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Tequila

Tequila is a distilled beverage made from the blue agave plant, primarily in the area surrounding the city of Tequila 65 km northwest of Guadalajara, and in the Jaliscan Highlands (Los Altos de Jalisco) of the central western Mexican state of Jalisco.

The red volcanic soils in the region of Tequila are well suited for growing the blue agave, and more than 300 million plants are harvested there each year. Agave grows differently depending on the region. Blue agaves grown in the highlands Los Altos region are larger and sweeter in aroma and taste. Agaves harvested in the valley region have a more herbaceous fragrance and flavor. Due to its historical and cultural importance, the region near Tequila was declared a UNESCO World Heritage Site in 2006, the Agave Landscape and Ancient Industrial Facilities of Tequila.

Mexican laws state that tequila can be produced only in the state of Jalisco and limited municipalities in the states of Guanajuato, Michoacán, Nayarit, and Tamaulipas. Tequila is recognized as a Mexican designation of origin product in more than 40 countries. It was protected through NAFTA in Canada and the United States until July 2020 and through bilateral agreements with individual countries such as Japan and Israel, and it has been a protected designation of origin product in the European Union since 1997.

Aside from its geographical distinction, tequila is differentiated from other varieties of mezcal in that all of the agave used in tequila is blue agave, and the methods of production are different. Tequila is commonly served neat in Mexico and as a shot with salt and lime around the world. Tequila must have between 35% and 55% alcohol content (70 and 110 U.S. proof).

Early history
The distillation technology to produce mezcal from agave heart juice was first introduced from the coastal regions of what was then Nueva Galicia (present-day Aguascalientes, Colima, Guanajuato, Jalisco, Nayarit, and Zacatecas) into the highland valleys of Amatitán, Tequila, Magdalena, and El Arenal in the mid-1700s. The mezcal produced in these regions became distinctive enough as to become known as "tequila" (after the town). Prior to the production of either tequila or mezcal, pulque had been brewed from agave for thousands of years.

Don Pedro Sánchez de Tagle is considered the first person to produce tequila. In 1595, King Philip II of Spain banned the planting of new vineyards in Mexico and other Spanish colonies due to the decline of wine trade with Spain. The main reason behind this is because Mexico is self-sufficient in producing its own wines. The king did this to maintain the market for Spanish products in the New World, and reap the taxes on wine exports. The Marquis of Altamira grabbed the opportunity of the neglected blue agave plants. He built his first Tequila factory in his Hacienda Cuisillos, one of the largest haciendas during that time and amassed a great fortune. The Marquis is now known today as the "Father of Tequila". Don Pedro also served as the Prior of the Consulado, which is the head of the largest corporation in Mexico.

Spain's King Carlos IV granted the Cuervo family the first legal license to commercially make tequila. Don Cenobio Sauza, founder of Sauza Tequila and Municipal President of the Village of Tequila from 1884 to 1885, was the first to export tequila to the United States, and shortened the name from "Tequila Extract" to just "Tequila" for the American markets. Don Cenobio's grandson Don Francisco Javier gained international attention for insisting that "there cannot be tequila where there are no agaves!" His efforts led to the principle that real tequila can come only from the State of Jalisco.

The first tequila distillery in the United States was opened in 1936 in Nogales, Arizona by Harry J. Karns, former Arizona state senator and Nogales Mayor.

In a move to take ownership of the term "tequila", the Mexican government declared the term to be its intellectual property in 1974.

Recent history

Although some tequilas have remained as family-owned brands, most well-known tequila brands are owned by large multinational corporations. However, over 100 distilleries make over 900 brands of tequila in Mexico and over 2,000 brand names have been registered (2009 statistics). Due to this, each bottle of tequila contains a serial number (NOM) denoting in which distillery the tequila was produced. In many cases, multiple different brands come from the same manufacturer.

In 2003, Mexico issued a proposal that would require all Mexican-made tequila be bottled in Mexico before being exported to other countries. The Mexican government said that bottling tequila in Mexico would guarantee its quality. Liquor companies in the United States said Mexico just wanted to create bottling jobs in their own country, and also claimed this rule would violate international trade agreements and was in discord with usual exporting practices worldwide. The proposal might have resulted in the loss of jobs at plants in California, Arkansas, Missouri, and Kentucky, because Mexican tequila exported in bulk to the United States is bottled in those plants. On January 17, 2006, the United States and Mexico signed an agreement allowing the continued bulk import of tequila into the United States. The agreement also created a "tequila bottlers registry" to identify approved bottlers of tequila and created an agency to monitor the registry.

The Tequila Regulatory Council of Mexico (TRCM) originally did not permit flavored tequila to carry the tequila name. In 2004, the Council decided to allow flavored tequila to be called tequila, with the exception of 100% agave tequila, which still cannot be flavored.

A new Norma Oficial Mexicana (NOM) for tequila (NOM-006-SCFI-2005) was issued in 2006 and, among other changes, introduced a class of tequila called extra añejo or "ultra-aged" which must be aged a minimum of three years.

A one-liter bottle of limited-edition premium tequila was sold for $225,000 in July 2006 in Tequila, Jalisco, by the company Tequila Ley .925. The bottle that contained the tequila was a two-kilo display of platinum and gold. The manufacturer received a certificate from The Guinness World Records for the most expensive bottle of tequila spirit ever sold.

In June 2013, a Chinese ban on the importation of premium (100% blue agave) tequila into China was lifted, following a state visit to Mexico by Chinese Communist Party general secretary Xi Jinping. The entry of premium tequila into the country was expected to increase tequila exports by 20 percent within the decade above the 170 million liters in 2013. Ramon Gonzalez, director of the Consejo Regulador del Tequila, estimates that each of the top 16 producers of tequila had invested up to $3 million to enter the Chinese market. On 30 August 2013, the first 70,380 bottles of premium tequila from ten brands arrived in Shanghai. The arrival happened during an event held at the House of Roosevelt, a well-known club located on The Bund– an area with a long tradition of importing alcoholic beverages in China.

The latest version of the tequila standard (NOM-006-SCFI-2012) updated the standard to specify that the silver class of tequila cannot contain additives, to allow the aging time for the ultra-aged class to be displayed on the label, and to prohibit the selling of bulk tequila through vending machines, and required registering the agave during the calendar year of its plantation and required annual updates.

In 2018, the Mexican government approved a proposal to celebrate the third Saturday of March as National Tequila Day.

Production
Planting, tending, and harvesting the agave plant remains a manual effort, largely unchanged by modern farm machinery and relying on centuries-old expertise. The people who harvest it, the jimadores, have intimate knowledge of how the plants should be cultivated, passed down from generation to generation.

By regularly trimming any quiotes (a stalk at the center of the plant that could grow several meters high if not cut back), the jimadores prevent the agave from flowering and dying early, allowing it to fully ripen. They also determine when each plant is ready to be harvested, and using a special knife called a coa (with a circular blade on a long pole), carefully cut away the leaves from the piña (the succulent core of the plant), which can average around 70 kg in the valley and 110 kg in the highlands. If harvested too late or too early, the piñas will not have the right amount of carbohydrates for fermentation.

After harvesting, the piñas are transported to ovens where they are slowly baked to break down their complex fructans into simple fructose. Then, the baked piñas are either shredded or mashed under a large stone wheel called a tahona. The pulp fiber, or bagazo, left behind is often reused as compost or animal feed, or can be used as fuel or processed into paper. Some producers like to add a small amount of bagazo back into their fermentation tanks for a stronger agave flavor in the final product.

The extracted agave juice is then poured into either large wooden or stainless steel vats for several days to ferment, resulting in a wort, or mosto, with low alcohol content. This wort is then distilled once to produce what is called ordinario, and then a second time to produce clear "silver" tequila. A minimum of two distillations is required by law. A few producers such as Casa Noble (for their "Crystal" expression) and Corzo (for their añejo expression) have experimented with distilling the product a third time, but this has not caught on as a trend, and some have said it removes too much of the agave flavor from the tequila. At this point the tequila is either bottled as silver tequila or it is pumped into wooden barrels to age, where it develops a mellower flavor and amber color.

The differences in taste between tequila made from valley and highland agave plants can be noticeable. Plants grown in the highlands often yield sweeter and fruitier-tasting tequila, while valley agaves give the tequila an earthier flavor.

Fermentation
Unlike other tequila production steps, fermentation is largely outside the control of human beings. Fermentation is the conversion of sugars and carbohydrates to alcohol through yeast under anaerobic conditions, meaning that oxygen is not present during the process. Fermentation is also carried out in a non-aseptic environment which increases the bacterial activity of tequila. The participation of microorganisms from the environment (yeasts and bacteria) makes fermentation a spontaneous process which gives rise to many byproducts that contribute to the flavor and aroma of tequila.

During the fermentation process, inoculum is added to the batch to accelerate the rate of fermentation. When inoculum is added, fermentation can take approximately 20 hours to 3 days. If inoculum is not added, fermentation could take up to 7 days. The rate of fermentation is a key factor in the quality and flavor of tequila produced. Worts fermented slowly are best because the amount of organoleptic compounds produced is greater. The alcohol content at the end of fermentation is between 4-9%.

Organoleptic compounds
Organoleptic compounds enhance flavor and aroma. These include fusel oil, methanol, aldehydes, organic acids and esters. Production of isoamyl and isobutyl alcohols begins after the sugar level is lowered substantially and continues for several hours after the alcoholic fermentation ends. In contrast, ethanol production begins in the first hours of the fermentation and ends with logarithmic yeast growth. The alcohol content in tequila is affected by three factors: the amount of isoamyl alcohol and isobutanol in the yeast strain, the carbon:nitrogen ratio (the higher the ratio, the more alcohol produced), and the temperature of fermentation.

The higher the temperature, the greater concentration of isobutyl and isoamyl alcohols produced. Although if temperatures are too high, this can cause the yeast to become less effective. Similarly, if the temperature is too low, the process occurs too slowly. This can become a large issue in Central Mexico, most precisely the city of Tequila, Jalisco, where most tequila is processed. The temperature in the city of Tequila can reach 31 °C. For this reason, tequila producers often use large stainless steel tanks for fermentation.

Yeast
The specific yeasts and the environments in which they act determine the resultant organoleptic combinations. The role of yeast is, through many enzymatic processes, to turn sugars and carbohydrates into alcohol. There are two steps, first in aerobic conditions, yeast is doubled in colony size every four hours. This process goes on for 24–48 hours. Next, yeast turns acetaldehyde into ethyl alcohol which is known as one of the organoleptic compounds produced in fermentation.

The two main categories of yeast used in tequila are commercial brewers yeast and yeast that comes from precultivated existing yeast that has been preserved. The use of either type of yeast can result in different end products of tequila. Traditional production uses so-called "open fermentation", relying on yeasts from the surrounding environment. A 2023 article analyzes the diversity of yeasts found in these uncontrolled conditions.

Alcohol content
Tequila is a distilled beverage that is made from the fermentation of the sugars in the blue agave plant once it has been cooked, the main sugar being fructose. Through the fermentation process, many factors influence the higher-order alcohols present in tequila, which include molecules such as isobutyl alcohol and isoamyl alcohol, along with the ethanol. Factors include the strain of yeast, the age of the agave plant itself, temperature, and the ratio of carbon to nitrogen.

The yeast strain used and the carbon-to-nitrogen ratio have the biggest influence on the production of higher-order alcohols; this is not surprising, as production of ethanol and higher-order alcohols is an intrinsic property of the metabolism of each strain. The type of yeast most commonly found in tequila is Saccharomyces cerevisiae, which can include many different strains. For example, CF1 agaves, a type of yeast, produces much more ethanol than a CF2 strain, as the two yeasts' metabolic mechanisms differ. Prevalence of certain strains of yeast may be influenced by agricultural practices. It was found that higher ratios of carbon to nitrogen resulted in greater production of higher-order alcohols such as isobutyl alcohol and isoamyl alcohol. The lower level of nitrogen in the fermentation process results in deamination reactions of amino acids, which in turn leads to the synthesis of higher alcohols. The Ehrlich pathway refers to this process in which alpha-keto acids are decarboxylated and transformed to aldehydes and to higher alcohols.

The temperature of the fermentation process also greatly affects the alcohol content of the resulting product. For example, a study conducted by Pinal et al. found that cultivating two strains at a temperature of 35 °C as compared to a temperature of 30 °C produced more isoamyl alcohol. The higher temperature appears to be favorable for the action of the yeast.

The age of the agave plant is also a factor: the older the plant, the greater the production of higher-order alcohols. It was shown in a study that the concentration of amyl alcohol increased by 30% as the plant aged. Conversely, a higher concentration of methanol is found when using younger plants. This change may be due to differences in agricultural practices with plants of different ages.

Color
Tequila's color ranges from clear to a brownish amber, depending on the aging process and the type of wood used for storage. The white (blanco) version of tequila, also nicknamed "silver", is the product with no aging, or may be aged less than two months. It is the purest form, as little has been done to change it. What is known as gold, joven or oro tequila is usually white tequila with the addition of grain alcohols and caramel color; however, some higher-end gold tequilas may be a blend of white and reposado. Rested (reposado) tequila is aged for a relatively short time (two months or more), and aged (añejo) tequila for a longer time (a year or more). All aging of tequila is done in wooden containers. The aging process can last three years or more, and can create or enhance flavors and aromas. It generally imparts a golden color.

Flavor and aroma
There are more than 300 known compounds in tequila, many of which are produced during the fermentation process, the raw material used, and to a lesser degree during the maturation. The components that make up tequila do not act individually to give tequila its distinctive flavor and aroma, but rather, depend on the interaction and quantity of each volatile compound. The volatile compounds responsible for the flavor and aroma profiles of the tequila are put into a category called organoleptic compounds and are known to increase in concentration with a slower fermentation process. The organoleptic compounds produced during fermentation include higher order alcohols, methanol, esters, carbonyls, terpenes, and furans.

Higher-order alcohols have a strong aroma, and the quantity present in each tequila depends on the carbon:nitrogen ratio and temperature during the cooking and fermentation processes. Some of the most common alcohols present other than ethanol are: isoamyl alcohol, isobutanol, and 1-propanol. Methanol is thought to be mainly generated through hydrolysis of methylated pectin which is naturally present in the agave plant, but there has been speculation that it is also partly produced from the enzymatic reactions of yeast strains containing pectin methyl esterase enzyme, which break up the methoxyl group from the pectin. Nearly 50 different esters are identified in tequila, which together give rise to the fruit-like flavors and smell. One of the most abundant esters is ethyl acetate which is synthesized during fermentation by the yeast Saccharomyces cerevisiae, using alcohol transferase enzyme which links acetic acid to ethanol.

In general, the longer the controlled fermentation period, the higher yield of esters produced. During the fermentation process, ethanol is oxidized and one of the main compounds produced are acetaldehydes, which adds the flavor necessary for the final product of tequila. For example, isovaleraldehyde seems to produce a sweet, cocoa, and chocolate-like flavor. 2 and 3-methylbutanal produce a malty flavor. The agave plant contains many phenolics such as vanillin and syringaldehydes which presents a strong and fruity or herbal aroma. It also contains eugenol which can deliver a hint of spicy flavor to the tequila. Since the production of tequila involves heating, Maillard browning reactions occur, and furans are produced during the thermal degradation of sugar. The most prominent furanic compounds include 2-furaldehyde and 5-methylfuraldehyde, which can contribute to the smoky flavor of tequila. Guaiacol also seems to contribute to Tequila's smoky flavor. Beta-demascenone contributes to the woody, floral taste of tequila.

Volatile compounds that contribute to the overall taste and aroma of tequila can be quantitatively assessed and evaluated by gas chromatography. Discrimination tests such as duo-trio and triangle tests are also used to evaluate the quality of the tequila.

Process
If silver or white (Blanco) tequila is the desired final product, distillation is the final process it undergoes. Rested (Reposado) or aged (Añejo) tequila must be matured in 200-liter (or larger) white oak barrels for at least two months for the former and 12 months for the latter. There are, however, more than 50 different companies producing tequila in the Mexican province of Jalisco, with different maturation times according to the variety of tequila and desired quality of the final product.

All companies producing tequila have their aging processes regulated and fiscalized by the Mexican government.

Chemistry
The maturation process causes four main chemical transformations to the tequila compounds: (1) decreasing of fusel oils by the char in barrels, which acts as an absorbing agent; (2) extraction of complex wood constituents by tequila, giving specific aroma and flavor to the final product; (3) reactions among the components of tequila, creating new chemical compounds; and (4) oxidation of the original contents of tequila and of those extracted from wood. The final results of these changes are increased concentrations of acids, esters and aldehydes, and a decrease in fusel oil concentration.

Reposado may be rested in oak barrels or casks as large as 20,000 liters (5,280 gallons), allowing for richer and more complex flavors. The preferred oak comes from the US, France, or Canada, and is usually white oak. Some companies char the wood to impart a smoky flavor or use barrels previously used with different kinds of alcohol (e.g. whiskey or wine). Some reposados can also be aged in new wood barrels to achieve the same woody flavor and smoothness, but in less time.

Añejos are often rested in barrels previously used to rest reposados. The barrels cannot be more than 600 liters (158 gallons), and most are in the 200-liter (52-gallon) range. Many of the barrels used are from whiskey distilleries in the US or Canada, and Jack Daniels barrels are especially popular. This treatment creates many of the aspects of the dark color and more complex flavors of the añejo tequila. After aging of at least one year, the añejo can be removed from the wood barrels and placed in stainless steel tanks to reduce the amount of evaporation that can occur in the barrels.

Threats to quality
TMA (tristeza y muerte de agave — "agave depression and death") is a blight that has reduced the production of the agave grown to produce tequila. This has resulted in lower production and higher prices throughout the early 21st century, and due to the long maturation of the plant, will likely continue to affect prices for years to come.

"Tequila worm" misconception
Only certain mezcals, usually from the state of Oaxaca, are ever sold con gusano (with worm). They are added as a marketing gimmick and are not traditional. The tequila regulatory council does not allow gusanos or scorpions (which are sometimes also added to mezcals) to be included in tequila bottles. The worm in some mezcals is actually the larval form of the moth Hypopta agavis, which lives on the agave plant. Finding one in the plant during processing indicates an infestation and, correspondingly, a lower-quality product. However, the misconception that tequilas may contain worms continues, despite effort and marketing to represent tequila as a premium liquor.

Norma Oficial Mexicana
The Norma Oficial Mexicana (NOM) applies to all processes and activities related to the supply of agave, production, bottling, marketing, information, and business practices linked to the distilled alcoholic beverage known as tequila. Tequila must be produced using agave of the species Tequilana Weber Blue variety, grown in the federal states and municipalities indicated in the Declaration.

Furthermore, the NOM establishes the technical specifications and legal requirements for the protection of the Appellation of Origin of "Tequila" in accordance with the current General Declaration of Protection of the Appellation of Origin of "Tequila", the Law, the Industrial Property Law, the Federal Consumer Protection Law and other related legal provisions.

All authentic, regulated tequilas will have a NOM identifier on the bottle. The important laws since 1990 were NOM-006-SCFI-1993, the later updates NOM-006-SCFI-1994 and NOM-006-SCFI-2005 and the most recent revision published on December 13, 2012, NOM-006-SCFI-2012.

The number after NOM is the distillery number, assigned by the government. NOM does not indicate the location of the distillery, merely the parent company or, in the case where a company leases space in a plant, the physical plant where the tequila was manufactured.

Storage condition
Unlike wine, whose character may change in taste over time and storage conditions, tequila does not change much once bottled, even without ideal storage conditions, much like most other distilled spirits such as whiskey, rum, or vodka. The quality characteristics (flavor, aroma, color, etc.) of a tequila are primarily determined during its aging in wood barrels. To maintain the utmost quality though, some conditions should be met: a constant and moderate temperature (60 to 65 °F), protection from direct sunlight, and maintenance of the integrity of the seal of the bottle. Improper storage conditions will have more effect on the taste of aged tequila rather than the un-aged version, due to tannins and other compounds introduced into the spirit from the aging barrel. For instance, if stored in improper conditions, the dark and more complex flavors of the añejo tequila are more likely to be tainted than the blanco or the silver tequila.

Once the bottle is opened, the tequila will be subject to oxidation which will continue to happen even if no more oxygen is introduced. In addition, if the bottle has more room for air, the process of oxidation occurs faster on the liquor remaining inside the bottle. Therefore, it may be the best to consume the tequila within one or two years after opening. For the most part, the change in quality of tequila is due to extreme conditions of improper storage, not due to oxidation.

Types
The two basic categories of tequila are mixtos and 100% agave. Mixtos use no less than 51% agave, with other sugars making up the remainder. Mixtos use both glucose and fructose sugars.

There are also four categories for tequila, depending on the aging period:
 * Blanco ("white") or plata  ("silver"): white spirit, unaged and bottled or stored immediately after distillation, or aged less than two months in stainless steel or neutral oak barrels
 * Reposado ("rested"): aged a minimum of two months, but less than a year in oak barrels of any size
 * Añejo ("aged" or "vintage"): aged a minimum of one year, but less than three years in small oak barrels
 * Extra Añejo ("extra aged" or "ultra aged"): aged a minimum of three years in oak barrels; this category was established in March 2006.

Brands
The Consejo Regulador del Tequila (Tequila Regulatory Council) reported 1377 registered brands from 150 producers for the year 2013.

Serving

In Mexico, the most traditional way to drink tequila is neat, without lime and salt. It is popular in some regions to drink fine tequila with a side of sangrita—a sweet, sour, and spicy drink typically made from orange juice, grenadine (or tomato juice), and hot chilli. Equal-sized shots of tequila and sangrita are sipped alternately, without salt or lime. Another popular drink in Mexico is the bandera (flag, in Spanish), named after the Flag of Mexico, it consists of three shot glasses, filled with lime juice (for the green), white tequila, and sangrita (for the red).

Outside Mexico, a single shot of tequila is often served with salt and a slice of lime or lemon. This is called tequila cruda and is sometimes referred to as "training wheels", "lick-sip-suck", or "lick-shoot-suck" (referring to the way in which the combination of ingredients is imbibed). The drinkers moisten the back of their hands below the index finger (usually by licking) and pour on the salt. Then the salt is licked off the hand, the tequila is drunk, and the fruit slice is quickly bitten. Groups of drinkers often do this simultaneously. Drinking tequila in this way is often erroneously called a Tequila Slammer, which is in fact a mix of tequila and carbonated drink. Though the traditional Mexican shot is tequila by itself, lime is the fruit of choice when a chaser must be used. The salt is believed to lessen the "burn" of the tequila and the sour fruit balances and enhances the flavor. In Germany and some other countries, tequila oro (gold) is often consumed with cinnamon on a slice of orange after, while tequila blanco (white) is consumed with salt and lime.

If the bottle of tequila does not state on the label that it is manufactured from 100% blue agave (no sugars added), then, by default, that tequila is a mixto (manufactured from at least 51% blue agave). Some tequila distilleries label their tequila as "made with blue agave" or "made from blue agave". However, the Tequila Regulatory Council has stated only tequilas distilled with 100% agave can be designated as "100% agave".

Many of the higher-quality, 100% agave tequilas do not impart significant alcohol burn, and drinking them with salt and lime is likely to remove much of the flavor. These tequilas are usually sipped from a snifter glass rather than a shot glass, and savoured instead of quickly gulped. Doing so allows the taster to detect subtler fragrances and flavors that would otherwise be missed.

Tequila glasses

When served neat (without any additional ingredients), tequila is most often served in a narrow shot glass called a caballito (little horse, in Spanish), but can often be found in anything from a snifter to a tumbler.

The Consejo Regulador del Tequila approved an "official tequila glass" in 2002 called the Ouverture Tequila glass, made by Riedel.

The margarita glass, frequently rimmed with salt or sugar, is a staple for the entire genre of tequila-based mixed drinks, including the margarita.

Cocktails
A variety of cocktails are made with tequila, including the margarita, a cocktail that helped make tequila popular in the United States. The traditional margarita uses tequila, Cointreau, and lime juice, though many variations exist. A popular cocktail in Mexico is the Paloma. Also, a number of martini variants involve tequila, and a large number of tequila drinks are made by adding fruit juice. These include the Tequila Sunrise and the Matador. Sodas and other carbonated drinks are a common mixer, as in the Tequila Slammer. Other popular cocktails are the Acapulco cocktail, Bloody Aztec Chimayó Cocktail, Mexican martini, Mojito Blanco and Vampiro.

IP protection and regulation outside of Mexico
The Mexican government took initiatives to protect tequila as early as the 1970s. Tequila became the country's first appellation of origin (AO) in 1974, and in 1978 it was internationally registered for protection under the Lisbon Agreement for the Protection of Appellations of Origin and their International Registration. The AO defines the area in which tequila can be produced, essentially the area home to the blue agave from which it is derived. Under the AO, selected municipalities within only five Mexican states can grow the agave and produce tequila. These municipalities were selected based on geographical factors (such as climate, altitude and soil characteristics) and human factors (the use of traditional techniques of producing tequila that have been passed on from generation to generation).

Further to the AO registration, the Mexican government holds all the rights over the use of the name tequila.

Along with the international protection, Mexico has also sought similar protection for tequila in other important countries and regions through a number of bilateral and multilateral agreements. Pursuant to Annex 313 of the North American Free Trade Agreement (NAFTA), Canada and the United States recognize tequila and mezcal as products originating from Mexico and, consequently, do not permit the sale of any product as tequila and/or mezcal unless they have been lawfully prepared in Mexico.

The Mexican government enforces the tequila intellectual property through the Mexican Institute of Industrial Property (IMPI). The use of the name tequila is authorized and administered only by IMPI. IMPI may prohibit any unauthorized use of the name that may create confusion for the consumers, or constitute an act of unfair competition.

Despite protection under the Lisbon Agreement and NAFTA, tequila still faced competition from some pseudo tequilas. During the mid-1990s, as much as 3.5 million liters of these liquors were sold annually in Europe. The European liquors were produced from sugars of agave-like plants and were illegally labeled as tequila. Following the NAFTA precedent, Mexico entered into a bilateral agreement with the European Union concerning the mutual recognition and protection of AO in the spirits sector. Under this agreement the EU recognizes tequila and mezcal as "denominations of origin", although enforcement has still been quite lax in Europe.

Mexico's tequila protection interests were prominent during the Chinese accession negotiations to the World Trade Organization. In accordance with WTO accession regulations, in order to gain considerations for WTO membership, China had to sign bilateral agreements with all existing members, including Mexico. Since China has a climatic condition suitable for blue agave plants, Mexico was worried about rival pseudo tequilas from China. The Mexican authorities took the opportunity of the bilateral negotiations with China to ensure that the latter accepted tequila as a geographically indicated product.

Canada
Under Canadian regulations (C.R.C., c.870, section B.02.90), any product labelled, advertised, or sold as Tequila must be manufactured in Mexico, as it would be for consumption in Mexico. However, once imported to Canada for sale, it is legal for Tequila to be diluted with distilled or otherwise purified water to adjust it to the desired strength and then sold.

United States
Similar to the law of Canada, the U.S. law (27 CFR 5.22 (g)) says that tequila must be "manufactured in Mexico in compliance with the laws of Mexico regulating the manufacture of Tequila for consumption in that country."

title: Tequila


text:
Open set

In mathematics, an open set is a generalization of an open interval in the real line.

In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point $P$, contains all points that are sufficiently near to $P$ (that is, all points whose distance to $P$ is less than some value depending on $P$).

More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology).

In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity, connectedness, and compactness, which were originally defined by means of a distance.

The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.

Motivation
Intuitively, an open set provides a method to distinguish two points. For example, if about one of two points in a topological space, there exists an open set not containing the other (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two points, or more generally two subsets, of a topological space are "near" without concretely defining a distance. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called metric spaces.

In the set of all real numbers, one has the natural Euclidean metric; that is, a function which measures the distance between two real numbers: $x^{2} + y^{2} = r^{2}$. Therefore, given a real number x, one can speak of the set of all points close to that real number; that is, within ε of x. In essence, points within ε of x approximate x to an accuracy of degree ε. Note that ε > 0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1 and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5). Clearly, these points approximate x to a greater degree of accuracy than when ε = 1.

The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. In particular, sets of the form (−ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−ε, ε)), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define R as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things in R are equally close to 0, while any item that is not in R is not close to 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collection of sets "around" (that is, containing) x, used to approximate x. Of course, this collection would have to satisfy certain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. For example, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Once we begin to define "smaller" sets containing x, we tend to approximate x to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy.

Definitions
Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.

Euclidean space
A subset $$U$$ of the Euclidean $x^{2} + y^{2} < r^{2}$-space $d(x, y) = |x − y|$ is open if, for every point $x$ in $$U$$, there exists a positive real number $ε$ (depending on $x$) such that any point in $n$ whose Euclidean distance from $x$ is smaller than $ε$ belongs to $$U$$. Equivalently, a subset $$U$$ of $R^{n}$ is open if every point in $$U$$ is the center of an open ball contained in $$U.$$

An example of a subset of $R^{n}$ that is not open is the closed interval $[0,1]$, since neither $R^{n}$ nor $R$ belongs to $[0,1]$ for any $0 - ε$, no matter how small.

Metric space
A subset U of a metric space $1 + ε$ is called open if, for any point x in U, there exists a real number ε > 0 such that any point $$y \in M$$ satisfying $ε > 0$ belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U.

This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological space
A topology $$\tau$$ on a set $X$ is a set of subsets of $X$ with the properties below. Each member of $$\tau$$ is called an open set.

 * $$X \in \tau$$ and $$\varnothing \in \tau$$
 * Any union of sets in $$\tau$$ belong to $$\tau$$: if $$\left\{ U_i : i \in I \right\} \subseteq \tau$$ then $$\bigcup_{i \in I} U_i \in \tau$$
 * Any finite intersection of sets in $$\tau$$ belong to $$\tau$$: if $$U_1, \ldots, U_n \in \tau$$ then $$U_1 \cap \cdots \cap U_n \in \tau$$

$X$ together with $$\tau$$ is called a topological space.

Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form $$\left( -1/n, 1/n \right),$$ where $$n$$ is a positive integer, is the set $$\{ 0 \}$$ which is not open in the real line.

A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.

Clopen sets and non-open and/or non-closed sets
A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset a closed subset. Such subsets are known as . Explicitly, a subset $$S$$ of a topological space $$(X, \tau)$$ is called  if both $$S$$ and its complement $$X \setminus S$$ are open subsets of $$(X, \tau)$$; or equivalently, if $$S \in \tau$$ and $$X \setminus S \in \tau.$$

In topological space $$(X, \tau),$$ the empty set $$\varnothing$$ and the set $$X$$ itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space. To see, it suffices to remark that, by definition of a topology, $$X$$ and $$\varnothing$$ are both open, and that they are also closed, since each is the complement of the other.

The open sets of the usual Euclidean topology of the real line $$\R$$ are the empty set, the open intervals and every union of open intervals.

 * The interval $$I = (0, 1)$$ is open in $$\R$$ by definition of the Euclidean topology. It is not closed since its complement in $$\R$$ is $$ I^\complement = (-\infty, 0] \cup [1, \infty),$$ which is not open; indeed, an open interval contained in $$I^\complement$$ cannot contain $(M, d)$, and it follows that $$I^\complement$$ cannot be a union of open intervals. Hence, $$I$$ is an example of a set that is open but not closed.
 * By a similar argument, the interval $$J = [0, 1]$$ is a closed subset but not an open subset.
 * Finally, neither $$K = [0, 1)$$ nor its complement $$\R \setminus K = (-\infty, 0) \cup [1, \infty)$$ are open (because they cannot be written as a union of open intervals); this means that $$K$$ is neither open nor closed.

If a topological space $$X$$ is endowed with the discrete topology (so that by definition, every subset of $$X$$ is open) then every subset of $$X$$ is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that $$\mathcal{U}$$ is an ultrafilter on a non-empty set $$X.$$ Then the union $$\tau := \mathcal{U} \cup \{ \varnothing \}$$ is a topology on $$X$$ with the property that non-empty proper subset $$S$$ of $$X$$ is  an open subset or else a closed subset, but never both; that is, if $$\varnothing \neq S \subsetneq X$$ (where $$S \neq X$$) then  of the following two statements is true: either (1) $$S \in \tau$$ or else, (2) $$X \setminus S \in \tau.$$ Said differently,  subset is open or closed but the   subsets that are both (i.e. that are clopen) are $$\varnothing$$ and $$X.$$

Regular open sets
A subset $$S$$ of a topological space $$X$$ is called a  if $$\operatorname{Int} \left( \overline{S} \right) = S$$ or equivalently, if $$\operatorname{Bd} \left( \overline{S} \right) = \operatorname{Bd} S$$, where $$\operatorname{Bd} S$$, $$\operatorname{Int} S$$, and $$\overline{S}$$ denote, respectively, the topological boundary, interior, and closure of $$S$$ in $$X$$. A topological space for which there exists a base consisting of regular open sets is called a . A subset of $$X$$ is a regular open set if and only if its complement in $$X$$ is a regular closed set, where by definition a subset $$S$$ of $$X$$ is called a  if $$\overline{\operatorname{Int} S} = S$$ or equivalently, if $$\operatorname{Bd} \left( \operatorname{Int} S \right) = \operatorname{Bd} S.$$ Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general, the converses are true.

Properties
The union of any number of open sets, or infinitely many open sets, is open. The intersection of a finite number of open sets is open.

A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed.

Uses
Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces.

Every subset A of a topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A.

A function $$f : X \to Y$$ between two topological spaces $$X$$ and $$Y$$ is if the preimage of every open set in $$Y$$ is open in $$X.$$ The function $$f : X \to Y$$ is called if the image of every open set in $$X$$ is open in $$Y.$$

An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

"Open" is defined relative to a particular topology
Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greater clarity, we refer to a set X endowed with a topology $$\tau$$ as "the topological space X" rather than "the topological space $$(X, \tau)$$", despite the fact that all the topological data is contained in $$\tau.$$ If there are two topologies on the same set, a set U that is open in the first topology might fail to be open in the second topology. For example, if X is any topological space and Y is any subset of X, the set Y can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from the original topology on X." This potentially introduces new open sets: if V is open in the original topology on X, but $$V \cap Y$$ isn't open in the original topology on X, then $$V \cap Y$$ is open in the subspace topology on Y.

As a concrete example of this, if U is defined as the set of rational numbers in the interval $$(0, 1),$$ then U is an open subset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point x in U, there exists a positive number a such that all points within distance a of x are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is positive a such that all  points within distance a of x are in U (because U contains no non-rational numbers).

Generalizations of open sets
Throughout, $$(X, \tau)$$ will be a topological space.

A subset $$A \subseteq X$$ of a topological space $$X$$ is called:

 ' if $$A ~\subseteq~ \operatorname{int}_X \left( \operatorname{cl}_X \left( \operatorname{int}_X A \right) \right)$$, and the complement of such a set is called '. ', ', or  if it satisfies any of the following equivalent conditions:  $$A ~\subseteq~ \operatorname{int}_X \left( \operatorname{cl}_X A \right).$$ There exists subsets $$D, U \subseteq X$$ such that $$U$$ is open in $$X,$$ $$D$$ is a dense subset of $$X,$$ and $$A = U \cap D.$$ There exists an open (in $$X$$) subset $$U \subseteq X$$ such that $$A$$ is a dense subset of $$U.$$  The complement of a preopen set is called .   if $$A ~\subseteq~ \operatorname{int}_X \left( \operatorname{cl}_X A \right) ~\cup~ \operatorname{cl}_X \left( \operatorname{int}_X A \right)$$. The complement of a b-open set is called . ' or ' if it satisfies any of the following equivalent conditions:

$$A ~\subseteq~ \operatorname{cl}_X \left( \operatorname{int}_X \left( \operatorname{cl}_X A \right) \right)$$

$$ \operatorname{cl}_X A$$ is a regular closed subset of $$X.$$

There exists a preopen subset $$U$$ of $$X$$ such that $$U \subseteq A \subseteq \operatorname{cl}_X U.$$

 The complement of a β-open set is called .

 if it satisfies any of the following equivalent conditions:

Whenever a sequence in $$X$$ converges to some point of $$A,$$ then that sequence is eventually in $$A.$$ Explicitly, this means that if $$x_{\bull} = \left( x_i \right)_{i=1}^{\infty}$$ is a sequence in $$X$$ and if there exists some $$a \in A$$ is such that $$x_{\bull} \to x$$ in $$(X, \tau),$$ then $$x_{\bull}$$ is eventually in $$A$$ (that is, there exists some integer $$i$$ such that if $$j \geq i,$$ then $$x_j \in A$$).

$$A$$ is equal to its  in $$X,$$ which by definition is the set
 * $$\begin{alignat}{4}

\operatorname{SeqInt}_X A
 * &= \{ a \in A ~:~ \text{ whenever a sequence in } X \text{ converges to } a \text{ in } (X, \tau), \text{ then that sequence is eventually in } A \} \\

&= \{ a \in A ~:~ \text{ there does NOT exist a sequence in } X \setminus A \text{ that converges in } (X, \tau) \text{ to a point in } A \} \\ \end{alignat} $$

 The complement of a sequentially open set is called . A subset $$S \subseteq X$$ is sequentially closed in $$X$$ if and only if $$S$$ is equal to its , which by definition is the set $$\operatorname{SeqCl}_X S$$ consisting of all $$x \in X$$ for which there exists a sequence in $$S$$ that converges to $$x$$ (in $$X$$).

' and is said to have ' if there exists an open subset $$U \subseteq X$$ such that $$A \bigtriangleup U$$ is a meager subset, where $$\bigtriangleup$$ denotes the symmetric difference.
 if $$A ~\subseteq~ \operatorname{cl}_X \left( \operatorname{int}_X A \right)$$ or, equivalently, $$\operatorname{cl}_X A = \operatorname{cl}_X \left( \operatorname{int}_X A \right)$$. The complement in $$X$$ of a semi-open set is called a  set.
 if for each $$x \in A$$ there exists some semiopen subset $$U$$ of $$X$$ such that $$x \in U \subseteq \operatorname{sCl}_X U \subseteq A.$$

' (resp. ') if its complement in $$X$$ is a θ-closed (resp. ) set, where by definition, a subset of $$X$$ is called ' (resp. ') if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point $$x \in X$$ is called a ' (resp. a ') of a subset $$B \subseteq X$$ if for every open neighborhood $$U$$ of $$x$$ in $$X,$$ the intersection $$B \cap \operatorname{cl}_X U$$ is not empty (resp. $$B \cap \operatorname{int}_X\left( \operatorname{cl}_X U \right)$$ is not empty).

 * The subset $$A \subseteq X$$ is said to have the Baire property in the restricted sense if for every subset $$E$$ of $$X$$ the intersection $$A\cap E$$ has the Baire property relative to $$E$$.

 * The  (in $$X$$) of a subset $$A \subseteq X,$$ denoted by $$\operatorname{sCl}_X A,$$ is the intersection of all semi-closed subsets of $$X$$ that contain $$A$$ as a subset.

Using the fact that
 * $$A ~\subseteq~ \operatorname{cl}_X A ~\subseteq~ \operatorname{cl}_X B$$    and     $$\operatorname{int}_X A ~\subseteq~ \operatorname{int}_X B ~\subseteq~ B$$

whenever two subsets $$A, B \subseteq X$$ satisfy $$A \subseteq B,$$ the following may be deduced:

 * Every α-open subset is semi-open, semi-preopen, preopen, and b-open.
 * Every b-open set is semi-preopen (i.e. β-open).
 * Every preopen set is b-open and semi-preopen.
 * Every semi-open set is b-open and semi-preopen.

Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set. Preopen sets need not be semi-open and semi-open sets need not be preopen.

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space $$(X, \tau)$$ forms a topology on $$X$$ that is finer than $$\tau.$$

A topological space $$X$$ is Hausdorff if and only if every compact subspace of $$X$$ is θ-closed. A space $$X$$ is totally disconnected if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the  of every preopen subset is open.

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

title: Open set


text:
Disjoint sets

In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint.

Generalizations

This definition of disjoint sets can be extended to families of sets and to indexed families of sets. By definition, a collection of sets is called a family of sets (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An $$\left(A_i\right)_{i \in I},$$ is by definition a set-valued function (that is, it is a function that assigns a set $$A_i$$ to every element $$i \in I$$ in its domain) whose domain $$I$$ is called its  (and elements of its domain are called ).

There are two subtly different definitions for when a family of sets $$\mathcal{F}$$ is called pairwise disjoint. According to one such definition, the family is disjoint if each two sets in the family are either identical or disjoint. This definition would allow pairwise disjoint families of sets to have repeated copies of the same set. According to an alternative definition, each two sets in the family must be disjoint; repeated copies are not allowed. The same two definitions can be applied to an indexed family of sets: according to the first definition, every two distinct indices in the family must name sets that are disjoint or identical, while according to the second, every two distinct indices must name disjoint sets. For example, the family of sets { {0, 1, 2}, {3, 4, 5}, {6, 7, 8}, ... } is disjoint according to both definitions, as is the family { {..., −2, 0, 2, 4, ...}, {..., −3, −1, 1, 3, 5} } of the two parity classes of integers. However, the family $$(\{n + 2k \mid k\in\mathbb{Z}\})_{n \in \{0, 1, \ldots, 9\}}$$ with 10 members has five repetitions each of two disjoint sets, so it is pairwise disjoint under the first definition but not under the second.

Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two infinite sets whose intersection is a finite set may be said to be almost disjoint.

In topology, there are various notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods. Similarly, in a metric space, positively separated sets are sets separated by a nonzero distance.

Intersections
Disjointness of two sets, or of a family of sets, may be expressed in terms of intersections of pairs of them.

Two sets A and B are disjoint if and only if their intersection $$A\cap B$$ is the empty set. It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.

If a collection contains at least two sets, the condition that the collection is disjoint implies that the intersection of the whole collection is empty. However, a collection of sets may have an empty intersection without being disjoint. Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. For instance, the three sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. Also the empty family of sets is pairwise disjoint.

A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the closed intervals of the real numbers form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint.

Disjoint unions and partitions
A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X. Every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. Disjoint-set data structures and partition refinement are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two.

A disjoint union may mean one of two things. Most simply, it may mean the union of sets that are disjoint. But if two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make them disjoint before forming the union of the modified sets. For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set. For families of more than two sets, one may similarly replace each element by an ordered pair of the element and the index of the set that contains it.

title: Disjoint sets


text:
Good-Bye to All That

Good-Bye to All That is an autobiography by Robert Graves which first appeared in 1929, when the author was 34 years old. "It was my bitter leave-taking of England," he wrote in a prologue to the revised second edition of 1957, "where I had recently broken a good many conventions". The title may also point to the passing of an old order following the cataclysm of the First World War; the supposed inadequacies of patriotism, the interest of some in atheism, feminism, socialism and pacifism, the changes to traditional married life, and not least the emergence of new styles of literary expression, are all treated in the work, bearing as they did directly on Graves's life. The unsentimental and frequently comic treatment of the banalities and intensities of the life of a British army officer in the First World War gave Graves fame, notoriety and financial security, but the book's subject is also his family history, childhood, schooling and, immediately following the war, early married life; all phases bearing witness to the "particular mode of living and thinking" that constitute a poetic sensibility.

Laura Riding, Graves's lover, is credited with being a "spiritual and intellectual midwife" to the work. Graves, in a 1969 interview, claimed that he "entirely rewrote" the book—"every single sentence"—when it was reissued in the 1950s, suggesting that the process of co-writing The Reader Over Your Shoulder had made him more conscious of, and determined to rectify, deficiencies in his own style.

Pre-war life
Graves undertook climbing, stating "the sport made all others seem trivial." His first climb was Crib y Ddysgl, followed by climbs on Crib Goch and Y Lliwedd.

Graves goes on to claim, "In English preparatory and public schools romance is necessarily homosexual. The opposite sex is despised and treated as something obscene.  Many boys never recover from this perversion.  For every one born homosexual, at least ten permanent pseudo-homosexuals are made by the public school system: nine of these ten as honourably chaste and sentimental as I was."

Wartime experiences
A large part of the book is taken up by his experience of the First World War, in which Graves served as a lieutenant, then captain in the Royal Welch Fusiliers, with Siegfried Sassoon. Good-Bye to All That provides a detailed description of trench warfare, including the tragic incompetence of the Battle of Loos, including the use of gas, and the bitter fighting in the first phase of the Somme Offensive. At one point Graves agrees with his C.S.M., "Of course, it's murder, you bloody fool, And there's nothing else for it, is there?"

Graves claimed, "At least one in three of my generation at school died; because they all took commissions as soon as they could, most of them in the infantry and Royal Flying Corps. The average life expectancy of an infantry subaltern on the Western Front was, at some stages of the War, only about three months; by which time he had been either wounded or killed."

Regarding trench conditions and Cuinchy-bred rats, Graves stated, "They came up from the canal, fed on the plentiful corpses, and multiplied exceedingly."

Wounds
In the Somme engagement, Graves was wounded while leading his men through the cemetery at Bazentin-le-petit church on 20 July 1916. The wound initially appeared so severe that military authorities erroneously reported to his family that he had died. While mourning his death, Graves's family received word from him that he was alive, and put an announcement to that effect in the newspapers. Graves later regretted omitting from the book the name of the soldier who had rescued him, Owen Roberts. The two met again fifty years later in a hospital ward to which both had been admitted for surgery, after which Graves signed Roberts's copy of the book, giving Roberts full credit for saving his life.

Reputed atrocities
Graves also discussed atrocities committed during the war in Good-Bye to All That. He wrote that among his fellow troops, Allied atrocity propaganda, such as reports of the rape of Belgium, was widely disbelieved (defining "atrocities" in the book as wartime sexual violence, mutilation and torture instead of summary executions). Graves also noted that if "the atrocity-list had to include the accidental-on-purpose bombing or machine-gunning of civilians from the air, the Allies were now committing as many atrocities as the Germans." Observing French and Belgian civilians showing British soldiers body parts allegedly mutilated by German troops, he argued that these were more likely the result of indiscriminate shelling.

Though the German use of serrated knives and British deployment of expanding bullets were regarded by the other side as "atrocious", Graves claimed that the opportunity for soldiers on both sides to commit "true atrocities" only occurred when escorting prisoners of war to the rear lines. "Nearly every instructor in the mess", he wrote, "could quote specific instances of prisoners having been murdered on the way back. The commonest motives were, it seems, revenge for the death of friends or relatives, jealousy of the prisoner's trip to a comfortable prison camp in England, military enthusiasm, fear of being suddenly overpowered by the prisoners or, more simply, impatience with the escorting job." Similarly, "If a German patrol found a wounded man, they were likely as not to cut his throat." However, if POWs arrived at their destination, they were treated well during interrogations.

In the book, Graves stated that Australian and Canadian troops had the worst reputation for atrocities against German POWs. He recounted two first-hand anecdotes from a Scottish-Canadian and an Australian, who told him how they murdered German prisoners while escorting them using Mills bombs. Canadian soldiers were motivated to commit atrocities against POWs due to the story of "The Crucified Soldier", which Graves and his fellow soldiers also refused to believe. He also added that the use of "semi-civilized coloured troops in Europe was, from the German point of view, we knew, one of the chief Allied atrocities. We sympathized."

Postwar trauma
Graves was severely traumatised by his war experience. After being wounded in the lung by a shell blast, he endured a squalid five-day train journey with unchanged bandages. During initial military training in England, he received an electric shock from a telephone that had been hit by lightning, which caused him for the next twelve years to stammer and sweat badly if he had to use one. Upon his return home, he describes being haunted by ghosts and nightmares.

According to Graves, "My particular disability was neurasthenia." He went on to say, "Shells used to come bursting on my bed at midnight ... strangers in daytime would assume the faces of friends who had been killed." Offered a chance to rejoin George Mallory in climbing, Graves declined, "I could never again now deliberately take chances with my life."

Critical responses
Siegfried Sassoon and his friend Edmund Blunden (whose First World War service had been in a different regiment) took umbrage at the contents of the book. Sassoon's complaints mostly related to Graves's depiction of him and his family, whereas Blunden had read the memoirs of J. C. Dunn and found them at odds with Graves in some places. The two men took Blunden's copy of Good-Bye to All That and made marginal notes contradicting some of the text. That copy survives and is held by the New York Public Library. Graves's father, Alfred Perceval Graves, also incensed at some aspects of Graves's book, wrote a riposte to it titled To Return to All That.

It was included in Robert McCrum's Guide to the 100 greatest nonfiction books in English published by The Guardian.

title: Good-Bye to All That


text:
Frederick I, Elector of Brandenburg

Frederick (Middle High German: Friderich, Standard German: Friedrich; 21 September 1371 – 20 September 1440) was the last Burgrave of Nuremberg from 1397 to 1427 (as Frederick VI), Margrave of Brandenburg-Ansbach from 1398, Margrave of Brandenburg-Kulmbach from 1420, and Elector of Brandenburg (as Frederick I) from 1415 until his death. He became the first member of the House of Hohenzollern to rule the Margraviate of Brandenburg.

Biography
Frederick was born in Nuremberg, the second-born son of Burgrave Frederick V (1333–1398) and the Wettin princess Elisabeth of Meissen. He entered early into the service of his brother-in-law, the Habsburg duke Albert III of Austria. After Albert's death in 1395, he fought on the side of the Luxembourg king Sigismund of Hungary against invading Ottoman forces. He and his elder brother John, husband of Sigismund's sister Margaret of Bohemia, fought in the 1396 Battle of Nicopolis where they suffered a disastrous defeat.

After Frederick returned to Nuremberg, he divided the Franconian inheritance from his father with his brother John, who received the Principality of Bayreuth, while Frederick kept the Principality of Ansbach; the brothers ruled jointly in the Burgraviate of Nuremberg. At first Frederick tried to mediate in the imperial confusion between Sigismund's elder half-brother, the German king Wenceslaus, and the party of Elector Rupert III of the Palatinate, but from September 1399 he fought on the side of Rupert, husband of his sister Elizabeth, nonetheless. Wenceslaus was eventually deposed and Rupert elected King of the Romans on 21 August 1400.

Frederick resumed his rule of Ansbach in 1409 and after heavy feuding, entered into the service of King Sigismund. In 1410, the death of Rupert, King of the Germans, left the throne of the Holy Roman Empire vacant. Sigismund enlisted Frederick's help in obtaining the throne. At the time, Jobst of Moravia ruled Brandenburg and thus was one of the prince-electors who had the right to vote for the new king. However, Sigismund disputed Jobst's claim to Brandenburg and his right to vote in the imperial election. Sigismund claimed these rights for himself and designated Frederick to represent him as elector of Brandenburg in the imperial election of 20 September 1410. While Sigismund won this initial vote, Jobst of Moravia won the support of a majority of electors in an election in October 1410 and himself claimed the imperial throne. Jobst's death under suspicious circumstances in January 1411 cleared the way for Sigismund's recovery of Brandenburg and his undisputed election as king of the empire later that year. In gratitude for Frederick's services, King Sigismund made him Oberster Hauptmann and Verwalter der Marken (1411). With an iron hand Frederick fought against the rebellious nobility of the March of Brandenburg (in particular, the Quitzow family) and, in the end, restored security. Frederick also became a member of the Parakeet Society and of the League of Constance.

At the Council of Constance (30 April 1415) Sigismund granted Frederick the titles of Margrave and Prince-elector of Brandenburg. On 21 October 1415 the Brandenburg states meeting in a Landtag asked him to rule in Berlin. The king awarded him the formal enfeoffment of the margravate on 18 April 1417. As Frederick did not agree with the forcible action of Sigismund against the Hussites, relations between them cooled.

Constant feuding with the nobility of Brandenburg led Frederick to withdraw to his castle at Cadolzburg in 1425 and transfer the regency of the mark to his son John in 1426 (Frederick, however, remained elector). After 1427 he organized the imperial war against the Hussites. He is best known for joining the papal legate Julian Cesarini in August 1431 to attack the Hussites in the disastrous Battle of Domažlice. He subsequently provided substantial assistance in the mediation of the Compacta of Prague at the Council of Basel (30 November 1433).

Upon his death in 1440, Frederick was succeeded as elector by his second-eldest son, Frederick II.

Family and children
He married Elisabeth of Bavaria-Landshut (1383–13 November 1442, Ansbach), daughter of Duke Frederick of Bavaria-Landshut and Maddalena Visconti in 1401. Their children were:
 * 1) Elisabeth (1403–31 October 1449, Liegnitz), married:
 * 2) in Konstanz 1418 Duke Louis II of Brieg and Legnica (1380/5–1436);
 * 3) in 1438 Duke Wenzel I of Teschen (1413/18–1474).
 * 4) John "the Alchemist" (1405–1465), Margrave of Brandenburg-Kulmbach.
 * 5) married in 1416 Princess Barbara of Saxe-Wittenberg (1405–1465)
 * 6) Cecilia (c. 1405–4 January 1449), married:
 * 7) in Berlin 30 May 1423 Duke William III of Brunswick-Lüneburg (1392–1482).
 * 8) Margaret (1410–27 July 1465, Landshut), married:
 * 9) in 1423 to Duke Albert V, Duke of Mecklenburg (1397–1423);
 * 10) in Ingolstadt 20 July 1441 to Louis VIII, Duke of Bavaria (1403–1445);
 * 11) in 1446 to Count Martin of Waldenfels (d. 1471).
 * 12) Magdalene (c. 1412 –27 October 1454, Scharnebeck), married:
 * 13) in Tangermünde 3 July 1429 to Duke Frederick of Brunswick-Lüneburg (1418–1478).
 * 14) Frederick II (1413–1471), Elector of Brandenburg
 * 15) married in 1441 Princess Catherine of Saxony (1421–1476)
 * 16) Albert Achilles, (1414–1486), Elector of Brandenburg, married:
 * 17) in 1446 Princess Margarete of Baden (1431–1457)
 * 18) in 1458 Princess Anna of Saxony (1437–1512)
 * 19) Sofie, born and died 1417.
 * 20) Dorothea (9 February 1420–19 January 1491, Rehna), married:
 * 21) in 1432 Duke Henry IV, Duke of Mecklenburg (1417–1477)
 * 22) Frederick "the Fat" (c. 1424–6 October 1463, Tangermünde), Lord of Altmark, married:
 * 23) in 1449 Princess Agnes of Pomerania (1436–1512)

title: Frederick I, Elector of Brandenburg


text:
Frederick II, Elector of Brandenburg

Frederick II of Brandenburg (Friedrich II.) (19 November 1413 – 10 February 1471), nicknamed "the Iron" (der Eiserne) and sometimes "Irontooth" (Eisenzahn), was a Prince-elector of the Margraviate of Brandenburg from 1440 until his abdication in 1470, and was a member of the House of Hohenzollern.

Biography
Frederick II was born at Tangermünde Castle (German: Burg Tangermünde), when Tangermünde was within the Margraviate of Brandenburg, to Frederick I, Brandenburg's first Hohenzollern ruler, and his wife Elizabeth, daughter of Frederick, Duke of Bavaria-Landshut, and Maddalena Visconti. The latter was a daughter of Bernabò Visconti and Beatrice della Scala. As the second son, his brothers included John the Alchemist and Albert Achilles, both of whom also ruled Brandenburg as margraves.

In 1421, at age 8, Frederick was betrothed to Hedwig Jagiellon, but she died on 8 December 1431, before the marriage could take place.

When Frederick I retired in 1437, he compensated his incapable eldest son John with the Principality of Bayreuth while Frederick II assumed the government of Brandenburg. Unlike his father, he turned away from imperial politics and concentrated on his efforts to pacify the nobility and towns of the electorate. Quarrels with the city of Berlin began in 1440 with his plans to build a new residence on the Cölln island of the Spree river. In 1448 the Berliner Unwille (indignation) against the cession of the city's territory for an electoral stronghold culminated in open revolt, when the citizens flooded the excavation of the future Stadtschloss. Nevertheless, Frederick II prevailed, had the palace built and the city's rights decisively curtailed.

In 1453, Frederick II, in his position as the Margrave of Brandenburg, was sent a letter from Rhodes, penned by the Grand Master of the Knights of St John, notifying him of the Fall of Constantinople and requesting he send them support so as to allow them to continue to combat the Turkish threat. His response is unknown.

In 1454 and 1455, he concluded the Treaties of Cölln and Mewe and thereby re-gained the Neumark (New March) from the Teutonic Order state. Weary of the long struggle with the Duchy of Pomerania, he abdicated in 1470 in favour of his younger brother Albert Achilles, he retired to the Bayreuth Principality and died one year later in Neustadt an der Aisch.

Family and children
On 11 June 1441, Frederick II married Catherine of Saxony (1421 – August 23, 1476), a daughter of Elector Frederick I of Saxony and Katharina of Brunswick-Lüneburg. They had three children (and Frederick had one bastard):

 * Dorothea of Brandenburg (c. 1446 – March, 1519), married Duke John V of Saxe-Lauenburg on 12 February 1464.
 * Margaret of Brandenburg (c. 1450 – 1489), married Bogislaw X, Duke of Pomerania.
 * Johan of Brandenburg (December 23, 14{06-16} - 1454)
 * Erasmus von Brandenburg (c. 1452 - 1465).

title: Frederick II, Elector of Brandenburg


text:
Prototile

In mathematics, a prototile is one of the shapes of a tile in a tessellation.

Definition
A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others. If $S$ is the set of tiles in a tessellation, a set $R$ of shapes is called a set of prototiles if no two shapes in $R$ are congruent to each other, and every tile in $S$ is congruent to one of the shapes in $R$.

It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality, so the number of prototiles is well defined. A tessellation is said to be monohedral if it has exactly one prototile.

Aperiodicity
A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. In March 2023, four researchers, Chaim Goodman-Strauss, David Smith, Joseph Samuel Myers and Craig S. Kaplan, announced the discovery of an aperiodic monohedral prototile (monotile) and a proof that the tile discovered by David Smith is an aperiodic monotile, i.e. a solution to a longstanding open einstein problem.

In higher dimensions, the problem had been solved earlier: the Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, and cannot tile space periodically.

title: Prototile


text:
Large Magellanic Cloud

The Large Magellanic Cloud (LMC) is a dwarf galaxy and satellite galaxy of the Milky Way. At a distance of around 50 kpc,  the LMC is the second- or third-closest galaxy to the Milky Way, after the Sagittarius Dwarf Spheroidal (c. 16 kpc away) and the possible dwarf irregular galaxy called the Canis Major Overdensity. Based on the D25 isophote at the B-band (445 nm wavelength of light), the Large Magellanic Cloud is about 9.86 kpc across. It is roughly one-hundredth the mass of the Milky Way and is the fourth-largest galaxy in the Local Group, after the Andromeda Galaxy (M31), the Milky Way, and the Triangulum Galaxy (M33).

The LMC is classified as a Magellanic spiral. It contains a stellar bar that is geometrically off-center, suggesting that it was once a barred dwarf spiral galaxy before its spiral arms were disrupted, likely by tidal interactions from the nearby Small Magellanic Cloud (SMC) and the Milky Way's gravity. The LMC is predicted to merge with the Milky Way in approximately 2.4 billion years.

With a declination of about −70°, the LMC is visible as a faint "cloud" from the southern hemisphere of the Earth and from as far north as 20° N. It straddles the constellations Dorado and Mensa and has an apparent length of about 10° to the naked eye, 20 times the Moon's diameter, from dark sites away from light pollution.

History of observation
Both the Large and Small Magellanic Clouds have been easily visible for southern nighttime observers well back into prehistory. It has been claimed that the first known written mention of the Large Magellanic Cloud was by the Persian astronomer 'Abd al-Rahman al-Sufi Shirazi (later known in Europe as "Azophi"), which he referred to as Al Bakr, the White Ox, in his Book of Fixed Stars around 964 AD. However, this seems to be a misunderstanding of a reference to some stars south of Canopus which he admits he has not seen.

The first confirmed recorded observation was in 1503–1504 by Amerigo Vespucci in a letter about his third voyage. He mentioned "three Canopes [sic], two bright and one obscure"; "bright" refers to the two Magellanic Clouds, and "obscure" refers to the Coalsack.

Ferdinand Magellan sighted the LMC on his voyage in 1519 and his writings brought it into common Western knowledge. The galaxy now bears his name. The galaxy and southern end of Dorado are in the current epoch at opposition on about 5 December when thus visible from sunset to sunrise from equatorial points such as Ecuador, the Congos, Uganda, Kenya and Indonesia and for part of the night in nearby months. Above about 28° south, such as most of Australia and South Africa, the galaxy is always sufficiently above the horizon to be considered properly circumpolar, thus during spring and autumn the cloud is also visible much of the night, and the height of winter in June nearly coincides with closest proximity to the Sun's apparent position.

Measurements with the Hubble Space Telescope, announced in 2006, suggest the Large and Small Magellanic Clouds may be moving too quickly to be orbiting the Milky Way.

Astronomers discovered a new black hole inside the Large Magellanic Cloud in November 2021 using the European Southern Observatory's Very Large Telescope in Chile. Astronomers claim its gravity is influenced by a nearby star, which is about five times the mass of the Sun.

Geometry

The Large Magellanic Cloud has a prominent central bar and spiral arm. The central bar seems to be warped so that the east and west ends are nearer the Milky Way than the middle. In 2014, measurements from the Hubble Space Telescope made it possible to determine a rotation period of 250 million years.

The LMC was long considered to be a planar galaxy that could be assumed to lie at a single distance from the Solar System. However, in 1986, Caldwell and Coulson found that field Cepheid variables in the northeast lie closer to the Milky Way than those in the southwest. From 2001 to 2002 this inclined geometry was confirmed by the same means, by core helium-burning red clump stars, and by the tip of the red giant branch. All three papers find an inclination of 35°, where a face-on galaxy has an inclination of 0°. Further work on the structure of the LMC using the kinematics of carbon stars showed that the LMC's disk is both thick and flared, likely due to interactions with the SMC. Regarding the distribution of star clusters in the LMC, Schommer et al. measured velocities for 80 clusters and found that the LMC's cluster system has kinematics consistent with the clusters moving in a disk-like distribution. These results were confirmed by Grocholski et al., who calculated distances to a sample of clusters and showed that the cluster system is distributed in the same plane as the field stars.

Distance
Distance to the LMC has been calculated using standard candles; Cepheid variables are one of the most popular. These have been shown to have a relationship between their absolute luminosity and the period over which their brightness varies. However the variable of metallicity may also need to be taken as a component of this as consensus is this likely affects their period-luminosity relations. Unfortunately, those in the Milky Way typically used to calibrate the relation are more metal-rich than those found in the LMC.

Modern 8-meter-class optical telescopes have discovered eclipsing binaries throughout the Local Group. Parameters of these systems can be measured without mass or compositional assumptions. The light echoes of supernova 1987A are also geometric measurements, without any stellar models or assumptions.

In 2006, the Cepheid absolute luminosity was re-calibrated using Cepheid variables in the galaxy Messier 106 that cover a range of metallicities. Using this improved calibration, they find an absolute distance modulus of $$(m - M)_0 = 18.41$$, or 48 kpc. This distance has been confirmed by other authors.

By cross-correlating different measurement methods, one can bound the distance; the residual errors are now less than the estimated size parameters of the LMC.

The results of a study using late-type eclipsing binaries to determine the distance more accurately was published in the scientific journal Nature in March 2013. A distance of 49.97 kpc with an accuracy of 2.2% was obtained.

Features

Like many irregular galaxies, the LMC is rich in gas and dust, and is currently undergoing vigorous star formation activity. It holds the Tarantula Nebula, the most active star-forming region in the Local Group.

The LMC has a wide range of galactic objects and phenomena that make it known as an "astronomical treasure-house, a great celestial laboratory for the study of the growth and evolution of the stars", per Robert Burnham Jr. Surveys of the galaxy have found roughly 60 globular clusters, 400 planetary nebulae and 700 open clusters, along with hundreds of thousands of giant and supergiant stars. Supernova 1987A—the nearest supernova in recent years—was in the Large Magellanic Cloud. The Lionel-Murphy SNR (N86) nitrogen-abundant supernova remnant was named by astronomers at the Australian National University's Mount Stromlo Observatory, acknowledging Australian High Court Justice Lionel Murphy's interest in science and its perceived resemblance to his large nose.

A bridge of gas connects the Small Magellanic Cloud (SMC) with the LMC, which evinces tidal interaction between the galaxies. The Magellanic Clouds have a common envelope of neutral hydrogen, indicating that they have been gravitationally bound for a long time. This bridge of gas is a star-forming site.

X-ray sources
No X-rays above background were detected from either cloud during the September 20, 1966, Nike-Tomahawk rocket flight nor that of two days later. The second took off from Johnston Atoll at 17:13 UTC and reached an apogee of 160 km, with spin-stabilization at 5.6 rps. The LMC was not detected in the X-ray range 8–80 keV.

Another was launched from same atoll at 11:32 UTC on October 29, 1968, to scan the LMC for X-rays. The first discrete X-ray source in Dorado was at RA 05:20:0 Dec -69°, and it was the Large Magellanic Cloud. This X-ray source extended over about 12° and is consistent with the Cloud. Its emission rate between 1.5–10.5 keV for a distance of 50 kpc is $1$/s. An X-ray astronomy instrument was carried aboard a Thor missile launched from the same atoll on September 24, 1970, at 12:54 UTC and altitudes above 300 km, to search for the Small Magellanic Cloud and to extend observation of the LMC. The source in the LMC appeared extended and contained star ε Dor. The X-ray luminosity (Lx) over the range 1.5–12 keV was $1.38$ ($4 ergs$).

The Large Magellanic Cloud (LMC) appears in the constellations Mensa and Dorado. LMC X-1 (the first X-ray source in the LMC) is at RA 05:40:05 Dec -69°45′51″, and is a high-mass X-ray binary (star system) source (HMXB). Of the first five luminous LMC X-ray binaries: LMC X-1, X-2, X-3, X-4 and A 0538–66 (detected by Ariel 5 at A 0538–66), LMC X-2 is the one that is a bright low-mass X-ray binary system (LMXB) in the LMC.

DEM L316 in the Cloud consists of two supernova remnants. Chandra X-ray spectra show that the hot gas shell on the upper left has an abundance of iron. This implies that the upper-left SNR is the product of a Type Ia supernova; much lower such abundance in the lower remnant belies a Type II supernova.

A 16 ms X-ray pulsar is associated with SNR 0538-69.1. SNR 0540-697 was resolved using ROSAT.

title: Large Magellanic Cloud