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Monster group

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000   = 246·320·59·76·112·133·17·19·23·29·31·41·47·59·71    ≈ 8.

The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the happy family, and the remaining six exceptions pariahs.

It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in Scientific American.

History
The monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months, the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed. Griess constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative algebra over the real numbers; he first announced his construction in Ann Arbor on January 14, 1980. In his 1982 paper, he referred to the monster as the Friendly Giant, but this name has not been generally adopted. John Conway and Jacques Tits subsequently simplified this construction.

Griess's construction showed that the monster exists. Thompson showed that its uniqueness (as a simple group satisfying certain conditions coming from the classification of finite simple groups) would follow from the existence of a 196,883-dimensional faithful representation. A proof of the existence of such a representation was announced by Norton, though he never published the details. Griess, Meierfrankenfeld, and Segev gave the first complete published proof of the uniqueness of the monster (more precisely, they showed that a group with the same centralizers of involutions as the monster is isomorphic to the monster).

The monster was a culmination of the development of sporadic simple groups and can be built from any two of three subquotients: the Fischer group Fi24, the baby monster, and the Conway group Co1.

The Schur multiplier and the outer automorphism group of the monster are both trivial.

Representations
The minimal degree of a faithful complex representation is 47 × 59 × 71 = 196,883, hence is the product of the three largest prime divisors of the order of M. The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful complex representation.

The smallest faithful permutation representation of the monster is on   97,239,461,142,009,186,000 = 24·37·53·74·11·132·29·41·59·71    ≈ 1020

points.

The monster can be realized as a Galois group over the rational numbers, and as a Hurwitz group.

The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A100 and SL20(2) are far larger but easy to calculate with as they have "small" permutation or linear representations. Alternating groups, such as A100, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type, such as SL20(2), have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby monster, with a representation of dimension 4370).

Computer construction
Martin Seysen has implemented a fast Python package named mmgroup, which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by Robert A. Wilson in 2013. The mmgroup software package has been used to find two new maximal subgroups of the monster group.

Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.

Wilson asserts that the best description of the monster is to say, "It is the automorphism group of the monster vertex algebra". This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".

Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. Let V be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is 31+12.2.Suz.2, where Suz is the Suzuki group. Elements of the monster are stored as words in the elements of H and an extra generator T. It is reasonably quick to calculate the action of one of these words on a vector in V. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors u and v whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element g of the monster by finding the smallest i > 0 such that giu = u and giv = v. This and similar constructions (in different characteristics) were used to find some of the non-local maximal subgroups of the monster group.

Subquotients

The monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together. The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.

Maximal subgroups
The monster has 46 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A12.

The 46 classes of maximal subgroups of the monster are given by the following table. Previous unpublished work of Wilson et. al had purported to rule out any almost simple subgroups with non-abelian simple socles of the form U3(4), L2(8), and L2(16). However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U3(4). The same authors had previously found a new maximal subgroup of the form L2(13) and confirmed that there are no maximal subgroups with socle L2(8) or L2(16), thus completing the classification in the literature.

Note that tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups in this table were incorrectly omitted from some previous lists.

McKay's E8 observation
There are also connections between the monster and the extended Dynkin diagrams $$\tilde E_8,$$ specifically between the nodes of the diagram and certain conjugacy classes in the monster, known as McKay's E8 observation. This is then extended to a relation between the extended diagrams $$\tilde E_6, \tilde E_7, \tilde E_8$$ and the groups 3.Fi24, 2.B, and M, where these are (3/2/1-fold central extensions) of the Fischer group, baby monster group, and monster. These are the sporadic groups associated with centralizers of elements of type 1A, 2A, and 3A in the monster, and the order of the extension corresponds to the symmetries of the diagram. See ADE classification: trinities for further connections (of McKay correspondence type), including (for the monster) with the rather small simple group PSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4 known as Bring's curve.

Moonshine
The monster group is one of two principal constituents in the monstrous moonshine conjecture by Conway and Norton, which relates discrete and non-discrete mathematics and was finally proved by Richard Borcherds in 1992.

In this setting, the monster group is visible as the automorphism group of the monster module, a vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the monster Lie algebra, a generalized Kac–Moody algebra.

Many mathematicians, including Conway, have seen the monster as a beautiful and still mysterious object. Conway said of the monster group: "There's never been any kind of explanation of why it's there, and it's obviously not there just by coincidence. It's got too many intriguing properties for it all to be just an accident." Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, "I can explain what Monstrous Moonshine is in one sentence, it is the voice of God."

title: Monster group


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Bungee jumping

Bungee jumping, also spelled bungy jumping, is an activity that involves a person jumping from a great height while connected to a large elastic cord. The launching pad is usually erected on a tall structure such as a building or crane, a bridge across a deep ravine, or on a natural geographic feature such as a cliff. It is also possible to jump from a type of aircraft that has the ability to hover above the ground, such as a hot-air-balloon or helicopter. The thrill comes from the free-falling and the rebound. When the person jumps, the cord stretches and the jumper flies upwards again as the cord recoils, and continues to oscillate up and down until all the kinetic energy is dissipated.

Early tethered jumping
The land diving (Sa: Gol) of Pentecost Island in Vanuatu is an ancient ritual in which young men jump from tall wooden platforms with vines tied to their ankles as a test of their courage and passage into manhood. Unlike in modern bungee-jumping, land-divers intentionally hit the ground, but the vines absorb sufficient force to make the impact non-lethal. The land-diving ritual on Pentecost has been claimed as an inspiration by A. J. Hackett, prompting calls from the islanders' representatives for compensation for what they view as the unauthorised appropriation of their cultural property.

A tower 4000 ft high with a system to drop a "car" suspended by a cable of "best rubber" was proposed for the Chicago World Fair, 1892–1893. The car, seating two hundred people, would have been shoved from a platform on the tower and then would have bounced to a stop. The designer engineer suggested that for safety the ground below "be covered with eight feet of feather bedding". The proposal was declined by the Fair's organizers.

Etymology
The word "bungee" originates from West Country dialect of the English language, meaning "Anything thick and squat", as defined by James Jennings in his book "Observations of Some of the Dialects in The West of England" published 1825. In 1928, the word started to be used for a rubber eraser.

The Oxford English Dictionary records early use of the phrase in 1938 relating to launching of gliders using an elasticated cord, and also as "A long nylon-cased rubber band used for securing luggage".

"Bungy" is the usual spelling in New Zealand and other countries.

Modern sport
The first modern bungee jumps were made on 1 April 1979 from the 250 ft Clifton Suspension Bridge in Bristol, England, by David Kirke and Simon Keeling, members of the Oxford University Dangerous Sports Club, and Geoff Tabin, a professional climber who tied the ropes for the jump. The students had come up with the idea after discussing the "vine jumping" ritual of Vanuatu. The jumpers were arrested shortly after, but continued with jumps in the US from the Golden Gate Bridge and the Royal Gorge Bridge. The last jump was sponsored by and televised on the American programme That's Incredible, spreading the concept worldwide. By 1982, Kirk and Keelling were jumping from mobile cranes and hot air balloons.

Colorado climbers Mike Munger and Charlie Fowler may have bungee-jumped earlier in Eldorado Springs, CO in 1977. Both were cutting-edge alpinists, preparing for a trip to Monte Fitzroy in Patagonia by simulating long falls onto a springy, 150 ft nylon climbing rope. They scrambled up to a large tree at the top of the 700 ft wall, above a severely overhanging climb appropriately named "Diving Board", and tied one end of the rope into the tree. With a piece of flat seat belt webbing around his waist and some homemade leg loops, Mike tied into the other end of the rope and, after no small amount of trepidation, he jumped. He then ascended the rope mechanically to the tree and untied. Then Charlie tied in and jumped. The total fall was about 130 ft. Organised commercial bungee jumping began with the New Zealander, A. J. Hackett, who made his first jump from Auckland's Greenhithe Bridge in 1986. During the following years, Hackett performed a number of jumps from bridges and other structures (including the Eiffel Tower), building public interest in the sport, and opening the world's first permanent commercial bungee site, the Kawarau Bridge Bungy at the Kawarau Gorge Suspension Bridge near Queenstown in the South Island of New Zealand. Hackett remains one of the largest commercial operators, with concerns in several countries.

Several million successful jumps have taken place since 1980. This safety record is attributable to bungee operators rigorously conforming to standards and guidelines governing jumps, such as double checking calculations and fittings for every jump. As with any sport, injuries can still occur (see below), and there have been fatalities. A relatively common mistake in fatality cases is to use a cord that is too long. The cord should be substantially shorter than the height of the jumping platform to allow it room to stretch. When the cord becomes taut and then is stretched, the tension in the cord progressively increases. Initially the tension is less than the jumper's weight and the jumper continues to accelerate downwards. At some point, the tension equals the jumper's weight and the acceleration is temporarily zero. With further stretching, the jumper has an increasing upward acceleration and at some point has zero vertical velocity before recoiling upward. See also potential energy for a discussion of the spring constant and the force required to distort bungee cords and other spring-like objects.

The Bloukrans River Bridge was the first bridge to be used as a bungee jump launch spot in Africa when Face Adrenalin introduced bungee jumping to the African continent in 1990. Bloukrans Bridge Bungy has been operated commercially by Face Adrenalin since 1997, and is the highest commercial bridge bungy in the world.

In 2008, Carl Dionisio of Durban performed a 30 meter bungy jump attached to a cord made of 18,500 condoms. He currently runs the only Ocean Touch bungy jump in the World at Calheta Beach in Madeira, Portugal, and claims to be the only person operating in the bungy industry single-handed. He holds the world record for being the only person to bungy jump while driving a tower crane at the same time, something that he has done hundreds of times since 2017.

Equipment

The elastic rope first used in bungee jumping, and still used by many commercial operators, is factory-produced braided shock cord. This special bungee cord consists of many latex strands enclosed in a tough outer cover. The outer cover may be applied when the latex is pre-stressed, so that the cord's resistance to extension is already significant at the cord's natural length. This gives a harder, sharper bounce. The braided cover also provides significant durability benefits. Other operators, including A. J. Hackett and most southern-hemisphere operators, use unbraided cords with exposed latex strands. These give a softer, longer bounce and can be home-produced.

Accidents where participants became detached led many commercial operators to use a body harness, if only as a backup for an ankle attachment. Body harnesses generally derive from climbing equipment rather than parachute equipment.

The highest jump

Milad tower bungee jumping with a height of 280 meters is the highest jumping platform in the world

In August 2005, AJ Hackett added a SkyJump to the Macau Tower, making it the world's highest jump at 233 m. The SkyJump did not qualify as the world's highest bungee as it is not strictly speaking a bungee jump, but instead what is referred to as a 'Decelerator-Descent' jump, using a steel cable and decelerator system, rather than an elastic rope. On 17 December 2006, the Macau Tower started operating a proper bungee jump, which became the "Highest Commercial Bungee Jump in the World" according to the Guinness Book of Records. The Macau Tower Bungy has a "Guide cable" system that limits swing (the jump is very close to the structure of the tower itself) but does not have any effect on the speed of descent, so this still qualifies the jump for the World Record.

Kushma Bungee Jump is the world's second-highest bungee jump with a height of 228 m. It is located in the gorge of Kaligandaki River and world-first natural canyon bungee jump. Another commercial bungee jump currently in operation is just 13 m smaller, at 220 m. This jump, made without guide ropes, is from the top of the Verzasca Dam near Locarno, Switzerland. It appears in the opening scene of the James Bond film GoldenEye. The 216 m Bloukrans Bridge Bungy in South Africa and the Verzasca Dam jumps are pure freefall swinging bungee from a single cord.

Guinness only records jumps from fixed objects to guarantee the accuracy of the measurement. John Kockleman however recorded a 2200 ft bungee jump from a hot air balloon in California in 1989. In 1991 Andrew Salisbury jumped from 9000 ft from a helicopter over Cancun for a television program and with Reebok sponsorship. The full stretch was recorded at 3157 ft. He landed safely under parachute.

One commercial jump higher than all others is at the Royal Gorge Bridge in Colorado. The height of the platform is 321 m. However, this jump is rarely available, as part of the Royal Gorge Go Fast Games—first in 2005, then again in 2007. Previous to this the record was held in West Virginia, USA, by New Zealander Chris Allum, who bungee jumped 823 ft from the New River Gorge Bridge on "Bridge Day" 1992 to set a world's record for the longest bungee jump from a fixed structure.

Catapult
In "Catapult" (Reverse Bungee or Bungee Rocket), the jumper starts on the ground. The jumper is secured and the cord is stretched, then released and shooting the jumper up into the air. This is often achieved using either a crane or a hoist attached to a (semi-)perma structure. This simplifies the action of stretching the cord and later lowering the participant to the ground.

Trampoline
"Bungy Trampoline" uses elements from bungy and trampolining. The participant begins on a trampoline and is fitted into a body harness, which is attached via bungy cords to two high poles on either side of the trampoline. As they begin to jump, the bungy cords are tightened, allowing a higher jump than could normally be made from a trampoline alone.

Running
"Bungee Running" involves no jumping as such. It merely consists of, as the name suggests, running along a track (often inflatable) with a bungee cord attached. One often has a velcro-backed marker that marks how far the runner got before the bungee cord pulled back. This activity can often be found at fairs and carnivals and is often most popular with children.

Ramp
Bungee jumping off a ramp. Two rubber cords – the "bungees" – are tied around the participant's waist to a harness. Those bungee cords are linked to steel cables along which they can slide due to stainless pulleys. The participants bicycle, sled or ski before jumping.

SCAD diving
SCAD diving (Suspended Catch Air Device) is similar to bungee jumping in that the participant is dropped from a height, but in this variation there is no cord; instead the participant free-falls into a net. Untrained SCAD divers use a special free fall harness to ensure the correct falling position. Free-style SCAD divers do not use harnesses. The landing into the huge airtube framed net is extremely soft and forgiving. The SCAD was invented by MONTIC Hamburg, Germany in 1997.

Risk of injury
Bungee jumping injuries may be divided into those that occur after jumping secondary to equipment mishap or tragic accident, and those that occur regardless of safety measures.

In the first instance, injury can happen if the safety harness fails, the cord length is miscalculated, or the cord is not properly connected to the jump platform. In 1986, a man died during rehearsals for a bungee jumping stunt on a BBC television programme, because the cord sprang loose from a carabiner clip.

Injuries that occur despite safety measures generally relate to the abrupt rise in upper body intravascular pressure during bungee cord recoil. Eyesight damage is the most frequently reported complication. Impaired eyesight secondary to retinal haemorrhage may be transient or take several weeks to resolve. In one case, a 26-year-old woman's eyesight was still impaired after 7 months. Whiplash injuries may occur as the jumper is jolted on the bungee cord and in at least one case, this has led to quadriplegia secondary to a broken neck. Very serious injury can also occur if the jumper's neck or body gets entangled in the cord. More recently, carotid artery dissection leading to a type of stroke after bungee jumping has also been described.

In popular culture
In the film GoldenEye and its associated videogame, James Bond makes a jump over the edge of a dam in Russia (in reality the dam is in Switzerland: Verzasca Dam, and the jump was genuine, not an animated special effect). The jump in the dam later makes an appearance as a Roadblock task in the 14th season of the reality competition series The Amazing Race.

A fictional proto-bungee jump is a plot point in the Michael Chabon novel The Amazing Adventures of Kavalier and Clay.

In the film Selena, in which Jennifer Lopez plays Selena Quintanilla-Perez, her character is shown bungee jumping at a carnival. This actual event took place shortly before Selena's murder on 31 March 1995.

In Valiant (comics) #171 (January 8, 1966), the two boys from Worrag island in "The Wild Wonders" in a circus story, jump from high up and seem ready to crash to their deaths, but are stopped by elasticated ropes tied to an ankle of each one.

In the video game Aero the Acro-Bat, Aero will perform bungee jumping to obtain items like keys to open gates in a level.

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

title: Bungee jumping


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Water polo

Water polo is a competitive team sport played in water between two teams of seven players each. The game consists of four quarters in which the teams attempt to score goals by throwing the ball into the opposing team's goal. The team with the most goals at the end of the game wins the match. Each team is made up of six field players and one goalkeeper. Excluding the goalkeeper, players participate in both offensive and defensive roles. It is typically played in an all-deep pool where players cannot touch the bottom.

A game consists mainly of the players swimming to move about the pool, treading water (mainly using the eggbeater kick), passing the ball, and shooting at the goal. Teamwork, tactical thinking and awareness are also highly important aspects. Water polo is a highly physical and demanding sport and has frequently been cited as one of the most difficult to play.

Special equipment for water polo includes a water polo ball, a ball of varying colors which floats on the water; numbered and coloured caps; and two goals, which either float in the water or are attached to the sides of the pool.

The game is thought to have originated in Scotland in the mid-19th century; specifically, William Wilson is thought to have developed it in the 1870s as a sort of "water rugby". The game further developed with the formation of the London Water Polo League and has since expanded, becoming popular in parts of Europe, the United States, Brazil, China, Canada and Australia.

History

The history of water polo as a team sport began as a demonstration of strength and swimming skill in mid-19th century England and Scotland, where water sports and racing exhibitions were a feature of county fairs and festivals. Men's water polo was among the first team sports introduced at the modern Olympic games in 1900. The present-day game involves teams of seven players (plus up to six substitutes), with a water polo ball similar in size to a soccer ball but constructed of air-tight nylon.

One of the earliest recorded viewings of water polo was conducted at the 4th Open Air Fete of the London Swimming Club, held at the Crystal Palace, London on 15 September 1873. Another antecedent of the modern game of water polo was a game of water 'handball' played at Bournemouth on 13 July 1876. This was a game between 12 members of the Premier Rowing Club, with goals being marked by four flags placed in the water near to the midpoint of Bournemouth Pier. The game started at 6:00 pm and lasted for 15 minutes (when the ball burst) watched by a large crowd; with plans being made for play on a larger scale the following week.

The rules of water polo were originally developed in the late nineteenth century in Great Britain by William Wilson. Wilson is believed to have been the First Baths Master of the Arlington Baths Club in Glasgow. The first games of 'aquatic football' were played at the Arlington in the late 1800s (the club was founded in 1870), with a ball constructed of India rubber. This "water rugby" came to be called "water polo" based on the English pronunciation of the Balti word for ball, pulu. Early play allowed brute strength, wrestling and holding opposing players underwater to recover the ball. Players held underwater for lengthy periods usually surrendered possession. The goalie stood outside the playing area and defended the goal by jumping in on any opponent attempting to score by placing the ball on the deck.

Geography
Water polo is now popular in many countries around the world, notably Europe (particularly in Spain, France, Netherlands, Germany, Italy, Croatia, Hungary, Serbia, Montenegro, Greece and Romania), Australia, Brazil, Canada and the United States.

Some countries have two principal competitions: a more prestigious league which is typically a double round-robin tournament restricted to the elite clubs, and a cup which is a single-elimination tournament open to both the elite and lesser clubs.

Rules
The rules of water polo cover the play, procedures, equipment and officiating of water polo. These rules are similar throughout the world, although slight variations to the rules occur regionally and depending on the governing body. Governing bodies of water polo include FINA, the international governing organization; LEN, which governs international European matches; the NCAA, which governs collegiate matches in the United States; the NFHS, which governs high schools in the US, and the IOC, which governs Olympic events.

Positions
There are seven players in the water from each team at one time. There are six players that play out and one goalkeeper. Unlike most common team sports, there is little positional play; field players will often fill several positions throughout the game as situations demand. These positions usually consist of a center forward, a center back, the two wing players and the two drivers. Players who are skilled in all positions of offense or defense are called utility players. Utility players tend to come off of the bench, though this is not absolute. Certain body types are more suited for particular positions, and left-handed players are especially coveted on the right-hand side of the field, allowing teams to launch two-sided attacks.

Offense
The offensive positions include: one center forward (also called a "set", "hole-set", "center", "setter", "hole", or "2-meter man", located on or near the 2-meter, roughly in the center of the goal), two wings (located on or near the 2-meter, just outside of the goal posts, respectively), two drivers (also called "flats", located on or near the 5-meter, roughly at the goal posts, respectively), and one "point" (usually just behind the 5 meter, roughly in the center of the goal, respectively), positioned farthest from the goal. The wings, drivers and point are often called the perimeter players; while the hole-set directs play. There is a typical numbering system for these positions in U.S. NCAA men's division one polo. Beginning with the offensive wing to the opposing goalie's right side is called one. The flat in a counter clockwise from one is called two. Moving along in the same direction the point player is three, the next flat is four, the final wing is five, and the hole set is called six. Additionally, the position in which a player is can give advantages based on a player's handedness, to improve a shooting or passing angle (for example, the right wing is often left handed).

The center sets up in front of the opposing team's goalie and scores the most individually (especially during lower level play where flats do not have the required strength to effectively shoot from outside or to penetrate and then pass to teammates like the point guard in basketball, or center midfield player in soccer). The center's position nearest to the goal allows explosive shots from close-range.

Defense
Defensive positions are often the same, but just switched from offence to defence. For example, the centre forward or hole set, who directs the attack on offence, on defence is known as "hole D" (also known as set guard, hole guard, hole check, pit defence or two-metre defence), and guards the opposing team's centre forward (also called the hole). Defence can be played man-to-man or in zones, such as a 2–4 (four defenders along the goal line). It can also be played as a combination of the two in what is known as an "M drop" defence, in which the point defender moves away ("sloughs off") his man into a zone in order to better defend the centre position. In this defence, the two wing defenders split the area furthest from the goal, allowing them a clearer lane for the counter-attack if their team recovers the ball.

Goalkeeper

The goalkeeper has the main role in blocking shots against the goal as well as guiding and informing their defense of imposing threats and gaps in the defense. The goalkeeper usually begins the offensive play by passing the ball across the pool to an attacker. It is not unusual for a goalkeeper to make an assisting pass to a goal on a break away.

The goalkeeper is given several privileges above those of the other players, but only within the five-meter area in front of their own goal:
 * The ability to punch the ball with a clenched fist,
 * The ability to touch the ball with two hands.

In general, a foul that would cause an ejection of a field player might bring on a five-metre shot on the goalkeeper. Also, if a goalkeeper pushes the ball under water, the action will not be punished with a turnover like with field players, but with a penalty shot.

Player positioning
The most basic positional set up is known as a "3–3", so called because there are two lines in front of the opponent's goal. Another set up, used more by professional teams, is known as an "arc", "umbrella", or "mushroom"; perimeter players form the shape of an arc around the goal, with the hole set as the handle or stalk. Yet another option for offensive set is called a 4–2 or double hole; there are two center forward offensive players in front of the goal. Double hole is most often used in "man up" situations, or when the defense has only one skilled "hole D", or to draw in a defender and then pass out to a perimeter player for a shot ("kick out").

Another, albeit less common offense, is the "motion c", sometimes nicknamed "washing machine offence", in which two "weak-side" (to the right of the goal for right-handed players) perimeter players set up as a wing and a flat. The remaining four players swim in square pattern in which a player swims from the point to the hole and then out to the strong side wing. The wing moves to the flat and the flat to the point. The weak side wing and flat then control the tempo of play and try to make passes into the player driving towards the centre forward who can then either shoot or pass. This form of offence is used when no dominate hole set is available, or the hole defence is too strong. It is also seen much more often in women's water polo where teams may lack a player of sufficient size or strength to set up in the centre forward. The best advantage to this system is it makes man-coverage much more difficult for the defender and allows the offence to control the game tempo better once the players are "set up". The main drawback is this constant motion can be very tiring as well as somewhat predictable as to where the next pass is going to go.

Advancing the ball
When the offence takes possession of the ball, the strategy is to advance the ball down the field of play and to score a goal. Players can move the ball by throwing it to a teammate or swimming with the ball in front of them (dribbling). If an attacker uses their arm to push away a defending player and free up space for a pass or shot, the referee will rule a turnover and the defence will take possession of the ball. If an attacker advances inside the 2-metre line without the ball or before the ball is inside the 2-metre area, they are ruled offside and the ball is turned over to the defence. This is often overlooked if the attacker is well to the side of the pool or when the ball is at the other side of the pool.

Setting the ball
The key to the offence is to accurately pass (or "set") the ball into the centre forward or hole set, positioned directly in front of the goal ("the hole"). Any field player may throw the hole set a "wet pass". A wet pass is one that hits the water just outside the hole set's reach. A dry pass may also be used. This is where the hole set receives the ball directly in his hand and then attempts a shot at the cage. This pass is much more difficult because if the pass is not properly caught, the officials will be likely to call an offensive foul resulting in a change of ball possession. The hole set attempts to take possession of the ball [after a wet pass], to shoot at the goal, or to draw a foul from his defender. A minor foul is called if his defender (called the "hole D") attempts to impede movement before the hole set has possession. The referee indicates the foul with one short whistle blow and points one hand to the spot of the foul and the other hand in the direction of the attack of the team to whom the free throw has been awarded. The hole set then has a "reasonable amount of time" (typically about three seconds; there is no FINA rule on this issue) to re-commence play by making a free pass to one of the other players. The defensive team cannot hinder the hole set until the free throw has been taken, but the hole set cannot shoot a goal once the foul has been awarded until the ball has been played by at least one other player. If the hole set attempts a goal without the free throw, the goal is not counted and the defence takes possession of the ball, unless the shot is made outside the 5-metre line. As soon as the hole set has a free pass, the other attacking players attempt to swim (or drive) away from their defenders towards the goal. The players at the flat position will attempt to set a screen (also known as a pick) for the driver. If a driver gets free from a defender, the player calls for the pass from the hole set and attempts a shot at the goal.

Man-Up (5 on 6)
If a defender interferes with a free throw, holds or sinks an attacker who is not in possession or splashes water into the face of an opponent, the defensive player is excluded from the game for twenty seconds, known as a 'kick out' or an ejection. The attacking team typically positions 4 players on the 2 metre line, and 2 players on 5 metre line (4–2), passing the ball around until an open player attempts a shot. Other formations include a 3–3 (two lines of three attackers each) or arc (attackers make an arc in front of the goal and one offensive player sits in the 'hole' or 'pit' in front of the goal). The five defending players try to pressure the attackers, block shots and prevent a goal being scored for the 20 seconds while they are a player down. The other defenders can only block the ball with one hand to help the goalkeeper. The defensive player is allowed to return immediately if the offence scores, or if the defence recovers the ball before the twenty seconds expires.

Defense strategy
On defence, the players work to regain possession of the ball and to prevent a goal in their own net. The defence attempts to knock away or steal the ball from the offense or to commit a foul in order to stop an offensive player from taking a goal shot. The defender attempts to stay between the attacker and the goal, a position known as inside water.

Goalkeeper
Even with good backup from the rest of the defenders, stopping attacks can prove very difficult if the goalkeeper remains in the middle of the goal. The most defensible position is along a semicircular line connecting the goalposts and extending out in the centre. Depending on the ball carrier's location, the goalkeeper is positioned along that semicircle roughly a metre out of the goal to reduce the attacker's shooting angle. The goalkeeper stops using their hands to tread water once the opponent enters at about the 7-metre mark and starts to lift their upper body using the eggbeater technique to prepare to block the shot. Finally, the goalkeeper tries to block the ball down, which is often hard for the longer reaches, but prevents an offensive rebound and second shot. As is the case with other defensive players, a goalkeeper who aggressively fouls an attacker in position to score can be charged with a penalty shot for the other team. The goalkeeper can also be ejected for twenty seconds if a major foul is committed. Also, inside the five metre mark, the goalie can swing at the ball with a closed fist without being penalised.

Advantage rule
If an offensive player, such as the centre forward, has possession of the ball in front of the goal, the defensive player tries to steal the ball or to keep the centre from shooting or passing. If the defender cannot achieve these aims, he may commit a foul intentionally. The hole set then is given a free throw but must pass off the ball to another offensive player, rather than making a direct shot at the goal. Defensive perimeter players may also intentionally cause a minor foul and then move toward the goal, away from their attacker, who must take a free throw. This technique, called sloughing, allows the defense an opportunity to double-team the hole set and possibly steal the inbound pass. The referee may refrain from declaring a foul, if in his judgment this would give the advantage to the offender's team. This is known as the Advantage Rule.

Injuries
Water polo is a contact sport, with little protective gear besides swimsuits and caps with ear protectors, and thus injuries are common. Among the most frequent serious injuries are those affecting the head and shoulders. Those induced to the head are usually caused by elbows or the ball itself, while shoulder injuries are a result of grabbing and pushing while throwing the ball or simply of repetitive overexertion of joints and muscles when taking hard shots. The hands and fingers are vulnerable areas, due to contact when opponents attempt to steal the ball, or when players block shots. Other injuries take place underwater, such as leg and groin injuries, as many actions cannot be seen from above the surface and not much padding is used to protect the players.

Sunburn is a common minor injury in outdoor matches. Players often don't apply sunscreen as it makes their skin, and hence the ball, slippery; FINA and most state governing bodies forbid the use of copious sunscreen to make the body harder for the opposing team to grip.

Variations
Inner tube water polo is a style of water polo in which players, excluding the goalkeeper, are required to float in inner tubes. By floating in an inner tube players expend less energy than traditional water polo players, not having to tread water. This allows casual players to enjoy water polo without undertaking the intense conditioning required for conventional water polo.

Surf polo, another variation of water polo, is played on surfboards. First played on the beaches of Waikiki in Hawaii in the 1930s and 1940s, it is credited to Louis Kahanamoku, Duke Kahanamoku's brother.

Canoe polo or kayak polo is one of the eight disciplines of canoeing pursued in the UK, known simply as "polo" by its aficionados. Polo combines paddling and ball handling skills with a contact team game, where tactics and positional play are as important as the speed and fitness of the individual athletes.

Flippa ball is a precursor variant intended for younger and beginner players to learn the basics of polo. It is played in shallow water and permits touching the bottom of the pool. Players rotate positions after each score.

Water polo equipment
Little player equipment is needed to play water polo. Items required in water polo include:
 * Ball: A water polo ball is constructed of waterproof material to allow it to float on the water. The cover is textured to give players additional grip. The size of the ball is different for men's, women's and junior games.
 * Caps: A water polo cap is used to protect the players' heads and ears, and to make them identifiable from afar. Home team field players wear numbered dark-colored caps; Visiting team field players wear numbered white caps. Both starting goalkeepers wear red caps (sometimes quartered), numbered "1" (substitute goalies' caps are numbered either "13" for FINA international play or "15" for NCAA play) Caps are fitted with ear protectors.

 * Goals: Two goals are needed in order to play water polo. These can either be put on the side of the pool, or in the pool using floaters.
 * Mouthguard: A mouthguard is not mandatory in most tournaments but is recommended.
 * Swimwear: Male water polo players wear either swim briefs or jammers (thigh-length trunks). Female players must wear a one-piece swimsuit. Suit-grabbing fouls are common, so players often wear tight-fitting suits, and may layer on several suits at a time for additional security. Many swimwear labels also sell specialized water polo suits that feature reinforced stitching and tougher fabric. Female water polo suits are generally one-piece outfits which do not have open backs, but zip securely up the back so as to not have straps that can be easily grabbed.

Summer Olympics
Men's water polo at the Olympics was the first team sport introduced at the 1900 games, along with cricket, rugby, football, polo (with horses), rowing and tug of war. Women's water polo became an Olympic sport at the 2000 Sydney Olympic Games after political protests from the Australian women's team.

One of the most historically known matches often referred to as the Blood in the Water match, was a 1956 Summer Olympics semi-final match between Hungary and the Soviet Union, played in Melbourne. As the athletes left for the games, the Hungarian revolution began, and the Soviet army crushed the uprising. The Hungarians defeated the Soviets 4–0 before the game was called off in the final minute to prevent angry Hungarians in the crowd reacting to Valentin Prokopov punching Ervin Zádor.

Other tournaments
Every 2 to 4 years since 1973, a men's Water Polo World Championship is organized within the FINA World Aquatics Championships. Women's water polo was added in 1986. A second tournament series, the FINA Water Polo World Cup, has been held every other year since 1979. In 2002, FINA organised the sport's first international league, the FINA Water Polo World League.

There is also a European Water Polo Championship that is held every other year.

Professional water polo is played in many Southern and Eastern European countries like Croatia, Greece, Hungary, Italy, Montenegro, Russia, Serbia, Spain, etc. with the LEN Euroleague tournament played amongst the best teams.

There is also a World Club Water Polo Challenge.

title: Water polo


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Atlas (moon)

Atlas is an inner satellite of Saturn which was discovered by Richard Terrile in 1980 from Voyager photos and was designated S/1980 S 28. In 1983 it was officially named after Atlas of Greek mythology, because it "holds the rings on its shoulders" like the Titan Atlas held the sky up above the Earth. It is also designated Saturn XV.

Atlas is the closest satellite to the sharp outer edge of the A ring, and was long thought to be a shepherd satellite for this ring. However, now it is known that the outer edge of the ring is instead maintained by a 7:6 orbital resonance with the larger but more distant moons Janus and Epimetheus. In 2004 a faint, thin ring, temporarily designated R/2004 S 1, was discovered in the Atlantean orbit.

High-resolution images taken in June 2005 by Cassini revealed Atlas to have a roughly spherical centre surrounded by a large, smooth equatorial ridge. The most likely explanation for this unusual and prominent structure is that ring material swept up by the moon accumulates on the moon, with a strong preference for the equator due to the ring's thinness. The size of the equatorial ridge is comparable with the expected Roche lobe of the moon, which means that for any additional particles impacting the equator, the centrifugal force will nearly overcome Atlas's tiny gravity, and they will probably be lost.

Atlas is significantly perturbed by Prometheus and to a lesser degree by Pandora, leading to excursions in longitude of up to 600 km (~0.25°) away from the precessing Keplerian orbit with a rough period of about 3 years. Because the orbits of Prometheus and Pandora are chaotic, it is suspected that Atlas's may be as well.

title: Atlas (moon)


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Atlas (mythology)

In Greek mythology, Atlas (Ἄτλας, Átlās) is a Titan condemned to hold up the heavens or sky for eternity after the Titanomachy. Atlas also plays a role in the myths of two of the greatest Greek heroes: Heracles (Hercules in Roman mythology) and Perseus. According to the ancient Greek poet Hesiod, Atlas stood at the ends of the earth in the extreme west. Later, he became commonly identified with the Atlas Mountains in northwest Africa and was said to be the first King of Mauretania (modern-day Morocco and, much later, including west Algeria, not to be confused with the modern-day country of Mauritania). Atlas was said to have been skilled in philosophy, mathematics, and astronomy. In antiquity, he was credited with inventing the first celestial sphere. In some texts, he is even credited with the invention of astronomy itself.

Atlas was the son of the Titan Iapetus and the Oceanid Asia or Clymene. He was a brother of Epimetheus and Prometheus. He had many children, mostly daughters, the Hesperides, the Hyades, the Pleiades, and the nymph Calypso who lived on the island Ogygia.

The term "atlas" has been used to describe a collection of maps since the 16th century when Flemish geographer Gerardus Mercator published his work in honour of the mythological Titan.

The "Atlantic Ocean" is derived from "Sea of Atlas". The name of Atlantis mentioned in Plato's Timaeus' dialogue derives from "Atlantis nesos" (Ἀτλαντὶς νῆσος), literally meaning "Atlas's Island".

Etymology
The etymology of the name Atlas is uncertain. Virgil took pleasure in translating etymologies of Greek names by combining them with adjectives that explained them: for Atlas his adjective is durus, "hard, enduring", which suggested to George Doig that Virgil was aware of the Greek τλῆναι "to endure"; Doig offers the further possibility that Virgil was aware of Strabo's remark that the native North African name for this mountain was Douris. Since the Atlas mountains rise in the region inhabited by Berbers, it has been suggested that the name might be taken from one of the Berber languages, specifically from the word ádrār "mountain".

Traditionally historical linguists etymologize the Ancient Greek word Ἄτλας (genitive: Ἄτλαντος) as comprised from copulative α- and the Proto-Indo-European root 'to uphold, support' (whence also τλῆναι), and which was later reshaped to an nt-stem. However, Robert S. P. Beekes argues that it cannot be expected that this ancient Titan carries an Indo-European name, and he suggests instead that the word is of Pre-Greek origin, as such words often end in -ant.

War and punishment
Atlas and his brother Menoetius sided with the Titans in their war against the Olympians, the Titanomachy. When the Titans were defeated, many of them (including Menoetius) were confined to Tartarus, but Zeus condemned Atlas to stand at the western edge of the earth and hold up the sky on his shoulders. Thus, he was Atlas Telamon, "enduring Atlas", and became a doublet of Coeus, the embodiment of the celestial axis around which the heavens revolve.

A common misconception today is that Atlas was forced to hold the Earth on his shoulders, but Classical art shows Atlas holding the celestial spheres, not the terrestrial globe; the solidity of the marble globe borne by the renowned Farnese Atlas may have aided the conflation, reinforced in the 16th century by the developing usage of atlas to describe a corpus of terrestrial maps.

Encounter with Perseus
The Greek poet Polyidus c. 398 BC tells a tale of Atlas, then a shepherd, encountering Perseus who turned him to stone. Ovid later gives a more detailed account of the incident, combining it with the myth of Heracles. In this account Atlas is not a shepherd, but a king. According to Ovid, Perseus arrives in Atlas's Kingdom and asks for shelter, declaring he is a son of Zeus. Atlas, fearful of a prophecy that warned of a son of Zeus stealing his golden apples from his orchard, refuses Perseus hospitality. In this account, Atlas is turned not just into stone by Perseus, but an entire mountain range: Atlas's head the peak, his shoulders ridges and his hair woods. The prophecy did not relate to Perseus stealing the golden apples but to Heracles, another son of Zeus, and Perseus's great-grandson.

Encounter with Heracles
One of the Twelve Labours of the hero Heracles was to fetch some of the golden apples that grow in Hera's garden, tended by Atlas's reputed daughters, the Hesperides (which were also called the Atlantides), and guarded by the dragon Ladon. Heracles went to Atlas and offered to hold up the heavens while Atlas got the apples from his daughters.

Upon his return with the apples, however, Atlas attempted to trick Heracles into carrying the sky permanently by offering to deliver the apples himself, as anyone who purposely took the burden must carry it forever, or until someone else took it away. Heracles, suspecting Atlas did not intend to return, pretended to agree to Atlas's offer, asking only that Atlas take the sky again for a few minutes so Heracles could rearrange his cloak as padding on his shoulders. When Atlas set down the apples and took the heavens upon his shoulders again, Heracles took the apples and ran away.

In some versions, Heracles instead built the two great Pillars of Hercules to hold the sky away from the earth, liberating Atlas much as he liberated Prometheus.

Other mythological characters named Atlas
Besides the Titan, there are other mythological characters who were also called Atlas:

King of Atlantis

According to Plato, the first king of Atlantis was also named Atlas, but that Atlas was a son of Poseidon and the mortal woman Cleito. The works of Eusebius and Diodorus also give an Atlantean account of Atlas. In these accounts, Atlas' father was Uranus and his mother was Gaia. His grandfather was Elium "King of Phoenicia" who lived in Byblos with his wife Beruth. Atlas was raised by his sister, Basilia.

King of Mauretania
Atlas was also a legendary king of Mauretania, the land of the Mauri in antiquity roughly corresponding with modern Morocco and Algeria. In the 16th century, Gerardus Mercator put together the first collection of maps to be called an "Atlas" and devoted his book to the "King of Mauretania".

Atlas became associated with Northwest Africa over time. He had been connected with the Hesperides, or "Nymphs", which guarded the golden apples, and Gorgons both of which were said to live beyond Ocean in the extreme west of the world since Hesiod's Theogony. Diodorus and Palaephatus mention that the Gorgons lived in the Gorgades, islands in the Aethiopian Sea. The main island was called Cerna, and modern-day arguments have been advanced that these islands may correspond to Cape Verde due to Phoenician exploration. The Northwest Africa region emerged as the canonical home of the King via separate sources. In particular, according to Ovid, after Perseus turns Atlas into a mountain range, he flies over Aethiopia, the blood of Medusa's head giving rise to Libyan snakes. By the time of the Roman Empire, the habit of associating Atlas's home to a chain of mountains, the Atlas Mountains, which were near Mauretania and Numidia, was firmly entrenched.

Other
The identifying name Aril is inscribed on two 5th-century BC Etruscan bronze items: a mirror from Vulci and a ring from an unknown site. Both objects depict the encounter with Atlas of Hercle—the Etruscan Heracles—identified by the inscription; they represent rare instances where a figure from Greek mythology was imported into Etruscan mythology, but the name was not. The Etruscan name Aril is etymologically independent.

Genealogy
Sources describe Atlas as the father, by different goddesses, of numerous children, mostly daughters. Some of these are assigned conflicting or overlapping identities or parentage in different sources.

 * By Hesperis:
 * The Hesperides
 * By Pleione (or Aethra ):
 * The Hyades
 * A son, Hyas
 * The Pleiades
 * By one or more unspecified goddesses:
 * Calypso
 * Dione
 * Maera

Hyginus, in his Fabulae, adds an older Atlas who is the son of Aether and Gaia.

Cultural influence
Atlas' best-known cultural association is in cartography. The first publisher to associate the Titan Atlas with a group of maps was the print-seller Antonio Lafreri, who included a depiction of the Titan on the engraved title-page he applied to his ad hoc assemblages of maps, Tavole Moderne di Geografia de la Maggior parte del Mondo di Diversi Autori (1572). However, Lafreri did not use the word "Atlas" in the title of his work; this was an innovation of Gerardus Mercator, who named his work Atlas Sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati (1585 – 1595), using the word Atlas as a dedication specifically to honour the Titan Atlas, in his capacity as King of Mauretania, a learned philosopher, mathematician, and astronomer.

In psychology, Atlas is used metaphorically to describe the personality of someone whose childhood was characterized by excessive responsibilities.

Ayn Rand's political dystopian novel Atlas Shrugged (1957) references the popular misconception of Atlas holding up the entire world on his back by comparing the capitalist and intellectual class as being "modern Atlases" who hold the modern world up at great expense to themselves.

title: Atlas (mythology)


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Atlas

An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a continent or region of Earth.

Atlases have traditionally been bound into book form, but today, many atlases are in multimedia formats. In addition to presenting geographical features and political boundaries, many atlases often feature geopolitical, social, religious, and economic statistics. They also have information about the map and places in it.

Etymology
The use of the word "atlas" in a geographical context dates from 1595 when the German-Flemish geographer Gerardus Mercator published Atlas Sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura ("Atlas or cosmographical meditations upon the creation of the universe and the universe as created"). This title provides Mercator's definition of the word as a description of the creation and form of the whole universe, not simply as a collection of maps. The volume that was published posthumously one year after his death is a wide-ranging text but, as the editions evolved, it became simply a collection of maps and it is in that sense that the word was used from the middle of the 17th century. The neologism coined by Mercator was a mark of his respect for the Titan Atlas, the "King of Mauretania", whom he considered to be the first great geographer.

History of atlases
The first work that contained systematically arranged maps of uniform size representing the first modern atlas was prepared by Italian cartographer Pietro Coppo in the early 16th century; however, it was not published at that time, so it is conventionally not considered the first atlas. Rather, that title is awarded to the collection of maps Theatrum Orbis Terrarum by the Brabantian cartographer Abraham Ortelius printed in 1570.

Atlases published nowadays are quite different from those published in the 16th–19th centuries. Unlike today, most atlases were not bound and ready for the customer to buy, but their possible components were shelved separately. The client could select the contents to their liking, and have the maps coloured/gilded or not. The atlas was then bound. Thus, early printed atlases with the same title page can be different in contents.

States began producing national atlases in the 19th century.

Types of atlases
A travel atlas is made for easy use during travel, and often has spiral bindings, so it may be folded flat. National atlases in Europe are typically printed at a scale of 1:250,000 to 1:500,000; city atlases are 1:20,000 to 1:25,000, doubling for the central area (for example, Geographers' A-Z Map Company's A–Z atlas of London is 1:22,000 for Greater London and 1:11,000 for Central London). A travel atlas may also be referred to as a road map.

A desk atlas is made similar to a reference book. It may be in hardback or paperback form.

There are atlases of the other planets (and their satellites) in the Solar System.

Atlases of anatomy exist, mapping out organs of the human body or other organisms.

Selected atlases
Some cartographically or commercially important atlases are:

17th century and earlier:
 * Theatrum Orbis Terrarum 1570 atlas by Abraham Ortelius
 * Atlas Sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura (Mercator, Duisburg, in present-day Germany, 1595)
 * Atlas Novus (Joan Blaeu, Netherlands, 1635–1658)
 * Atlas Maior (Blaeu, Netherlands, 1662–1667)
 * Cartes générales de toutes les parties du monde (France, 1658–1676)
 * Dell'Arcano del Mare (Robert Dudley, England/Italy, 1645–1661)
 * Piri Reis map (Piri Reis, Ottoman Empire, 1570–1612)
 * Theatrum Orbis Terrarum (Ortelius, Netherlands, 1570–1612)
 * Klencke Atlas (1660; one of the world's largest books)
 * Britannia (1675), John Ogilby (1600–1676), first to be printed at a specific scale (1:63,360 or one inch to one mile

18th century
 * Atlas Nouveau (Amsterdam, 1742)
 * Britannia Depicta (London, 1720)
 * Cary's New and Correct English Atlas (London, 1787)

19th century:
 * Andrees Allgemeiner Handatlas (Germany, 1881–1939; in the UK as Times Atlas of the World, 1895)
 * Rand McNally Atlas (United States, 1881–present)
 * Stielers Handatlas (Germany, 1817–1944)
 * Times Atlas of the World (United Kingdom, 1895–present)

20th century:
 * Atlante Internazionale del Touring Club Italiano (Italy, 1927–1978)
 * Atlas Linguisticus (Austria, 1934)
 * Atlas Mira (Soviet Union/Russia, 1937–present)
 * Geographers' A–Z Street Atlas (United Kingdom, 1938–present)
 * Gran Atlas Aguilar (Spain, 1969/1970)
 * The Historical Atlas of China (China)
 * National Geographic Atlas of the World (United States, 1963–present)
 * Pergamon World Atlas (1962/1968)

21st century:
 * North American Environmental Atlas

title: Atlas


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Atlas (anatomy)

In anatomy, the atlas (C1) is the most superior (first) cervical vertebra of the spine and is located in the neck.

The bone is named for Atlas of Greek mythology, just as Atlas bore the weight of the heavens, the first cervical vertebra supports the head. However, the term atlas was first used by the ancient Romans for the seventh cervical vertebra (C7) due to its suitability for supporting burdens. In Greek mythology, Atlas was condemned to bear the weight of the heavens as punishment for rebelling against Zeus. Ancient depictions of Atlas show the globe of the heavens resting at the base of his neck, on C7. Sometime around 1522, anatomists decided to call the first cervical vertebra the atlas. Scholars believe that by switching the designation atlas from the seventh to the first cervical vertebra Renaissance anatomists were commenting that the point of man’s burden had shifted from his shoulders to his head--that man’s true burden was not a physical load, but rather, his mind.

The atlas is the topmost vertebra and the axis (the vertebra below it) forms the joint connecting the skull and spine. The atlas and axis are specialized to allow a greater range of motion than normal vertebrae. They are responsible for the nodding and rotation movements of the head.

The atlanto-occipital joint allows the head to nod up and down on the vertebral column. The dens acts as a pivot that allows the atlas and attached head to rotate on the axis, side to side.

The atlas's chief peculiarity is that it has no body, which has fused with the next vertebra. It is ring-like and consists of an anterior and a posterior arch and two lateral masses.

The atlas and axis are important neurologically because the brainstem extends down to the axis.

Anterior arch
The anterior arch forms about one-fifth of the ring: its anterior surface is convex, and presents at its center the anterior tubercle for the attachment of the longus colli muscles and the anterior longitudinal ligament; posteriorly it is concave, and marked by a smooth, oval or circular facet (fovea dentis), for articulation with the odontoid process (dens) of the axis.

The upper and lower borders respectively give attachment to the anterior atlantooccipital membrane and the anterior atlantoaxial ligament; the former connects it with the occipital bone above, and the latter with the axis below.

Posterior arch

The posterior arch forms about two-fifths of the circumference of the ring: it ends behind in the posterior tubercle, which is the rudiment of a spinous process and gives origin to the recti capitis posteriores minores and the ligamentum nuchae. The diminutive size of this process prevents any interference with the movements between the atlas and the skull.

The posterior part of the arch presents above and behind a rounded edge for the attachment of the posterior atlantooccipital membrane, while immediately behind each superior articular process is the superior vertebral notch (sulcus arteriae vertebralis). This is a groove that is sometimes converted into a foramen by ossification of the posterior atlantooccipital membrane to create a delicate bony spiculum which arches backward from the posterior end of the superior articular process. This anatomical variant is known as an arcuate foramen.

This groove transmits the vertebral artery, which, after ascending through the foramen in the transverse process, winds around the lateral mass in a direction backward and medially to enter the vertebrobasilar circulation through the foramen magnum; it also transmits the suboccipital nerve (first spinal nerve)

On the under surface of the posterior arch, behind the inferior articular facets, are two shallow grooves, the inferior vertebral notches. The lower border gives attachment to the posterior atlantoaxial ligament, which connects it with the axis.

Lateral masses
The lateral masses are the most bulky and solid parts of the atlas, in order to support the weight of the head.

Each carries two articular facets, a superior and an inferior.

 * The superior facets are of large size, oval, concave, and approach each other in front, but diverge behind: they are directed upward, medially, and a little backward, each forming a cup for the corresponding condyle of the occipital bone, and are admirably adapted to the nodding movements of the head. Not infrequently they are partially subdivided by indentations which encroach upon their margins.
 * The inferior articular facets are circular in form, flattened or slightly convex and directed downward and medially, articulating with the axis, and permitting the rotatory movements of the head.

Vertebral foramen
Just below the medial margin of each superior facet is a small tubercle, for the attachment of the transverse atlantal ligament which stretches across the ring of the atlas and divides the vertebral foramen into two unequal parts:

 * the anterior or smaller receiving the odontoid process of the axis
 * the posterior transmitting the spinal cord (medulla spinalis) and its membranes

This part of the vertebral canal is of considerable size, much greater than is required for the accommodation of the spinal cord.

Transverse processes
The transverse processes are large; they project laterally and downward from the lateral masses, and serve for the attachment of muscles which assist in rotating the head. They are long, and their anterior and posterior tubercles are fused into one mass; the foramen transversarium is directed from below, upward and backward.

Development
The atlas is usually ossified from three centers.

Of these, one appears in each lateral mass about the seventh week of fetal life, and extends backward; at birth, these portions of bone are separated from one another behind by a narrow interval filled with cartilage.

Between the third and fourth years they unite either directly or through the medium of a separate center developed in the cartilage.

At birth, the anterior arch consists of cartilage; in this a separate center appears about the end of the first year after birth, and joins the lateral masses from the sixth to the eighth year.

The lines of union extend across the anterior portions of the superior articular facets.

Occasionally there is no separate center, the anterior arch being formed by the forward extension and ultimate junction of the two lateral masses; sometimes this arch is ossified from two centers, one on either side of the middle line.

Variations
Accessory transverse foramen of the atlas is present in 1.4–12.5% across the population.

Foramen arcual] or a bony bridge above the vertebral artery on the posterior arch of the atlas may be present. This foramen has an overall prevalence of 9.1%. Arch defects refer to the condition where a gap or cleft exists at the anterior arch or posterior arch of the atlas. The prevalence of the posterior arch defect and anterior arch defect was 0.95% and 0.087%, respectively. The anterior arch defect may be presented along with posterior arch defect, a condition known as combined arch defect or bipartite atlas.

Transverse processes
Upper surface:

 * rectus capitis anterior – occipital bone (inferior surface of the base)
 * rectus capitis lateralis – occipital bone (beneath the jugular process)
 * obliquus capitis superior – occipital bone (between the superior and inferior nuchal lines)

Interior and dorsal part:

 * obliquus capitis inferior – spinous process of the axis

Lower surface:

 * splenius cervicis (part) – spinous processes of T02–T05
 * levator scapulae (part) – superior part of medial border of the scapula
 * intertransversarius posterior cervicis – transverse process of the axis (posterior tubercle)
 * intertransversarius anterior cervicis – transverse process of the axis (anterior tubercle)

Posterior tubercle
Upper surface:

 * rectus capitis posterior minor – occipital bone (medial part of the interior nuchal line, and the surface between it and the foramen magnum)

Lower surface:

 * interspinalis cervicis – spinous process of the axis

Anterior arch

 * longus colli (superior oblique) – transverse processes of C03–C05.

Clinical significance
There are 5 types of C1 fractures referred to as the Levine Classification of Atlas Fractures

Type 1: Isolated bony apophysis (transverse process fracture

Type 2: Isolated posterior arch fractures

Type 3: Isolated anterior arch fracture

Type 4: Comminuted fracture of the lateral mass

Type 5: Bilateral burst fracture (AKA Jefferson Fracture)

A break in the first vertebra is referred to as a Jefferson fracture.

Craniocervical junction misalignment is also suspected as a factor in neurodegenerative diseases where altered CSF flow plays a part in the pathological process.

Hyperextension (Whiplash) Injury

A rear-end traffic collision or a poorly performed rugby tackle can both result in the head being whipped back on the shoulders, causing whiplash. In minor cases, the anterior longitudinal ligament of the spine is damaged which is acutely painful for the patient.

In more severe cases, fractures can occur to any of the cervical vertebrae as they are suddenly compressed by rapid deceleration. Again, since the vertebral foramen is large there is less chance of spinal cord involvement.

The worst-case scenario for these injuries is that dislocation or subluxation of the cervical vertebrae occurs. This often happens at the C2 level, where the body of C2 moves anteriorly with respect to C3. Such an injury may well lead to spinal cord involvement, and as a consequence quadriplegia or death may occur. More commonly, subluxation occurs at the C6/C7 level (50% of cases).

title: Atlas (anatomy)


text:
Atlas Mountains

The Atlas Mountains are a mountain range in the Maghreb in North Africa. It separates the Sahara Desert from the Mediterranean Sea and the Atlantic Ocean; the name "Atlantic" is derived from the mountain range, which stretches around 2500 km through Morocco, Algeria and Tunisia. The range's highest peak is Toubkal, which is in central Morocco, with an elevation of 4167 m. The Atlas Mountains are primarily inhabited by Berber populations.

The terms for 'mountain' are Adrar and adras in some Berber languages, and these terms are believed to be cognates of the toponym Atlas. The mountains are home to a number of animals and plants which are mostly found within Africa but some of which can be found in Europe. Many of these species are endangered and a few are already extinct. The weather is generally cool but summers are sunny, and the average temperature there is 25 °C. The Atlas Mountains have earned a reputation as a trekkers’ den, attracting adventurers year-round.

Geology

The basement rock of most of Africa was formed during the Precambrian supereon and is much older than the Atlas Mountains lying on the continent. The Atlas was formed during three subsequent phases of Earth's geology.

The first tectonic deformation phase involves only the Anti-Atlas, which was formed in the Paleozoic Era (~300 million years ago) as the result of continental collisions. North America, Europe and Africa were connected millions of years ago. The Anti-Atlas Mountains are believed to have originally been formed as part of the Alleghenian orogeny. These mountains were formed when Africa and America collided and were once a chain rivaling today's Himalayas. Today, the remains of this chain can be seen in the Fall Line region in the Eastern United States. Some remnants can also be found in the later formed Appalachians in North America.

A second phase took place during the Mesozoic Era (before ~66 My). It consisted of a widespread extension of the Earth's crust that rifted and separated the continents mentioned above. This extension was responsible for the formation of many thick intracontinental sedimentary basins including the present Atlas. Most of the rocks forming the surface of the present High Atlas were deposited under the ocean at that time.

In the Paleogene and Neogene Periods (~66 million to ~1.8 million years ago), the mountain chains that today constitute the Atlas were uplifted, as the land masses of Europe and Africa collided at the southern end of the Iberian Peninsula. Such convergent tectonic boundaries occur where two plates slide towards each other forming a subduction zone (if one plate moves underneath the other), and/or a continental collision (when the two plates contain continental crust). In the case of the Africa-Europe collision, it is clear that tectonic convergence is partially responsible for the formation of the High Atlas, as well as for the closure of the Strait of Gibraltar and the formation of the Alps and the Pyrenees.

However, there is a lack of evidence for the nature of the subduction in the Atlas region, or for the thickening of the Earth's crust generally associated with continental collisions. One of the most striking features of the Atlas to geologists is the relative small amount of crustal thickening and tectonic shortening despite the important altitude of the mountain range. Recent studies suggest that deep processes rooted in the Earth's mantle may have contributed to the uplift of the High and Middle Atlas.

Natural resources
The Atlas are rich in natural resources. There are deposits of iron ore, lead ore, copper, silver, mercury, rock salt, phosphate, marble, anthracite coal and natural gas among other resources.

Subranges
The range can be divided into four general regions:

 * Anti-Atlas, High Atlas and Middle Atlas (Morocco)
 * Tell Atlas (Morocco, Algeria, Tunisia)
 * Aurès Mountains (Algeria, Tunisia)
 * Saharan Atlas (Algeria)

Anti-Atlas
The Anti-Atlas extends from the Atlantic Ocean in the southwest of Morocco toward the northeast to the heights of Ouarzazate and further east to the city of Tafilalt (altogether a distance of approximately 500 km). In the south it borders the Sahara. The easternmost point of the anti-Atlas is the Jbel Saghro range and its northern boundary is flanked by sections of the High Atlas range. It includes the Djebel Siroua, a massif of volcanic origin with the highest summit of the range at 3,304 m. The Jebel Bani is a much lower range running along the southern side of the Anti Atlas.

High Atlas

The High Atlas in central Morocco rises in the west at the Atlantic coast and stretches in an eastern direction to the Moroccan-Algerian border. It has several peaks over 4000 m, including the highest summit in North Africa, Toubkal (4167 m), and further east Ighil m'Goun (4071 m), the second major summit of the range. At the Atlantic and to the southwest, the range drops abruptly and makes a transition to the coast and the Anti-Atlas range. To the north, in the direction of Marrakesh, the range descends less abruptly. On the heights of Ouarzazate the massif is cut through by the Draa Valley which opens southward. It is mainly inhabited by Berber people, who live in small villages and cultivate the high plains of the Ourika Valley. Near Barrage Cavagnac there is a hydroelectric dam that has created the artificial lake Lalla Takerkoust. The lake serves also as a source for fish for the local fishermen.

The largest villages and towns of the area are Ouarzazate, Tahannaout, Amizmiz, Imlil, Tin Mal and Ijoukak.

Middle Atlas
The Middle Atlas is completely in Morocco and is the northernmost of its three main Atlas ranges. The range lies north of the High Atlas, separated by the Moulouya and Oum Er-Rbia rivers, and south of the Rif mountains, separated by the Sebou River. To the west are the main coastal plains of Morocco with many of the major cities and, to the east, the high barren plateau that lies between the Saharan and Tell Atlas. The high point of the range is the jbel Bou Naceur (3340 m). The Middle Atlas experiences more rain than the ranges to the south, making it an important water catchment for the coastal plains and important for biodiversity. It is home to the majority of the world's population of Barbary macaque.

Saharan Atlas
The Saharan Atlas of Algeria runs east of the High Atlas, crossing Algeria from the Moroccan border and into Tunisia. The Aures Mountains are often presented as being the easternmost part of the Saharan Atlas. Though not as high as the High Atlas, they reach similar altitudes as the Tell Atlas range that runs to the north of them and closer to the coast. The highest peak in the range, outside of the Aures Mountains, is the 2236 m high Djebel Aissa. They mark the northern edge of the Sahara Desert. The mountains see some rainfall and are better suited to agriculture than the plateau region to the north. Today, most of the population of the region are Berbers (Imazighen).

Tell Atlas

The Tell Atlas is a mountain chain over 1500 km in length, belonging to the Atlas mountain ranges and stretching from Eastern Morocco to Tunisia, and through Algeria. It parallels the Mediterranean coast and joins with the Saharan Atlas in Eastern Algeria and Tunisia. The highest summit of the Tell Atlas is the 2308 m Lalla Khadidja in the Djurdjura range of Kabylia. The western end of the Tell Atlas merges with the Middle Atlas range in Morocco.

The area immediately to the south of the Tell Atlas is the high plateau of the Hautes Plaines, with lakes in the wet season and salt flats in the dry. The eastern half of the Tell Atlas has the most humid climate of North Africa, with annual precipitation reaching well above 1000 mm, and sometimes over 1500 mm like in the Collo Peninsula or near Ain Draham. An important amount of snow falls on the summits in winter.

Aurès
The Aurès Mountains are the easternmost portion of the Atlas mountain range. It covers parts of Algeria and Tunisia. The Aurès natural region is named after the range.

Flora and fauna

Flora in the mountains include the Atlas cedar, evergreen oak and many semi-evergreen oaks such as the Algerian oak.

Animals that live in the area include the Barbary macaque (misnamed as the Barbary ape), Barbary leopard, Barbary stag, Barbary sheep, Atlas Mountain badger, Cuvier's gazelle, North African boar, striped hyena, Red fox, northern bald ibis, Algerian nuthatch, dipper, and Atlas mountain viper.

Many animals used to inhabit the Atlas mountains such as the Atlas bear, North African elephant, North African aurochs, bubal hartebeest and Atlas wild ass, but these subspecies are all extinct. Barbary lions are currently extinct in the wild, but descendants exist in captivity.

title: Atlas Mountains


text:
Atlas (architecture)

In European architectural sculpture, an atlas (also known as an atlant, or atlante or atlantid; plural atlantes) is a support sculpted in the form of a man, which may take the place of a column, a pier or a pilaster. The Roman term for such a sculptural support is telamon (plural telamones or telamons).

The term atlantes is the Greek plural of the name Atlas—the Titan who was forced to hold the sky on his shoulders for eternity. The alternative term, telamones, also is derived from a later mythological hero, Telamon, one of the Argonauts, who was the father of Ajax.

The caryatid is the female precursor of this architectural form in Greece, a woman standing in the place of each column or pillar. Caryatids are found at the treasuries at Delphi and the Erechtheion on the Acropolis at Athens for Athene. They usually are in an Ionic context and represented a ritual association with the goddesses worshiped within. The Atlante is typically life-size or larger; smaller similar figures in the decorative arts are called terms. The body of many Atlantes turns into a rectangular pillar or other architectural feature around the waist level, a feature borrowed from the term. The pose and expression of Atlantes very often show their effort to bear the heavy load of the building, which is rarely the case with terms and caryatids. The herma or herm is a classical boundary marker or wayside monument to a god which is usually a square pillar with only a carved head on top, about life-size, and male genitals at the appropriate mid-point. Figures that are rightly called Atlantes may sometimes be described as herms.

Atlantes express extreme effort in their function, heads bent forward to support the weight of the structure above them across their shoulders, forearms often lifted to provide additional support, providing an architectural motif. Atlantes and caryatids were noted by the Roman late Republican architect Vitruvius, whose description of the structures, rather than surviving examples, transmitted the idea of atlantes to the Renaissance architectural vocabulary.

Origin
Not only did the Caryatids precede them, but similar architectural figures already had been made in ancient Egypt out of monoliths. Atlantes originated in Greek Sicily and in Magna Graecia, Southern Italy. The earliest surviving atlantes are fallen ones from the Early Classical Greek temple of Zeus, the Olympeion, in Agrigento, Sicily. Atlantes also played a significant role in Mannerist and Baroque architecture.

During the eighteenth and nineteenth centuries, the designs of many buildings featured glorious atlantes that looked much like Greek originals. Their inclusion in the final design for the portico of the Hermitage Museum in St. Petersburg that was built for Tsar Nicholas I of Russia in the 1840’s made the use of atlantes especially fashionable. The Hermitage portico incorporates ten enormous atlantes, approximately three times life-size, carved from Serdobol granite, which were designed by Johann Halbig and executed by the sculptor Alexander Terebenev.

Mesoamerica
Similar carved stone columns or pillars in the shape of fierce men at some sites of Pre-Columbian Mesoamerica are typically called Atlantean figures. These figures are considered to be "massive statues of Toltec warriors".

Examples

 * Basilica di Santa Croce, Lecce, Italy
 * Casa degli Omenoni, Milan, Italy
 * Church of St. Georg, Hamburg, Germany
 * Dům U Čtyř mamlasů, Brno, Czech Republic
 * Hermitage Museum, St. Petersburg, Russia
 * House in Kanałowa Str. 17, Poznań, Poland
 * Palazzo Davia Bargellini, Bologna, Italy
 * Pavilion Vendôme, Aix-en-Provence, France
 * Porta Nuova, Palermo, Italy
 * Sanssouci, Potsdam, Germany
 * Sunshine Marketplace, Victoria, Australia
 * Temple of Olympian Zeus, Valle dei Templi, Agrigento, Italy
 * Tyszkiewicz Palace, Warsaw, Poland
 * Zwinger Palace, Germany
 * Wayne County Courthouse, Wooster, Ohio, United States

title: Atlas (architecture)


text:
Taiwan Capitalization Weighted Stock Index

The Taiwan Stock Exchange Capitalization Weighted Stock Index, TWSE Capitalization Weighted Stock Index, or TAIEX is a stock market index for companies traded on the Taiwan Stock Exchange (TWSE). TAIEX covers all of the listed stocks excluding preferred stocks, full-delivery stocks and newly listed stocks, which are listed for less than one calendar month. It was first published in 1967 by TWSE with 1966 being the base year with a value of 100.

The Taiwanese stock market experienced a "bubble economy" from 1986 to 1990. When in 1986 the index was around 1000 points, it reached a peak of over 12,000 points in February 1990 then plummeted to around 2500 points by October 1990.

Annual Returns
The following table shows the annual development of the TAIEX since 1966.


text:
Talk:Albedo

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Added Archives
Here is a list of archives/accessdates that I added to this article. -- Tim 1357  talk 04:40, 25 September 2016 (UTC)

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Snow
I attempted a grammar cleanup of the section of snow and sea water albedo. I'd appreciate it if someone could take a look at my edits and advise me on how I could further improve the article, or alternatively, if I missed something important with my edit. It seemed to me that the section was a little ad hoc. I tried to make it flow better. I am a failed undergraduate Physics major of several decades, but I still remember my Feynman. -- Alan Canon Alan Canon (talk) 01:22, 15 January 2018 (UTC)

I would like to know more about the reasoning behind the variability in snow albedo. Why is the method of using tarpaulins so expensive, I feel like there could be a bit more explanation regarding this method. However, I find it very interesting that this method can be used and reduces ice melt by so much. Maybe there could be another section dedicated to more information about other countries using this method as well, if there are any. — Preceding unsigned comment added by Envijustice25 (talk • contribs) 16:22, 30 March 2024 (UTC)

"Illumination" Subsection
There are two major errors in this part of the article: "Although the albedo–temperature effect is best known in colder, whiter regions on Earth, the maximum albedo is actually found in the tropics where year-round illumination is greater. The maximum is additionally in the northern hemisphere, varying between three and twelve degrees north.[24]"

The polar regions are, by far, the highest-albedo regions on Earth [1 ] as a result of their snow cover and angle of incidence. The source merely states that the tropical region is a maximum, not that it is the global maximum. Second, as this article's introduction clearly states, albedo is a rate of radiation reflected to radiation received. Thus, the fact that the tropics receive greater year-round illumination is irrelevant. On that note, I don't understand why this section exists at all.

If there isn't any good reason to keep this section, I'd like to delete it. For now, though, I intend to edit that section to correct the two errors I mentioned above. If I've got some sort of conceptual misunderstanding of the subject that makes my edits erroneous, please let me know. Matthew V. Milone (talk) 16:37, 5 July 2018 (UTC)

Unclear sentence in the lead
I've just done some readability improvements to the lead. There is one sentence remaining in the lead that lights up in red with the readability tool, and I have no clue on how to improve it (as I don't fully understand it): While bi-hemispherical reflectance is calculated for a single angle of incidence (i.e., for a given position of the Sun), albedo is the directional integration of reflectance over all solar angles in a given period.. Can someone improve it please? EMsmile (talk) 13:40, 4 March 2024 (UTC)

What should we add from this article?
https://www.nature.com/articles/s41467-024-46577-1 is licensed under a Creative Commons Attribution 4.0 International License. I guess Fig 1 - anything else?

Chidgk1 (talk) 08:47, 14 May 2024 (UTC)

 * Thanks for pointing this out. There is probably quite a bit of content that could be woven into this article. And also into climate change mitigation and afforestation? E.g. this sentence in the conclusions jumped at me "It also further underscores the inadvisability of afforestation of native grasslands, which can negatively impact biodiversity". Pinging User:ASRASR and User:InformationToKnowledge. - Is it a disadvantage though that this is a primary source and we are meant to use secondary sources rather? (I am fine with using primary sources for something like this). EMsmile (talk) 10:36, 6 June 2024 (UTC)

Removed image from the lead
I've just removed this image (see on the right) from the lead that had recently been added by a student. The image does not look professional to me, e.g. icebergs don't have that shape, why have those polar bears etc. Also, albedo is not just about ice but also forests. EMsmile (talk) 13:58, 14 June 2024 (UTC)

Better lead image needed?
The current lead image (left) from Greeland might not be ideal because it zooms in on the climate change aspect, rather than explaining albedo in general. Also the caption is perhaps a bit lengthy. EMsmile (talk) 13:58, 14 June 2024 (UTC) EMsmile (talk) 13:58, 14 June 2024 (UTC)

title: Talk:Albedo


text:
Atlas (topology)

In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.

Charts
The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism $$\varphi$$ from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair $$(U, \varphi)$$.

When a coordinate system is chosen in the Euclidean space, this defines coordinates on $$U$$: the coordinates of a point $$P$$ of $$U$$ are defined as the coordinates of $$\varphi(P).$$ The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame.

Formal definition of atlas
An atlas for a topological space $$M$$ is an indexed family $$\{(U_{\alpha}, \varphi_{\alpha}) : \alpha \in I\}$$ of charts on $$M$$ which covers $$M$$ (that is, $\bigcup_{\alpha\in I} U_{\alpha} = M$ ). If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then $$M$$ is said to be an n-dimensional manifold.

The plural of atlas is atlases, although some authors use atlantes.

An atlas $$\left( U_i, \varphi_i \right)_{i \in I}$$ on an $$n$$-dimensional manifold $$M$$ is called an adequate atlas if the following conditions hold:

 * The image of each chart is either $$\R^n$$ or $$\R_+^n$$, where $$\R_+^n$$ is the closed half-space,
 * $$\left( U_i \right)_{i \in I}$$ is a locally finite open cover of $$M$$, and
 * $M = \bigcup_{i \in I} \varphi_i^{-1}\left( B_1 \right)$, where $$B_1$$ is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas. Moreover, if $$\mathcal{V} = \left( V_j \right)_{j \in J}$$ is an open covering of the second-countable manifold $$M$$, then there is an adequate atlas $$\left( U_i, \varphi_i \right)_{i \in I}$$ on $$M$$, such that $$\left( U_i\right)_{i \in I}$$ is a refinement of $$\mathcal{V}$$.

Transition maps
A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

To be more precise, suppose that $$(U_{\alpha}, \varphi_{\alpha})$$ and $$(U_{\beta}, \varphi_{\beta})$$ are two charts for a manifold M such that $$U_{\alpha} \cap U_{\beta}$$ is non-empty. The transition map $$ \tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})$$ is the map defined by $$\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.$$

Note that since $$\varphi_{\alpha}$$ and $$\varphi_{\beta}$$ are both homeomorphisms, the transition map $$ \tau_{\alpha, \beta}$$ is also a homeomorphism.

More structure
One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth. Alternatively, one could require that the transition maps have only k continuous derivatives in which case the atlas is said to be $$ C^k $$.

Very generally, if each transition function belongs to a pseudogroup $$ \mathcal G$$ of homeomorphisms of Euclidean space, then the atlas is called a $$\mathcal G$$-atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.

title: Atlas (topology)