Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
In economics and other social sciences, preference is the ordering of alternatives based on their relative utility, a process which results in an optimal "choice". The character of the individual preferences is determined purely by taste factors, independent of considerations of prices, income, or availability of goods. With the help of the scientific method many practical decisions of life can be modelled, resulting in testable predictions about human behavior. Although economists are not interested in the underlying causes of the preferences in themselves, they are interested in the theory of choice because it serves as a background for empirical demand analysis. In 1926 Ragnar Frisch developed for the first time a mathematical model of preferences in the context of economic demand and utility functions. Up to economists had developed an elaborated theory of demand that omitted primitive characteristics of people; this omission ceased when, at the end of the 19th and the beginning of the 20th century, logical positivism predicated the need of theoretical concepts to be related with observables.
Whereas economists in the 18th and 19th centuries felt comfortable theorizing about utility, with the advent of logical positivism in the 20th century, they felt that it needed more of an empirical structure. Because binary choices are directly observable, it appealed to economists; the search for observables in microeconomics is taken further by revealed preference theory. Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function; this has been achieved by mapping it to the mathematical index called utility. Von Neumann and Morgenstern 1944 book "Games and Economic Behaviour" treated preferences as a formal relation whose properties can be stated axiomatically; these type of axiomatic handling of preferences soon began to influence other economists: Marschak adopted it by 1950, Houthakker employed it in a 1950 paper, Kenneth Arrow perfected it in his 1951 book "Social Choice and Individual Values".
Gérard Debreu, influenced by the ideas of the Bourbaki group, championed the axiomatization of consumer theory in the 1950s, the tools he borrowed from the mathematical field of binary relations have become mainstream since then. Though the economics of choice can be examined either at the level of utility functions or at the level of preferences, to move from one to the other can be useful. For example, shifting the conceptual basis from an abstract preference relation to an abstract utility scale results in a new mathematical framework, allowing new kinds of conditions on the structure of preference to be formulated and investigated. Another historical turnpoint can be traced back to 1895, when Georg Cantor proved in a theorem that if a binary relation is linearly ordered it is isomorphically embeddable in the ordered real numbers; this notion would become influential for the theory of preferences in economics: by the 1940s prominent authors such as Paul Samuelson, would theorize about people having weakly ordered preferences.
Suppose the set of all states of the world is X and an agent has a preference relation on X. It is common to mark the weak preference relation by ⪯, so that x ⪯ y means "the agent wants y at least as much as x" or "the agent weakly prefers y to x"; the symbol ∼ is used as a shorthand to the indifference relation: x ∼ y ⟺, which reads "the agent is indifferent between y and x". The symbol ≺ is used as a shorthand to the strong preference relation: x ≺ y ⟺, which reads "the agent prefers y to x". In everyday speech, the statement "x is preferred to y" is understood to mean that someone chooses x over y. However, decision theory rests on more precise definitions of preferences given that there are many experimental conditions influencing people's choices in many directions. Suppose a person is confronted with a mental experiment that she must solve with the aid of introspection, she is offered apples and oranges, is asked to verbally choose one of the two. A decision scientist observing this single event would be inclined to say that whichever is chosen is the preferred alternative.
Under several repetitions of this experiment, if the scientist observes that apples are chosen 51% of the time it would mean that x ≻ y. If half of the time oranges are chosen x ∼ y. If 51% of the time she chooses oranges it means that y ≻ x. Preference is here being identified with a greater frequency of choice; this experiment implicitly assumes. Otherwise, out of 100 repetitions, some of them will give as a result that neither apples, oranges or ties are chosen; these few cases of uncertainty will ruin any preference information resulting from the frequency attributes of the other valid cases. However, this example was used
Robert John Aumann is an Israeli-American mathematician, a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel, he holds a visiting position at Stony Brook University, is one of the founding members of the Stony Brook Center for Game Theory. Aumann received the Nobel Memorial Prize in Economic Sciences in 2005 for his work on conflict and cooperation through game-theory analysis, he shared the prize with Thomas Schelling. Aumann was born in Frankfurt am Main and fled to the United States with his family in 1938, two weeks before the Kristallnacht pogrom, he attended a yeshiva high school in New York City. He graduated from the City College of New York in 1950 with a B. Sc. in Mathematics. He received his M. Sc. in 1952, his Ph. D. in Mathematics both from the Massachusetts Institute of Technology. His doctoral dissertation, Asphericity of Alternating Linkages, concerned knot theory, his advisor was Jr..
In 1956 he joined the Mathematics faculty of the Hebrew University of Jerusalem and has been a visiting professor at Stony Brook University since 1989. He has held visiting professorship at the University of California, Stanford University, Universite Catholique de Louvain. Aumann's greatest contribution was in the realm of repeated games, which are situations in which players encounter the same situation over and over again. Aumann was the first to define the concept of correlated equilibrium in game theory, a type of equilibrium in non-cooperative games, more flexible than the classical Nash equilibrium. Furthermore, Aumann has introduced the first purely formal account of the notion of common knowledge in game theory, he collaborated with Lloyd Shapley on the Aumann–Shapley value. He is known for his agreement theorem, in which he argues that under his given conditions, two Bayesian rationalists with common prior beliefs cannot agree to disagree. Aumann and Maschler used game theory to analyze Talmudic dilemmas.
They were able to solve the mystery about the "division problem", a long-standing dilemma of explaining the Talmudic rationale in dividing the heritage of a late husband to his three wives depending on the worth of the heritage compared to its original worth. The article in that matter was dedicated to a son of Aumann, killed during the 1982 Lebanon War, while serving as a tank gunner in the Israel Defense Forces's armored corps; these are some of the themes of Aumann's Nobel lecture, named "War and Peace": War is not irrational, but must be scientifically studied in order to be understood, conquered. Aumann is a member in the Professors for a right-wing political group. Aumann opposed the disengagement from Gaza in 2005 claiming it is a crime against Gush Katif settlers and a serious threat to the security of Israel. Aumann draws on a case in game theory called the Blackmailer Paradox to argue that giving land to the Arabs is strategically foolish based on the mathematical theory. By presenting an unyielding demand, the Arab states force Israel to "yield to blackmail due to the perception that it will leave the negotiating room with nothing if it is inflexible".
As a result of his political views, his use of his research to justify them, the decision to give him the Nobel prize was criticized in the European press. A petition to cancel his prize garnered signatures from 1,000 academics worldwide. In a speech to a religious Zionist youth movement, Bnei Akiva, Aumann claimed that Israel is in "deep trouble", he revealed his belief that the anti-Zionist Satmar Jews might have been right in their condemnation of the original Zionist movement. "I fear the Satmars were right", he said, quoted a verse from Psalm 127: "Unless the Lord builds a house, its builders toil on it in vain." Aumann feels that the historical Zionist establishment failed to transmit its message to its successors, because it was secular. The only way that Zionism can survive, according to Aumann, is. In 2008, Aumann joined the new political party Ahi led by Yitzhak Levy. Aumann has entered the controversy of Bible codes research. In his position as both a religious Jew and a man of science, the codes research holds special interest to him.
He has vouched for the validity of the "Great Rabbis Experiment" by Doron Witztum, Eliyahu Rips, Yoav Rosenberg, published in Statistical Science. Aumann not only arranged for Rips to give a lecture on Torah codes in the Israel Academy of Sciences and Humanities, but sponsored the Witztum-Rips-Rosenberg paper for publication in the Proceedings of the National Academy of Sciences; the Academy requires a member to sponsor any publication in its Proceedings. In 1996, a committee consisting of Robert J. Aumann, Dror Bar-Natan, Hillel Furstenberg, Isaak Lapides, Rips, was formed to examine the results, reported by H. J. Gans regarding the existence of "encoded" text in the bible foretelling events that took place many years after the Bible was written; the committee performed two additional tests in the spirit of the Gans experiments. Both tests failed to confirm the existence of the putative code. After a long analysis of the experiment and the dynamics of the controversy, stating for example that "almost everybody included made up thei
Jean-François Mertens was a Belgian game theorist and mathematical economist. Mertens contributed to economic theory in regards to order-book of market games, cooperative games, noncooperative games, repeated games, epistemic models of strategic behavior, refinements of Nash equilibrium. In cooperative game theory he contributed to the solution concepts called the core and the Shapley value. Regarding repeated games and stochastic games, Mertens 1982 and 1986 survey articles, his 1994 survey co-authored with Sylvain Sorin and Shmuel Zamir, are compendiums of results on this topic, including his own contributions. Mertens made contributions to probability theory and published articles on elementary topology. Mertens and Zamir implemented John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types, they constructed a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs.
They showed that any subspace can be approximated arbitrarily by a finite subspace, the usual tactic in applications. Repeated games with incomplete information, were pioneered by Maschler. Two of Jean-François Mertens's contributions to the field are the extensions of repeated two person zero-sum games with incomplete information on both sides for both the type of information available to players and the signalling structure. Information: Mertens extended the theory from the independent case where the private information of the players is generated by independent random variables, to the dependent case where correlation is allowed. Signalling structures: the standard signalling theory where after each stage both players are informed of the previous moves played, was extended to deal with general signalling structure where after each stage each player gets a private signal that may depend on the moves and on the state. In those set-ups Jean-François Mertens provided an extension of the characterization of the minmax and maxmin value for the infinite game in the dependent case with state independent signals.
Additionally with Shmuel Zamir, Jean-François Mertens showed the existence of a limiting value. Such a value can be thought either as the limit of the values v n of the n stage games, as n goes to infinity, or the limit of the values v λ of the λ -discounted games, as agents become more patient and λ → 1. A building block of Mertens and Zamir's approach is the construction of an operator, now referred to as the MZ operator in the field in their honor. In continuous time, the MZ operator becomes an infinitesimal operator at the core of the theory of such games. Unique solution of a pair of functional equations and Zamir showed that the limit value may be a transcendental function unlike the maxmin or the minmax. Mertens found the exact rate of convergence in the case of game with incomplete information on one side and general signalling structure. A detailed analysis of the speed of convergence of the n-stage game value to its limit has profound links to the central limit theorem and the normal law, as well as the maximal variation of bounded martingales.
Attacking the study of the difficult case of games with state dependent signals and without recursive structure and Zamir introduced new tools on the introduction based on an auxiliary game, reducing down the set of strategies to a core that is'statistically sufficient.'Collectively Jean-François Mertens's contributions with Zamir provide the foundation for a general theory for two person zero sum repeated games that encompasses stochastic and incomplete information aspects and where concepts of wide relevance are deployed as for example reputation, bounds on rational levels for the payoffs, but tools like splitting lemma and approachability. While in many ways Mertens's work here goes back to the von Neumann original roots of game theory with a zero-sum two person set up, vitality and innovations with wider application have been pervasive. Stochastic games were introduced by Lloyd Shapley in 1953; the first paper studied the discounted two-person zero-sum stochastic game with finitely many states and actions and demonstrates the existence of a value and stationary optimal strategies.
The study of the undiscounted case evolved in the following three decades, with solutions of special cases by Blackwell and Ferguson in 1968 and Kohlberg in 1974. The existence of an undiscounted value in a strong sense, both a uniform value and a limiting average value, was proved in 1981 by Jean-François Mertens and Abraham Neyman; the study of the non-zero-sum with a general state and action spaces attracted much attention, Mertens and Parthasarathy proved a general existence result under the condition that the transitions, as a function of the state and actions, are norm continuous in the actions. Mertens had the idea to use linear competitive economies as an order book to model limit orders and generalize double auctions to a multivariate set up. Acceptable relative prices of players are conveyed by their linear preferences, money can be one of the goods
Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
An extensive-form game is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, their payoffs for all possible game outcomes. Extensive-form games allow for the representation of incomplete information in the form of chance events modeled as "moves by nature"; some authors in introductory textbooks define the extensive-form game as being just a game tree with payoffs, add the other elements in subsequent chapters as refinements. Whereas the rest of this article follows this gentle approach with motivating examples, we present upfront the finite extensive-form games as constructed here; this general definition was introduced by Harold W. Kuhn in 1953, who extended an earlier definition of von Neumann from 1928. Following the presentation from Hart, an n-player extensive-form game thus consists of the following: A finite set of n players A rooted tree, called the game tree Each terminal node of the game tree has an n-tuple of payoffs, meaning there is one payoff for each player at the end of every possible play A partition of the non-terminal nodes of the game tree in n+1 subsets, one for each player, with a special subset for a fictitious player called Chance.
Each player's subset of nodes is referred to as the "nodes of the player". Each node of the Chance player has a probability distribution over its outgoing edges; each set of nodes of a rational player is further partitioned in information sets, which make certain choices indistinguishable for the player when making a move, in the sense that: there is a one-to-one correspondence between outgoing edges of any two nodes of the same information set—thus the set of all outgoing edges of an information set is partitioned in equivalence classes, each class representing a possible choice for a player's move at some point—, every path in the tree from the root to a terminal node can cross each information set at most once the complete description of the game specified by the above parameters is common knowledge among the playersA play is thus a path through the tree from the root to a terminal node. At any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution.
At any rational player's node, the player must choose one of the equivalence classes for the edges, which determines one outgoing edge except the player doesn't know which one is being followed. A pure strategy for a player thus consists of a selection—choosing one class of outgoing edges for every information set. In a game of perfect information, the information sets are singletons. It's less evident, it is assumed that each player has a von Neumann–Morgenstern utility function defined for every game outcome. The above presentation, while defining the mathematical structure over which the game is played, elides however the more technical discussion of formalizing statements about how the game is played like "a player cannot distinguish between nodes in the same information set when making a decision"; these can be made precise using epistemic modal logic. A perfect information two-player game over a game tree can be represented as an extensive form game with outcomes. Examples of such games include tic-tac-toe and infinite chess.
A game over an expectminimax tree, like that of backgammon, has no imperfect information but has moves of chance. For example, poker has both moves of imperfect information. A complete extensive-form representation specifies: the players of a game for every player every opportunity they have to move what each player can do at each of their moves what each player knows for every move the payoffs received by every player for every possible combination of moves The game on the right has two players: 1 and 2; the numbers by every non-terminal node indicate. The numbers by every terminal node represent the payoffs to the players; the labels by every edge of the graph are the name of the action. The initial node belongs to player 1. Play according to the tree is as follows: player 1 chooses between U and D; the payoffs are as specified in the tree. There are four outcomes represented by the four terminal nodes of the tree:, and; the payoffs associated with each outcome are as follows, and. If player 1 plays D, player 2 will play U' to maximise their payoff and so player 1 will only receive 1.
However, if player 1 plays U, player 2 maximises their payoff by playing D' and player 1 receives 2. Player 1 prefers 2 to 1 and s
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1