text:
Alloy

An alloy is a mixture of chemical elements of which in most cases at least one is a metallic element, although it is also sometimes used for mixtures of elements; herein only metallic alloys are described. Most alloys are metallic and show good electrical conductivity, ductility, opacity, and luster, and may have properties that differ from those of the pure elements such as increased strength or hardness. In some cases, an alloy may reduce the overall cost of the material while preserving important properties. In other cases, the mixture imparts synergistic properties such as corrosion resistance or mechanical strength.

In an alloy, the atoms are joined by metallic bonding rather than by covalent bonds typically found in chemical compounds. The alloy constituents are usually measured by mass percentage for practical applications, and in atomic fraction for basic science studies. Alloys are usually classified as substitutional or interstitial alloys, depending on the atomic arrangement that forms the alloy. They can be further classified as homogeneous (consisting of a single phase), or heterogeneous (consisting of two or more phases) or intermetallic. An alloy may be a solid solution of metal elements (a single phase, where all metallic grains (crystals) are of the same composition) or a mixture of metallic phases (two or more solutions, forming a microstructure of different crystals within the metal).

Examples of alloys include red gold (gold and copper), white gold (gold and silver), sterling silver (silver and copper), steel or silicon steel (iron with non-metallic carbon or silicon respectively), solder, brass, pewter, duralumin, bronze, and amalgams.

Alloys are used in a wide variety of applications, from the steel alloys, used in everything from buildings to automobiles to surgical tools, to exotic titanium alloys used in the aerospace industry, to beryllium-copper alloys for non-sparking tools.

Characteristics

An alloy is a mixture of chemical elements, which forms an impure substance (admixture) that retains the characteristics of a metal. An alloy is distinct from an impure metal in that, with an alloy, the added elements are well controlled to produce desirable properties, while impure metals such as wrought iron are less controlled, but are often considered useful. Alloys are made by mixing two or more elements, at least one of which is a metal. This is usually called the primary metal or the base metal, and the name of this metal may also be the name of the alloy. The other constituents may or may not be metals but, when mixed with the molten base, they will be soluble and dissolve into the mixture. The mechanical properties of alloys will often be quite different from those of its individual constituents. A metal that is normally very soft (malleable), such as aluminium, can be altered by alloying it with another soft metal, such as copper. Although both metals are very soft and ductile, the resulting aluminium alloy will have much greater strength. Adding a small amount of non-metallic carbon to iron trades its great ductility for the greater strength of an alloy called steel. Due to its very-high strength, but still substantial toughness, and its ability to be greatly altered by heat treatment, steel is one of the most useful and common alloys in modern use. By adding chromium to steel, its resistance to corrosion can be enhanced, creating stainless steel, while adding silicon will alter its electrical characteristics, producing silicon steel.

Like oil and water, a molten metal may not always mix with another element. For example, pure iron is almost completely insoluble with copper. Even when the constituents are soluble, each will usually have a saturation point, beyond which no more of the constituent can be added. Iron, for example, can hold a maximum of 6.67% carbon. Although the elements of an alloy usually must be soluble in the liquid state, they may not always be soluble in the solid state. If the metals remain soluble when solid, the alloy forms a solid solution, becoming a homogeneous structure consisting of identical crystals, called a phase. If as the mixture cools the constituents become insoluble, they may separate to form two or more different types of crystals, creating a heterogeneous microstructure of different phases, some with more of one constituent than the other. However, in other alloys, the insoluble elements may not separate until after crystallization occurs. If cooled very quickly, they first crystallize as a homogeneous phase, but they are supersaturated with the secondary constituents. As time passes, the atoms of these supersaturated alloys can separate from the crystal lattice, becoming more stable, and forming a second phase that serves to reinforce the crystals internally.

Some alloys, such as electrum—an alloy of silver and gold—occur naturally. Meteorites are sometimes made of naturally occurring alloys of iron and nickel, but are not native to the Earth. One of the first alloys made by humans was bronze, which is a mixture of the metals tin and copper. Bronze was an extremely useful alloy to the ancients, because it is much stronger and harder than either of its components. Steel was another common alloy. However, in ancient times, it could only be created as an accidental byproduct from the heating of iron ore in fires (smelting) during the manufacture of iron. Other ancient alloys include pewter, brass and pig iron. In the modern age, steel can be created in many forms. Carbon steel can be made by varying only the carbon content, producing soft alloys like mild steel or hard alloys like spring steel. Alloy steels can be made by adding other elements, such as chromium, molybdenum, vanadium or nickel, resulting in alloys such as high-speed steel or tool steel. Small amounts of manganese are usually alloyed with most modern steels because of its ability to remove unwanted impurities, like phosphorus, sulfur and oxygen, which can have detrimental effects on the alloy. However, most alloys were not created until the 1900s, such as various aluminium, titanium, nickel, and magnesium alloys. Some modern superalloys, such as incoloy, inconel, and hastelloy, may consist of a multitude of different elements.

An alloy is technically an impure metal, but when referring to alloys, the term impurities usually denotes undesirable elements. Such impurities are introduced from the base metals and alloying elements, but are removed during processing. For instance, sulfur is a common impurity in steel. Sulfur combines readily with iron to form iron sulfide, which is very brittle, creating weak spots in the steel. Lithium, sodium and calcium are common impurities in aluminium alloys, which can have adverse effects on the structural integrity of castings. Conversely, otherwise pure-metals that contain unwanted impurities are often called "impure metals" and are not usually referred to as alloys. Oxygen, present in the air, readily combines with most metals to form metal oxides; especially at higher temperatures encountered during alloying. Great care is often taken during the alloying process to remove excess impurities, using fluxes, chemical additives, or other methods of extractive metallurgy.

Theory
Alloying a metal is done by combining it with one or more other elements. The most common and oldest alloying process is performed by heating the base metal beyond its melting point and then dissolving the solutes into the molten liquid, which may be possible even if the melting point of the solute is far greater than that of the base. For example, in its liquid state, titanium is a very strong solvent capable of dissolving most metals and elements. In addition, it readily absorbs gases like oxygen and burns in the presence of nitrogen. This increases the chance of contamination from any contacting surface, and so must be melted in vacuum induction-heating and special, water-cooled, copper crucibles. However, some metals and solutes, such as iron and carbon, have very high melting-points and were impossible for ancient people to melt. Thus, alloying (in particular, interstitial alloying) may also be performed with one or more constituents in a gaseous state, such as found in a blast furnace to make pig iron (liquid-gas), nitriding, carbonitriding or other forms of case hardening (solid-gas), or the cementation process used to make blister steel (solid-gas). It may also be done with one, more, or all of the constituents in the solid state, such as found in ancient methods of pattern welding (solid-solid), shear steel (solid-solid), or crucible steel production (solid-liquid), mixing the elements via solid-state diffusion.

By adding another element to a metal, differences in the size of the atoms create internal stresses in the lattice of the metallic crystals; stresses that often enhance its properties. For example, the combination of carbon with iron produces steel, which is stronger than iron, its primary element. The electrical and thermal conductivity of alloys is usually lower than that of the pure metals. The physical properties, such as density, reactivity, Young's modulus of an alloy may not differ greatly from those of its base element, but engineering properties such as tensile strength, ductility, and shear strength may be substantially different from those of the constituent materials. This is sometimes a result of the sizes of the atoms in the alloy, because larger atoms exert a compressive force on neighboring atoms, and smaller atoms exert a tensile force on their neighbors, helping the alloy resist deformation. Sometimes alloys may exhibit marked differences in behavior even when small amounts of one element are present. For example, impurities in semiconducting ferromagnetic alloys lead to different properties, as first predicted by White, Hogan, Suhl, Tian Abrie and Nakamura.

Unlike pure metals, most alloys do not have a single melting point, but a melting range during which the material is a mixture of solid and liquid phases (a slush). The temperature at which melting begins is called the solidus, and the temperature when melting is just complete is called the liquidus. For many alloys there is a particular alloy proportion (in some cases more than one), called either a eutectic mixture or a peritectic composition, which gives the alloy a unique and low melting point, and no liquid/solid slush transition.

Heat treatment

Alloying elements are added to a base metal, to induce hardness, toughness, ductility, or other desired properties. Most metals and alloys can be work hardened by creating defects in their crystal structure. These defects are created during plastic deformation by hammering, bending, extruding, et cetera, and are permanent unless the metal is recrystallized. Otherwise, some alloys can also have their properties altered by heat treatment. Nearly all metals can be softened by annealing, which recrystallizes the alloy and repairs the defects, but not as many can be hardened by controlled heating and cooling. Many alloys of aluminium, copper, magnesium, titanium, and nickel can be strengthened to some degree by some method of heat treatment, but few respond to this to the same degree as does steel.

The base metal iron of the iron-carbon alloy known as steel, undergoes a change in the arrangement (allotropy) of the atoms of its crystal matrix at a certain temperature (usually between 1500 F and 1600 F, depending on carbon content). This allows the smaller carbon atoms to enter the interstices of the iron crystal. When this diffusion happens, the carbon atoms are said to be in solution in the iron, forming a particular single, homogeneous, crystalline phase called austenite. If the steel is cooled slowly, the carbon can diffuse out of the iron and it will gradually revert to its low temperature allotrope. During slow cooling, the carbon atoms will no longer be as soluble with the iron, and will be forced to precipitate out of solution, nucleating into a more concentrated form of iron carbide (Fe3C) in the spaces between the pure iron crystals. The steel then becomes heterogeneous, as it is formed of two phases, the iron-carbon phase called cementite (or carbide), and pure iron ferrite. Such a heat treatment produces a steel that is rather soft. If the steel is cooled quickly, however, the carbon atoms will not have time to diffuse and precipitate out as carbide, but will be trapped within the iron crystals. When rapidly cooled, a diffusionless (martensite) transformation occurs, in which the carbon atoms become trapped in solution. This causes the iron crystals to deform as the crystal structure tries to change to its low temperature state, leaving those crystals very hard but much less ductile (more brittle).

While the high strength of steel results when diffusion and precipitation is prevented (forming martensite), most heat-treatable alloys are precipitation hardening alloys, that depend on the diffusion of alloying elements to achieve their strength. When heated to form a solution and then cooled quickly, these alloys become much softer than normal, during the diffusionless transformation, but then harden as they age. The solutes in these alloys will precipitate over time, forming intermetallic phases, which are difficult to discern from the base metal. Unlike steel, in which the solid solution separates into different crystal phases (carbide and ferrite), precipitation hardening alloys form different phases within the same crystal. These intermetallic alloys appear homogeneous in crystal structure, but tend to behave heterogeneously, becoming hard and somewhat brittle.

In 1906, precipitation hardening alloys were discovered by Alfred Wilm. Precipitation hardening alloys, such as certain alloys of aluminium, titanium, and copper, are heat-treatable alloys that soften when quenched (cooled quickly), and then harden over time. Wilm had been searching for a way to harden aluminium alloys for use in machine-gun cartridge cases. Knowing that aluminium-copper alloys were heat-treatable to some degree, Wilm tried quenching a ternary alloy of aluminium, copper, and the addition of magnesium, but was initially disappointed with the results. However, when Wilm retested it the next day he discovered that the alloy increased in hardness when left to age at room temperature, and far exceeded his expectations. Although an explanation for the phenomenon was not provided until 1919, duralumin was one of the first "age hardening" alloys used, becoming the primary building material for the first Zeppelins, and was soon followed by many others. Because they often exhibit a combination of high strength and low weight, these alloys became widely used in many forms of industry, including the construction of modern aircraft.

Mechanisms

When a molten metal is mixed with another substance, there are two mechanisms that can cause an alloy to form, called atom exchange and the interstitial mechanism. The relative size of each element in the mix plays a primary role in determining which mechanism will occur. When the atoms are relatively similar in size, the atom exchange method usually happens, where some of the atoms composing the metallic crystals are substituted with atoms of the other constituent. This is called a substitutional alloy. Examples of substitutional alloys include bronze and brass, in which some of the copper atoms are substituted with either tin or zinc atoms respectively.

In the case of the interstitial mechanism, one atom is usually much smaller than the other and can not successfully substitute for the other type of atom in the crystals of the base metal. Instead, the smaller atoms become trapped in the interstitial sites between the atoms of the crystal matrix. This is referred to as an interstitial alloy. Steel is an example of an interstitial alloy, because the very small carbon atoms fit into interstices of the iron matrix.

Stainless steel is an example of a combination of interstitial and substitutional alloys, because the carbon atoms fit into the interstices, but some of the iron atoms are substituted by nickel and chromium atoms.

Meteoric iron
The use of alloys by humans started with the use of meteoric iron, a naturally occurring alloy of nickel and iron. It is the main constituent of iron meteorites. As no metallurgic processes were used to separate iron from nickel, the alloy was used as it was. Meteoric iron could be forged from a red heat to make objects such as tools, weapons, and nails. In many cultures it was shaped by cold hammering into knives and arrowheads. They were often used as anvils. Meteoric iron was very rare and valuable, and difficult for ancient people to work.

Bronze and brass

Iron is usually found as iron ore on Earth, except for one deposit of native iron in Greenland, which was used by the Inuit. Native copper, however, was found worldwide, along with silver, gold, and platinum, which were also used to make tools, jewelry, and other objects since Neolithic times. Copper was the hardest of these metals, and the most widely distributed. It became one of the most important metals to the ancients. Around 10,000 years ago in the highlands of Anatolia (Turkey), humans learned to smelt metals such as copper and tin from ore. Around 2500 BC, people began alloying the two metals to form bronze, which was much harder than its ingredients. Tin was rare, however, being found mostly in Great Britain. In the Middle East, people began alloying copper with zinc to form brass. Ancient civilizations took into account the mixture and the various properties it produced, such as hardness, toughness and melting point, under various conditions of temperature and work hardening, developing much of the information contained in modern alloy phase diagrams. For example, arrowheads from the Chinese Qin dynasty (around 200 BC) were often constructed with a hard bronze-head, but a softer bronze-tang, combining the alloys to prevent both dulling and breaking during use.

Amalgams
Mercury has been smelted from cinnabar for thousands of years. Mercury dissolves many metals, such as gold, silver, and tin, to form amalgams (an alloy in a soft paste or liquid form at ambient temperature). Amalgams have been used since 200 BC in China for gilding objects such as armor and mirrors with precious metals. The ancient Romans often used mercury-tin amalgams for gilding their armor. The amalgam was applied as a paste and then heated until the mercury vaporized, leaving the gold, silver, or tin behind. Mercury was often used in mining, to extract precious metals like gold and silver from their ores.

Precious metals

Many ancient civilizations alloyed metals for purely aesthetic purposes. In ancient Egypt and Mycenae, gold was often alloyed with copper to produce red-gold, or iron to produce a bright burgundy-gold. Gold was often found alloyed with silver or other metals to produce various types of colored gold. These metals were also used to strengthen each other, for more practical purposes. Copper was often added to silver to make sterling silver, increasing its strength for use in dishes, silverware, and other practical items. Quite often, precious metals were alloyed with less valuable substances as a means to deceive buyers. Around 250 BC, Archimedes was commissioned by the King of Syracuse to find a way to check the purity of the gold in a crown, leading to the famous bath-house shouting of "Eureka!" upon the discovery of Archimedes' principle.

Pewter
The term pewter covers a variety of alloys consisting primarily of tin. As a pure metal, tin is much too soft to use for most practical purposes. However, during the Bronze Age, tin was a rare metal in many parts of Europe and the Mediterranean, so it was often valued higher than gold. To make jewellery, cutlery, or other objects from tin, workers usually alloyed it with other metals to increase strength and hardness. These metals were typically lead, antimony, bismuth or copper. These solutes were sometimes added individually in varying amounts, or added together, making a wide variety of objects, ranging from practical items such as dishes, surgical tools, candlesticks or funnels, to decorative items like ear rings and hair clips.

The earliest examples of pewter come from ancient Egypt, around 1450 BC. The use of pewter was widespread across Europe, from France to Norway and Britain (where most of the ancient tin was mined) to the Near East. The alloy was also used in China and the Far East, arriving in Japan around 800 AD, where it was used for making objects like ceremonial vessels, tea canisters, or chalices used in shinto shrines.

Iron

The first known smelting of iron began in Anatolia, around 1800 BC. Called the bloomery process, it produced very soft but ductile wrought iron. By 800 BC, iron-making technology had spread to Europe, arriving in Japan around 700 AD. Pig iron, a very hard but brittle alloy of iron and carbon, was being produced in China as early as 1200 BC, but did not arrive in Europe until the Middle Ages. Pig iron has a lower melting point than iron, and was used for making cast-iron. However, these metals found little practical use until the introduction of crucible steel around 300 BC. These steels were of poor quality, and the introduction of pattern welding, around the 1st century AD, sought to balance the extreme properties of the alloys by laminating them, to create a tougher metal. Around 700 AD, the Japanese began folding bloomery-steel and cast-iron in alternating layers to increase the strength of their swords, using clay fluxes to remove slag and impurities. This method of Japanese swordsmithing produced one of the purest steel-alloys of the ancient world.

While the use of iron started to become more widespread around 1200 BC, mainly because of interruptions in the trade routes for tin, the metal was much softer than bronze. However, very small amounts of steel, (an alloy of iron and around 1% carbon), was always a byproduct of the bloomery process. The ability to modify the hardness of steel by heat treatment had been known since 1100 BC, and the rare material was valued for the manufacture of tools and weapons. Because the ancients could not produce temperatures high enough to melt iron fully, the production of steel in decent quantities did not occur until the introduction of blister steel during the Middle Ages. This method introduced carbon by heating wrought iron in charcoal for long periods of time, but the absorption of carbon in this manner is extremely slow thus the penetration was not very deep, so the alloy was not homogeneous. In 1740, Benjamin Huntsman began melting blister steel in a crucible to even out the carbon content, creating the first process for the mass production of tool steel. Huntsman's process was used for manufacturing tool steel until the early 1900s.

The introduction of the blast furnace to Europe in the Middle Ages meant that people could produce pig iron in much higher volumes than wrought iron. Because pig iron could be melted, people began to develop processes to reduce carbon in liquid pig iron to create steel. Puddling had been used in China since the first century, and was introduced in Europe during the 1700s, where molten pig iron was stirred while exposed to the air, to remove the carbon by oxidation. In 1858, Henry Bessemer developed a process of steel-making by blowing hot air through liquid pig iron to reduce the carbon content. The Bessemer process led to the first large scale manufacture of steel.

Steel is an alloy of iron and carbon, but the term alloy steel usually only refers to steels that contain other elements— like vanadium, molybdenum, or cobalt—in amounts sufficient to alter the properties of the base steel. Since ancient times, when steel was used primarily for tools and weapons, the methods of producing and working the metal were often closely guarded secrets. Even long after the Age of Enlightenment, the steel industry was very competitive and manufacturers went through great lengths to keep their processes confidential, resisting any attempts to scientifically analyze the material for fear it would reveal their methods. For example, the people of Sheffield, a center of steel production in England, were known to routinely bar visitors and tourists from entering town to deter industrial espionage. Thus, almost no metallurgical information existed about steel until 1860. Because of this lack of understanding, steel was not generally considered an alloy until the decades between 1930 and 1970 (primarily due to the work of scientists like William Chandler Roberts-Austen, Adolf Martens, and Edgar Bain), so "alloy steel" became the popular term for ternary and quaternary steel-alloys.

After Benjamin Huntsman developed his crucible steel in 1740, he began experimenting with the addition of elements like manganese (in the form of a high-manganese pig-iron called spiegeleisen), which helped remove impurities such as phosphorus and oxygen; a process adopted by Bessemer and still used in modern steels (albeit in concentrations low enough to still be considered carbon steel). Afterward, many people began experimenting with various alloys of steel without much success. However, in 1882, Robert Hadfield, being a pioneer in steel metallurgy, took an interest and produced a steel alloy containing around 12% manganese. Called mangalloy, it exhibited extreme hardness and toughness, becoming the first commercially viable alloy-steel. Afterward, he created silicon steel, launching the search for other possible alloys of steel.

Robert Forester Mushet found that by adding tungsten to steel it could produce a very hard edge that would resist losing its hardness at high temperatures. "R. Mushet's special steel" (RMS) became the first high-speed steel. Mushet's steel was quickly replaced by tungsten carbide steel, developed by Taylor and White in 1900, in which they doubled the tungsten content and added small amounts of chromium and vanadium, producing a superior steel for use in lathes and machining tools. In 1903, the Wright brothers used a chromium-nickel steel to make the crankshaft for their airplane engine, while in 1908 Henry Ford began using vanadium steels for parts like crankshafts and valves in his Model T Ford, due to their higher strength and resistance to high temperatures. In 1912, the Krupp Ironworks in Germany developed a rust-resistant steel by adding 21% chromium and 7% nickel, producing the first stainless steel.

Others
Due to their high reactivity, most metals were not discovered until the 19th century. A method for extracting aluminium from bauxite was proposed by Humphry Davy in 1807, using an electric arc. Although his attempts were unsuccessful, by 1855 the first sales of pure aluminium reached the market. However, as extractive metallurgy was still in its infancy, most aluminium extraction-processes produced unintended alloys contaminated with other elements found in the ore; the most abundant of which was copper. These aluminium-copper alloys (at the time termed "aluminum bronze") preceded pure aluminium, offering greater strength and hardness over the soft, pure metal, and to a slight degree were found to be heat treatable. However, due to their softness and limited hardenability these alloys found little practical use, and were more of a novelty, until the Wright brothers used an aluminium alloy to construct the first airplane engine in 1903. During the time between 1865 and 1910, processes for extracting many other metals were discovered, such as chromium, vanadium, tungsten, iridium, cobalt, and molybdenum, and various alloys were developed.

Prior to 1910, research mainly consisted of private individuals tinkering in their own laboratories. However, as the aircraft and automotive industries began growing, research into alloys became an industrial effort in the years following 1910, as new magnesium alloys were developed for pistons and wheels in cars, and pot metal for levers and knobs, and aluminium alloys developed for airframes and aircraft skins were put into use. The Doehler Die Casting Co. of Toledo, Ohio were known for the production of Brastil, a high tensile corrosion resistant bronze alloy.

title: Alloy


text:
Artistic revolution

Throughout history, forms of art have gone through periodic abrupt changes called artistic revolutions. Movements have come to an end to be replaced by a new movement markedly different in striking ways.

Scientific and technological
1 Not all artistic revolutions were political. Sometimes, science and technological innovations have brought about unforeseen transformations in the works of artists. The stylistic revolution known as Impressionism, by painters eager to more accurately capture the changing colors of light and shadow, is inseparable from discoveries and inventions in the mid-19th century in which the style was born.

Michel Eugène Chevreul, a French chemist hired as director of dyes at a French tapestry works, began to investigate the optical nature of color in order to improve color in fabrics. Chevreul realized It was the eye, and not the dye, that had the greatest influence on color, and from this, he revolutionized color theory by grasping what came to be called the law of simultaneous contrast: that colors mutually influence one another when juxtaposed, each imposing its own complementary color on the other. The French painter Eugène Delacroix, who had been experimenting with what he called broken tones, embraced Chevreul's book, The Law of Contrast of Color (1839) with its explanations of how juxtaposed colors can enhance or diminish each other, and his exploration of all the visible colors of the spectrum. Inspired by Chevreul's 1839 treatise, Delacroix passed his enthusiasm on to the young artists who were inspired by him. It was Chevreul who led the Impressionists to grasp that they should apply separate brushstrokes of pure color to a canvas and allow the viewer's eye to combine them optically.

They were aided greatly in this by innovations in oil paint itself. Since the Renaissance, painters had to grind pigment, add oil and thus create their own paints; these time-consuming paints also quickly dried out, making studio painting a necessity for large works, and limiting painters to mix one or two colors at a time and fill in an entire area using just that one color before it dried out. In 1841, a little-known American painter named John G. Rand invented a simple improvement without which the Impressionist movement could not have occurred: the small, flexible tin tube with removable cap in which oil paints could be stored. Oil paints kept in such tubes stayed moist, usable, and portable. For the first time since the Renaissance, painters were not trapped by the time frame of how quickly oil paint dried.

Paints in tubes could be easily loaded up and carried out into the real world, to directly observe the play of color and natural light, in shadow and movement, to paint in the moment. Selling the oil paint in tubes also brought about the arrival of dazzling new pigments - chrome yellow, cadmium blue - invented by 19th century industrial chemists. The tubes freed the Impressionists to paint quickly, and across an entire canvas, rather than carefully delineated single-color sections at a time; in short, to sketch directly in oil - racing across the canvas in every color that came to hand and thus inspiring their name of "impressionists" - since such speedy, bold brushwork and dabs of separate colors made contemporary critics think their paintings were mere impressions, not finished paintings, which were to have no visible brush marks at all, seamless under layers of varnish.

Pierre-Auguste Renoir said, “Without colors in tubes, there would be no Cézanne, no Monet, no Pissarro, and no Impressionism.”

Finally, the careful, hyper-realistic techniques of French neo-classicism were seen as stiff and lifeless when compared to the remarkable new vision of the world as seen through the new invention of photography by the mid-1850s. It was not merely that the increasing ability of this new invention, particularly by the French inventor Daguerre, made the realism of the painted image redundant as he deliberately competed in the Paris diorama with large-scale historical paintings. The neo-classical subject matter, limited by Academic tradition to Greek and Roman legends, historical battles and Biblical stories, seemed oppressively clichéd and limited to artists eager to explore the actual world in front of their own eyes revealed by the camera - daily life, candid groupings of everyday people doing simple things, Paris itself, rural landscapes and most particularly the play of captured light - not the imaginary lionizing of unseen past events. Early photographs influenced Impressionist style by its use of asymmetry, cropping and most obviously the blurring of motion, as inadvertently captured in the very slow speeds of early photography.

Edgar Degas, Claude Monet, Pierre-Auguste Renoir - in their framing, use of color, light and shadow, subject matter - put these innovations to work to create a new language of visual beauty and meaning.

Faking revolution: the CIA and Abstract Expressionism
Their initial break with realism into an exploration of light, color and the nature of paint was brought to an ultimate conclusion by the abstract expressionists who broke away from recognizable content of any kind into works of pure shape, color and painterliness which emerged at the end of the Second World War. At first thought of as primitive, inept works - as in "my four year old could do that - these works were misunderstood and neglected until given critical and support by the rise of art journalists and critics who championed their work in the 1940s and 50s, expressing the power of such work in aesthetic terms the artists themselves seldom used, or even understood. Jackson Pollock who pioneered splatter painting, dispensing with a paint brush altogether, soon became lionized as the angry young man in a large spread in Life magazine.

In fact, in a deliberate, secret and successful effort to separate artistic revolutions from political ones, abstract expressionists like Pollock, Robert Motherwell, Willem de Kooning and Mark Rothko, while seemingly difficult, pathbreaking artists, were in fact secretly supported for twenty years by the Central Intelligence Agency (CIA) in a Cold War policy begun in 1947 to prove that the United States could foster more artistic freedom than the Soviet bloc. "It was recognized that Abstract Expressionism was the kind of art that made Socialist Realism look even more stylized and rigid and confined than it was," said former CIA case worker Donald Jameson, who finally broke the silence on this program in 1995. Ironically, the covert CIA support for these radical works was required because an attempt to use government funds for a European tour of these works during the Truman administration led to a public uproar in conservative McCarthy-era America, with Truman famously remarking, "If that's art, I'm a Hottentot." Thus, the program was hidden under the guise of fabricated foundations and the support of wealthy patrons who were actually using CIA funds, not their own, to sponsor traveling exhibitions of American abstract expressionists all over the world, publish books and articles praising them and to purchase and exhibit abstract expressionist works in major American and British museums. Thomas Braden, in charge of these cultural programs for the CIA, in the early years of the Cold War, had formerly been executive secretary of the Museum of Modern Art, America's leading institution for 20th century art and the charges of collusion between the two echoed for many years after this program was revealed, though most of the artists involved had no idea they were being used in this way and were furious when they found out.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Artistic revolution


text:
Agrarianism

Agrarianism is a social and political philosophy that promotes subsistence agriculture, family farming, widespread property ownership, and political decentralization. Adherents of agrarianism tend to value traditional bonds of local community over urban modernity. Agrarian political parties sometimes aim to support the rights and sustainability of small farmers and poor peasants against the wealthy in society.

Philosophy
Some scholars suggest that agrarianism values rural society as superior to urban society and the independent farmer as superior to the paid worker, and sees farming as a way of life that can shape the ideal social values. It stresses the superiority of a simpler rural life as opposed to the complexity of city life. For example, M. Thomas Inge defines agrarianism by the following basic tenets:
 * Farming is the sole occupation that offers total independence and self-sufficiency.
 * Urban life, capitalism, and technology destroy independence and dignity and foster vice and weakness.
 * The agricultural community, with its fellowship of labor and co-operation, is the model society.
 * The farmer has a solid, stable position in the world order. They have "a sense of identity, a sense of historical and religious tradition, a feeling of belonging to a concrete family, place, and region, which are psychologically and culturally beneficial." The harmony of their life checks the encroachments of a fragmented, alienated modern society.
 * Cultivation of the soil "has within it a positive spiritual good" and from it the cultivator acquires the virtues of "honor, manliness, self-reliance, courage, moral integrity, and hospitality." They result from a direct contact with nature and, through nature, a closer relationship to God. The agrarian is blessed in that they follow the example of God in creating order out of chaos.

History
The philosophical roots of agrarianism include European and Chinese philosophers. The Chinese school of Agriculturalism (农家/農家) was a philosophy that advocated peasant utopian communalism and egalitarianism. In societies influenced by Confucianism, the farmer was considered an esteemed productive member of society, but merchants who made money were looked down upon. That influenced European intellectuals like François Quesnay, an avid Confucianist and advocate of China's agrarian policies, in forming the French agrarian philosophy of physiocracy. The physiocrats, along with the ideas of John Locke and the Romantic Era, formed the basis of modern European and American agrarianism.

Jeffersonian democracy

The United States president Thomas Jefferson was an agrarian who based his ideas about the budding American democracy around the notion that farmers are "the most valuable citizens" and the truest republicans. Jefferson and his support base were committed to American republicanism, which they saw as being in opposition to aristocracy and corruption, and which prioritized virtue, exemplified by the "yeoman farmer", "planters", and the "plain folk". In praising the rural farmfolk, the Jeffersonians felt that financiers, bankers and industrialists created "cesspools of corruption" in the cities and should thus be avoided.

The Jeffersonians sought to align the American economy more with agriculture than industry. Part of their motive to do so was Jefferson's fear that the over-industrialization of America would create a class of wage slaves who relied on their employers for income and sustenance. In turn, these workers would cease to be independent voters as their vote could be manipulated by said employers. To counter this, Jefferson introduced, as scholar Clay Jenkinson noted, "a graduated income tax that would serve as a disincentive to vast accumulations of wealth and would make funds available for some sort of benign redistribution downward" and tariffs on imported articles, which were mainly purchased by the wealthy. In 1811, Jefferson, writing to a friend, explained: "these revenues will be levied entirely on the rich... . the rich alone use imported articles, and on these alone the whole taxes of the general government are levied. the poor man ... pays not a farthing of tax to the general government, but on his salt."

There is general agreement that the substantial United States' federal policy of offering land grants (such as thousands of gifts of land to veterans) had a positive impact on economic development in the 19th century.

Agrarian socialism
Agrarian socialism is a form of agrarianism that is anti-capitalist in nature and seeks to introduce socialist economic systems in their stead.

Zapatismo
Notable agrarian socialists include Emiliano Zapata who was a leading figure in the Mexican Revolution. As part of the Liberation Army of the South, his group of revolutionaries fought on behalf of the Mexican peasants, whom they saw as exploited by the landowning classes. Zapata published Plan of Ayala, which called for significant land reforms and land redistribution in Mexico as part of the revolution. Zapata was killed and his forces crushed over the course of the Revolution, but his political ideas lived on in the form of Zapatismo.

Zapatismo would form the basis for neozapatismo, the ideology of the Zapatista Army of National Liberation. Known as Ejército Zapatista de Liberación Nacional or EZLN in Spanish, EZLN is a far-left libertarian socialist political and militant group that emerged in the state of Chiapas in southmost Mexico in 1994. EZLN and Neozapatismo, as explicit in their name, seek to revive the agrarian socialist movement of Zapata, but fuse it with new elements such as a commitment to indigenous rights and community-level decision making.

Subcommander Marcos, a leading member of the movement, argues that the peoples' collective ownership of the land was and is the basis for all subsequent developments the movement sought to create: ...When the land became property of the peasants ... when the land passed into the hands of those who work it ... [This was] the starting point for advances in government, health, education, housing, nutrition, women's participation, trade, culture, communication, and information ...[it was] recovering the means of production, in this case, the land, animals, and machines that were in the hands of large property owners."

Maoism
Maoism, the far-left ideology of Mao Zedong and his followers, places a heavy emphasis on the role of peasants in its goals. In contrast to other Marxist schools of thought which normally seek to acquire the support of urban workers, Maoism sees the peasantry as key. Believing that "political power grows out of the barrel of a gun", Maoism saw the Chinese Peasantry as the prime source for a Marxist vanguard because it possessed two qualities: (i) they were poor, and (ii) they were a political blank slate; in Mao's words, "A clean sheet of paper has no blotches, and so the newest and most beautiful words can be written on it". During the Chinese Civil War and the Second Sino-Japanese War, Mao and the Chinese Communist Party made extensive use of peasants and rural bases in their military tactics, often eschewing the cities.

Following the eventual victory of the Communist Party in both wars, the countryside and how it should be run remained a focus for Mao. In 1958, Mao launched the Great Leap Forward, a social and economic campaign which, amongst other things, altered many aspects of rural Chinese life. It introduced mandatory collective farming and forced the peasantry to organize itself into communal living units which were known as people's communes. These communes, which consisted of 5,000 people on average, were expected to meet high production quotas while the peasants who lived on them adapted to this radically new way of life. The communes were run as co-operatives where wages and money were replaced by work points. Peasants who criticised this new system were persecuted as "rightists" and "counter-revolutionaries". Leaving the communes was forbidden and escaping from them was difficult or impossible, and those who attempted it were subjected to party-orchestrated "public struggle sessions," which further jeopardized their survival. These public criticism sessions were often used to intimidate the peasants into obeying local officials and they often devolved into little more than public beatings.

On the communes, experiments were conducted in order to find new methods of planting crops, efforts were made to construct new irrigation systems on a massive scale, and the communes were all encouraged to produce steel backyard furnaces as part of an effort to increase steel production. However, following the Anti-Rightist Campaign, Mao had instilled a mass distrust of intellectuals into China, and thus engineers often were not consulted with regard to the new irrigation systems and the wisdom of asking untrained peasants to produce good quality steel from scrap iron was not publicly questioned. Similarly, the experimentation with the crops did not produce results. In addition to this the Four Pests Campaign was launched, in which the peasants were called upon to destroy sparrows and other wild birds that ate crop seeds, in order to protect fields. Pest birds were shot down or scared away from landing until they dropped from exhaustion. This campaign resulted in an ecological disaster that saw an explosion of the vermin population, especially crop-eating insects, which was consequently not in danger of being killed by predators.

None of these new systems were working, but local leaders did not dare to state this, instead, they falsified reports so as not to be punished for failing to meet the quotas. In many cases they stated that they were greatly exceeding their quotas, and in turn, the Chinese state developed a completely false sense of success with regard to the commune system.

All of this culminated in the Great Chinese Famine, which began in 1959, lasted 3 years, and saw an estimated 15 to 30 million Chinese people die. A combination of bad weather and the new, failed farming techniques that were introduced by the state led to massive shortages of food. By 1962, the Great Leap Forward was declared to be at an end.

In the late 1960s and early 1970s, Mao once again radically altered life in rural China with the launching of the Down to the Countryside Movement. As a response to the Great Chinese Famine, the Chinese President Liu Shaoqi began "sending down" urban youths to rural China in order to recover its population losses and alleviate overcrowding in the cities. However, Mao turned the practice into a political crusade, declaring that the sending down would strip the youth of any bourgeois tendencies by forcing them to learn from the unprivileged rural peasants. In reality, it was the Communist Party's attempt to reign in the Red Guards, who had become uncontrollable during the course of the Cultural Revolution. 10% of the 1970 urban population of China was sent out to remote rural villages, often in Inner Mongolia. The villages, which were still poorly recovering from the effects of the Great Chinese Famine, did not have the excess resources that were needed to support the newcomers. Furthermore, the so-called "sent-down youth" had no agricultural experience and as a result, they were unaccustomed to the harsh lifestyle that existed in the countryside, and their unskilled labor in the villages provided little benefit to the agricultural sector. As a result, many of the sent-down youth died in the countryside. The relocation of the youths was originally intended to be permanent, but by the end of the Cultural Revolution, the Communist Party relented and some of those who had the capacity to return to the cities were allowed to do so.

In imitation of Mao's policies, the Khmer Rouge of Cambodia (who were heavily funded and supported by the People's Republic of China) created their own version of the Great Leap Forward which was known as "Maha Lout Ploh". With the Great Leap Forward as its model, it had similarly disastrous effects, contributing to what is now known as the Cambodian genocide. As a part of the Maha Lout Ploh, the Khmer Rouge sought to create an entirely agrarian socialist society by forcibly relocating 100,000 people to move from Cambodia's cities into newly created communes. The Khmer Rouge leader, Pol Pot sought to "purify" the country by setting it back to "Year Zero", freeing it from "corrupting influences". Besides trying to completely de-urbanize Cambodia, ethnic minorities were slaughtered along with anyone else who was suspected of being a "reactionary" or a member of the "bourgeoisie", to the point that wearing glasses was seen as grounds for execution. The killings were only brought to an end when Cambodia was invaded by the neighboring socialist nation of Vietnam, whose army toppled the Khmer Rouge. However, with Cambodia's entire society and economy in disarray, including its agricultural sector, the country still plunged into renewed famine due to vast food shortages. However, as international journalists began to report on the situation and send images of it out to the world, a massive international response was provoked, leading to one of the most concentrated relief efforts of its time.

Notable agrarian parties
Peasant parties first appeared across Eastern Europe between 1860 and 1910, when commercialized agriculture and world market forces disrupted traditional rural society, and the railway and growing literacy facilitated the work of roving organizers. Agrarian parties advocated land reforms to redistribute land on large estates among those who work it. They also wanted village cooperatives to keep the profit from crop sales in local hands and credit institutions to underwrite needed improvements. Many peasant parties were also nationalist parties because peasants often worked their land for the benefit of landlords of different ethnicity.

Peasant parties rarely had any power before World War I but some became influential in the interwar era, especially in Bulgaria and Czechoslovakia. For a while, in the 1920s and the 1930s, there was a Green International (International Agrarian Bureau) based on the peasant parties in Bulgaria, Czechoslovakia, Poland, and Serbia. It functioned primarily as an information center that spread the ideas of agrarianism and combating socialism on the left and landlords on the right and never launched any significant activities.

Bulgaria
In Bulgaria, the Bulgarian Agrarian National Union (BZNS) was organized in 1899 to resist taxes and build cooperatives. BZNS came to power in 1919 and introduced many economic, social, and legal reforms. However, conservative forces crushed BZNS in a 1923 coup and assassinated its leader, Aleksandar Stamboliyski (1879–1923). BZNS was made into a communist puppet group until 1989, when it reorganized as a genuine party.

Czechoslovakia
In Czechoslovakia, the Republican Party of Agricultural and Smallholder People often shared power in parliament as a partner in the five-party pětka coalition. The party's leader, Antonín Švehla (1873–1933), was prime minister several times. It was consistently the strongest party, forming and dominating coalitions. It moved beyond its original agrarian base to reach middle-class voters. The party was banned by the National Front after the Second World War.

France
In France, the Hunting, Fishing, Nature, Tradition party is a moderate conservative, agrarian party, reaching a peak of 4.23% in the 2002 French presidential election. It would later on become affiliated to France's main conservative party, Union for a Popular Movement. More recently, the Resistons! movement of Jean Lassalle espoused agrarianism.

Hungary
In Hungary, the first major agrarian party, the small-holders party was founded in 1908. The party became part of the government in the 1920s but lost influence in the government. A new party, the Independent Smallholders, Agrarian Workers and Civic Party was established in 1930 with a more radical program representing larger scale land redistribution initiatives. They implemented this program together with the other coalition parties after WWII. However, after 1949 the party was outlawed when a one-party system was introduced. They became part of the government again 1990–1994, and 1998–2002 after which they lost political support. The ruling Fidesz party has an agrarian faction, and promotes agrarian interest since 2010 with the emphasis now placed on supporting larger family farms versus small-holders.

Ireland
In the late 19th century, the Irish National Land League aimed to abolish landlordism in Ireland and enable tenant farmers to own the land they worked on. The "Land War" of 1878–1909 led to the Irish Land Acts, ending absentee landlords and ground rent and redistributing land among peasant farmers.

Post-independence, the Farmers' Party operated in the Irish Free State from 1922, folding into the National Centre Party in 1932. It was mostly supported by wealthy farmers in the east of Ireland.

Clann na Talmhan (Family of the Land; also called the National Agricultural Party) was founded in 1938. They focused more on the poor smallholders of the west, supporting land reclamation, afforestation, social democracy and rates reform. They formed part of the governing coalition of the Government of the 13th Dáil and Government of the 15th Dáil. Economic improvement in the 1960s saw farmers vote for other parties and Clann na Talmhan disbanded in 1965.

Kazakhstan
In Kazakhstan, the Peasants' Union, originally a communist organization, was formed as one of first agrarian parties in independent Kazakhstan and would win four seats in the 1994 legislative election. The Agrarian Party of Kazakhstan, led by Romin Madinov, was founded in 1999, which favored the privatization of agricultural land, developments towards rural infrastructure, as well as changes in the tax system in agrarian economy. The party would go on to win three Mäjilis seats in the 1999 legislative election and eventually unite with the Civic Party of Kazakhstan to form the pro-government Agrarian-Industrial Union of Workers (AIST) bloc that would be chaired by Madinov for the 2004 legislative election, with the AIST bloc winning 11 seats in the Mäjilis. From there, the bloc remained short-lived as it would merge with the ruling Nur Otan party in 2006.

Several other parties in Kazakhstan over the years have embraced agrarian policies in their programs in an effort to appeal towards a large rural Kazakh demographic base, which included Amanat, ADAL, and Respublica.

Since late 2000s, the "Auyl" People's Democratic Patriotic Party remains the largest and most influential agrarian-oriented party in Kazakhstan, as its presidential candidate Jiguli Dairabaev had become the second-place frontrunner in the 2022 presidential election after sweeping 3.4% of the vote. In the 2023 legislative election, the Auyl party for the first time was represented the parliament after winning nine seats in the lower chamber Mäjilis. The party raises rural issues in regard to decaying villages, rural development and the agro-industrial complex, the issues of social security of the rural population, and has consistently opposed the ongoing rural flight in Kazakhstan.

Latvia
In Latvia, the Union of Greens and Farmers is supportive of traditional small farms and perceives them as more environmentally friendly than large-scale farming: Nature is threatened by development, while small farms are threatened by large industrial-scale farms.

Lithuania
In Lithuania, the government led by the Lithuanian Farmers and Greens Union was in power between 2016 and 2020.

Poland
In Poland, the Polish People's Party (Polskie Stronnictwo Ludowe, PSL) traces its tradition to an agrarian party in Austro-Hungarian-controlled Galician Poland. After the fall of the communist regime, PSL's biggest success came in 1993 elections, where it won 132 out of 460 parliamentary seats. Since then, PSL's support has steadily declined, until 2019, when they formed Polish Coalition with an anti- establishment, direct democracy Kukiz'15 party, and managed to get 8.5% of popular vote. Moreover, PSL tends to get much better results in local elections. In 2014 elections they have managed to get 23.88% of votes.

The right-wing Law and Justice party has also become supportive of agrarian policies in recent years and polls show that most of their support comes from rural areas. AGROunia resembles the features of agrarianism.

Romania
In Romania, older party parties from Transylvania, Moldavia, and Wallachia merged to become the National Peasants' Party (PNȚ) in 1926. Iuliu Maniu (1873–1953) was a prime minister with an agrarian cabinet from 1928 to 1930 and briefly in 1932–1933, but the Great Depression made proposed reforms impossible. The communist administration dissolved the party in 1947 (along with other historical parties such as the National Liberal Party), but it reformed in 1989 after they fell from power.

The reformed party, which also incorporated elements of Christian democracy in its ideology, governed Romania as part of the Romanian Democratic Convention (CDR) between 1996 and 2000.

Serbia
In Serbia, Nikola Pašić (1845–1926) and his People's Radical Party dominated Serbian politics after 1903. The party also monopolized power in Yugoslavia from 1918 to 1929. During the dictatorship of the 1930s, the prime minister was from that party.

Ukraine
In Ukraine, the Radical Party of Oleh Lyashko has promised to purify the country of oligarchs "with a pitchfork". The party advocates a number of traditional left-wing positions (a progressive tax structure, a ban on agricultural land sale and eliminating the illegal land market, a tenfold increase in budget spending on health, setting up primary health centres in every village) and mixes them with strong nationalist sentiments.

United Kingdom
In land law the heyday of English, Irish (and thus Welsh) agrarianism was c. 1500 to 1603, led by the Tudor royal advisors, who sought to maintain a broad pool of agricultural commoners from which to draw military men, against the interests of larger landowners who sought enclosure (meaning complete private control of common land, over which by custom and common law lords of the manor always enjoyed minor rights). The heyday was eroded by hundreds of Acts of Parliament to expressly permit enclosure, chiefly from 1650 to the 1810s. Politicians standing strongly as reactionaries to this included the Levellers, those anti-industrialists (Luddites) going beyond opposing new weaving technology and, later, radicals such as William Cobbett.

A high level of net national or local self-sufficiency has a strong base in campaigns and movements. In the 19th century such empowered advocates included Peelites and most Conservatives. The 20th century saw the growth or start of influential non-governmental organisations, such as the National Farmers' Union of England and Wales, Campaign for Rural England, Friends of the Earth (EWNI) and of the England Wales, Scottish and Northern Irish political parties prefixed by and focussed on Green politics. The 21st century has seen decarbonisation already in electricity markets. Following protests and charitable lobbying local food has seen growing market share, sometimes backed by wording in public policy papers and manifestos. The UK has many sustainability-prioritising businesses, green charity campaigns, events and lobby groups ranging from espousing allotment gardens (hobby community farming) through to a clear policy of local food and/or self-sustainability models.

Australia
Historian F.K. Crowley finds that:

"Australian farmers and their spokesman have always considered that life on the land is inherently more virtuous, as well as more healthy, more important and more productive, than life in the towns and cities....The farmers complained that something was wrong with an electoral system which produced parliamentarians who spent money beautifying vampire-cities instead of developing the interior."

The National Party of Australia (formerly called the Country Party), from the 1920s to the 1970s, promulgated its version of agrarianism, which it called "countrymindedness". The goal was to enhance the status of the graziers (operators of big sheep stations) and small farmers and justified subsidies for them.

New Zealand
The New Zealand Liberal Party aggressively promoted agrarianism in its heyday (1891–1912). The landed gentry and aristocracy ruled Britain at this time. New Zealand never had an aristocracy but its wealthy landowners largely controlled politics before 1891. The Liberal Party set out to change that by a policy it called "populism." Richard Seddon had proclaimed the goal as early as 1884: "It is the rich and the poor; it is the wealthy and the landowners against the middle and labouring classes. That, Sir, shows the real political position of New Zealand." The Liberal strategy was to create a large class of small landowning farmers who supported Liberal ideals. The Liberal government also established the basis of the later welfare state such as old age pensions and developed a system for settling industrial disputes, which was accepted by both employers and trade unions. In 1893, it extended voting rights to women, making New Zealand the first country in the world to do so.

To obtain land for farmers, the Liberal government from 1891 to 1911 purchased 3,100,000 acres of Maori land. The government also purchased 1,300,000 acres from large estate holders for subdivision and closer settlement by small farmers. The Advances to Settlers Act (1894) provided low-interest mortgages, and the agriculture department disseminated information on the best farming methods. The Liberals proclaimed success in forging an egalitarian, anti-monopoly land policy. The policy built up support for the Liberal Party in rural North Island electorates. By 1903, the Liberals were so dominant that there was no longer an organized opposition in Parliament.

North America
The United States and Canada both saw a rise of Agrarian-oriented parties in the early twentieth century as economic troubles motivated farming communities to become politically active. It has been proposed that different responses to agrarian protest largely determined the course of power generated by these newly energized rural factions. According to Sociologist Barry Eidlin:"'In the United States, Democrats adopted a co-optive response to farmer and labor protest, incorporating these constituencies into the New Deal coalition. In Canada, both mainstream parties adopted a coercive response, leaving these constituencies politically excluded and available for an independent left coalition.'"These reactions may have helped determine the outcome of agrarian power and political associations in the US and Canada.

Kansas
Economic desperation experienced by farmers across the state of Kansas in the nineteenth century spurred the creation of The People's Party in 1890, and soon-after would gain control of the governor's office in 1892. This party, consisting of a mix of Democrats, Socialists, Populists, and Fusionists, would find itself buckling from internal conflict regarding the unlimited coinage of silver. The Populists permanently lost power in 1898.

Oklahoma
Oklahoma farmers considered their political activity during the early twentieth century due to the outbreak of war, depressed crop prices, and an inhibited sense of progression towards owning their own farms. Tenancy had been reportedly as high as 55% in Oklahoma by 1910. These pressures saw agrarian counties in Oklahoma supporting Socialist policies and politics, with the Socialist platform proposing a deeply agrarian-radical platform:"...the platform proposed a 'Renters and Farmer's Program' which was strongly agrarian radical in its insistence upon various measures to put land into 'The hands of the actual tillers of the soil.' Although it did not propose to nationalize privately owned land, it did offer numerous plans to enlarge the state's public domain, from which land would be rented at prevailing share rents to tenants until they had paid rent equal to the land's value. The tenant and his children would have the right of occupancy and use, but the 'title' would remind in the 'commonwealth', an arrangement that might be aptly termed 'Socialist fee simple'. They proposed to exempt from taxation all farm dwellings, animals, and improvements up to the value of $1,000. The State Board of Agriculture would encourage 'co-operative societies' of farmers to make plans f or the purchase of land, seed, tools, and for preparing and selling produce. In order to give farmers essential services at cost, the Socialists called for the creation of state banks and mortgage agencies, crop insurance, elevators, and warehouses."This agrarian-backed Socialist party would win numerous offices, causing a panic within the local Democratic party. This agrarian-Socialist movement would be inhibited by voter suppression laws aimed at reducing the participation of voters of color, as well as national wartime policies intended to disrupt political elements considered subversive. This party would peak in power in 1914.

Back-to-the-land movement
Agrarianism is similar to but not identical with the back-to-the-land movement. Agrarianism concentrates on the fundamental goods of the earth, on communities of more limited economic and political scale than in modern society, and on simple living, even when the shift involves questioning the "progressive" character of some recent social and economic developments. Thus, agrarianism is not industrial farming, with its specialization on products and industrial scale.

Agrarian values

 * Brass, Tom. Peasants, Populism and Postmodernism: The Return of the Agrarian Myth (2000)
 * * Inge, M. Thomas. Agrarianism in American Literature (1969)
 * Kolodny, Annette. The Land before Her: Fantasy and Experience of the American Frontiers, 1630–1860 (1984). online edition
 * Marx, Leo. The Machine in the Garden: Technology and the Pastoral Ideal in America (1964).
 * Murphy, Paul V. The Rebuke of History: The Southern Agrarians and American Conservative Thought (2000)
 * Parrington, Vernon. Main Currents in American Thought (1927), 3-vol online
 * Thompson, Paul, and Thomas C. Hilde, eds. The Agrarian Roots of Pragmatism (2000)
 * Murphy, Paul V. The Rebuke of History: The Southern Agrarians and American Conservative Thought (2000)
 * Parrington, Vernon. Main Currents in American Thought (1927), 3-vol online
 * Thompson, Paul, and Thomas C. Hilde, eds. The Agrarian Roots of Pragmatism (2000)
 * Thompson, Paul, and Thomas C. Hilde, eds. The Agrarian Roots of Pragmatism (2000)

Europe

 * Bell, John D. Peasants in Power: Alexander Stamboliski and the Bulgarian Agrarian National Union, 1899–1923(1923)
 * Donnelly, James S. Captain Rock: The Irish Agrarian Rebellion of 1821–1824 (2009)
 * Donnelly, James S. Irish Agrarian Rebellion, 1760–1800 (2006)
 * Gross, Feliks, ed. European Ideologies: A Survey of 20th Century Political Ideas (1948) pp. 391–481 online edition, on Russia and Bulgaria
 * Kubricht, Andrew Paul. "The Czech Agrarian Party, 1899-1914: a study of national and economic agitation in the Habsburg monarchy" (PhD thesis, Ohio State University Press, 1974)
 * Narkiewicz, Olga A. The Green Flag: Polish Populist Politics, 1867–1970 (1976).
 * Oren, Nissan. Revolution Administered: Agrarianism and Communism in Bulgaria (1973), focus is post 1945
 * Stefanov, Kristian. ''Between Ideological Loyalty and Political Adaptation: 'The Agrarian Question' in the Development of Bulgarian Social Democracy, 1891–1912, East European Politics, Societies and Cultures, Is. 4, 2023.
 * Paine, Thomas. Agrarian Justice (1794)
 * Roberts, Henry L. Rumania: Political Problems of an Agrarian State (1951).
 * Paine, Thomas. Agrarian Justice (1794)
 * Roberts, Henry L. Rumania: Political Problems of an Agrarian State (1951).
 * Roberts, Henry L. Rumania: Political Problems of an Agrarian State (1951).

North America

 * Goodwyn, Lawrence. The Populist Moment: A Short History of the Agrarian Revolt in America (1978), 1880s and 1890s in U.S.
 * Lipset, Seymour Martin. Agrarian socialism: the Coöperative Commonwealth Federation in Saskatchewan (1950), 1930s-1940s
 * McConnell, Grant. The decline of agrarian democracy(1953), 20th century U.S.
 * Mark, Irving. Agrarian conflicts in colonial New York, 1711–1775 (1940)
 * Ochiai, Akiko. Harvesting Freedom: African American Agrarianism in Civil War Era South Carolina (2007)
 * Robison, Dan Merritt. Bob Taylor and the agrarian revolt in Tennessee (1935)
 * Stine, Harold E. The agrarian revolt in South Carolina;: Ben Tillman and the Farmers' Alliance (1974)
 * Summerhill, Thomas. Harvest of Dissent: Agrarianism in Nineteenth-Century New York (2005)
 * Szatmary, David P. Shays' Rebellion: The Making of an Agrarian Insurrection (1984), 1787 in Massachusetts
 * Woodward, C. Vann.  Tom Watson: Agrarian Rebel (1938) online edition
 * Stine, Harold E. The agrarian revolt in South Carolina;: Ben Tillman and the Farmers' Alliance (1974)
 * Summerhill, Thomas. Harvest of Dissent: Agrarianism in Nineteenth-Century New York (2005)
 * Szatmary, David P. Shays' Rebellion: The Making of an Agrarian Insurrection (1984), 1787 in Massachusetts
 * Woodward, C. Vann.  Tom Watson: Agrarian Rebel (1938) online edition

Global South

 * Brass, Tom (ed.). New Farmers' Movements in India (1995) 304 pages.
 * Handy, Jim. Revolution in the Countryside: Rural Conflict and Agrarian Reform in Guatemala, 1944–1954 (1994)
 * Paige, Jeffery M.  Agrarian revolution: social movements and export agriculture in the underdeveloped world (1978) 435 pages excerpt and text search
 * Sanderson, Steven E.  Agrarian populism and the Mexican state: the struggle for land in Sonora (1981)
 * Stokes, Eric. The Peasant and the Raj: Studies in Agrarian Society and Peasant Rebellion in Colonial India (1980)
 * Tannenbaum, Frank. The Mexican Agrarian Revolution (1930)
 * Paige, Jeffery M.  Agrarian revolution: social movements and export agriculture in the underdeveloped world (1978) 435 pages excerpt and text search
 * Sanderson, Steven E.  Agrarian populism and the Mexican state: the struggle for land in Sonora (1981)
 * Stokes, Eric. The Peasant and the Raj: Studies in Agrarian Society and Peasant Rebellion in Colonial India (1980)
 * Tannenbaum, Frank. The Mexican Agrarian Revolution (1930)
 * Tannenbaum, Frank. The Mexican Agrarian Revolution (1930)

title: Agrarianism


text:
Atomic

Atomic may refer to:

 * Of or relating to the atom, the smallest particle of a chemical element that retains its chemical properties
 * Atomic physics, the study of the atom
 * Atomic Age, also known as the "Atomic Era"
 * Atomic scale, distances comparable to the dimensions of an atom
 * Atom (order theory), in mathematics
 * Atomic (cocktail), a champagne cocktail
 * Atomic (magazine), an Australian computing and technology magazine
 * Atomic Skis, an Austrian ski producer

Music

 * Atomic (band), a Norwegian jazz quintet
 * Atomic (Lit album), 2001
 * Atomic (Mogwai album), 2016
 * Atomic, an album by Rockets, 1982
 * Atomic (EP), by Labrinth, 2013
 * "Atomic" (song), by Blondie, 1979
 * "Atomic", a song by Tiger Army from Tiger Army III: Ghost Tigers Rise

title: Atomic


text:
Angle

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius, and may be a negative number. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

History and etymology
The word angle comes from the Latin word angulus, meaning "corner". Cognate words include the Greek ἀγκύλος (ankylοs) meaning "crooked, curved" and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes, who regarded an angle as a deviation from a straight line; the second, angle as quality, by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.

Identifying angles
In mathematical expressions, it is common to use Greek letters ( α, β , γ , θ , φ , . . . ) as variables denoting the size of some angle (the symbol $π$ is typically not used for this purpose to avoid confusion with the constant denoted by that symbol). Lower case Roman letters (a, b, c, . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples.

The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted $∠BAC$ or $$\widehat{\rm BAC}$$. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A").

In other ways, an angle denoted as, say, $∠BAC$ might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see ). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, $∠BAC$ always refers to the anticlockwise (positive) angle from B to C about A and $∠CAB$ the anticlockwise (positive) angle from C to B about A.

Individual angles
There is some common terminology for angles, whose measure is always non-negative (see ):
 * An angle equal to 0° or not turned is called a zero angle.
 * An angle smaller than a right angle (less than 90°) is called an acute angle ("acute" meaning "sharp").
 * An angle equal to $1⁄4$ turn (90° or $$π$⁄2$ radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
 * An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle ("obtuse" meaning "blunt").
 * An angle equal to $1⁄2$ turn (180° or $π$ radians) is called a straight angle.
 * An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.
 * An angle equal to 1 turn (360° or 2$π$ radians) is called a full angle, complete angle, round angle or perigon.
 * An angle that is not a multiple of a right angle is called an oblique angle.

The names, intervals, and measuring units are shown in the table below:

Vertical and adjacent angle pairs

When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other. • A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.

The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
 * All straight angles are equal.
 * Equals added to equals are equal.
 * Equals subtracted from equals are equal.

When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.

Adjacentangles.svg

• Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called complementary, supplementary, and explementary angles (see below). A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.

Combining angle pairs
The angle addition postulate states that if B is in the interior of angle AOC, then

$$ m\angle \mathrm{AOC} = m\angle \mathrm{AOB} + m\angle \mathrm{BOC} $$

I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

Three special angle pairs involve the summation of angles:

• Complementary angles are angle pairs whose measures sum to one right angle ($1⁄4$ turn, 90°, or $1⁄4$ radians). If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of a triangle is 180 degrees, and the right angle accounts for 90 degrees.

The adjective complementary is from the Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle.

The difference between an angle and a right angle is termed the complement of the angle.

If angles A and B are complementary, the following relationships hold: $ \begin{align} & \sin^2A + \sin^2B = 1 & & \cos^2A + \cos^2B = 1 \\[3pt] & \tan A = \cot B & & \sec A = \csc B \end{align}$

(The tangent of an angle equals the cotangent of its complement, and its secant equals the cosecant of its complement.)

The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".

Angle obtuse acute straight.svg

• Two angles that sum to a straight angle ($1⁄4$ turn, 180°, or $π$ radians) are called supplementary angles.

If the two supplementary angles are adjacent (i.e., have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles. However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.

If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.

The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.

In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.

Conjugate Angles.svg

• Two angles that sum to a complete angle (1 turn, 360°, or 2$π$ radians) are called explementary angles or conjugate angles.

The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle.

Polygon-related angles

 * An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle, that is, a reflex angle.  In Euclidean geometry, the measures of the interior angles of a triangle add up to $π$ radians, 180°, or $1⁄2$ turn; the measures of the interior angles of a simple convex quadrilateral add up to 2$π$ radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)$π$ radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)$1⁄2$ turn.
 * The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.  In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
 * In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).
 * In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.
 * In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.
 * Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle. This conflicts with the above usage.

Plane-related angles

 * The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.
 * The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.

Measuring angles
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

To measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle: $$ \theta = \frac{s}{r} \, \mathrm{rad}. $$ Conventionally, in mathematics and the SI, the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted.

The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form $1⁄2$, where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians):

$$ \theta = \frac{k}{2\pi} \cdot \frac{s}{r}. $$

The value of $π$ thus defined is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s/r is unaltered.

Units

Throughout history, angles have been measured in various units. These are known as angular units, with the most contemporary units being the degree ( ° ), the radian (rad), and the gradian (grad), though many others have been used throughout history. Most units of angular measurement are defined such that one turn (i.e., the angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n. Two exceptions are the radian (and its decimal submultiples) and the diameter part.

In the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis.

The following table list some units used to represent angles.

Signed angles

It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference.

In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are anticlockwise, and negative cycles are clockwise.

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation, which is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

Equivalent angles

 * Angles that have the same measure (i.e., the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all right angles are equal in measure).
 * Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.
 * The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative). Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo $1⁄2$ turn, 180°, or $θ$ radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

Related quantities
For an angular unit, it is definitional that the angle addition postulate holds. Some quantities related to angles where the angle addition postulate does not hold include:
 * The slope or gradient is equal to the tangent of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.
 * The spread between two lines is defined in rational geometry as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
 * Although done rarely, one can report the direct results of trigonometric functions, such as the sine of the angle.

Angles between curves
The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.

Bisecting and trisecting angles
The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge but could only trisect certain angles. In 1837, Pierre Wantzel showed that this construction could not be performed for most angles.

Dot product and generalisations
In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula

$$ \mathbf{u} \cdot \mathbf{v} = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .$$

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

Inner product
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product $$ \langle \cdot, \cdot \rangle $$, i.e.

$$ \langle \mathbf{u}, \mathbf{v} \rangle = \cos(\theta)\ \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .$$

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

$$ \operatorname{Re} \left( \langle \mathbf{u}, \mathbf{v} \rangle \right) = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .$$

or, more commonly, using the absolute value, with

$$ \left| \langle \mathbf{u}, \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .$$

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces $$\operatorname{span}(\mathbf{u})$$ and $$\operatorname{span}(\mathbf{v})$$ spanned by the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ correspondingly.

Angles between subspaces
The definition of the angle between one-dimensional subspaces $$\operatorname{span}(\mathbf{u})$$ and $$\operatorname{span}(\mathbf{v})$$ given by

$$ \left| \langle \mathbf{u}, \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| $$

in a Hilbert space can be extended to subspaces of finite dimensions. Given two subspaces $$ \mathcal{U} $$, $$ \mathcal{W} $$ with $$ \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l $$, this leads to a definition of $$k$$ angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry
In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,

$$ \cos \theta = \frac{g_{ij} U^i V^j}{\sqrt{ \left| g_{ij} U^i U^j \right| \left| g_{ij} V^i V^j \right|}}. $$

Hyperbolic angle
A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by Leonhard Euler in Introduction to the Analysis of the Infinite (1748).

Angles in geography and astronomy
In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the center of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the center of the Earth, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.

In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.

Astronomers also measure objects' apparent size as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can convert such an angular measurement into a distance/size ratio.

Other astronomical approximations include: These measurements depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.
 * 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth.
 * 1° is the approximate width of the little finger at arm's length.
 * 10° is the approximate width of a closed fist at arm's length.
 * 20° is the approximate width of a handspan at arm's length.

In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

title: Angle


text:
Asa

Asa may refer to:

People and fictional characters

 * Asa (name), a given name, including a list of people and fictional characters so named
 * Asa people, an ethnic group based in Tanzania
 * Aṣa, Nigerian-French singer, songwriter, and recording artist Bukola Elemide (born 1982)
 * Asa (rapper), Finnish rapper Matti Salo (born 1980)

Biblical and mythological figures

 * Asa of Judah, third king of the Kingdom of Judah and the fifth king of the House of David
 * Ása or Æsir, Norse gods

Places

 * Asa, Hardoi Uttar Pradesh, India, a village
 * Asu, South Khorasan, Iran, also spelled Asa, a village
 * Asa, Kwara State, Nigeria, a local government area
 * Asa River (Japan), a tributary of the Tama River in Tokyo, Japan
 * Asa (Kazakhstan), a river
 * Asa River (Venezuela), a river in Venezuela

Other uses

 * Asa (album), the sixth studio album by the German Viking metal band Falkenbach
 * Asa (raga), a peculiar musical raga in Gurmat Sangeet tradition
 * ASA carriage control characters, simple printing command characters used to control the movement of paper through line printers
 * Asa language, spoken by the Asa people of Tanzania
 * Asa Station, a railway station in San'yō-Onoda, Yamaguchi, Japan
 * Asa (railway station), Jambyl Region, Kazakhstan
 * Naboot, also called asa, a quarterstaff constructed of palm wood or rattan
 * Asha, romanized as aṣ̌a, a Zoroastrian concept
 * "Asa", a song by Kitt Wakeley featuring Starr Parodi from An Adoption Story, 2022
 * “Asa”, 2024 single by Snazzy the Optimist

title: Asa