# Regular Polygon

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For other uses, see Polygon (disambiguation).

Some polygons of different kinds

In geometry a polygon ( /ˈpɒlɪɡɒn/) is a flat shape consisting of straight lines that are joined to form a closed chain or circuit. A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The word "polygon" derives from the Greek πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". (The word γόνυ gónu, with a short o, is unrelated and means "knee".) Today a polygon is more usually understood in terms of sides. The basic geometrical notion has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the closed polygonal chain and with simple polygons which do not self-intersect, and may define a polygon accordingly. Geometrically two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge – 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. Contents * 1 Classification * 1.1 Number of sides * 1.2 Convexity and types of non-convexity * 1.3 Symmetry * 1.4 Miscellaneous * 2 Properties * 2.1 Angles * 2.2 Area and centroid * 2.2.1 Self-intersecting polygons * 2.3 Degrees of freedom * 2.4 Product of distances from a vertex to other vertices of a regular polygon * 3 Generalizations of polygons * 4 Naming polygons * 4.1 Constructing higher names * 5 History * 6 Polygons in nature * 7 Polygons in computer graphics * 8 See also * 9 References * 9.1 Bibliography * 9.2 Notes * 10 External links| Classification

Some different types of polygon

Number of sides

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 180°. * 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 180°. * 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 Grünbaum 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:...

References: Bibliography

* Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948).

* Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).

* Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)

Notes

3. ^ A.M. Lopshits (1963). Computation of areas of oriented figures. D C Heath and Company: Boston, MA.

4. ^ Dergiades,Nikolaos, "An elementary proof of the isoperimetric inequality", Forum Mathematicorum 2, 2002, 129-130.

6. ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.

7. ^ Sepkoski, David (2005). "Nominalism and constructivism in seventeenth-century mathematical philosophy". Historia Mathematica 32: 33-59. Retrieved 18 April 2012.

8. ^ Gottfried Martin (1955), Kant 's Metaphysics and Theory of Science, Manchester University Press, p. 22.

9. ^ David Hume, The Philosophical Works of David Hume, Volume 1, Black and Tait, 1826, p. 101.

10. ^ Gibilisco, Stan (2003). Geometry demystified (Online-Ausg. ed.). New York: McGraw-Hill. ISBN 978-0-07-141650-4.

11. ^ Darling, David J., The universal book of mathematics: from Abracadabra to Zeno 's paradoxes, John Wiley & Sons, 2004. Page 249. ISBN 0-471-27047-4.

12. ^ Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.

13. ^ McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.

14. ^ Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0-582-28157-1.

15. ^ Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.

16. ^ Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.

17. ^ Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.

18. ^ Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.

19. ^ Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9.

20. ^ Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.

21. ^ Coxeter, H.S.M.; Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p.114

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