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Interesting And Arguable Invention Essay Topics
Thursday, August 27, 2020
Natural Technological Environment of Subway Sandwich Essay Example
Characteristic Technological Environment of Subway Sandwich Paper There are incorporate, all new and renovated stores are presently utilizing vitality productive lighting sparing power just as low stream water spigots sparing an expected 182. 1 million gallons of water yearly and todayââ¬â¢s most of the bundling utilized in metro eatery is made with reused substance and 100% can be reused or treated the soil. Furthermore, metros are moving to these bundling details universally. These days there are 14 Subway Echo Restaurant are accessible and working. These are the stores that manufactured and intended to be more vitality proficient, moderate water and diminish resource. Tram is attempting to give their establishment by having the best answers for assist them with getting enhanced their tasks concentrating on vitality proficiency, asset protection, squander decrease and sanitation. Vitality Efficiency is structure and the idea of our item offering our activities are reasonably vitality proficient however as vitality costs keep on raising it is critical to improve the vitality productivity of our eateries. Lighting and electronic signage are the most significant segments of the eateries and are the enormous clients of vitality in the store. In the course of recent years, they have been improving their vitality effectiveness by moved all new and renovated eateries to high proficiency lighting and made LED outside signage discretionary in 2008 and afterward they are started to change to LED lighting for sandwich units. Metro is directing the utilization of extra vitality sparing hardware, highlights and procedures, for example, LED inside lighting. We will compose a custom exposition test on Natural Technological Environment of Subway Sandwich explicitly for you for just $16.38 $13.9/page Request now We will compose a custom paper test on Natural Technological Environment of Subway Sandwich explicitly for you FOR ONLY $16.38 $13.9/page Recruit Writer We will compose a custom article test on Natural Technological Environment of Subway Sandwich explicitly for you FOR ONLY $16.38 $13.9/page Recruit Writer So on water protection, numerous locales around the globe are starting to encounter fluctuating degrees of water worry because of increments in populace, climate wear out changes or sufficient water the executives framework that can bring about lacking gracefully of safe drinking water. For a long time, reusing and treating the soil containers have been accessible for franchiserââ¬â¢s for planning to reuse manure. Metro cafés are required to keep up a litter free condition and we urge our clients to reuse. Tram are worked with their providers to improve how our items are bundled to decrease squander streams for breads, meats, cheeses, sauces, produce, paper items and cleaning items utilized in our cafés. For instance, the sauces and the vast majority of the fish are presently in film pockets which require less vitality to bundle and transport and evacuate 6 million pounds of crease from our waste stream every year. By evacuating the paper interline between our cut cheeses, an extra 450,000 pounds of paper has been expelled from our waste stream yearly. In today, franchisees have the choice to parcel a few fixings utilizing a scoop. This procedure dispenses with the need to pre-extents in food plate and expels extra bundling from our waste stream every year. Moreover bundling the bundling objective is to just utilizing the bundling that can have the option to reused or treated the soil where offices exists. The Subway café franchisees have the alternative to serve their feast in visitors with a reuse capable bin fixed with shop paper or lined plate. This will be the methodology limits the related waste stream and improves the supper introduction. Notwithstanding, the franchisees adjust to all the neighborhood enactment and having the recyclable paper packs that made with 100% reused paper or reuse capable sacks are accessible for those regions that limit or a plastic pack use. In extra, the reverberation eateries are Subway franchisees are business visionaries and they are numerous visionaries devoted to supporting their networks and the earth. Actually, Some have manufactured ECHO-Restaurants intended to save assets during the advancement procedure and set up for effective activities later on. Mechanical condition The sanitation and food quality are the top need for Subway. The objective for Subway is to serve food that is reliably the highest caliber of meats and feet principles from all when it is placed into a sandwich or plate of mixed greens. To guarantee that the food that been serve is in every case new and economical created they have well as rigid review process that they will be authorize all through their gracefully chain just as require anger that their suppliersââ¬â¢ work rehearses meet their Vendor Code of Conduct. By giving the exclusive requirements for sanitation to guarantee that the whole Subway client will getting the quality from the Subway brand. It begins with setting point of interest item mollifications followed by the review of tests that pulled from the dissemination to guarantee that particulars are being met. The Subway brand Office of Food Safety and Quality Enhancement group will survey all the Subway sanitation and quality projects and may giving suggestion for development if itââ¬â¢s required. This group will be cooperates with the merchants to accomplish the metro sanitation and the quality improvement objectives. Sanitation is one of the primary significant jobs in the Subway eatery. Sandwich Artists are required to consistently follow the exacting hand slashing strategies and appropriate glove utilization for all food taking care of procedures.
Saturday, August 22, 2020
Should Induced Abortion Be Prohibited To Protect Fetus' Human Rights Assignment
Should Induced Abortion Be Prohibited To Protect Fetus' Human Rights - Assignment Example In a worldwide setting we find that in 17 out of 18 countries surveyed far and wide, greater parts dismiss utilizing criminal punishments, for example, fines and detainment, as a way to forestall fetus removal. On normal over every one of the 18 nations, 52 percent favor leaving the matter of premature birth to the individual, while 42 percent figure their legislature should attempt to dishearten premature births. This shows precisely how warmed the discussion on instigated premature birth must be! When talking about initiated fetus removal the most noteworthy inquiry that rings a bell is would we be able to decide to murder undesirable pregnancies? Does the contract of human rights apply to people after birth or as ahead of schedule as an undeveloped organism? Pundits of premature birth contend that since life starts at origination, fetus removal is close to kill as a human life is taken. They accept that fetus removal is willfully ignorant of the holiness of human life. Then again supporters of incited premature birth legitimize that practically all premature births happen in the principal trimester when the embryo is totally subject to the mother and therefore can't be guaranteed as a different person. Additionally as the baby is a piece of the womanââ¬â¢s body it ought to be totally her choice whether to experience labor or select fetus removal. Individuals are separated into two gatherings; Pro-Life, comprehensive of those individuals who advocate the insurance of human undeveloped organisms and embryos and decision in favor of the banning of premature birth on the premise that is murder, and Pro-Choice, for example individuals who bolster the fundamental right of ladies to pick whether to proceed with pregnancy till labor. The single bargain that really allows explicit rights to embryos is the American Convention on Human Rights of 1969, marked by 24 Latin American nations, which says that human rights start right now of conception5. The United States of America has not marked this settlement. Anyway this arrangement doesn't authorize signatories to boycott fetus removal, as per the latest understanding. Presently letââ¬â¢s examine premature birth from a strict perspective. Religion in reality has the tendency to engage in most of discussions. Pastors are
Friday, August 21, 2020
Essay Writing Topics For University Students
Essay Writing Topics For University StudentsHave you ever wondered what are the essay writing topics for university students? Why should essay writing be difficult when the topic is academic in nature? Do you feel that there are essay writing topics for university students to consider for their school essays and papers? These are a few of the questions that come to mind when the topic of essays and papers for students at school is concerned.It all boils down to choosing the topic. When one is given the assignment of writing a research paper, it is a must that the topic or subject matter should be one that would be interesting and captivating for the reader. And since this is the case, the topic chosen should not be mundane and repetitive. For example, if the assignment was to research information about tropical forests, then it is imperative that the topic on which the research would be based should be interesting.The assignment would be completed after students have gone through the entire assignment outline. For essay writing topics for university students, the first task should be to go through the outline of the assignment. These outlines would include everything that needs to be considered in the topic area. The outline should be included to show the ideas of the student that needs to be tackled in the essay writing project.Then, students can browse the web and look for resources on well-known websites. These can be sites that would provide them with relevant information on the topic. The internet also allows access to online databases that would provide these students with detailed information on a variety of topics. There are some instances where online databases may contain essay writing topics for university students to consider for their essays and papers.Since there are online databases that offer resources and suggestions on essay writing topics for university students, these can be browsed and read over by the students as they get familiar with the se online resources. Sincethe topics would depend on the type of institution that the student would be going to, the topics may vary. But most of the topics would revolve around the university where the student is studying.Essay writing topics for university students should be decided upon before the student enters the paper writing session. The topic could be done as an introduction to the essay writing assignment. This is because a student cannot do well in an essay writing session without having a thorough idea of the topic.A research paper or essay writing assignment will be a bit tedious if it is not properly planned out. This is because the topic should not be such that it would be overwhelming and an intimidating topic. Also, the topic should not be a far off topic so that the writer would feel that he or she has already studied enough.Moreover, a good essay writing topics for university students will always be relevant and it should be so. This means that when the topic is c entered on a well-known event or subject matter, the topic is going to be more interesting and helpful to the writer. Therefore, it is imperative that the topic is well researched, discussed on in terms of research.
Monday, May 25, 2020
Analysis Of Raghu Nandan s Understanding Of An Intrapreneur
Intrapreneurial Intrapreneurial Opportunities are key components for companies actively involved in new strategic planning as they seek success in the near future. Raghu Nandanââ¬â¢s understanding of an intrapreneur is an individual or a group of individuals that work for a large company that take internal ideas and visions within a company and configures them into a higher gross profit for a company (Nandan, 2009). An intrapreneur is similar to an entrepreneur with the concept of a business idea that is created with innovation and designed to generate a revenue stream but the only difference is that and entrepreneur doesnââ¬â¢t operate under a large company. Customer Reach By performing a PEST analysis on State Farm insurance, it is clear that the auto insurance industry in New York is operating under many laws and restrictions when viewed from a political perspective. Since State Farm is a financial service, there is no physical product that is being created and manufactured. Financial service can be difficult to market because there is no product and discussed in the social element of the PEST analysis, insurance companies donââ¬â¢t always have a good reputation. It is import for intrapreneurs to develop excellent marketing and advertisements that will attracted many customers with intentions of company expansion. With advertisement in this industry, there should be many targeted markets. A great example of State Farm taking advantage of its intrapreneurial opportunities is
Friday, May 15, 2020
Discuss Miltonââ¬â¢s presentation of Satan in Paradise Lost...
Discuss Miltonââ¬â¢s presentation of Satan in Paradise Lost There has been considerable critical interest in the figure of Satan in Paradise Lost, and in the possibility that he may be the true hero of the epic poem. The opening of the poem finds Milton in a tough spot: writing an epic poem without an epic hero in sight. In order to achieve a rationally balanced poem, he wants to let the first half rise from Hell through Chaos and towards Heaven, thereby balancing the fall of humankind in the following Garden scenes. Since Satan is the only point of view (other than God above) that witnesses all of these early scenes, he must be transformed into the hero for the first few books. Besides allowing Milton to add new scenes to the storyâ⬠¦show more contentâ⬠¦For example in Book I line 118, Satan speaks of having learned from experience, and from profiting from that experience in future actions. We are accustomed to this Period Pieces Imran Hussain process in our lives, but we may easily fail to see that Satanââ¬â¢s experience and his capacity to learn from it is intimately linked with his fallen state. Satan seems to be like us because he demonstrates aspects of being fallen, as humans do. Another characteristic of Satan which is particularly evident in the opening two books of the poem is his desire to rouse his fallen troops through flattery, adulation and by painting for them a more positive picture of what they have just experienced than is strictly true. S.A.J. Bradley (1982: P.12) in his introduction to his translation of Genesis B says of the relationship between Genesis B and Paradise Lost: ââ¬ËBoth poets as a direct consequence for opting for an epic heroic genre, risk counterproductively investing the rebel angel with an admirable dignity and heroic appeal which are inherent in the traditional diction and manner of the genreââ¬â¢. We should note Bradleyââ¬â¢s reference here to the ââ¬Ëoptingââ¬â¢, that is making a choice: it is all too easy to forget that Paradise Lost is the product of much deliberate choice on Miltonââ¬â¢s part. Bradley goes on toShow MoreRelatedFrankenstein Study Guide14107 Words à |à 57 Pagesrelate by theme or topic to the main reading. Each work in the Library has a two-part Study Guide that contains a variety of resources for both you and your students. Use the Guide to plan your instruction of the work and enrich your classroom presentations. In For the Teacher you will find these time-saving instructor aids: â⬠¢ About the Work: pertinent background information on the work and a detailed synopsis of its plot. â⬠¢ Media Links: annotated listings of audio, visual, electronic, and print resources
Wednesday, May 6, 2020
Bill of Rights Four Freedoms - 1528 Words
The first amendment of the Bill of Rights to the constitution guarantees four freedoms: freedom of religion, speech, press and assembly. The Bill of Rights was passed on December 15, 1791. Since then, the freedoms have been debated, discussed, and fought and died for. Many have immigrated to America to receive those freedoms. The Founding Fathers knew what they were doing because they believed in power of ideas and debate, not censorship. The first amendment to the constitution is important because it prohibits congress from establishing an official religion in the U.S., gives Americans the freedom to say what is on their minds, and gives the right to petition and assemble peacefully. The First Amendment of the Constitution of the Unitedâ⬠¦show more contentâ⬠¦During the Red Scare of the early 1920s, thousands were deported for their political views. During the McCarthy period, the infamous blacklist ruined lives and careers. Today, the creators, producers and distributors of popular culture are often blamed for the nations deep social problems. Calls for censorship threaten to erode free speech (American Civil Liberties union). All you ever hear about in the news lately is people getting in trouble for speech. Many Americans embrace freedom of speech for the same reasons they embrace other aspects of individualism. Freedom of speech is the right to defiantly, robustly and irreverently speak ones mind just because it is ones mind. Freedom of speech is thus bonded in special and unique ways to the human capacity to think, imagine and create. Conscience and consciousness are the sacred precincts of mind and soul. Freedom of speech is intimately linked to freedom of thought, to that central capacity to reason and wonder, hope and believe, that largely defines our humanity (Smolla). The right to assemble allows people to gather peacefully for protests to bring about changes. A woman named Diane Pretty from the United Kingdom is paralyzed from the neck down. She has had Neuron disease since 1999 and is confined to a wheelchair. Her disease isShow MoreRelatedWhat is the Bill of Rights?1440 Words à |à 6 PagesThe Bill of Rights Essay ââ¬Å"Rightful liberty is unobstructed action according to our will within limits drawn around us by the equal rights of others.I do not add ââ¬Ëwithin the limits of the law because law is often but the tyrantsââ¬â¢ will,and always so when it violates the rights of the individualâ⬠-Thomas Jefferson. The Constitution was created because of the ineffectiveness of the Articles of Confederation did not simply protecting the rights of the people which the Founding Fathers was concernedRead MoreB1 World War II and Its Aftermath UA Essay1683 Words à |à 7 PagesAnalysis Task 1: Rooseveltââ¬â¢s Four Freedoms In this task, you will analyze President Franklin Rooseveltââ¬â¢s ââ¬Å"Four Freedomsâ⬠speech, and answer short-answer questions about that speech. You will then write a brief essay to turn into your teacher. a. Consider that President Roosevelet delivered the ââ¬Å"Four Freedomsâ⬠speech on January 6, 1941. What is the historical context of this speech in relation to World War II? Type your response here: President Franklin Rooseveltââ¬â¢s four freedoms speech talks about AmericaRead MoreTheu.s. Bill Of Rights And The Un Universal Declaration Of Human Rights1039 Words à |à 5 PagesContrast the Views of Rights in the U.S. Bill of Rights and the UN Universal Declaration of Human Rights The U.S. Bill of Rights and the UN Universal Declaration of Human Rights are documents that were created to tell the rights we as human beings are entitled to have. The documents were designed to keep order so we can live productive lives. The documents stop us from being taken advantage of and extorted by others. The U.S. Bill of Rights and the UN Universal Declaration of Human Rights offers us a voiceRead MoreThe American Civil Liberties Union1639 Words à |à 7 PagesBefore the Bill of Rights, in 1787 the delegates of the thirteen states all got together in Philadelphia to write up the U.S Constitution. But this was only the first attempt at the constitution because there were flaws in it that needed to be fixed. This first constitution only relayed what the government could do and not what it couldnââ¬â¢t do. This was a problem because people in the government could do anything because there was no say in what they were not allowed to do. Another flaw was that thisRead MorePr otecting Life, Liberty, and Property in the Bill of Rights848 Words à |à 4 PagesBill of Rights Essay ââ¬Å" The government is morally obligated to serve people and protect their rights such as life, liberty, and property. ââ¬Å" The Constitution had no effect against the Articles of Confederation. The Constitution was created to make a stronger central government to protect the rights of its people. The social contract protects the natural rights, while the Constitution protects every other rights, like adding an extra layer of protection. The writers intention of the Bill of RightsRead MoreFederalism, Separation Of Powers, Checks And Balances And The Bill Of Rights1208 Words à |à 5 Pages Federalism, separation of powers, checks and balances and the bill of rights are the four main elements that the constitution is made up of. All four of these elements are what made the constitution strong, lasting over two hundred years. Three separate branches were created that had their own specific powers. The government was then given the power to tax and enforce its laws. The struct ure of how this one document has governed is what makes the constitution unique. The U.S. constitution establishedRead MoreThe United States Bill Of Rights882 Words à |à 4 PagesThe United States Bill of Rights was created in September 25, 1789 and ratified December 15, 1791. The Bill of Rights are the first ten amendments to the Constitution that were established to defend our rights as individuals and as American citizens. The Bill of Rights describes the rights of its people. The first four articles of the amendments deal specifically with the balance of power between the federal government and state government. There were some people who opposed to the ConstitutionRead MoreBill Of Rights, The Center Of The American Constitution984 Words à |à 4 Pages Bill of Rights Bill of rights, the center of the American constitution. The reason that why Bill of rights can be established is the anti federalist thought the constitution only stipulated that power from government, and the citizen can not get any power from it. In 1788, at the beginning of the United States, all the citizens want the freedom and democracy, then The ââ¬Å"Bill of Rightsâ⬠is here. The birth of the Bill of Rights, is the most basic principles of personal freedom and limitedRead MoreEssay on The First Amendment876 Words à |à 4 Pages The First Amendment is the first section of the Bill of Rights and is often considered the most important part of the U.S Constitution because it guarantees the citizens of United States the essential personal freedoms of religion, speech, press, peaceful assembly and the freedom to petition the Government. Thanks to the rights granted by the First Amendment, Americans are able to live in a country where they can freely express themselves, speak their mind, pray without interference, protest inRea d MoreU.s. Constitution Vs. Georgia Constitution859 Words à |à 4 PagesGeorgia Constitution Bill of Rights A Bill of Rights recognizes and lists the rights individuals have and protects those rights from governmental interference, unless of course there is a valid reason for government action to take place. While the Bill of Rights for the Georgia Constitution and the United States Constitution do just that, they do have some distinctive similarities and differences. For example, the major difference I see is the fact that the Georgia Bill of Rights is found in Article
Tuesday, May 5, 2020
Regular Polygon free essay sample
Geometrically two edges meeting at a corner are required to form an angle that is not straight (180à °); otherwise, the line segments will be considered parts of a single edge ââ¬â however mathematically, such corners may sometimes be allowed. In fields relating to computation, the term polygon has taken on a slightly altered meaning derived from the way the shape is stored and manipulated in computer graphics (image generation). Some other generalizations of polygons are described below. Polygons are primarily classified by the number of sides. See table below. Convexity and types of non-convexity Polygons may be characterized by their convexity or type of non-convexity: * Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. Equivalently, all its interior angles are less than 180à °. * Non-convex: a line may be found which meets its boundary more than twice. In other words, it contains at least one interior angle with a measure larger than 180à °. * Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. * Concave: Non-convex and simple. * Star-shaped: the whole interior is visible from a single point, without crossing any edge. The polygon must be simple, and may be convex or concave. * Self-intersecting: the boundary of the polygon crosses itself. Branko Grunbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this 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. Symmetry * Equiangular: all its corner angles are equal. * Cyclic: all corners lie on a single circle. * Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular. Equilateral: all edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex. ) [1] * Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral. * Tangential: all sides are tangent to an inscribed circle. Regular: A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon. Miscellaneous * Rectilinear: a polygon whose sides meet at right angles, i. e. , all its interior angles are 90 or 270 degrees. * Monotone with respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice. Properties Euclidean geometry is assumed throughout. Angles Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are: * Interior angle ââ¬â The sum of the interior angles of a simple n-gon is (n ? 2)? radians or (n ? 2)180 degrees. This is because any simple n-gon can be considered to be made up of (n ? 2) triangles, each of which has an angle sum of ? radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is radians or 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. Exterior angle ââ¬â Tracing around a convex n-gon, the angle turned at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360à °. 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 (the total amount one rotates at the vertices) can be any integer multiple d of 360à °, e. g. 20à ° for a pentagram and 0à ° for an angular eight, where d is the density or starriness of the polygon. See also orbit (dynamics). The exterior angle is the supplementary angle to the interior angle. From this the sum of the interior angles can be easily confirmed, even if some interior angles are more than 180à °: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. The area formula is derived by taking each edge AB, and calculating the (signed) area of triangle ABO with a vertex at the origin O, by taking the cross-product (which gives the area of a parallelogram) and dividing by 2. As one wraps around the polygon, these triangles with positive and negative area will overlap, and the areas between the origin and the polygon will be cancelled out and sum to 0, while only the area inside the reference triangle remains. This is why the formula is called the Surveyors Formula, since the surveyor is at the origin; if going counterclockwise, positive area is added when going from left to right and negative area is added when going from right to left, from the perspective of the origin. The formula was described by Meister[citation needed] in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Greens theorem. à The formula is The formula was described by Lopshits in 1963. [3] If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Picks theorem gives a simple formula for the polygons area based on the numbers of interior and boundary grid points. In every polygon with perimeter p and area A , the isoperimetric inequality holds. [4] If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem. The area of a regular polygon is also given in terms of the radius r of its inscribed circle and its perimeter p by . This radius is also termed its apothem and is often represented as a. The area of a regular n-gon with side s inscribed in a unit circle is . The area of a regular n-gon in terms of the radius r of its circumscribed circle and its perimeter p is given by . The area of a regular n-gon, inscribed in a unit-radius circle, with side s and interior angle ? can also be expressed trigonometrically as . The sides of a polygon do not in general determine the area. 5] However, if the polygon is cyclic the sides do determine the area. Of all n-gons with given sides, the one with the largest area is cyclic. Of all n-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic). [6] Self-intersecting polygons The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer: * Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region. For example the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure. * Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon, or to the area of a simple polygon having the same outline as the self-intersecting one (or, in the case of the cross-quadrilateral, the two simple triangles). Degrees of freedom An n-gon has 2n degrees of freedom, including 2 for position, 1 for rotational orientation, and 1 for overall size, so 2n ? 4 for shape. In the case of a line of symmetry the latter reduces to n ? 2. Let k ? 2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n ? 2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n ? 1 degrees of freedom. Product of distances from a vertex to other vertices of a regular polygon For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. Generalizations of polygons In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners). An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an apeirogon (infinite polygon) is unbounded because it goes on for ever so you can never reach any bounding end point. The modern mathematical understanding is to describe such a structural sequence in terms of an abstract polygon which is a partially ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope. A geometric polygon is understood to be a realization of the associated abstract polygon; this involves some mapping of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. So when we talk about polygons we must be careful to explain what kind we are talking about. A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. Add another arc of a different great circle and you have a digon. Tile the sphere with digons and you have a polyhedron called a hosohedron. Take just one great circle instead, run it all the way around, and add just one corner point, and you have a monogon or henagon ââ¬â although many authorities do not regard this as a proper polygon. Other realizations of these polygons are possible on other surfaces, but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate. The idea of a polygon has been generalized in various ways. Here is a short list of some degenerate cases (or special cases, depending on your point of view): * Digon: Interior angle of 0à ° in the Euclidean plane. See remarks above re. on the sphere. * Interior angle of 180à °: In the plane this gives an apeirogon (see below), on the sphere a dihedron * A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygons of the regular polyhedra are classic examples. * A spherical polygon is a circuit of sides and corners on the surface of a sphere.
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