Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, is used to refer to an object, invariant under some transformations. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; this article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people. The opposite of symmetry is asymmetry, which refers to a violation of symmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion; this means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry if there is a line going through it which divides it into two pieces that are mirror images of each other.

An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry. An object has helical symmetry if it can be translated and rotated in three-dimensional space along a line known as a screw axis. An object contracted. Fractals exhibit a form of scale symmetry, where smaller portions of the fractal are similar in shape to larger portions. Other symmetries include glide reflection rotoreflection symmetry. A dyadic relation R = S × S is symmetric if for each element a, b in S, whenever it is true that Rab, it is true that Rba. Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary Mary is the same age as Paul. In propositional logic, symmetric binary logical connectives include and, or and if and only if, while the connective if is not symmetric. Other symmetric logical connectives include nand and nor. Generalizing from geometrical symmetry in the previous section, one can say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object.

The set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, Galois groups in Galois theory. In statistics, symmetry manifests as symmetric probability distributions, as skewness—the asymmetry of distributions. Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations; this concept has become one of the most powerful tools of theoretical physics, as it has become evident that all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his read 1972 article More is Different that "it is only overstating the case to say that physics is the study of symmetry." See Noether's theorem. Important symmetries in physics include discrete symmetries of spacetime.

In biology, the notion of symmetry is used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. Animals that move in one direction have upper and lower sides and tail ends, therefore a left and a right; the head becomes specialized with a mouth and sense organs, the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs remain asymmetric. Plants and sessile animals such as sea anemones have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, sea lilies. In biology, the notion of symmetry is used as in physics, to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the cha

Michael Carbajal vs. Humberto González I

Michael Carbajal vs. Humberto González I was a professional boxing match contested on March 13, 1993 at the Las Vegas Hilton for the IBF and WBC junior flyweight world championships; the fight made boxing history in many ways. While this pair of world champions fought three times, their first fight is the one that writers and fans remember with fondness. In boxing, there are certain types of "rivalries"; when a Puerto Rican fighter faces a Mexican, for example, in a major title fight, most of the time the fight itself is considered a big boxing event. The same can be said about Mexicans fighting against a fellow Mexican for a world title; the Mexican versus Mexican-American rivalries fall among boxing fans' favorite rivalries. A good example is the sustained rivalry between Bobby Chacon, from California, Rafael Limón, from Mexico. A more recent example is Oscar De La Hoya against Julio César Chávez; when Michael Carbajal, from Phoenix, became a professional boxer in 1989, Humberto González, from Mexico, was a seasoned professional.

Carbajal's promoter, Bob Arum, told Ring magazine that Carbajal would be the first junior flyweight to earn one million dollars for one fight. Carbajal won the IBF world junior flyweight title in 1990, joining WBC world champion Gonzalez as co-champions of that division. After Carbajal won the title, fan excitement about a possible match between the two began to grow, but Gonzalez lost his title to Rolando Pascua on October of that year, he had to regain it from Melchor Cob Castro, who took over as champion after beating Pascua. Meanwhile, Carbajal had begun a streak of defenses against a number of important challengers; when Gonzalez regained his WBC belt, he retained it against a long list of challengers. The fact that both boxers were on what boxing fans call a "clash course" brought another element of excitement whenever fans would talk about a future match between Carbajal and Gonzalez. Knowing that this was a well talked about match-up, promoter Arum began to work towards making it happen.

In what could be called a rare case of boxing unity, both the WBC and IBF agreed to have their champions fight a unification bout. The date set was March 17, 1993; the fight received much media attention both in the United States and in Latin America, with Carbajal and Gonzalez being featured on the cover of multiple magazines on the months preceding the bout. Arum proved half prophetic in his guess that Carbajal would be the first junior flyweight to earn one million dollars in a fight: Both Carbajal and Gonzalez got a guarantee of one million dollars, after they paid their due fees, Carbajal-Gonzalez I became the first time in history that a fight fought at the Flyweight division or a smaller one garnered one boxer that amount of money, it was the first time that a junior flyweight world title bout was shown as the main event of a Pay Per View fight card. The fight took place on the date. Carbajal and Gonzalez fought that night in front of an estimated crowd of 6,400 in the Hilton Center at the Las Vegas Hilton.

Among the people watching the fight in person were many Hollywood stars, very rare for a junior flyweight bout. Gonzalez set the pace in round one with his cross, stunning Carbajal as he counter-punched. Gonzalez proved slick for Carbajal, who had trouble finding his rival, but nonetheless, fought back furiously. In round two, a Gonzalez right to Carbajal's chin sent the Arizona native down for a short count. Carbajal started landing stronger punches by round three, but Gonzalez's ability to counter-punch Carbajal's attack seemed to be the factor that would dictate the fight. By the end of the round, Carbajal sported a cut over one of his eyes. Gonzalez and Carbajal kept trading punches in round four, but it was apparent that Gonzalez was faring better than Carbajal. Carbajal's corner team had begun to grow impatient by this point. Gonzalez had Carbajal knocked out in round five; that round might have been the round that turned the fight in Carbajal's favor. A straight right landed flush on Carbajal's chin, Carbajal struggled to stay inside the fight's ring as he fell, grabbing to the rope with his left hand.

Gonzalez was now eager to finish the fight by knockout, Carbajal realized he was behind on points so badly, that he had to try to win by knockout as well. Fighting with desperation, Carbajal got to Gonzalez, who by now was tired, in round six, connecting strong punches to the head and causing swelling around one of Gonzalez's eyes. Still, Carbajal felt; as both fighters were looking for a defining, knockout punch they traded punches in the seventh round, as they say in boxing, "toe to toe". During one of these exchanges, Carbajal unexpectedly landed a right uppercut to Gonzalez's chin, leaving him paralyzed on his feet; as Gonzalez did not move from his stand for the next couple of seconds, Carbajal took advantage and moved to the right, adding a left to Gonzalez's face that would make Gonzalez fall to the canvas on his back. The referee counted Gonzalez out. Michael Carbajal defended the IBF and WBC world junior flyweight titles twice before his second fight with Gonzalez; the pair fought three fights, with Gonzalez winning the next two, one in Los Angeles and another one in Mexico City, but both of Gonzalez's wins came by controversial decision and the two sequels to fight one between Carbajal and Gonzalez were considered to be boring fights by fans and writers alike.

Gonzalez retired after losing the IBF and WBC belts to Saman Sorjaturong in 1995. Carbajal, for his part, kept

Arnold Schölzel

Arnold Angelus Schölzel is a German editor and former defector the editor-in-chief of the far-left newspaper Junge Welt. Prior to 1989, Schölzel worked at Humboldt University in East Berlin, was an informant for the East German domestic intelligence agency, the Stasi. Arnold Angelus Schölzel was born on October 21, 1947, grew up in Bremen, West Germany, at age 16 became a member of the Social Democratic Party of Germany, the largest moderate left-wing political party in Germany. On August 13, 1967, the anniversary of construction beginning of the Berlin Wall five years earlier, Schölzel decided to desert from the West German Army, defected to communist East Germany. Schölzel worked as a laborer in Leipzig until he began studying philosophy at Humboldt University in East Berlin in 1970, earning his diploma in 1974 and a doctorate in 1982 becoming employed as a research assistant at the university. Schölzel was recruited by the East German government under the Socialist Unity Party as an informant for the country's domestic intelligence agency, the Stasi, receiving the code name "André Holzer".

At Humboldt University, Schölzel infiltrated opposition groups where he reported on the activities of numerous individuals, including Wolfgang Templin. Stefan Wolle of the Research Network on the Communist State at the Free University of Berlin described Schölzel as an informant who "...with real enthusiasm and great perfidiousness, he went behind the backs of the people with whom he was friends." He maintained being an informant for the Stasi until 1989, shortly before East Germany was dissolved the following year. After the Reunification of Germany, Schölzel was suspended from his job at Humboldt University in 1991 and dishonourably discharged in 1994, due to his activities as an informant for the Stasi. Schölzel's informant activities were the subject of a documentary film, Verraten – sechs Freunde und ein Spitzel, broadcast by ARD in 2007. In the film, Arnold Schölzel acknowledged having been an informant for the Stasi, when asked by film-maker Inga Wolfram why he had betrayed his friends, he replied: "Well, you betrayed 17 million people", referring to the population of East Germany.

In 1997, Schölzel became the feuilleton editor of Junge Welt, the largest far-left newspaper in Germany, in February 2000 he became editor-in-chief. The newspaper is under observation by the Bundesamt für Verfassungsschutz, who categorized it as left-wing extremist, was described by the conservative German newspaper Die Welt as being dominated by former members of the Stasi. Arnold Schölzel in the German National Library catalogue