in Norway reported nine fatalities over the 11-year period from 1995 to 2005, or one in every 2,317 jumps. However, at that site, one in every 254 jumps over that period resulted in a nonfatal accident. BASE jumping is one of the most dangerous recreational activities in the world, with a fatality and injury rate 43 times higher than that of parachuting from a plane. , the BASE Fatality List records 480 deaths for BASE jumping since April 1981.
Beauty Beauty is commonly described as a feature of objects that makes them pleasurable to perceive. Such objects include landscapes, sunsets, humans and works of art. Beauty, art and taste are the main subjects of aesthetics, one of the fields of study within philosophy. As a positive aesthetic value, it is contrasted with ugliness as its negative counterpart. One difficulty in understanding beauty is that it has both objective and subjective aspects: it is seen as a property of things but also as depending on the emotional response of observers. Because of its subjective side, beauty is said to be "in the eye of the beholder". It has been argued that the ability on the side of the subject needed to perceive and judge beauty, sometimes referred to as the "sense of taste", can be trained and that the verdicts of experts coincide in the long run. This suggests the standards of validity of judgments of beauty are intersubjective, i.e. dependent on a group of judges, rather than fully subjective or objective. Conceptions of beauty aim to capture what is essential to all beautiful things. "Classical conceptions" define beauty in terms of the relation between the beautiful object as a whole and its parts: the parts should stand in the right proportion to each other and thus compose an integrated harmonious whole. "Hedonist conceptions" see a necessary connection between pleasure and beauty, e.g. that for an object to be beautiful is for it to cause disinterested pleasure. Other conceptions include defining beautiful objects in terms of their value, of a loving attitude toward them or of their function.
Overview. Beauty, together with art and taste, is the main subject of aesthetics, one of the major branches of philosophy. Beauty is usually categorized as an aesthetic property besides other properties, like grace, elegance or the sublime. As a positive aesthetic value, beauty is contrasted with ugliness as its negative counterpart. Beauty is often listed as one of the three fundamental concepts of human understanding besides truth and goodness. Objectivists or realists see beauty as an objective or mind-independent feature of beautiful things, which is denied by "subjectivists". The source of this debate is that judgments of beauty seem to be based on subjective grounds, namely our feelings, while claiming universal correctness at the same time. This tension is sometimes referred to as the "antinomy of taste". Adherents of both sides have suggested that a certain faculty, commonly called a "sense of taste", is necessary for making reliable judgments about beauty. David Hume, for example, suggests that this faculty can be trained and that the verdicts of experts coincide in the long run.
Beauty is mainly discussed in relation to "concrete objects" accessible to sensory perception. It has been suggested that the beauty of a thing supervenes on the sensory features of this thing. It has also been proposed that abstract objects like stories or mathematical proofs can be beautiful. Beauty plays a central role in works of art and nature. An influential distinction among beautiful things, according to Immanuel Kant, is that between "adherent" beauty ("pulchritudo adhaerens") and "free beauty" ("pulchritudo vaga"). A thing has adherent beauty if its beauty depends on the conception or function of this thing, unlike free or absolute beauty. Examples of adherent beauty include an ox which is beautiful as an ox but not beautiful as a horse or a photograph which is beautiful, because it depicts a beautiful building but that lacks beauty generally speaking because of its low quality. Objectivism and subjectivism. Judgments of beauty seem to occupy an intermediary position between objective judgments, e.g. concerning the mass and shape of a grapefruit, and subjective likes, e.g. concerning whether the grapefruit tastes good. Judgments of beauty differ from the former because they are based on subjective feelings rather than objective perception. But they also differ from the latter because they lay claim on universal correctness. This tension is also reflected in common language. On the one hand, we talk about beauty as an objective feature of the world that is ascribed, for example, to landscapes, paintings or humans. The subjective side, on the other hand, is expressed in sayings like "beauty is in the eye of the beholder".
These two positions are often referred to as "objectivism" (or realism) and "subjectivism". "Objectivism" is the traditional view, while subjectivism developed more recently in western philosophy. Objectivists hold that beauty is a mind-independent feature of things. On this account, the beauty of a landscape is independent of who perceives it or whether it is perceived at all. Disagreements may be explained by an inability to perceive this feature, sometimes referred to as a "lack of taste". Subjectivism, on the other hand, denies the mind-independent existence of beauty. Influential for the development of this position was John Locke's distinction between primary qualities, which the object has independent of the observer, and secondary qualities, which constitute powers in the object to produce certain ideas in the observer. When applied to beauty, there is still a sense in which it depends on the object and its powers. But this account makes the possibility of genuine disagreements about claims of beauty implausible, since the same object may produce very different ideas in distinct observers. The notion of "taste" can still be used to explain why different people disagree about what is beautiful, but there is no objectively right or wrong taste, there are just different tastes.
The problem with both the objectivist and the subjectivist position in their extreme form is that each has to deny some intuitions about beauty. This issue is sometimes discussed under the label "antinomy of taste". It has prompted various philosophers to seek a unified theory that can take all these intuitions into account. One promising route to solve this problem is to move from subjective to "intersubjective theories", which hold that the standards of validity of judgments of taste are intersubjective or dependent on a group of judges rather than objective. This approach tries to explain how genuine disagreement about beauty is possible despite the fact that beauty is a mind-dependent property, dependent not on an individual but a group. A closely related theory sees beauty as a "secondary" or "response-dependent property". On one such account, an object is beautiful "if it causes pleasure by virtue of its aesthetic properties". The problem that different people respond differently can be addressed by combining response-dependence theories with so-called "ideal-observer theories": it only matters how an ideal observer would respond. There is no general agreement on how "ideal observers" are to be defined, but it is usually assumed that they are experienced judges of beauty with a fully developed sense of taste. This suggests an indirect way of solving the "antinomy of taste": instead of looking for necessary and sufficient conditions of beauty itself, one can learn to identify the qualities of good critics and rely on their judgments. This approach only works if unanimity among experts was ensured. But even experienced judges may disagree in their judgments, which threatens to undermine ideal-observer theories.
Conceptions. Classical. The "classical conception" (see Classicism) defines beauty in terms of the relation between the beautiful object "as a whole" and its "parts": the parts should stand in the right proportion to each other and thus compose an integrated harmonious whole. On this account, which found its most explicit articulation in the Italian Renaissance, the beauty of a human body, for example, depends, among other things, on the right proportion of the different parts of the body and on the overall symmetry. One problem with this conception is that it is difficult to give a general and detailed description of what is meant by "harmony between parts" and raises the suspicion that defining beauty through harmony results in exchanging one unclear term for another one. Some attempts have been made to dissolve this suspicion by searching for "laws of beauty", like the golden ratio. 18th century philosopher Alexander Baumgarten, for example, saw laws of beauty in analogy with laws of nature and believed that they could be discovered through empirical research. As of 2003, these attempts have failed to find a general definition of beauty and several authors take the opposite claim that such laws cannot be formulated, as part of their definition of beauty.
Hedonism. A very common element in many conceptions of beauty is its relation to pleasure. Hedonism makes this relation part of the definition of beauty by holding that there is a necessary connection between pleasure and beauty, e.g. that for an object to be beautiful is for it to cause pleasure or that the experience of beauty is always accompanied by pleasure. This account is sometimes labeled as "aesthetic hedonism" in order to distinguish it from other forms of hedonism. An influential articulation of this position comes from Thomas Aquinas, who treats beauty as "that which pleases in the very apprehension of it". Immanuel Kant explains this pleasure through a harmonious interplay between the faculties of understanding and imagination. A further question for hedonists is how to explain the relation between beauty and pleasure. This problem is akin to the Euthyphro dilemma: is something beautiful because we enjoy it or do we enjoy it because it is beautiful? Identity theorists solve this problem by denying that there is a difference between beauty and pleasure: they identify beauty, or the appearance of it, with the experience of aesthetic pleasure.
Hedonists usually restrict and specify the notion of pleasure in various ways in order to avoid obvious counterexamples. One important distinction in this context is the difference between "pure" and "mixed pleasure". Pure pleasure excludes any form of pain or unpleasant feeling while the experience of mixed pleasure can include unpleasant elements. But beauty can involve mixed pleasure, for example, in the case of a beautifully tragic story, which is why mixed pleasure is usually allowed in hedonist conceptions of beauty. Another problem faced by hedonist theories is that we take pleasure from many things that are not beautiful. One way to address this issue is to associate beauty with a special type of pleasure: "aesthetic" or "disinterested pleasure". A pleasure is disinterested if it is indifferent to the existence of the beautiful object or if it did not arise owing to an antecedent desire through means-end reasoning. For example, the joy of looking at a beautiful landscape would still be valuable if it turned out that this experience was an illusion, which would not be true if this joy was due to seeing the landscape as a valuable real estate opportunity. Opponents of hedonism usually concede that many experiences of beauty are pleasurable but deny that this is true for all cases. For example, a cold jaded critic may still be a good judge of beauty because of her years of experience but lack the joy that initially accompanied her work. One way to avoid this objection is to allow responses to beautiful things to lack pleasure while insisting that all beautiful things merit pleasure, that aesthetic pleasure is the only appropriate response to them.
Others. G. E. Moore explained beauty in regard to intrinsic value as "that of which the admiring contemplation is good in itself". This definition connects beauty to experience while managing to avoid some of the problems usually associated with subjectivist positions since it allows that things may be beautiful even if they are never experienced. Another subjectivist theory of beauty comes from George Santayana, who suggested that we project pleasure onto the things we call "beautiful". So in a process akin to a category mistake, one treats one's subjective pleasure as an objective property of the beautiful thing. Other conceptions include defining beauty in terms of a loving or longing attitude toward the beautiful object or in terms of its usefulness or function. In 1871, functionalist Charles Darwin explained beauty as result of accumulative sexual selection in "The Descent of Man and Selection in Relation to Sex". In philosophy. Greco-Roman tradition. The classical Greek noun that best translates to the English-language words "beauty" or "beautiful" was κάλλος, "kallos", and the adjective was καλός, "kalos". This is also translated as "good" or "of fine quality" and thus has a broader meaning than mere physical or material beauty. Similarly, "kallos" was used differently from the English word beauty in that it first and foremost applied to humans and bore an erotic connotation. The Koine Greek word for beautiful was ὡραῖος, "hōraios", an adjective etymologically coming from the word ὥρα, "hōra", meaning "hour". In Koine Greek, beauty was thus associated with "being of one's hour". Thus, a ripe fruit (of its time) was considered beautiful, whereas a young woman trying to appear older or an older woman trying to appear younger would not be considered beautiful. In Attic Greek, "hōraios" had many meanings, including "youthful" and "ripe old age". Another classical term in use to describe beauty was "" (Latin).
Beauty for ancient thinkers existed both in , which is the material world as it is, and as embodied in the spirit, which is the world of mental formations. Greek mythology mentions Helen of Troy as the most beautiful woman. Ancient Greek architecture is based on this view of symmetry and proportion. Pre-Socratic. In one fragment of Heraclitus's writings ("Fragment 106") he mentions beauty, this reads: "To God all things are beautiful, good, right..." The earliest Western theory of beauty can be found in the works of early Greek philosophers from the pre-Socratic period, such as Pythagoras, who conceived of beauty as useful for a moral education of the soul. He wrote of how people experience pleasure when aware of a certain type of formal situation present in reality, perceivable by sight or through the ear and discovered the underlying mathematical ratios in the harmonic scales in music. The Pythagoreans conceived of the presence of beauty in universal terms, which is, as existing in a cosmological state, they observed beauty in "the heavens". They saw a strong connection between mathematics and beauty. In particular, they noted that objects proportioned according to the golden ratio seemed more attractive.
Classical period. The classical concept of beauty is one that exhibits perfect proportion (Wolfflin). In this context, the concept belonged often within the discipline of mathematics. An idea of spiritual beauty emerged during the classical period, beauty was something embodying divine goodness, while the demonstration of behaviour which might be classified as beautiful, from an inner state of morality which is aligned to the good. The writing of Xenophon shows a conversation between Socrates and Aristippus. Socrates discerned differences in the conception of the beautiful, for example, in inanimate objects, the effectiveness of execution of design was a deciding factor on the perception of beauty in something. By the account of Xenophon, Socrates found beauty congruent with that to which was defined as the morally good, in short, he thought beauty coincident with the good.
Plato also discusses beauty in his work "Phaedrus", and identifies Alcibiades as beautiful in "Parmenides". He considered beauty to be the Idea (Form) above all other Ideas. Platonic thought synthesized beauty with the divine. Scruton (cited: Konstan) states Plato states of the idea of beauty, of it (the idea), being something inviting desirousness (cf. ), and, promotes an intellectual (cf. ) of desire. For Alexander Nehamas, it is only the locating of desire to which the sense of beauty exists, in the considerations of Plato. Aristotle defines beauty in Metaphysics as having order, symmetry and definiteness "which the mathematical sciences exhibit to a special degree". He saw a relationship between the beautiful ("to kalon") and virtue, arguing that "Virtue aims at the beautiful." Roman. In "De Natura Deorum", Cicero wrote: "the splendour and beauty of creation", in respect to this, and all the facets of reality resulting from creation, he postulated these to be a reason to see the existence of a God as creator.
Western Middle Ages. In the Middle Ages, Catholic philosophers like Thomas Aquinas included beauty among the transcendental attributes of being. In his Summa Theologica, Aquinas described the three conditions of beauty as: integritas (wholeness), consonantia (harmony and proportion), and claritas (a radiance and clarity that makes the form of a thing apparent to the mind). In the Gothic Architecture of the High and Late Middle Ages, light was considered the most beautiful revelation of God, which was heralded in design. Examples are the stained glass of Gothic Cathedrals including Notre-Dame de Paris and Chartres Cathedral. St. Augustine said of beauty "Beauty is indeed a good gift of God; but that the good may not think it a great good, God dispenses it even to the wicked." Renaissance. Classical philosophy and sculptures of men and women produced according to the Greek philosophers' tenets of ideal human beauty were rediscovered in Renaissance Europe, leading to a re-adoption of what became known as a "classical ideal". In terms of female human beauty, a woman whose appearance conforms to these tenets is still called a "classical beauty" or said to possess a "classical beauty", whilst the foundations laid by Greek and Roman artists have also supplied the standard for male beauty and female beauty in western civilization as seen, for example, in the "Winged Victory of Samothrace". During the Gothic era, the classical aesthetical canon of beauty was rejected as sinful. Later, Renaissance and Humanist thinkers rejected this view, and considered beauty to be the product of rational order and harmonious proportions. Renaissance artists and architects (such as Giorgio Vasari in his "Lives of Artists") criticised the Gothic period as irrational and barbarian. This point of view of Gothic art lasted until Romanticism, in the 19th century. Vasari aligned himself to the classical notion and thought of beauty as defined as arising from proportion and order.
Age of Reason. The Age of Reason saw a rise in an interest in beauty as a philosophical subject. For example, Scottish philosopher Francis Hutcheson argued that beauty is "unity in variety and variety in unity". He wrote that beauty was neither purely subjective nor purely objective—it could be understood not as "any Quality suppos'd to be in the Object, which should of itself be beautiful, without relation to any Mind which perceives it: For Beauty, like other Names of sensible Ideas, properly denotes the "Perception" of some mind; ... however we generally imagine that there is something in the Object just like our Perception." Immanuel Kant believed that there could be no "universal criterion of the beautiful" and that the experience of beauty is subjective, but that an object is judged to be beautiful when it seems to display "purposiveness"; that is, when its form is perceived to have the character of a thing designed according to some principle and fitted for a purpose. He distinguished "free beauty" from "merely adherent beauty", explaining that "the first presupposes no concept of what the object ought to be; the second does presuppose such a concept and the perfection of the object in accordance therewith." By this definition, free beauty is found in seashells and wordless music; adherent beauty in buildings and the human body.
The Romantic poets, too, became highly concerned with the nature of beauty, with John Keats arguing in "Ode on a Grecian Urn" that: Western 19th and 20th century. In the Romantic period, Edmund Burke postulated a difference between beauty in its classical meaning and the sublime. The concept of the sublime, as explicated by Burke and Kant, suggested viewing Gothic art and architecture, though not in accordance with the classical standard of beauty, as sublime. The 20th century saw an increasing rejection of beauty by artists and philosophers alike, culminating in postmodernism's anti-aesthetics. This is despite beauty being a central concern of one of postmodernism's main influences, Friedrich Nietzsche, who argued that the Will to Power was the Will to Beauty. In the aftermath of postmodernism's rejection of beauty, thinkers have returned to beauty as an important value. American analytic philosopher Guy Sircello proposed his New Theory of Beauty as an effort to reaffirm the status of beauty as an important philosophical concept. He rejected the subjectivism of Kant and sought to identify the properties inherent in an object that make it beautiful. He called qualities such as vividness, boldness, and subtlety "properties of qualitative degree" (PQDs) and stated that a PQD makes an object beautiful if it is not—and does not create the appearance of—"a property of deficiency, lack, or defect"; and if the PQD is strongly present in the object.
Elaine Scarry argues that beauty is related to justice. Beauty is also studied by psychologists and neuroscientists in the field of experimental aesthetics and neuroesthetics respectively. Psychological theories see beauty as a form of pleasure. Correlational findings support the view that more beautiful objects are also more pleasing. Some studies suggest that higher experienced beauty is associated with activity in the medial orbitofrontal cortex. This approach of localizing the processing of beauty in one brain region has received criticism within the field. Philosopher and novelist Umberto Eco wrote "" (2004) and "On Ugliness" (2007). The narrator of his novel "The Name of the Rose" follows Aquinas in declaring: "three things concur in creating beauty: first of all integrity or perfection, and for this reason, we consider ugly all incomplete things; then proper proportion or consonance; and finally clarity and light", before going on to say "the sight of the beautiful implies peace". Mike Phillips has described Umberto Eco's "On Beauty" as "incoherent" and criticized him for focusing only on Western European history and devoting none of his book to Eastern European, Asian, or African history. Amy Finnerty described Eco's work "On Ugliness" favorably.
Chinese philosophy. Chinese philosophy has traditionally not made a separate discipline of the philosophy of beauty. Confucius identified beauty with goodness, and considered a virtuous personality to be the greatest of beauties: In his philosophy, "a neighborhood with a "ren" man in it is a beautiful neighborhood." Confucius's student Zeng Shen expressed a similar idea: "few men could see the beauty in some one whom they dislike." Mencius considered "complete truthfulness" to be beauty. Zhu Xi said: "When one has strenuously implemented goodness until it is filled to completion and has accumulated truth, then the beauty will reside within it and will not depend on externals." Human attributes. The word "beauty" is often used as a countable noun to describe a beautiful woman. The characterization of a person as "beautiful", whether on an individual basis or by community consensus, is often based on some combination of "inner beauty", which includes psychological factors such as personality, intelligence, grace, politeness, charisma, integrity, congruence and elegance, and "outer beauty" (i.e. physical attractiveness) which includes physical attributes which are valued on an aesthetic basis.
Standards of beauty have changed over time, based on changing cultural values. Historically, paintings show a wide range of different standards for beauty. A strong indicator of physical beauty is "averageness". When images of human faces are averaged together to form a composite image, they become progressively closer to the "ideal" image and are perceived as more attractive. This was first noticed in 1883, when Francis Galton overlaid photographic composite images of the faces of vegetarians and criminals to see if there was a typical facial appearance for each. When doing this, he noticed that the composite images were more attractive as compared to any of the individual images. Researchers have replicated the result under more controlled conditions and found that the computer-generated, mathematical average of a series of faces is rated more favorably than individual faces. It is argued that it is evolutionarily advantageous that sexual creatures are attracted to mates who possess predominantly common or average features, because it suggests the absence of genetic or acquired defects.
Since the 1970s there has been increasing evidence that a preference for beautiful faces emerges early in infancy, and is probably innate, and that the rules by which attractiveness is established are similar across different genders and cultures. A feature of beautiful women which has been explored by researchers is a waist–hip ratio of approximately 0.70. As of 2004, physiologists had shown that women with hourglass figures were more fertile than other women because of higher levels of certain female hormones, a fact that may subconsciously condition males choosing mates. In 2008, other commentators have suggested that this preference may not be universal. For instance, in some non-Western cultures in which women have to do work such as finding food, men tend to have preferences for higher waist-hip ratios. Exposure to the thin ideal in mass media, such as fashion magazines, directly correlates with body dissatisfaction, low self-esteem, and the development of eating disorders among female viewers. Further, the widening gap between individual body sizes and societal ideals continues to breed anxiety among young girls as they grow, highlighting the dangerous nature of beauty standards in society.
Western concept. A study using Chinese immigrants and Hispanic, Black and White American citizens found that their ideals of female beauty were not significantly different. Participants in the study rated Asian and Latina women as more attractive than White and Black women, and it was found that Asian and Latina women had more of the attributes that were considered attractive for women. Exposure to Western media did not influence or improve the Asian men's ratings of White women. One study found that East Asian women in the United States are closer to the ideal figure promoted in Western media, and that East Asian women conform to both Western and Eastern influences in the United States. East Asian men were found to be more impacted by Western beauty ideals then East Asian women, in the United States. East Asian men felt as though their bodies were not large enough and therefore deviated from the Western norm. East Asian men and white Western women were found to have the highest levels of body dissatisfaction in the United States. A study of African American and South Asian women found that some had internalized a white beauty ideal that placed light skin and straight hair at the top.
Eurocentric standards for men include tallness, leanness, and muscularity, which have been idolized through American media, such as in Hollywood films and magazine covers. In of the United States, African Americans have historically been subjected to beauty ideals that often do not reflect their own appearance, which can lead to issues of low self-esteem. African-American philosopher Cornel West elaborates that, "much of black self-hatred and self-contempt has to do with the refusal of many black Americans to love their own black bodies-especially their black noses, hips, lips, and hair." According to Patton (2006), the stereotype of African-American women's inferiority (relative to other races of women) maintains a system of oppression based on race and gender that operates to the detriment of women of all races, and also black men. In the 1960s, the black is beautiful cultural movement sought to dispel the notion of a Eurocentric concept of beauty. Much criticism has been directed at models of beauty which depend solely upon Western ideals of beauty, as seen, for example, in the Barbie franchise. Criticisms of Barbie are often centered around concerns that children consider Barbie a role model of beauty and will attempt to emulate her. One of the most common criticisms of Barbie is that she promotes an unrealistic idea of body image for a young woman, leading to a risk that girls who attempt to emulate her will become anorexic.
As of 1998, these criticisms of the lack of diversity in such franchises as the Barbie model of beauty in Western culture, had led to a dialogue to create non-exclusive models of Western ideals in body type for young girls who do not match the thinness ideal that Barbie represents. Mattel responded to these criticisms. In East Asian cultures, familial pressures and cultural norms shape beauty ideals. A 2017 experimental study concluded that Asian cultural idealization of "fragile" girls was impacting Asian American women's lifestyle, eating, and appearance choices. Effects on society. Researchers have found that good-looking students get higher grades from their teachers than students with an ordinary appearance. Some studies using mock criminal trials have shown that physically attractive "defendants" are less likely to be convicted—and if convicted are likely to receive lighter sentences—than less attractive ones (although the opposite effect was observed when the alleged crime was swindling, perhaps because jurors perceived the defendant's attractiveness as facilitating the crime). Studies among teens and young adults, such as those of psychiatrist and self-help author Eva Ritvo show that skin conditions have a profound effect on social behavior and opportunity.
How much money a person earns may also be influenced by physical beauty. One study found that people low in physical attractiveness earn 5 to 10 percent less than ordinary-looking people, who in turn earn 3 to 8 percent less than those who are considered good-looking. In the market for loans, the least attractive people are less likely to get approvals, although they are less likely to default. In the marriage market, women's looks are at a premium, but men's looks do not matter much. The impact of physical attractiveness on earnings varies across races, with the largest beauty wage gap among black women and black men. Conversely, being very unattractive increases the individual's propensity for criminal activity for a number of crimes ranging from burglary to theft to selling illicit drugs. Discrimination against others based on their appearance is known as lookism.
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the Scottish botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant "Clarkia pulchella" immersed in water. In 1900, the French mathematician Louis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (Théorie de la spéculation), prepared under the supervision of Henri Poincaré. Then, in 1905, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions.
The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter". The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).
History. The Roman philosopher-poet Lucretius' scientific poem "On the Nature of Things" () has a remarkable description of the motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms: Although the mingling, tumbling motion of dust particles is caused largely by air currents, the glittering, jiggling motion of small dust particles is caused chiefly by true Brownian dynamics; Lucretius "perfectly describes and explains the Brownian movement by a wrong example". While Jan Ingenhousz described the irregular motion of coal dust particles on the surface of alcohol in 1785, the discovery of this phenomenon is often credited to the botanist Robert Brown in 1827. Brown was studying pollen grains of the plant "Clarkia pulchella" suspended in water under a microscope when he observed minute particles, ejected by the pollen grains, executing a jittery motion. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained.
The mathematics of much of stochastic analysis including the mathematics of Brownian motion was introduced by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a analysis of the stock and option markets. However this work was largely unknown until the 1950s. Albert Einstein (in one of his 1905 papers) provided an explanation of Brownian motion in terms of atoms and molecules at a time when their existence was still debated. Einstein proved the relation between the probability distribution of a Brownian particle and the diffusion equation. These equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1908, leading to his Nobel prize. Norbert Wiener gave the first complete and rigorous mathematical analysis in 1923, leading to the underlying mathematical concept being called a Wiener process. The instantaneous velocity of the Brownian motion can be defined as , when , where is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with optical tweezers) was measured successfully. The velocity data verified the Maxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle.
Statistical mechanics theories. Einstein's theory. There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities. In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas. In accordance to Avogadro's law, this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the Avogadro number, and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the molar mass of the gas by the Avogadro constant. The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second.
He regarded the increment of particle positions in time formula_1 in a one-dimensional ("x") space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable (formula_2) with some probability density function formula_3 (i.e., formula_4 is the probability density for a jump of magnitude formula_2, i.e., the probability density of the particle incrementing its position from formula_6 to formula_7 in the time interval formula_1). Further, assuming conservation of particle number, he expanded the number density formula_9 (number of particles per unit volume around formula_6) at time formula_11 in a Taylor series, formula_12 where the second equality is by definition of formula_13. The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation: formula_14 Where the coefficient after the Laplacian, the second moment of probability of displacement formula_2, is interpreted as mass diffusivity "D":
formula_16 Then the density of Brownian particles at point at time satisfies the diffusion equation: formula_17 Assuming that "N" particles start from the origin at the initial time "t" = 0, the diffusion equation has the solution formula_18 This expression (which is a normal distribution with the mean formula_19 and variance formula_20 usually called Brownian motion formula_21) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by formula_22 This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.
The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium. In his original treatment, Einstein considered an osmotic pressure experiment, but the same conclusion can be reached in other ways. Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of , where is the mass of the particle, is the acceleration due to gravity, and is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius is formula_23, where is the dynamic viscosity of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution
formula_24 where is the difference in density of particles separated by a height difference, of formula_25, is the Boltzmann constant (the ratio of the universal gas constant, , to the Avogadro constant, ), and is the absolute temperature. Dynamic equilibrium is established because the more that particles are pulled down by gravity, the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by Fick's law, formula_26 where . Introducing the formula for , we find that formula_27 In a state of dynamical equilibrium, this speed must also be equal to . Both expressions for are proportional to , reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical charged particles of charge in a uniform electric field of magnitude , where is replaced with the electrostatic force . Equating these two expressions yields the Einstein relation for the diffusivity, independent of or or other such forces:
formula_28 Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the Boltzmann constant as , and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant , the temperature , the viscosity , and the particle radius , the Avogadro constant can be determined. The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J. Thomson in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a concentration gradient given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other". An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888 in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by Stokes's law. He writes formula_29 for the diffusion coefficient , where formula_30 is the osmotic pressure and is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the ideal gas law per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's. The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path.
At first, the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906 and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who found displacements 3 times greater than Einstein's formula predicted. But Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909. The confirmation of Einstein's theory constituted empirical progress for the kinetic theory of heat. In essence, Einstein showed that the motion can be predicted directly from the kinetic model of thermal equilibrium. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the second law of thermodynamics as being an essentially statistical law. Smoluchowski model. Smoluchowski's theory of Brownian motion starts from the same premise as that of Einstein and derives the same probability distribution for the displacement of a Brownian particle along the in time . He therefore gets the same expression for the mean squared displacement: However, when he relates it to a particle of mass moving at a velocity which is the result of a frictional force governed by Stokes's law, he finds
formula_31 where is the viscosity coefficient, and is the radius of the particle. Associating the kinetic energy formula_32 with the thermal energy , the expression for the mean squared displacement is times that found by Einstein. The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt." Smoluchowski attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. If the probability of gains and losses follows a binomial distribution, formula_33 with equal probabilities of 1/2, the mean total gain is formula_34 If is large enough so that Stirling's approximation can be used in the form formula_35 then the expected total gain will be formula_36 showing that it increases as the square root of the total population. Suppose that a Brownian particle of mass is surrounded by lighter particles of mass which are traveling at a speed . Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be . This ratio is of the order of . But we also have to take into consideration that in a gas there will be more than 1016 collisions in a second, and even greater in a liquid where we expect that there will be 1020 collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 108 to 1010 collisions in one second, then velocity of the Brownian particle may be anywhere between . Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.
These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, , which depends on the collisions that tend to accelerate and decelerate it. The larger is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, will be equal, on the average, to the kinetic energy of the surrounding fluid particle, In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion. The model assumes collisions with where is the test particle's mass and the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of . If is the number of collisions from the right and the number of collisions from the left then after collisions the particle's velocity will have changed by . The multiplicity is then simply given by:
formula_37 and the total number of possible states is given by . Therefore, the probability of the particle being hit from the right times is: formula_38 As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. Also, there would be a distribution of different possible s instead of always just one in a realistic situation. Langevin equation. The diffusion equation yields an approximation of the time evolution of the probability density function associated with the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on short timescales. The time evolution of the position of the Brownian particle over all time scales described using the Langevin equation, an equation that involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle. In Langevin dynamics and Brownian dynamics, the Langevin equation is used to efficiently simulate the dynamics of molecular systems that exhibit a strong Brownian component.
Astrophysics: star motion within galaxies. In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity of the massive object, of mass , is related to the rms velocity formula_39 of the background stars by formula_40 where formula_41 is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both formula_39 and . The Brownian velocity of Sgr A*, the supermassive black hole at the center of the Milky Way galaxy, is predicted from this formula to be less than 1 km s−1. Mathematics. In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics.
The Wiener process is characterized by four facts: formula_44 denotes the normal distribution with expected value and variance . The condition that it has independent increments means that if formula_45 then formula_46 and formula_47 are independent random variables. In addition, for some filtration formula_48 is formula_49 measurable for all An alternative characterisation of the Wiener process is the so-called "Lévy characterisation" that says that the Wiener process is an almost surely continuous martingale with and quadratic variation A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent formula_50 random variables. This representation can be obtained using the Kosambi–Karhunen–Loève theorem. The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant.
A d-dimensional Gaussian free field has been described as "a d-dimensional-time analog of Brownian motion." Statistics. The Brownian motion can be modeled by a random walk. In the general case, Brownian motion is a Markov process and described by stochastic integral equations. Lévy characterisation. The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous -valued stochastic process to actually be -dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion. Let be a continuous stochastic process on a probability space taking values in . Then the following are equivalent: Spectral content. The spectral content of a stochastic process formula_51 can be found from the power spectral density, formally defined as formula_52 where formula_53 stands for the expected value. The power spectral density of Brownian motion is found to be formula_54 where is the diffusion coefficient of . For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e.,
formula_55 which for an individual realization of a Brownian motion trajectory, it is found to have expected value formula_56 formula_57 and variance formula_58 formula_59 For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density but its coefficient of variation formula_60 tends to This implies the distribution of formula_61 is broad even in the infinite time limit. Riemannian manifold. Brownian motion is usually considered in one or more dimensions of Euclidean space, such as that of three dimensions. In some applications of Brownian motion, it is natural to consider the motion of particles on a surface (or higher dimensional Riemannian manifold). Consider the case of a sphere, for example. A particle confined to the surface of a sphere can be subject to the random bombardment of molecules also on the sphere, such as might happen at the surface of a soap bubble. Among the questions to be answered in studying Brownian motion on such surfaces is what is "recurrence", whether a particle trajectory will return to the same value after a finite time.
A discretized model for this state of affairs is that a particle can, at any time, move in a "uniformly" random direction on the surface, and the displacement is determined by a "radial" Gaussian. That is, the distribution of geodesic distance in any direction is Gaussian. Formulating this in the continuum limit requires some care, however. In more analytic terms, the infinitesimal generator (and hence characteristic operator) of a Brownian motion on is easily calculated to be , where denotes the Laplace operator. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. This observation is useful in defining Brownian motion on an -dimensional Riemannian manifold : a Brownian motion on is defined to be a diffusion on whose characteristic operator formula_62 in local coordinates , , is given by , where is the Laplace–Beltrami operator given in local coordinates by formula_63 where in the sense of the inverse of a square matrix. Narrow escape.
The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.
Barcelona Barcelona ( ; ; ) is a city on the northeastern coast of Spain. It is the capital and largest city of the autonomous community of Catalonia, as well as the second-most populous municipality of Spain. With a population of 1.6 million within city limits, its urban area extends to numerous neighbouring municipalities within the province of Barcelona and is home to around 5.3 million people, making it the fifth most populous urban area of the European Union after Paris, the Ruhr area, Madrid and Milan. It is one of the largest metropolises on the Mediterranean Sea, located on the coast between the mouths of the rivers Llobregat and Besòs, bounded to the west by the Serra de Collserola mountain range. According to tradition, Barcelona was founded by either the Phoenicians or the Carthaginians, who had trading posts along the Catalonian coast. In the Middle Ages, Barcelona became the capital of the County of Barcelona. After joining with the Kingdom of Aragon to form the composite monarchy of the Crown of Aragon, Barcelona, which continued to be the capital of the Principality of Catalonia, became the most important city in the Crown of Aragon and its main economic and administrative centre, only to be overtaken by Valencia, wrested from Moorish control by the Catalans, shortly before the dynastic union between the Crown of Castile and the Crown of Aragon in 1516. Barcelona became the centre of Catalan separatism, briefly becoming part of France during the 17th century Reapers' War and again in 1812 until 1814 under Napoleon. Experiencing industrialization and several workers movements during the 19th and early 20th century, it became the capital of autonomous Catalonia in 1931 and it was the epicenter of the revolution experienced by Catalonia during the Spanish Revolution of 1936, until its capture by the fascists in 1939. After the Spanish transition to democracy in the 1970s, Barcelona once again became the capital of an autonomous Catalonia.
Barcelona has a rich cultural heritage and is today an important cultural centre and a major tourist destination. Particularly renowned are the architectural works of Antoni Gaudí and Lluís Domènech i Montaner, which have been designated UNESCO World Heritage Sites. The city is home to two of the most prestigious universities in Spain: the University of Barcelona and Pompeu Fabra University. The headquarters of the Union for the Mediterranean are located in Barcelona. The city is known for hosting the 1992 Summer Olympics as well as world-class conferences and expositions. In addition, many international sport tournaments have been played here. Barcelona is a major cultural, economic, and financial centre in southwestern Europe, as well as the main biotech hub in Spain. As a leading world city, Barcelona's influence in global socio-economic affairs qualifies it for global city status (Beta +). Barcelona is a transport hub, with the Port of Barcelona being one of Europe's principal seaports and busiest European passenger port, an international airport, Barcelona–El Prat Airport, which handles over 50-million passengers per year, an extensive motorway network, and a high-speed rail line with a link to France and the rest of Europe.
Names. The name "Barcelona" comes from the ancient Iberian "Baŕkeno", attested in an ancient coin inscription found on the right side of the coin in Iberian script as , in Ancient Greek sources as , "Barkinṓn"; and in Latin as "Barcino", "Barcilonum" and "Barcenona". Other sources suggest that the city may have been named after the Carthaginian general Hamilcar Barca, who was supposed to have founded the city in the 3rd century BC, but there is no evidence its name in antiquity, "Barcino", was connected with the Barcid family of Hamilcar. During the Middle Ages, the city was variously known as "Barchinona", "Barçalona", "Barchelonaa", and "Barchenona". An abbreviated form sometimes used by locals for the city is "Barna". "Barça" is only applied to the local football club FC Barcelona, not to the city. Another common abbreviation is 'BCN', which is also the IATA airport code of the Barcelona-El Prat Airport. The city is referred to as the "Ciutat Comtal" in Catalan and "Ciudad Condal" in Spanish (i.e., "Comital City" or "City of Counts"), owing to its past as the seat of the Count of Barcelona.
History. Legendary founding. The origin of the earliest settlement at the site of present-day Barcelona is unclear. The ruins of an early settlement have been found, including different tombs and dwellings dating to earlier than 5000 BC. In Greek mythology, the founding of Barcelona had been attributed to the mythological Hercules. Punic Barcelona. According to tradition, Barcelona was founded by Punic (Phoenician) settlers, who had trading posts along the Catalonian coast. In particular, some historians attribute the foundation of the city directly to the historical Carthaginian general, Hamilcar Barca, father of Hannibal, who supposedly named the city "Barcino" after his family in the 3rd century BC, but this theory has been questioned. Archeological evidence in the form of coins from the 3rd century BC have been found on the hills at the foot of Montjuïc with the name Bárkeno written in an ancient script in the Iberian language. Thus, we can conclude that the Laietani, an ancient Iberian (pre-Roman) people of the Iberian peninsula, who inhabited the area occupied by the city of Barcelona around 3–2 BC, called the area Bàrkeno, which means "The Place of the Plains" (Barrke = plains/terrace).
Roman Barcelona. In about 15 BC, the Romans redrew the town as a "castrum" (Roman military camp) centred on the "Mons Taber", a little hill near the Generalitat (Catalan Government) and city hall buildings. The Roman Forum, at the crossing of the Cardo Maximus and Decumanus Maximus, was approximately placed where current Plaça de Sant Jaume is. Thus, the political centre of the city, Catalonia, and its domains has remained in the same place for over 2,000 years. Under the Romans, it was a colony with the surname of "Faventia", or, in full, "Colonia Faventia Julia Augusta Pia Barcino" or "Colonia Julia Augusta Faventia Paterna Barcino". Pomponius Mela mentions it among the small towns of the district, probably as it was eclipsed by its neighbour "Tarraco" (modern Tarragona), but it may be gathered from later writers that it gradually grew in wealth and consequence, favoured as it was with a beautiful situation and an excellent harbour. It enjoyed immunity from imperial burdens. The city minted its own coins; some from the era of Galba survive.
Important Roman vestiges are displayed in Plaça del Rei underground, as a part of the Barcelona City History Museum (MUHBA); the typically Roman grid plan is still visible today in the layout of the historical centre, the "Barri Gòtic" (Gothic Quarter). Some remaining fragments of the Roman walls have been incorporated into the cathedral. The cathedral, Catedral Basílica Metropolitana de Barcelona, is also sometimes called "La Seu", which simply means cathedral (and see, among other things) in Catalan. It is said to have been founded in 343. Medieval Barcelona. The city was conquered by the Visigoths in the early 5th century, becoming for a few years the capital of all Hispania. After being conquered by the Umayyads in the early 8th century, it was conquered after a siege in 801 by Charlemagne's son Louis, who made Barcelona the seat of the Carolingian "Hispanic March" ("Marca Hispanica"), a buffer zone ruled by the Count of Barcelona. The Counts of Barcelona became increasingly independent and expanded their territory to include much of modern Catalonia, although in 985, Barcelona was sacked by the army of Almanzor. The sack was so traumatic that most of Barcelona's population was either killed or enslaved. In 1137, Aragon and the County of Barcelona merged in dynastic union by the marriage of Ramon Berenguer IV and Petronilla of Aragon, their titles finally borne by only one person when their son Alfonso II of Aragon ascended to the throne in 1162. His territories were later to be known as the Crown of Aragon, which conquered many overseas possessions and ruled the western Mediterranean Sea with outlying territories in Naples and Sicily and as far as Athens in the 13th century.
Barcelona also had a substantial Jewish community at the time, then the largest Jewish community in the Crown of Aragon. Called "the Call," for the many small streets that defined the area, it later became enclosed. Montjuïc or Montjuich, in medieval Latin and Catalan, meaning "Jewish Mountain" and the birthplace of the city, is the site of a medieval Jewish cemetery, Jews continued to live in Barcelona until the Massacre of 1391 diminished their numbers. The Spanish Inquisition forced the remaining Jews who refused to convert to Christianity to be burned at the stake, or sell their property and leave. Barcelona was the leading slave trade centre of the Crown of Aragon up until the 15th century, when it was eclipsed by Valencia. It initially fed from eastern and Balkan slave stock later drawing from a Maghribian and, ultimately, Subsaharan pool of slaves. The Bank or Taula de canvi de Barcelona, often viewed as the oldest public bank in Europe, was established by the city magistrates in 1401. It originated from necessities of the state, as did the Bank of Venice (1402) and the Bank of Genoa (1407).
Barcelona under the Spanish monarchy. In the beginning of the Early Modern period, Barcelona lost political primacy, but the economy managed to achieve a balance between production capacity and imports. In the context of the wider early recovery of Catalonia from the 17th-century crisis in the second half of the century, increasing maritime activity since 1675 doubled traffic in the port of Barcelona compared to figures from the beginning of the 17th century. In the late 17th and early 18th century, Barcelona repeatedly endured the effects of war, including the 1691 bombing, the sieges of 1697, 1704, 1705, 1706, and the 1713 blockade and ensuing 1714 siege and assault. In the 18th century, the population grew from 30,000 to about 100,000 inhabitants, as the city became one of the key mercantile centres in the Western Mediterranean, with inland influence up to Zaragoza, and to the south up to Alicante. A fortress was built at Montjuïc that overlooked the harbour. Much of Barcelona was negatively affected by the Napoleonic wars, but the start of industrialization saw the fortunes of the province improve.
Transforming the city. In the mid-1850s, Barcelona was struggling with population density as it became an industrial, port city and European capital. The city's density was at 856 people per hectare, more than double that of Paris. Mortality rates were on the rise and any outbreaks of disease would devastate the population. To solve the issue, a civil engineer named Ildefons Cerdà proposed a plan for a new district known as the Eixample. The citizens of Barcelona had begun to demolish the medieval wall surrounding and constricting the city. Cerdà thought it best to transform the land outside the walls into an area characterized by a scientific approach to urbanization. His proposal consisted of a grid of streets to unite the old city and surrounding villages. There would also be wide streets to allow people to breathe clean air, gardens in the centre of each street block, integration of rich and poor giving both groups access to the same services, and smooth-flowing traffic. Urban quality, egalitarianism, hygiene, sunlight, and efficiency were all major keys for Cerdà's vision.
Urban quality, egalitarianism, hygiene, sunlight, and efficiency were all major keys for Cerdà's vision. Not everything he imagined would be realized within the Eixample district, but the iconic octagonal superblocks with chamfered corners for better visibility are his direct brainchild and remain immensely helpful even 170 years later. The district and its ideals were not appreciated at the time. The city council awarded the design of the extension plan to another architect. The Spanish government was the one to step in and impose Cerdà's plan, laying the groundwork for many more tensions between the Spanish and Catalan administrations. Regardless, some of the upper class citizens of Barcelona were excited by the new plan and began a race to build "the biggest, tallest, most attractive house" in the district. Their interest and money fueled the rich diversity that we now see in the district's architecture. In the end, Cerdà's ideas would have a lasting impact on Barcelona's development, earning it international recognition for its highly efficient approach to urban planning and design.
The Spanish Civil War and the Franco period. Shortly after the establishment of the Second Spanish Republic, in 1931, it became the capital of the autonomous region of Catalonia. During the Spanish Civil War, the city, and Catalonia in general, were resolutely Republican. Many enterprises and public services were collectivized by the CNT and UGT unions. As the power of the Republican government and the Generalitat diminished, much of the city was under the effective control of anarchist groups. The anarchists lost control of the city to their own allies, the Communists and official government troops, after the street fighting of the Barcelona May Days. The fall of the city on 26 January 1939, caused a mass exodus of civilians who fled to the French border. The resistance of Barcelona to Franco's "coup d'état" was to have lasting effects after the defeat of the Republican government. The autonomous institutions of Catalonia were abolished, and the use of the Catalan language in public life was suppressed. Barcelona remained the second largest city in Spain, at the heart of a region which was relatively industrialized and prosperous, despite the devastation of the civil war. The result was a large-scale immigration from poorer regions of Spain (particularly Andalusia, Murcia and Galicia), which in turn led to rapid urbanization.
Late twentieth century. In 1992, Barcelona hosted the Summer Olympics. The after-effects of this are credited with driving major changes in what had, up until then, been a largely industrial city. As part of the preparation for the games, industrial buildings along the sea-front were demolished and of beach were created. New construction increased the road capacity of the city by 17%, the sewage handling capacity by 27% and the amount of new green areas and beaches by 78%. Between 1990 and 2004, the number of hotel rooms in the city doubled. Perhaps more importantly, the outside perception of the city was changed making, by 2012, Barcelona the 12th most popular city destination in the world and the 5th amongst European cities. Recent history. The death of Franco in 1975 brought on a period of democratization throughout Spain. Pressure for change was particularly strong in Barcelona, which considered that it had been punished during nearly forty years of Francoism for its support of the Republican government. Massive, but peaceful, demonstrations on 11 September 1977 assembled over a million people in the streets of Barcelona to call for the restoration of Catalan autonomy. It was granted less than a month later.
The development of Barcelona was promoted by two events in 1986: Spanish accession to the European Community, and particularly Barcelona's designation as host city of the 1992 Summer Olympics. The process of urban regeneration has been rapid, and accompanied by a greatly increased international reputation of the city as a tourist destination. The increased cost of housing has led to a slight decline (−16.6%) in the population over the last two decades of the 20th century as many families move out into the suburbs. This decline has been reversed since 2001, as a new wave of immigration (particularly from Latin America and from Morocco) has gathered pace. In 1987, an ETA car bombing at Hipercor killed 21 people. On 17 August 2017, a van was driven into pedestrians on La Rambla, killing 14 and injuring at least 100, one of whom later died. Other attacks took place elsewhere in Catalonia. The Prime Minister of Spain, Mariano Rajoy, called the attack in Barcelona a jihadist attack. Amaq News Agency attributed indirect responsibility for the attack to the Islamic State of Iraq and the Levant (ISIL). During the 2010s, Barcelona became the focus city for the ongoing Catalan independence movement, its consequent standoff between the regional and national government and later protests.
In July 2023, Barcelona was announced as the UNESCO-UIA World Capital of Architecture for the 2024–2026 term. This means it will be the hub for discussion around global challenges including culture, heritage, urban planning and architecture. In addition to being the capital through 2026, it will also host the UIA World Congress of Architects for that year. The honour is befitting of Barcelona, as its history is peppered with architectural achievement and various iconic styles and influences. From its ancient Roman roots, to the Gothic and Modernisme movements, Barcelona has thrived through the way it ties together architecture and culture. Geography. Location. Barcelona is located on the northeast coast of the Iberian Peninsula, facing the Mediterranean Sea, on a plain approximately wide limited by the mountain range of Collserola, the Llobregat river to the southwest and the Besòs river to the north. This plain covers an area of , of which are occupied by the city itself. It is south of the Pyrenees and the Catalan border with France.
Tibidabo, high, offers striking views over the city and is topped by the Torre de Collserola, a telecommunications tower that is visible from most of the city. Barcelona is peppered with small hills, most of them urbanized, that gave their name to the neighbourhoods built upon them, such as Carmel (), (es) () and Rovira (). The escarpment of Montjuïc (), situated to the southeast, overlooks the harbour and is topped by Montjuïc Castle, a fortress built in the 17–18th centuries to control the city as a replacement for the Ciutadella. Today, the fortress is a museum and Montjuïc is home to several sporting and cultural venues, as well as Barcelona's biggest park and gardens. The city borders on the municipalities of Santa Coloma de Gramenet and Sant Adrià de Besòs to the north; the Mediterranean Sea to the east; El Prat de Llobregat and L'Hospitalet de Llobregat to the south; and Sant Feliu de Llobregat, Sant Just Desvern, Esplugues de Llobregat, Sant Cugat del Vallès, and Montcada i Reixac to the west. The municipality includes two small sparsely-inhabited exclaves to the north-west.
Climate. According to the Köppen climate classification, Barcelona has a hot summer Mediterranean climate ("Csa"), with mild winters and warm to hot summers, while the rainiest seasons are autumn and spring. The rainfall pattern is characterized by a short (3 months) dry season in summer, as well as less winter rainfall than in a typical Mediterranean climate. However, both June and August are wetter than February, which is unusual for the Mediterranean climate. This subtype, labelled as "Portuguese" by the French geographer George Viers after the climate classification of Emmanuel de Martonne and found in the NW Mediterranean area (e.g. Marseille), can be seen as transitional to the humid subtropical climate ("Cfa") found in inland areas.
Barcelona averages 78 rainy days per year (≥ 1 mm), and annual average relative humidity is 72%, ranging from 69% in July to 75% in October. Rainfall totals are highest in late summer and autumn (September–November) and lowest in early and mid-summer (June–August), with a secondary winter minimum (February–March). Sunshine duration is 2,524 hours per year, from 138 (average 4.5 hours of sunshine a day) in December to 310 (average 10 hours of sunshine a day) in July. Demographics. According to Barcelona's City Council, the city's population was 1,608,746 people, on a land area of . It is the main component of an administrative area of Greater Barcelona, with a population of 3,218,071 in an area of (density 5,060 inhabitants/km2). The population of the urban area was 4,840,000. It is the central nucleus of the Barcelona metropolitan area, which relies on a population of 5,474,482. In 1900, Barcelona had a population of 533,000, which grew steadily but slowly until 1950, when it started absorbing a high number of people from other less-industrialized parts of Spain. Barcelona's population peaked in 1979 at 1,906,998, and fell throughout the 1980s and 1990s as more people sought a higher quality of life in outlying cities in the Barcelona Metropolitan Area. After bottoming out in 2000 with 1,496,266 residents, the city's population began to rise again as younger people started to return, causing a great increase in housing prices.
Languages spoken. Spanish is the most spoken language in Barcelona (according to the linguistic census held by the Government of Catalonia in 2013) and it is understood almost universally. Catalan is also very commonly spoken in the city: it is understood by 95% of the population, while 72.3% can speak it, 79% can read it, and 53% can write it. Knowledge of Catalan has increased significantly in recent decades thanks to a language immersion educational system. After Catalan and Spanish, the most spoken languages in Barcelona are those from North Africa, such as Amazigh and Arabic, followed by Bengali, Urdu, Panjabi, Mandarin Chinese, Romanian, English, Russian and Quechua, according to data collected by the University of Barcelona. Population density. Barcelona is one of the most densely populated cities in Europe. For the year 2008 the city council calculated the population to 1,621,090 living in the 102.2 km2 sized municipality, giving the city an average population density of 15,926 inhabitants per square kilometre with Eixample being the most populated district.
In the case of Barcelona though, the land distribution is extremely uneven. Half of the municipality or 50.2 km2, all of it located on the municipal edge is made up of the ten least densely populated neighbourhoods containing less than 10% of the city's population, the uninhabited Zona Franca industrial area and Montjuïc forest park. Leaving the remaining 90% or slightly below 1.5 million inhabitants living on the remaining at an average density close to 28,500 inhabitants per square kilometre. Of the 73 neighbourhoods in the city, 45 had a population density above 20,000 inhabitants per square kilometre with a combined population of 1,313,424 inhabitants living on 38.6 km2 at an average density of 33,987 inhabitants per square kilometre. The 30 most densely populated neighbourhoods accounted for 57.5% of the city population occupying only 22.7% of the municipality, or in other words, 936,406 people living at an average density of 40,322 inhabitants per square kilometre. The city's highest density is found at and around the neighbourhood of la Sagrada Família where four of the city's most densely populated neighbourhoods are located side by side, all with a population density above 50,000 inhabitants per square kilometre.
Age structure. In 1900, almost a third (28.9 percent) of the population were children (aged younger than 14 years). In 2017, this age group constituted only 12.7% of the population. In 2017, people aged between 15 and 24 years made up 9 percent of the population; those aged between 25 and 44 years made up 30.6 percent of the population; while those aged between 45 and 64 years formed 56.9% of all Barcelonans. In 1900, people aged 65 and older made up just 6.5 percent of the population. In 2017, this age group made up 21.5 percent of the population. Migration. In 2016, about 59% of the inhabitants of the city were born in Catalonia and 18.5% coming from the rest of the country. In addition to that, 22.5% of the population was born outside of Spain, a proportion which has more than doubled since 2001 and more than quintupled since 1996 when it was 8.6% and 3.9% respectively. The most important region of origin of migrants is Europe, with many coming from Italy (26,676) or France (13,506). Moreover, many migrants come from Latin American nations such as Bolivia, Ecuador or Colombia. Since the 1990s, and similar to other migrants, many Latin Americans have settled in northern parts of the city.
There exists a relatively large Pakistani community in Barcelona with up to twenty thousand nationals. The community consists of significantly more men than women. Many of the Pakistanis are living in Ciutat Vella. First Pakistani migrants came in the 1970s, with increasing numbers in the 1990s. Other significant migrant groups come from Asia as from China and the Philippines. There is a Japanese community clustered in Bonanova, Les Tres Torres, Pedralbes, and other northern neighbourhoods, and a Japanese international school serves that community. Religion. In 2007, most of the inhabitants stated they are Roman Catholic (208 churches). In a 2011 survey conducted by InfoCatólica, 49.5% of Barcelona residents of all ages identified themselves as Catholic. This was the first time that more than half of respondents did not identify themselves as Catholic Christians. The numbers reflect a broader trend in Spain whereby the numbers of self-identified Catholics have declined. In 2019, a survey by Centro de Investigaciones Sociológicas showed that 53.2% of residents in Barcelona identified themselves as Catholic (9.9% practising Catholics, 43.3% non-practising Catholics).
The province has the largest Muslim community in Spain, 322,698 people in Barcelona province are of Muslim religion. A considerable number of Muslims live in Barcelona due to immigration (169 locations, mostly professed by Moroccans in Spain). In 2014, 322,698 out of 5.5 million people in the province of Barcelona identified themselves as Muslim, which makes 5.6% of the total population. The city also has the largest Jewish community in Spain, with an estimated 3,500 Jews living in the city. There are also a number of other groups, including Evangelical (71 locations, mostly professed by Roma), Jehovah's Witnesses (21 Kingdom Halls), Buddhists (13 locations), and Eastern Orthodox. Economy. General information. The Barcelona metropolitan area comprises over 66% of the people of Catalonia, one of the richer regions in Europe and the fourth richest region per capita in Spain, with a GDP per capita amounting to €28,400 (16% more than the EU average). The greater Barcelona metropolitan area had a GDP amounting to $177 billion (equivalent to $34,821 in per capita terms, 44% more than the EU average), making it the 4th most economically powerful city by gross GDP in the European Union, and 35th in the world in 2009. Barcelona city had a very high GDP of €80,894 per head in 2004, according to Eurostat. Furthermore, Barcelona was Europe's fourth best business city and fastest improving European city, with growth improved by 17% per year .
Barcelona was the 24th most "livable city" in the world in 2015 according to lifestyle magazine "Monocle". Similarly, according to Innovation Analysts 2thinknow, Barcelona occupies 13th place in the world on "Innovation Cities™ Global Index". At the same time it is according to the Global Wealth and Lifestyle Report 2020 one of the most affordable cities in the world for a luxury lifestyle. Barcelona has a long-standing mercantile tradition. Less well known is that the city industrialized early, taking off in 1833, when Catalonia's already sophisticated textile industry began to use steam power. It became the first and most important industrial city in the Mediterranean basin. Since then, manufacturing has played a large role in its history. Borsa de Barcelona (Barcelona Stock Exchange) is the main stock exchange in the northeastern part of the Iberian Peninsula. Barcelona was recognized as the Southern European City of the Future for 2014/15, based on its economic potential, by "FDi Magazine" in their bi-annual rankings.
Trade fair and exhibitions. It also has several congress halls, notably Fira de Barcelona – the second largest trade fair and exhibition centre in Europe, that host a quickly growing number of national and international events each year (at present above 50). The total exhibition floor space of Fira de Barcelona venues is , not counting Gran Via centre on the Plaza de Europa. However, the Eurozone crisis and deep cuts in business travel affected the council's positioning of the city as a convention centre. An important business centre, the World Trade Center Barcelona, is located in Barcelona's Port Vell harbour. The city is known for hosting well as world-class conferences and expositions, including the 1888 "Exposición Universal de Barcelona", the 1929 Barcelona International Exposition (Expo 1929), the 2004 Universal Forum of Cultures and the 2004 World Urban Forum. Tourism. Barcelona was the 20th-most-visited city in the world by international visitors and the fifth most visited city in Europe after London, Paris, Istanbul and Rome, with 5.5 million international visitors in 2011. By 2015, both Prague and Milan had more international visitors. With its popular tree-lined pedestrian street, Les Rambles (Las Ramblas), Barcelona is ranked the most popular city to visit in Spain.
Barcelona is an internationally renowned tourist destination, with numerous recreational areas, one of the best beaches in the world, mild and warm climate, historical monuments, including eight UNESCO World Heritage Sites, 519 hotels including 35 five-star hotels, and developed tourist infrastructure. Due to its large influx of tourists each year, Barcelona, like many other tourism capitals, has to deal with pickpockets, with wallets and passports being commonly stolen items. Despite its moderate pickpocket rate, Barcelona is considered one of the safest cities in terms of security and personal safety, mainly because of a sophisticated policing strategy that has dropped crime by 32% in just over three years and has led it to be considered the 15th safest city in the world by "Business Insider" in 2016. While tourism produces economic benefits, according to one report, the city is "overrun [by] hordes of tourists". In early 2017, over 150,000 protesters warned that tourism is destabilizing the city. Slogans included "Tourists go home", "Barcelona is not for sale" and "We will not be driven out". By then, the number of visitors had increased from 1.7 million in 1990 to 32 million in a city with a population of 1.62 million, increasing the cost of rental housing for residents and overcrowding the public places. While tourists spent an estimated €30 billion in 2017, they are viewed by some as a threat to Barcelona's identity.
A May 2017 article in the British online daily "The Independent" included Barcelona among the "Eight Places That Hate Tourists the Most" and included a comment from Mayor Ada Colau, "We don't want the city to become a cheap souvenir shop", citing Venice as an example. To moderate the problem, the city has stopped issuing licenses for new hotels and holiday apartments; it also fined AirBnb €30,000. The mayor has suggested introducing a new tourist tax and setting a limit on the number of visitors. One industry insider, Justin Francis, founder of the Responsible Travel agency, stated that steps must be taken to limit the number of visitors that are causing an "overtourism crisis" in several major European cities. "Ultimately, residents must be prioritised over tourists for housing, infrastructure and access to services because they have a long-term stake in the city's success", he said. "Managing tourism more responsibly can help", Francis later told a journalist, "but some destinations may just have too many tourists, and Barcelona may be a case of that".
Manufacturing sector. Industry generates 21% of the total gross domestic product (GDP) of the region, with the energy, chemical and metallurgy industries accounting for 47% of industrial production. The Barcelona metropolitan area had 67% of the total number of industrial establishments in Catalonia as of 1997. Barcelona has long been an important European automobile manufacturing centre. Formerly there were automobile factories of AFA, Abadal, Actividades Industriales, Alvarez, America, Artés de Arcos, Balandrás, Baradat-Esteve, Biscúter, J. Castro, Clúa, David, Delfín, Díaz y Grilló, Ebro trucks, , Elizalde, Automóviles España, Eucort, Fenix, Fábrica Hispano, Auto Academia Garriga, Fábrica Española de Automóviles Hebe, Hispano-Suiza, Huracán Motors, Talleres Hereter, Junior SL, Kapi, La Cuadra, M.A., Automóviles Matas, Motores y Motos, Nacional Custals, National Pescara, Nacional RG, Nacional Rubi, Nacional Sitjes, Automóviles Nike, Orix, Otro Ford, Patria, Pegaso, PTV, Ricart, Ricart-España, Industrias Salvador, Siata Española, Stevenson, Romagosa y Compañía, Garaje Storm, Talleres Hereter, Trimak, Automóviles Victoria, Manufacturas Mecánicas Aleu.
Today, the headquarters and a large factory of SEAT (the largest Spanish automobile manufacturer) are in one of its suburbs. There is also a Nissan factory in the logistics and industrial area of the city. The factory of Derbi, a large manufacturer of motorcycles, scooters and mopeds, also lies near the city. As in other modern cities, the manufacturing sector has long since been overtaken by the services sector, though it remains very important. The region's leading industries are textiles, chemical, pharmaceutical, motor, electronic, printing, logistics, publishing, in telecommunications industry and culture the notable Mobile World Congress, and information technology services. Fashion. The traditional importance of textiles is reflected in Barcelona's drive to become a major fashion centre. There have been many attempts to launch Barcelona as a fashion capital, notably "Gaudi Home". Beginning in the summer of 2000, the city hosted the Bread & Butter urban fashion fair until 2009, when its organizers announced that it would be returning to Berlin. This was a hard blow for the city as the fair brought €100 m to the city in just three days.
From 2009, "The Brandery", an urban fashion show, was held in Barcelona twice a year until 2012. According to the Global Language Monitor's annual ranking of the world's top fifty fashion capitals Barcelona was named as the seventh most important fashion capital of the world right after Milan and before Berlin in 2015. Government and administrative divisions. As the capital of the autonomous community of Catalonia, Barcelona is the seat of the Catalan government, known as the "Generalitat de Catalunya"; of particular note are the executive branch, the parliament, and the High Court of Justice of Catalonia. The city is also the capital of the Province of Barcelona and the Barcelonès comarca (district). Barcelona is governed by a city council formed by 41 city councillors, elected for a four-year term by universal suffrage. As one of the two biggest cities in Spain, Barcelona is subject to a special law articulated through the "Carta Municipal" (Municipal Law). A first version of this law was passed in 1960 and amended later, but the current version was approved in March 2006.<ref name="ley 1/2006"></ref> According to this law, Barcelona's city council is organized in two levels: a political one, with elected city councillors, and one executive, which administers the programs and executes the decisions taken on the political level. This law also gives the local government a special relationship with the central government and it also gives the mayor wider prerogatives by the means of municipal executive commissions. It expands the powers of the city council in areas like telecommunications, city traffic, road safety and public safety. It also gives a special economic regime to the city's treasury and it gives the council a veto in matters that will be decided by the central government, but that will need a favourable report from the council.
The "Comissió de Govern" (Government Commission) is the executive branch, formed by 24 councillors, led by the Mayor, with 5 lieutenant-mayors and 17 city councillors, each in charge of an area of government, and 5 non-elected councillors. The plenary, formed by the 41 city councillors, has advisory, planning, regulatory, and fiscal executive functions. The six "Commissions del Consell Municipal" (City council commissions) have executive and controlling functions in the field of their jurisdiction. They are composed by a number of councillors proportional to the number of councillors each political party has in the plenary. The city council has jurisdiction in the fields of city planning, transportation, municipal taxes, public highways security through the "Guàrdia Urbana" (the municipal police), city maintenance, gardens, parks and environment, facilities (like schools, nurseries, sports centres, libraries, and so on), culture, sports, youth and social welfare. Some of these competencies are not exclusive, but shared with the Generalitat de Catalunya or the central Spanish government. In some fields with shared responsibility (such as public health, education or social services), there is a shared Agency or Consortium between the city and the Generalitat to plan and manage services.
The executive branch is led by a Chief Municipal Executive Officer which answers to the Mayor. It is made up of departments which are legally part of the city council and by separate legal entities of two types: autonomous public departments and public enterprises. The seat of the city council is on the Plaça de Sant Jaume, opposite the seat of Generalitat de Catalunya. Since the coming of the Spanish democracy, Barcelona had been governed by the PSC, first with an absolute majority and later in coalition with ERC and ICV. After the May 2007 election, the ERC did not renew the coalition agreement and the PSC governed in a minority coalition with ICV as the junior partner. After 32 years, on 22 May 2011, CiU gained a plurality of seats at the municipal election, gaining 15 seats to the PSC's 11. The PP hold 8 seats, ICV 5 and ERC 2. Districts. Since 1987, the city has been divided into 10 administrative districts ("districtes" in Catalan, "distritos" in Spanish): The districts are based mostly on historical divisions, and several are former towns annexed by the city of Barcelona in the 18th and 19th centuries that still maintain their own distinct character. Each district has its own council led by a city councillor. The composition of each district council depends on the number of votes each political party had in that district, so a district can be led by a councillor from a different party than the executive council.
Education. Barcelona has a well-developed higher education system of public universities. Most prominent among these are the University of Barcelona (established in 1450) and the more modern Pompeu Fabra University. Barcelona is also home to the Polytechnic University of Catalonia, and in the private sector the EADA Business School (founded in 1957), which became the first Barcelona institution to run manager training programmes for the business community. IESE Business School, as well as the largest private educational institution, the Ramon Llull University, which encompasses schools and institutes such as the ESADE Business School. The Autonomous University of Barcelona, another public university, is located in Bellaterra, a town in the Metropolitan Area. Toulouse Business School and the Open University of Catalonia (a private Internet-centred open university) are also based in Barcelona. The city has a network of public schools, from nurseries to high schools, under the responsibility of a consortium led by city council (though the curriculum is the responsibility of the Generalitat de Catalunya). There are also many private schools, some of them Roman Catholic. Most such schools receive a public subsidy on a per-student basis, are subject to inspection by the public authorities, and are required to follow the same curricular guidelines as public schools, though they charge tuition. Known as "escoles concertades", they are distinct from schools whose funding is entirely private ("escoles privades").
The language of instruction at public schools and "escoles concertades" is Catalan, as stipulated by the 2009 Catalan Education Act. Spanish may be used as a language of instruction by teachers of Spanish literature or language, and foreign languages by teachers of those languages. An experimental partial immersion programme adopted by some schools allows for the teaching of a foreign language (English, generally) across the curriculum, though this is limited to a maximum of 30% of the school day. No public school or "escola concertada" in Barcelona may offer 50% or full immersion programmes in a foreign language, nor does any public school or "escola concertada" offer International Baccalaureate programmes. Culture. Barcelona's cultural roots go back 2000 years. Since the arrival of democracy, the Catalan language (very much repressed during the dictatorship of Franco) has been promoted, both by recovering works from the past and by stimulating the creation of new works. Barcelona is designated as a world-class city by the Globalization and World Cities Study Group and Network. It has also been part of the UNESCO Creative Cities Network as a City of Literature since 2015.
Entertainment and performing arts. Barcelona has many venues for live music and theatre, including the world-renowned Gran Teatre del Liceu opera house, the Teatre Nacional de Catalunya, the Teatre Lliure and the Palau de la Música Catalana concert hall. Barcelona also is home to the Barcelona Symphony and Catalonia National Orchestra (Orquestra Simfònica de Barcelona i Nacional de Catalunya, usually known as OBC), the largest symphonic orchestra in Catalonia. In 1999, the OBC inaugurated its new venue in the brand-new Auditorium (L'Auditori). It performs around 75 concerts per season and its current director is Eiji Oue. It is home to the Barcelona Guitar Orchestra, directed by Sergi Vicente. The major thoroughfare of La Rambla is home to mime artists and street performers. Yearly, two major pop music festivals take place in the city, the Sónar Festival and the Primavera Sound Festival. The city also has a thriving alternative music scene, with groups such as The Pinker Tones receiving international attention. Barcelona is an international hub of highly active and diverse nightlife with bars, dance bars and nightclubs staying open well past midnight.
Media. "El Periódico de Catalunya", "La Vanguardia" and "Ara" are Barcelona's three major daily newspapers (the first two with Catalan and Spanish editions, "Ara" only in Catalan) while "Sport" and "El Mundo Deportivo" (both in Spanish) are the city's two major sports daily newspapers, published by the same companies. The city is also served by a number of smaller publications such as "Ara" and "El Punt Avui" (in Catalan), by nationwide newspapers with special Barcelona editions such as "El País" (in Spanish, with an online version in Catalan) and "El Mundo" (in Spanish), and by several free newspapers like "20 minutos" and "Què" (all bilingual). Barcelona's oldest and main online newspaper "VilaWeb" is also the oldest one in Europe (with Catalan and English editions). Several major FM stations include Catalunya Ràdio, RAC 1, RAC 105 and Cadena SER. Barcelona also has a local TV station, Betevé, owned by city council. The headquarters of Televisió de Catalunya, Catalonia's public network, are located in Sant Joan Despí, in Barcelona's metropolitan area.
Sports. Barcelona has a long sporting tradition and hosted the highly successful 1992 Summer Olympics as well as several matches during the 1982 FIFA World Cup (at the two stadiums). It has hosted about 30 sports events of international significance. FC Barcelona is a sports club best known worldwide for its football team, one of the largest and the wealthiest in the world. It has 74 national trophies (while finishing 46 times as runners-up) and 17 continental prizes (with being runners-up 11 times), including five UEFA Champions League trophies out of eight finals and three FIFA Club World Cup wins out of four finals. The club won six trophies in a calendar year in 2009, becoming one of only 2 male football teams in the world to win the coveted sextuple, apart from FC Bayern Munich in 2020. FC Barcelona also has professional teams in other sports like FC Barcelona Regal (basketball), FC Barcelona Handbol (handball), FC Barcelona Hoquei (roller hockey), FC Barcelona Ice Hockey (ice hockey), FC Barcelona Futsal (futsal) and FC Barcelona Rugby (rugby union), all at one point winners of the highest national or European competitions. The club's museum is the second most visited in Catalonia. The matches against cross-town rivals RCD Espanyol are of particular interest, but there are other Barcelonan football clubs in lower categories, like CE Europa and UE Sant Andreu. FC Barcelona's basketball team has a noted rivalry in the Liga ACB with nearby Joventut Badalona.
Barcelona has three UEFA elite stadiums: FC Barcelona's Camp Nou, the largest stadium in Europe with a capacity of 99,354; the publicly owned Estadi Olímpic Lluís Companys, with a capacity of 55,926; used for the 1992 Olympics; and Estadi Cornellà-El Prat, with a capacity of 40,500. Furthermore, the city has several smaller stadiums such as Mini Estadi (also owned by FC Barcelona) with a capacity of 15,000, Camp Municipal Narcís Sala with a capacity of 6,563 and Nou Sardenya with a capacity of 7,000. The city has a further three multifunctional venues for sports and concerts: the Palau Sant Jordi with a capacity of 12,000 to 24,000 (depending on use), the Palau Blaugrana with a capacity of 7,500, and the Palau dels Esports de Barcelona with a capacity of 3,500. Barcelona was the host city for the 2013 World Aquatics Championships, which were held at the Palau San Jordi. Several road running competitions are organized year-round in Barcelona: the Barcelona Marathon every March with over 10,000 participants in 2010, the Cursa de Bombers in April, the Cursa de El Corte Inglés in May (with about 60,000 participants each year), the Cursa de la Mercè, the Cursa Jean Bouin, the Milla Sagrada Família and the San Silvestre. There is also the Ultratrail Collserola which passes through the Collserola forest. The Open Seat Godó, a 50-year-old ATP World Tour 500 Series tennis tournament, is held annually in the facilities of the Real Club de Tenis Barcelona. Each year on Christmas Day, a 200-meter swimming race across the Old Port of Barcelona takes place. Near Barcelona, in Montmeló, the 107,000 capacity Circuit de Barcelona-Catalunya racetrack hosts the Formula One Spanish Grand Prix, the Catalan motorcycle Grand Prix, the Spanish GT Championship and races in the GP2 Series. Skateboarding and cycling are also very popular in Barcelona; in and around the city there are dozens of kilometers of bicycle paths.
Squatter movement. Barcelona is also home to numerous social centres and illegal squats that effectively form a shadow society mainly made up of the unemployed, immigrants, dropouts, anarchists, anti-authoritarians and autonomists. Peter Gelderloos estimates that there around 200 squatted buildings and 40 social centres across the city with thousands of inhabitants, making it one of the largest squatter movements in the world. He notes that they pirate electricity, internet and water allowing them to live on less than one euro a day. He argues that these squats embrace an anarcho-communist and anti-work philosophy, often freely fixing up new houses, cleaning, patching roofs, installing windows, toilets, showers, lights and kitchens. In the wake of austerity, the squats have provided a number of social services to the surrounding residents, including bicycle repair workshops, carpentry workshops, self-defense classes, free libraries, community gardens, free meals, computer labs, language classes, theatre groups, free medical care and legal support services. The squats help elderly residents avoid eviction and organize various protests throughout Barcelona. Notable squats include Can Vies and Can Masdeu.
Transport. Airports. Barcelona is served by Barcelona-El Prat Airport, about south-west of the centre of Barcelona. It is the second-largest airport in Spain, and the largest on the Mediterranean coast, which handled more than 50.17 million passengers in 2018, showing an annual upward trend. It is a main hub for Vueling Airlines and Ryanair, and also a focus for Iberia and Air Europa. The airport mainly serves domestic and European destinations, although some airlines offer destinations in Latin America, Asia and the United States. The airport is connected to the city by highway, metro (Airport T1 and Airport T2 stations), commuter train (Barcelona Airport railway station) and scheduled bus service. A new terminal (T1) has been built, and entered service on 17 June 2009. Some low-cost airlines, also use Girona-Costa Brava Airport, about to the north, Reus Airport, to the south, or Lleida-Alguaire Airport, about to the west, of the city. Sabadell Airport is a smaller airport in the nearby town of Sabadell, devoted to pilot training, aerotaxi and private flights.
Seaport. The Port of Barcelona has a 2000-year-old history and a great contemporary commercial importance. It is Europe's ninth largest container port, with a trade volume of 1.72 million TEU's in 2013. The port is managed by the Port Authority of Barcelona. Its are divided into three zones: Port Vell (the old port), the commercial port and the logistics port (Barcelona Free Port). The port is undergoing an enlargement that will double its size thanks to diverting the mouth of the Llobregat river to the south. The Barcelona harbour is the leading European cruiser port and a very important Mediterranean turnaround base. In 2013, 3.6 million pleasure cruise passengers used the Port of Barcelona. The Port Vell area also houses the Maremagnum (a commercial mall), a multiplex cinema, the IMAX Port Vell and one of Europe's largest aquariums – Aquarium Barcelona, containing 8,000 fish and 11 sharks contained in 22 basins filled with 4 million litres of sea water. The Maremagnum, being situated within the confines of the port, is the only commercial mall in the city that can open on Sundays and public holidays.
National and international rail. Barcelona is a major hub for the Spanish rail network. The city's main Inter-city rail station is Barcelona Sants railway station, whilst Estació de França terminus serves a secondary role handling suburban, regional and medium distance services. Freight services operate to local industries and to the Port of Barcelona. RENFE's AVE high-speed rail system, which is designed for speeds of , was extended from Madrid to Barcelona in 2008 in the form of the Madrid–Barcelona high-speed rail line. A shared RENFE-SNCF high-speed rail connecting Barcelona and France (Paris, Marseille and Toulouse, through Perpignan–Barcelona high-speed rail line) was launched in 2013. Both these lines serve Barcelona Sants terminal station. Metro and regional rail. Barcelona is served by an extensive local public transport network that includes a metro system, a bus network, a regional railway system, trams, funiculars, rack railways, a Gondola lift and aerial cable cars. These networks and lines are run by a number of different operators but they are integrated into a coordinated fare system, administered by the Autoritat del Transport Metropolità (ATM). The system is divided into fare zones (1 to 6) and various Integrated Travel Cards are available.
The Barcelona Metro network comprises twelve lines, identified by an "L" followed by the line number as well as by individual colours. The Metro largely runs underground; eight Metro lines are operated on dedicated track by the Transports Metropolitans de Barcelona (TMB), whilst four lines are operated by the (FGC) and some of them share tracks with RENFE commuter lines. In addition to the city Metro, several regional rail lines operated by RENFE's Rodalies de Catalunya run across the city, providing connections to outlying towns in the surrounding region. Tram. The city's two modern tram systems, Trambaix and Trambesòs, are operated by TRAMMET. A heritage tram line, the Tramvia Blau, also operates between the metro Line 7 and the Funicular del Tibidabo. Funicular and cable car. Barcelona's metro and rail system is supplemented by several aerial cable cars, funiculars and rack railways that provide connections to mountain-top stations. FGC operates the Funicular de Tibidabo up the hill of Tibidabo and the Funicular de Vallvidrera (FGC), while TMB runs the Funicular de Montjuïc up Montjuïc. The city has two aerial cable cars: the Montjuïc Cable Car, which serves Montjuïc castle, and the Port Vell Aerial Tramway that runs via Torre Jaume I and Torre Sant Sebastià over the port.
Bus. Buses in Barcelona are a major form of public transport, with extensive local, interurban and night bus networks. Most local services are operated by the TMB, although some other services are operated by a number of private companies, albeit still within the ATM fare structure. There are 21 night lines, 19 of them go through Plaça de Catalunya, where you can change to other lines. The frequency is 15/20 minutes between 22:20 and 06:00. A separate private bus line, known as Aerobús, links the airport with the city centre, with its own fare structure. The Estació del Nord (Northern Station), a former railway station which was renovated for the 1992 Olympic Games, now serves as the terminus for long-distance and regional bus services. Taxi. Barcelona has a metered taxi fleet governed by the Institut Metropolità del Taxi (Metropolitan Taxi Institute), composed of more than 10,000 cars. Most of the licences are in the hands of self-employed drivers. With their black and yellow livery, Barcelona's taxis are easily spotted, and can be caught from one of many taxi ranks, hailed on street, called by telephone or via app.
On 22 March 2007, Barcelona's City Council started the Bicing service, a bicycle service understood as a public transport. Once the user has their user card, they can take a bicycle from any of the more than 400 stations spread around the city and use it anywhere the urban area of the city, and then leave it at another station. The service has been a success, with 50,000 subscribed users in three months. Roads and highways. Barcelona lies on three international routes, including European route E15 that follows the Mediterranean coast, European route E90 to Madrid and Lisbon, and European route E09 to Paris. It is also served by a comprehensive network of motorways and highways throughout the metropolitan area, including A-2, A-7/AP-7, C-16, C-17, C-31, C-32, C-33, C-60. The city is circled by three half ring roads or bypasses, Ronda de Dalt (B-20) (on the mountain side), Ronda del Litoral (B-10) (along the coast) and Ronda del Mig (separated into two parts: Travessera de Dalt in the north and the Gran Via de Carles III), two partially covered fast highways with several exits that bypass the city.

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