PC Gamer is a magazine founded in the United Kingdom in 1993 devoted to PC gaming and published monthly by Future plc. The magazine has several regional editions, with the UK and US editions becoming the best selling PC games magazines in their respective countries; the magazine features news on developments in the video game industry, previews of new games, reviews of the latest popular PC games, along with other features relating to hardware, mods, "classic" games and various other topics. PC Gamer reviews are written by the magazine's editors and freelance writers, rate games on a percent scale. In the UK edition, no game has yet been awarded more than 96%. In the US edition, no game has yet received a rating higher than 98%. In the UK edition, the lowest numerical score was 2%, awarded to The 4th Golden Satellite Awards for Interactive Media Winner Big Brother 1; the sequel, Big Brother 2, was given an lower score of N/A%, the review explaining that " put as much effort into reviewing it as they did in making the game".
In issue 255, August 2013, the score of 2% was matched by the review of the re-released Leisure Suit Larry: Magna Cum Laude given 3% when it first launched. In the US edition, the lowest score awarded was 4%, given to Mad Dog McCree, unseating the lowest-rated game, Skydive!, given 5%. There are two main editions of PC Gamer, a British version and an American version, both are published by Future plc. Founded in the United Kingdom in November 1993, the American sister version was launched a year in June 1994. There are numerous local editions that use the materials of one of the two editions the British one, including a Malaysian and Russian edition; the Swedish edition, though rooted in its UK counterpart, has grown to be more independent due to the immense popularity of PC games compared to console games in Sweden, now produces most of its own material. An Australian edition was published monthly by Perth-based Conspiracy Publishing since August 1998, but it appears to have been discontinued in mid-late 2004.
A Spanish edition titled "PC Juegos y Jugadores" exists. Both American and British magazines are published thirteen times per year, although there are sometimes variations; the British edition of PC Gamer has been in constant monthly publication since 1993. Subscribers get a special edition of the magazine with no headlines on the front cover. Devoted to PC games, the magazine has a reputation for giving in-depth reviews; the magazine shipped with an accompanying 3.5-inch floppy disc. A CD demo disc was released alongside the floppy disk edition from issue 11 onwards with the first CD Gamer containing all the content from the previous 10 issues' floppy discs; the single CD was expanded to two CDs. An edition with a 9 GB DVD known as DVD Gamer ran alongside the 2CD edition for a couple of years, until production of the CD Gamer edition ceased as of issue 162; the UK Edition only came with a single double-sided DVD. In August 2011, the UK magazine announced it was to be discontinuing the disk as of issue 232, replacing it with more pages of content within the magazine and exclusive free gifts.
The magazine has many regular features. These include sections called ´Eyewitness´, ´Previews´, ´Send´, where letters from the readers are spread over 2 two page spreads, at least one special feature, which reports on gaming related issues such as the effect of PC gaming on the environment, a review section which reviews the latest released PC games and re-reviews titles that have been released on budget and ´Extra Life´ which reports on modding games and gaming culture and revisiting old games. There is a ´Systems´ section, which reviews and recommends hardware such as video cards and monitors; the back page of the magazine is entitled ´It's All Over´ and consists of game related artwork such as a version of Dalí's The Persistence of Memory featuring items from Portal. For a time, one of the magazine's features, ´Gamer Snap´, where amusing pictures sent in by readers were printed in the magazine, however the feature was discontinued and replaced with a ´Guess the game´ where readers sent in drawings of memorable scenes in video games drawn in Microsoft Paint.
The PC Gamer blog was started to coincide with the transfer of the PC Gamer UK site to become part of the Computer and Video Games network which incorporates all of Future plc's gaming magazines. The move brought some controversy, with many long-standing members of the forum leaving due to the new forum's cramped spacing and slow loading times; the introduction of a blog was seen as one of the redeeming features of the switch. The blog has since been updated with contributions from many of the magazine's staff; the topics discussed range from the controversy over violent video games, to the benefits of buying a PC over a console. In 2010, PC Gamer re-launched their website and blog by bringing together the online communities of both the US and UK magazines into one website; as a result, the PC Gamer blog now has contributions from both the US and UK magazines, all hosted at the new website along with the forums for both magazines. The PC Gamer UK podcast was started on 4 May 2007 and ran 93 episodes until its final episode, released on 5 July 2013.
It had a rotating cast made up of members of the staff including Chris Thursten, Tom Senior, Graham Smith, Tom Francis, Marsh Davies. The podca
PC Zone, founded in 1993, was the first magazine dedicated to games for IBM-compatible personal computers to be published in the United Kingdom. Earlier PC magazines such as PC Leisure, PC Format and PC Plus had covered games but only as part of a wider remit; the precursor to PC Zone was the award-winning multiformat title Zero. The magazine was published by Dennis Publishing Ltd. until 2004, when it was acquired by Future plc along with Computer And Video Games for £2.5m. In July 2010 it was announced by Future plc; the last issue of PC Zone went on sale 2 September 2010. PC Zone was first published in April 1993 and cost £3.95. Billed as the first UK magazine dedicated to PC games, it was sold with two accompanying floppy disks carrying game demonstrations; the first editor was Paul Lakin. The magazine was split into four sections: Reviews, Blueprints and Regulars. Among the first titles to be reviewed were Dune 2, Lemmings 2 and Stunt Island; the Blueprints section involved previews of new games and Features consisted of an article written about a specific area of gaming interest, such as gaming audio.
Regulars included a news bulletin, competitions and a Buyer's Guide which featured recommended games. In its original incarnation, PC Zone recognised that its audience consisted of males in their late twenties and older, adopted a tone suited to that audience; this was in contrast to contemporary multiformat and console magazines aimed at children and teenagers. During this period, the PC was not yet recognised as a games platform in the UK, an attitude PC Zone arguably helped to change by championing a succession of notable games such as Star Control II, Star Wars: X-Wing, Ultima Underworld and Doom. By 1995, under the editorship of John Davison, the magazine had adopted a tone which referenced lad culture, made fashionable by magazines such as FHM and Dennis Publishing stablemate Maxim; this period was marked by several moderately controversial episodes, including the accidental inclusion of a pornographic Doom modification on a covermounted CD-ROM, an article about the infamously bug-ridden Frontier 2: First Encounters illustrated with a large photograph of a piece of excrement wrapped with a bow, a joystick group test which featured a model dressed as a nun, a one-page comic by regular contributor Charlie Brooker, graphically depicting animal cruelty which resulted in the offending issue being withdrawn from W H Smith newsagents.
Towards the end of the decade, during the editorship of Chris Anderson, the magazine underwent another redesign and a stricter scoring methodology was introduced. For a twelve-month period it was rare for a game to score above 90%, although this was relaxed, resulting in controversial 94% and higher scores for Black & White, Unreal II and others, it was around this time that the magazine retired the long-running Mr Cursor column, a series of humorous, quasi-autobiographical anecdotes written by a thinly-disguised Duncan MacDonald intended to be a counterpoint to the jargon-heavy nature of much of the rest of the editorial. Anderson was succeeded by Dave Woods. Most of the regular recurring features used in the current version of the magazine were introduced during this period, Woods' final contribution was the redesign which marked the handover of the title to Future plc and the editorship to Jamie Sefton; each issue of PC zone came with a DVD-ROM containing game demos, mods, freeware software and patches among other things.
The DVD Zone sleeve would have unique codes which gave readers access to game betas, in-game content, among other things. A new format of PC Zone was introduced in October 2005 for issue #159. By issue #220, the magazine cost £5.99 and included several regular features including Supertest, where reviewers discussed which game is best in its genre. The Buyer's Guide developed from an indexed list of every game reviewed in the publication, along with closing comments; when the longevity of the magazine made this impractical it was pared down to just the best from each genre, becoming shorter with each redesign. As of issue #220, the leaders in each genre are: Shooters: Half-Life 2: 97% / 91% / 82% Strategy: Empire: Total War: 94% Action/Adventure: Grand Theft Auto IV: 91% MMOs: World of Warcraft: 95% Sport: Football Manager 2010: 88% Simulation: X3: Reunion: 92% RPGs: The Elder Scrolls IV: Oblivion: 95% Driving/Racing: GTR 2: 92% Oddball: Spore: 95%The oldest game in the Buyer's Guide was Deus Ex, reviewed issue #93 and given 94%.
PC Zone prided itself on its reviews scoring system, based on the idea that 50% was an average grade. As a result, many publishers accused the magazine of being too harsh. Games that scored 75-89% were given a Recommended Award. Few games only ten a year, received the latter distinction. Games scoring under 20% were given the PC Zone Dump award; as a combined result of its honest scoring system and its age, PC Zone managed to acquire many UK and world print exclusives in terms of news and reviews. PC Zone contained world ex
In mathematics and physics, a scalar field associates a scalar value to every point in a space – physical space. The scalar may either be a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, spin-zero quantum fields, such as the Higgs field; these fields are the subject of scalar field theory. Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U; the region U may be a set in some Euclidean space, Minkowski space, or more a subset of a manifold, it is typical in mathematics to impose further conditions on the field, such that it be continuous or continuously differentiable to some order.
A scalar field is a tensor field of order zero, the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. Physically, a scalar field is additionally distinguished by having units of measurement associated with it. In this context, a scalar field should be independent of the coordinate system used to describe the physical system—that is, any two observers using the same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as vector fields, which associate a vector to every point of a region, as well as tensor fields and spinor fields. More subtly, scalar fields are contrasted with pseudoscalar fields. In physics, scalar fields describe the potential energy associated with a particular force; the force is a vector field, which can be obtained as the gradient of the potential energy scalar field. Examples include: Potential fields, such as the Newtonian gravitational potential, or the electric potential in electrostatics, are scalar fields which describe the more familiar forces.
A temperature, humidity or pressure field, such as those used in meteorology. In quantum field theory, a scalar field is associated with spin-0 particles; the scalar field may be complex valued. Complex scalar fields represent charged particles; these include the charged Higgs field of the Standard Model, as well as the charged pions mediating the strong nuclear interaction. In the Standard Model of elementary particles, a scalar Higgs field is used to give the leptons and massive vector bosons their mass, via a combination of the Yukawa interaction and the spontaneous symmetry breaking; this mechanism is known as the Higgs mechanism. A candidate for the Higgs boson was first detected at CERN in 2012. In scalar theories of gravitation scalar fields are used to describe the gravitational field. Scalar-tensor theories represent the gravitational interaction through a scalar; such attempts are for example the Jordan theory as a generalization of the Kaluza–Klein theory and the Brans–Dicke theory. Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.
This field interacts Yukawa-like with the particles that get mass through it. Scalar fields are found within superstring theories as dilaton fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor. Scalar fields are supposed to cause the accelerated expansion of the universe, helping to solve the horizon problem and giving a hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless scalar fields in this context are known as inflatons. Massive scalar fields are proposed, using for example Higgs-like fields. Vector fields; some examples of vector fields include the electromagnetic field and the Newtonian gravitational field. Tensor fields, which associate a tensor to every point in space. For example, in general relativity gravitation is associated with the tensor field called Einstein tensor. In Kaluza–Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".
The dilaton scalar is found among the massless bosonic fields in string theory. Scalar field theory Vector-valued function
Railroad Tycoon II
Railroad Tycoon II is a train and business simulation video game for Microsoft Windows, Mac OS, PlayStation and Dreamcast in the Railroad Tycoon series. The Dreamcast version is a Gold Edition with improved graphics and gameplay. Railroad Tycoon II: Gold Edition was ported to Linux by Loki Software. Gameplay is displayed unlike the top-down view of Railroad Tycoon. Railroad Tycoon II is a railroad simulation that covers the entire history of railroads from inception to the present day and beyond; the player assumes the role of chairman of a railroad company. The player tries to make profits for investors and completes various other objectives while being hindered by rivals, random events such as train breakdowns, train robbers, economic swings, scripted events particular to the scenario. Most of the gameplay consists of building tracks and trains, which are used for hauling passengers and freight from one station to another. Delivery revenue can vary by time, demand, cargo type, economic state, station improvements and difficulty level.
Companies can connect to and use each other's track and stations, so revenue can be split. Expenses include the fuel and engine maintenance and management fee; the fuel cost depends on the cargo weight and the distance each engine runs. Engine maintenance depends on the engine's type; the elder engines cost more to keep them on the line. The track maintenance is calculated from track mileage; the player will determine what kind of cargo to put on/off at each station that the player adds to its routing. Way-points may be inserted to override default track selection where multiple paths are possible. There are many industries in the game, each can produce and/or convert specific cargoes. For example, coal mines produce coal, iron mines produce iron, and, in the advanced game, a steel mill can convert 1 load of iron plus 1 load of coal into 2 loads of steel; the players are encouraged to find a chain of production to make new cargo by hauling the right type of cargo to each step of the industries. By doing this right, the player can haul raw materials one way, "create" manufactured return cargo, make more money hauling finished products back the other way.
One of the key elements of game play relates to the player purchasing and operating a variety of locomotives, each of which possesses different attributes relating to speed, fuel type, preferred cargo, the ability to traverse hills and steep track grades. In general, the player should balance the cost of operating a train and the time required to transport cargo, with the balance being that the profit from cargo delivery outweighs train operating costs; the player may purchase various industrial plants to earn extra money based on amount of cargo received and delivered. Basic industry, such as bakeries, textile mills, tool and die factories, earns less profit than advanced factories such as canneries, steel mills and automobile plants, although the latter require multiple goods delivered to produce one final product. Furthermore, idle industries which do not produce goods will generate a negative profit, thus increasing overall operating costs and overhead; the following locomotives are available for game play.
Locomotives are only seen in specific time scenarios. In addition to the attributes described above, locomotives are subject to "mechanical reliability", a randomly generated factor influencing how a locomotive will break down. All locomotives begin their operational lives with a set chance of mechanical failure, with some types more prone to breakdowns than others. Breakdown percentage chance increases the older a locomotive becomes. A train may suffer a wreck, which destroys the locomotive and all hauled cargo; the game features one futuristic train, the "Mag-Lev TBX-1", portrayed as released in the future date of 2008. The train operates using Maglev technology which, as of 2015, has been prototyped in both Europe and Japan; the first successful Maglev train to be operated was the SCMaglev with speeds approaching 400 miles per hour. Gameplay includes financial manipulation of companies, issuing bonds, share repurchases, stock issues, manipulating dividends, merging with other companies and declaring bankruptcy.
These features are required in some scenarios and may be used for either great financial gain or total fiscal disaster. In normal financial mode, the player may buy or sell any companies' stocks at various prices depending upon the economy. On advanced settings, the broker may allow the player to buy on margin or short sell stocks if he has enough value of stocks or cash to rely on; the computer players may engage in these practices and will attempt to bankrupt the player when he is in debt. The economy may fluctuate during gameplay, with five economic states possible: Booming, Normal and Depression; when the economy is good, the revenue of hauled goods and stock prices will be higher. Bond interest rates will be adjusted with the economy level, including the interest received from cash and the interest paid for the bonds; the original game features eighteen missions, divided between North America and the rest of the world. The missions can be played on three difficulty levels, each includes three listed objectives.
Completing only the first objective awards the player a bronze medal, the first two a silver medal, all three a gold medal. The player may play any mission in each set as many times; the final score for the whole 18 mission campaign is calculated by the difficulty level and the number of each type o
Microsoft Windows is a group of several graphical operating system families, all of which are developed and sold by Microsoft. Each family caters to a certain sector of the computing industry. Active Windows families include Windows Embedded. Defunct Windows families include Windows Mobile and Windows Phone. Microsoft introduced an operating environment named Windows on November 20, 1985, as a graphical operating system shell for MS-DOS in response to the growing interest in graphical user interfaces. Microsoft Windows came to dominate the world's personal computer market with over 90% market share, overtaking Mac OS, introduced in 1984. Apple came to see Windows as an unfair encroachment on their innovation in GUI development as implemented on products such as the Lisa and Macintosh. On PCs, Windows is still the most popular operating system. However, in 2014, Microsoft admitted losing the majority of the overall operating system market to Android, because of the massive growth in sales of Android smartphones.
In 2014, the number of Windows devices sold was less than 25 %. This comparison however may not be relevant, as the two operating systems traditionally target different platforms. Still, numbers for server use of Windows show one third market share, similar to that for end user use; as of October 2018, the most recent version of Windows for PCs, tablets and embedded devices is Windows 10. The most recent versions for server computers is Windows Server 2019. A specialized version of Windows runs on the Xbox One video game console. Microsoft, the developer of Windows, has registered several trademarks, each of which denote a family of Windows operating systems that target a specific sector of the computing industry; as of 2014, the following Windows families are being developed: Windows NT: Started as a family of operating systems with Windows NT 3.1, an operating system for server computers and workstations. It now consists of three operating system subfamilies that are released at the same time and share the same kernel: Windows: The operating system for mainstream personal computers and smartphones.
The latest version is Windows 10. The main competitor of this family is macOS by Apple for personal computers and Android for mobile devices. Windows Server: The operating system for server computers; the latest version is Windows Server 2019. Unlike its client sibling, it has adopted a strong naming scheme; the main competitor of this family is Linux. Windows PE: A lightweight version of its Windows sibling, meant to operate as a live operating system, used for installing Windows on bare-metal computers, recovery or troubleshooting purposes; the latest version is Windows PE 10. Windows IoT: Initially, Microsoft developed Windows CE as a general-purpose operating system for every device, too resource-limited to be called a full-fledged computer. However, Windows CE was renamed Windows Embedded Compact and was folded under Windows Compact trademark which consists of Windows Embedded Industry, Windows Embedded Professional, Windows Embedded Standard, Windows Embedded Handheld and Windows Embedded Automotive.
The following Windows families are no longer being developed: Windows 9x: An operating system that targeted consumers market. Discontinued because of suboptimal performance. Microsoft now caters to the consumer market with Windows NT. Windows Mobile: The predecessor to Windows Phone, it was a mobile phone operating system; the first version was called Pocket PC 2000. The last version is Windows Mobile 6.5. Windows Phone: An operating system sold only to manufacturers of smartphones; the first version was Windows Phone 7, followed by Windows Phone 8, the last version Windows Phone 8.1. It was succeeded by Windows 10 Mobile; the term Windows collectively describes any or all of several generations of Microsoft operating system products. These products are categorized as follows: The history of Windows dates back to 1981, when Microsoft started work on a program called "Interface Manager", it was announced in November 1983 under the name "Windows", but Windows 1.0 was not released until November 1985.
Windows 1.0 was to achieved little popularity. Windows 1.0 is not a complete operating system. The shell of Windows 1.0 is a program known as the MS-DOS Executive. Components included Calculator, Cardfile, Clipboard viewer, Control Panel, Paint, Reversi and Write. Windows 1.0 does not allow overlapping windows. Instead all windows are tiled. Only modal dialog boxes may appear over other windows. Microsoft sold as included Windows Development libraries with the C development environment, which included numerous windows samples. Windows 2.0 was released in December 1987, was more popular than its predecessor. It features several improvements to the user memory management. Windows 2.03 changed the OS from tiled windows to overlapping windows. The result of this change led to Apple Computer filing a suit against Microsoft alleging infringement on Apple's copyrights. Windows 2.0
Video game genre
A video game genre is a classification assigned to a video game based on its gameplay interaction rather than visual or narrative differences. A video game genre is defined by a set of gameplay challenges and are classified independently of their setting or game-world content, unlike other works of fiction such as films or books. For example, a shooter game is still a shooter game, regardless of when it takes place; as with nearly all varieties of genre classification, the matter of any individual video game's specific genre is open to personal interpretation. Moreover, each individual game may belong to several genres at once; the first attempt to classify different genres of video games was made by Chris Crawford in his book The Art of Computer Game Design in 1984. In this book, Crawford focused on the player's experience and activities required for gameplay. Here, he stated that "the state of computer game design is changing quickly. We would therefore expect the taxonomy presented to become obsolete or inadequate in a short time."
Since among other genres, the platformer and 3D shooter genres, which hardly existed at the time, have gained a lot of popularity. As hardware capabilities have increased, new genres have become possible, with examples being increased memory, the move from 2D to 3D, new peripherals and location. Though genres were just interesting for game studies in the 1980s, the business of video games expanded in the 1990s and both smaller and independent publishers had little chance of surviving; because of this, games settled more into set genres that larger publishers and retailers could use for marketing. Due to "direct and active participation" of the player, video game genres differ from literary and film genres. Though one could state that Space Invaders is a science-fiction video game, such a classification "ignores the differences and similarities which are to be found in the player's experience of the game." In contrast to the visual aesthetics of games, which can vary it is argued that it is interactivity characteristics that are common to all games.
Descriptive names of genres take into account the goals of the game, the protagonist and the perspective offered to the player. For example, a first-person shooter is a game, played from a first-person perspective and involves the practice of shooting; the term "subgenre" may be used to refer to a category within a genre to further specify the genre of the game under discussion. Whereas "shooter game" is a genre name, "first-person shooter" and "third-person shooter" are common subgenres of the shooter genre. Other examples of such prefixes are real-time, turn based, side-scrolling; the target audience, underlying theme or purpose of a game are sometimes used as a genre identifier, such as with "games for girls," games for cats,"Christian game" and "Serious game" respectively. However, because these terms do not indicate anything about the gameplay of a video game, these are not considered genres. Video game genres vary in specificity, with popular video game reviews using genre names varying from "action" to "baseball."
In this practice, basic themes and more fundamental characteristics are used alongside each other. A game may combine aspects of multiple genres in such a way that it becomes hard to classify under existing genres. For example, because Grand Theft Auto III combined shooting and roleplaying in an unusual way, it was hard to classify using existing terms. Since the term Grand Theft Auto clone has been used to describe games mechanically similar to Grand Theft Auto III; the term roguelike has been developed for games that share similarities with Rogue. Elements of the role-playing genre, which focuses on storytelling and character growth, have been implemented in many different genres of video games; this is because the addition of a story and character enhancement to an action, strategy or puzzle video game does not take away from its core gameplay, but adds an incentive other than survival to the experience. According to some analysts, the count of each broad genre in the best selling physical games worldwide is broken down as follows.
The most popular genres are Shooter, Role-playing and Sports, with Platformer and Racing having both declined in the last decade. Puzzle games have declined when measured by sales, however, on mobile, where the majority of games are free-to-play, this genre remains the most popular worldwide. List of video game genres
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