PolytopeFrom Wikipedia, the free encyclopediaContents1 Base (group theory) 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Polygon 22.1 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Number of sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Convexity and non-convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.3 Equality and symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Area and centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Generalizations of polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Naming polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 Constructing higher names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Polygons in nature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Polygons in computer graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Polytope 123.1 Approaches to denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Important classes of polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Regular polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.2 Convex polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.3 Star polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Generalisations of a polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.1 Innite polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15iii CONTENTS3.4.2 Abstract polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.1 Self-dual polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Chapter 1Base (group theory)Let G be a nite permutation group acting on a set . A sequenceB= [1, 2, ..., k]of k distinct elements of is a base for G if the only element of Gwhich xes every i B pointwise is the identityelement of G .[1]Bases and strong generating sets are concepts of importance in computational group theory. A base and a stronggenerating set (together often called a BSGS) for a group can be obtained using the SchreierSims algorithm.[2]It is often benecial to deal with bases and strong generating sets as these may be easier to work with than the entiregroup. A group may have a small base compared to the set it acts on. In the worst case, the symmetric groupsand alternating groups have large bases (the symmetric group Sn has base size n 1), and there are often specializedalgorithms that deal with these cases.1.1 References[1] Dixon, John D. (1996), Permutation Groups, Graduate Texts in Mathematics 163, Springer, p. 76, ISBN 9780387945996.[2] Seress, kos (2003), Permutation Group Algorithms, Cambridge Tracts in Mathematics 152, Cambridge University Press,pp. 12, ISBN 9780521661034, Sims seminal idea was to introduce the notions of base and strong generating set.1Chapter 2PolygonFor other uses, see Polygon (disambiguation).In geometry, a polygon /pln/ is traditionally a plane gure that is bounded by a nite chain of straight lineSome polygons of dierent kinds: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (both), andself-intersecting with varying densities of dierent regions.segments closing in a loop to form a closed chain or circuit. These segments are called its edges or sides, and thepoints where two edges meet are the polygons vertices (singular: vertex) or corners.The interior of the polygon issometimes called its body. An n-gon is a polygon with n sides. A polygon is a 2-dimensional example of the moregeneral polytope in any number of dimensions.The word polygon derives from the Greek (pols) much, many and (gna) corner, angle(some references[1] indicate (gnu) knee as the possible origin of gon, but this is unfounded if we considerthat is written with and omicron while with an omega, and the Greek word for polygon ()happens to be written with an omega).The basic geometrical notion has been adapted in various ways to suit particular purposes. Mathematicians are oftenconcerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect,and they often dene a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating starpolygons. Geometrically two edges meeting at a corner are required to form an angle that is not straight (180);otherwise, the line segments may be considered parts of a single edge; however mathematically, such corners maysometimes be allowed. These and other generalizations of polygons are described below.2.1 Classication2.1.1 Number of sidesPolygons are primarily classied by the number of sides. See table below.2.1.2 Convexity and non-convexityPolygons may be characterized by their convexity or type of non-convexity:22.1. CLASSIFICATION 3SimpleConvex ConcaveCyclicEquilateral EquiangularRegular convex Regular starSome dierent types of polygonConvex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactlytwice. As a consequence, all its interior angles are less than 180. Equivalently, any line segment with endpointson 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 linesegment 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 a single point, without crossing any edge. The polygon mustbe simple, and may be convex or concave.Self-intersecting: the boundary of the polygon crosses itself. Branko Grnbaum calls these coptic, thoughthis term does not seem to be widely used. The term complex is sometimes used in contrast to simple, butthis usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert planeconsisting of two complex dimensions.Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.2.1.3 Equality and symmetryEquiangular: all corner angles are equal.4 CHAPTER 2. POLYGONCyclic: 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 also cyclic andequiangular.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 also equilateral andtangential.Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both cyclic and equilateral, or bothequilateral and equiangular. A non-convex regular polygon is called a regular star polygon.2.1.4 MiscellaneousRectilinear: the polygons 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.2.2 PropertiesEuclidean geometry is assumed throughout.2.2.1 AnglesAny polygon has as many corners as it has sides. 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 (n 2) radians or (n 2) 180 degrees.This is because any simple n-gon ( having " n sides ) 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 convexregular n-gon is (1 2n) radians or 180 360ndegrees. The interior angles of regular star polygons wererst studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regularpq -gon (a p-gon with central density q), each interior angle is(p2q)pradians or180(p2q)pdegrees.[2]Exterior angle The exterior angle is the supplementary angle to the interior angle. Tracing around a convexn-gon, the angle turned at a corner is the exterior or external angle. Tracing all the way around the polygonmakes one full turn, so the sum of the exterior angles must be 360. This argument can be generalized toconcave simple polygons, if external angles that turn in the opposite direction are subtracted from the totalturned. Tracing around an n-gon in general, the sum of the exterior angles (the total amount one rotates at thevertices) can be any integer multiple d of 360, e.g. 720 for a pentagram and 0 for an angular eight orantiparallelogram, where d is the density or starriness of the polygon. See also orbit (dynamics).2.2.2 Area and centroidSimple polygonsFor a non-self-intersecting (simple) polygon with n vertices xi, yi ( i = 1 to n), the area and centroid are given by:[3]A =12

n1i=0(xiyi+1 xi+1yi)

2.2. PROPERTIES 5Coordinates of a non-convex pentagon.Cx=16An1i=0(xi +xi+1)(xiyi+1 xi+1yi)Cy=16An1i=0(yi +yi+1)(xiyi+1 xi+1yi).To close the polygon, the rst and last vertices are the same, i.e., xn, yn = x0, y0. The vertices must be ordered accord-ing to positive or negative orientation (counterclockwise or clockwise, respectively); if they are ordered negatively,the value given by the area formula will be negative but correct in absolute value, but when calculating Cx and Cy ,the signed value of A (which in this case is negative) should be used. This is commonly called the Shoelace formulaor Surveyors formula.[4]The area A of a simple polygon can also be computed if the lengths of the sides, a1, a2, ..., an and the exterior angles,1, 2, ..., n are known, from:A =12(a1[a2 sin(1) +a3 sin(1 +2) + +an1 sin(1 +2 + +n2)]+a2[a3 sin(2) +a4 sin(2 +3) + +an1 sin(2 + +n2)]+ +an2[an1 sin(n2)]).The formula was described by Lopshits in 1963.[5]If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Picks theorem givesa 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 p2> 4A holds.[6]If any two simple polygons of equal area are given, then the rst can be cut into polygonal pieces which can bereassembled 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 byA =12 p r.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 isA =ns44 s2.The area of a regular n-gon in terms of the radius r of its circumscribed circle and its perimeter p is given byA =r2 p 1 p24n2r2.6 CHAPTER 2. POLYGONThe area of a regular n-gon, inscribed in a unit-radius circle, with side s and interior angle can also be expressedtrigonometrically asA =ns24cot n=ns24cotn 2= n sin n cos n= n sinn 2 cosn 2.The lengths of the sides of a polygon do not in general determine the area.[7] However, if the polygon is cyclic thesides do determine the area. Of all n-gons with given sides, the one with the largest area is cyclic. Of all n-gons witha given perimeter, the one with the largest area is regular (and therefore cyclic).[8]Self-intersecting polygonsThe area of a self-intersecting polygon can be dened in two dierent ways, each of which gives a dierent answer:Using the above methods for simple polygons, we allowthat particular regions within the polygon may have theirarea multiplied by a factor which we call the density of the region. For example the central convex pentagonin the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a gure8) have opposite-signed densities, and adding their areas together can give a total area of zero for the wholegure.Considering the enclosed regions as point sets, we can nd the area of the enclosed point set. This correspondsto the area of the plane covered by the polygon, or to the area of one or more simple polygons having the sameoutline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.2.3 Generalizations of polygonsThe idea of a polygon has been generalized in various ways. Some of the more important include:A spherical polygon is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allowsthe digon, a polygon having only two sides and two corners, which is impossible in a at plane. Sphericalpolygons play an important role in cartography (map making) and in Wythos construction of the uniformpolyhedra.A skew polygon does not lie in a at plane, but zigzags in three (or more) dimensions. The Petrie polygons ofthe regular polytopes are well known examples.An apeirogon is an innite sequence of sides and angles, which is not closed but has no ends because it extendsindenitely in both directions.A skew apeirogon is an innite sequence of sides and angles that do not lie in a at plane.A complex polygon is a conguration analogous to an ordinary polygon, which exists in the complex plane oftwo real and two imaginary dimensions.An abstract polygon is an algebraic partially ordered set representing the various elements (sides, vertices,etc.) and their connectivity. A real geometric polygon is said to be a realization of the associated abstractpolygon. Depending on the mapping, all the generalizations described here can be realized.2.4 Naming polygonsThe word polygon comes from Late Latin polygnum (a noun), from Greek (polygnon/polugnon),noun use of neuter of (polygnos/polugnos, the masculine adjective), meaning many-angled. Indi-vidual polygons are named (and sometimes classied) according to the number of sides, combining a Greek-derivednumerical prex with the sux -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are excep-tions.2.5. HISTORY 7Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for ex-ample 17-gon and 257-gon.[9]Exceptions exist for side counts that are more easily expressed in verbal form, or are used by non-mathematicians.Some special polygons also have their own names; for example the regular star pentagon is also known as thepentagram.2.4.1 Constructing higher namesTo construct the name of a polygon with more than 20 and less than 100 edges, combine the prexes as follows.[13]The kai term applies to 13-gons and higher was used by Kepler, and advocated by John H. Conway for clarity toconcatenated prex numbers in the naming of quasiregular polyhedra.[30]2.5 HistoryPolygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with thepentagram, a non-convex regular polygon (star polygon), appearing as early as the 7th century B.C. on a krater byAristonothos, found at Caere and now in the Capitoline Museum.[31][32]The rst known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14thcentury.[33]In 1952, Georey Colin Shephard generalized the idea of polygons to the complex plane, where each real dimensionis accompanied by an imaginary one, to create complex polygons.[34]2.6 Polygons in naturePolygons appear in rock formations, most commonly as the at facets of crystals, where the angles between the sidesdepend on the type of mineral from which the crystal is made.Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt, which may beseen at the Giants Causeway in Northern Ireland, or at the Devils Postpile in California.In biology, the surface of the wax honeycomb made by bees is an array of hexagons, and the sides and base of eachcell are also polygons.2.7 Polygons in computer graphicsA polygon in a computer graphics (image generation) system is a two-dimensional shape that is modelled and storedwithin its database. A polygon can be colored, shaded and textured, and its position in the database is dened by thecoordinates of its vertices (corners).Naming conventions dier from those of mathematicians:A simple polygon does not cross itself.a concave polygon is a simple polygon having at least one interior angle greater than 180.A complex polygon does cross itself.Any surface is modelled as a tessellation called meshed polygons. If a square mesh has n + 1 points (vertices) perside, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. Thereare (n + 1)2/ 2(n2) vertices per triangle. Where n is large, this approaches one half. Or, each vertex inside the squaremesh connects four edges (lines).The imaging system calls up the structure of polygons needed for the scene to be created from the database. This istransferred to active memory and nally, to the display system (screen, TV monitors etc.) so that the scene can be8 CHAPTER 2. POLYGONHistorical image of polygons (1699)viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission ofthe processed data to the display system. Although polygons are two-dimensional, through the system computer they2.8. SEE ALSO 9The Giants Causeway, in Northern Irelandare placed in a visual scene in the correct three-dimensional orientation.In computer graphics and computational geometry, it is often necessary to determine whether a given point P =(x0,y0) lies inside a simple polygon given by a sequence of line segments. This is called the Point in polygon test.2.8 See also2.9 References2.9.1 BibliographyCoxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948).Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).Grnbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461488. (pdf)2.9.2 Notes[1] Craig, John (1849). Anewuniversal etymological technological, and pronouncing dictionary of the English language. OxfordUniversity. p. 404., Extract of page 404[2] Kappra, Jay (2002). Beyond measure: a guided tour through nature, myth, and number. World Scientic. p. 258. ISBN978-981-02-4702-7.[3] Bourke, Paul (July 1988). Calculating The Area And Centroid Of A Polygon (PDF). Retrieved 6 Feb 2013.10 CHAPTER 2. POLYGON[4] Bart Braden (1986). The Surveyors Area Formula (PDF). The College Mathematics Journal 17 (4): 326337. doi:10.2307/2686282.[5] A.M. Lopshits (1963).Computation of areas of oriented gures. translators: J Massalski and C Mills, Jr. D C Heath andCompany: Boston, MA.[6] Dergiades,Nikolaos, An elementary proof of the isoperimetric inequality, Forum Mathematicorum 2, 2002, 129-130.[7] Robbins, Polygons inscribed in a circle, American Mathematical Monthly 102, JuneJuly 1995.[8] Chakerian, G. D. A Distorted View of Geometry. Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington,DC: Mathematical Association of America, 1979: 147.[9] Mathworld[10] Grunbaum, B.; Are your polyhedra the same as my polyhedra, Discrete and computational geometry: the Goodman-Pollack Festschrift, Ed. Aronov et al., Springer (2003), page 464.[11] Hass, Joel; Morgan, Frank (1996), Geodesic nets on the 2-sphere, Proceedings of the American Mathematical Society 124(12): 38433850, doi:10.1090/S0002-9939-96-03492-2, JSTOR 2161556, MR 1343696.[12] Coxeter, H.S.M.; Regular polytopes, Dover Edition (1973), Page 4.[13] Salomon, David (2011). The Computer Graphics Manual. Springer Science & Business Media. pp. 8890. ISBN 978-0-85729-886-7.[14] The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298[15] http://mathforum.org/dr.math/faq/faq.polygon.names.html[16] Sepkoski, David (2005). Nominalism and constructivism in seventeenth-century mathematical philosophy (PDF). His-toria Mathematica 32: 3359. doi:10.1016/j.hm.2003.09.002. Retrieved 18 April 2012.[17] Gottfried Martin (1955), Kants Metaphysics and Theory of Science, Manchester University Press, p. 22.[18] David Hume, The Philosophical Works of David Hume, Volume 1, Black and Tait, 1826, p. 101.[19] Gibilisco, Stan (2003). Geometry demystied (Online-Ausg. ed.). New York: McGraw-Hill. ISBN 978-0-07-141650-4.[20] Darling, David J., The universal book of mathematics: from Abracadabra to Zenos paradoxes, John Wiley & Sons, 2004.Page 249. ISBN 0-471-27047-4.[21] Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.[22] McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.[23] Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0-582-28157-1.[24] Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.[25] Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.[26] Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.[27] Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.[28] Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press,1993, p. 86, ISBN 0-8232-1486-9.[29] Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.[30] Naming Polygons and Polyhedra. Ask Dr. Math. The Math Forum - Drexel University. Retrieved 3 May 2015.[31] Heath, Sir Thomas Little (1981), A History of Greek Mathematics, Volume 1, Courier Dover Publications, p. 162, ISBN9780486240732. Reprint of original 1921 publication with corrected errata. Heath uses the spelling Aristonophus forthe vase painters name.[32] Cratere with the blinding of Polyphemus and a naval battle, Castellani Halls, Capitoline Museum, accessed 2013-11-11.Two pentagrams are visible near the center of the image,[33] Coxeter, H.S.M.; Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p.114[34] Shephard, G.C.; Regular complex polytopes, Proc. London Math. Soc. Series 3 Volume 2, 1952, pp 82-972.10. EXTERNAL LINKS 112.10 External linksWeisstein, Eric W., Polygon, MathWorld.What Are Polyhedra?, with Greek Numerical PrexesPolygons, types of polygons, and polygon properties, with interactive animationHow to draw monochrome orthogonal polygons on screens, by Herbert Glarnercomp.graphics.algorithms Frequently Asked Questions, solutions to mathematical problems computing 2Dand3D polygonsComparison of the dierent algorithms for Polygon Boolean operations, compares capabilities, speed andnumerical robustnessInterior angle sum of polygons: a general formula, Provides an interactive Java investigation that extends theinterior angle sum formula for simple closed polygons to include crossed (complex) polygonsChapter 3PolytopeNot to be confused with polytrope.In elementary geometry, a polytope is a geometric object with at sides, and may exist in any general number ofdimensions n as an n-dimensional polytope or n-polytope. For example a two-dimensional polygon is a 2-polytopeand a three-dimensional polyhedron is a 3-polytope.Some theories further generalize the idea to include such objects as unbounded (apeirotopes and tessellations), de-compositions or tilings of curved manifolds such as spherical polyhedra, and set-theoretic abstract polytopes.Polytopes in more than three dimensions were rst discovered by Ludwig Schli.The term polytop was coinedby the mathematician Hoppe, writing in German, and was introduced to English mathematicians in its present formby Alicia Boole Stott.3.1 Approaches to denitionThe term polytope is nowadays a broad term that covers a wide class of objects, and dierent denitions are attestedin mathematical literature. Many of these denitions are not equivalent, resulting in dierent sets of objects beingcalled polytopes. They represent dierent approaches to generalizing the convex polytopes to include other objectswith similar properties.The original approach broadly followed by Ludwig Schli, Thorold Gosset and others begins with the extensionby analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and threedimensions.[1]Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the developmentof topology and the treatment of a decomposition or CW-complex as analogous to a polytope.[2] In this approach, apolytope may be regarded as a tessellation or decomposition of some given manifold. An example of this approachdenes a polytope as a set of points that admits a simplicial decomposition. In this denition, a polytope is the unionof nitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection,their intersection is a vertex, edge, or higher dimensional face of the two.[3] However this denition does not allowstar polytopes with interior structures, and so is restricted to certain areas of mathematics.The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface,ignoring its interior.[4] In this light convex polytopes in p-space are equivalent to tilings of the (p1)-sphere, whileothers may be tilings of other elliptic, at or toroidal (p1)-surfaces see elliptic tiling and toroidal polyhedron. Apolyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets (cells)are polyhedra, and so forth.The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards indimension, with an (edge) seen as a 1-polytope bounded by a point pair, and a point or vertex as a 0-polytope. Thisapproach is used for example in the theory of abstract polytopes.In certain elds of mathematics, polytope and polyhedron are used in a dierent sense: a polyhedron is the genericobject in any dimension (which is referred to as polytope on this Wikipedia article) and polytope means a boundedpolyhedron.[5] This terminology is typically used for polytopes and polyhedra that are convex. With this terminology,a convex polyhedron is the intersection of a nite number of halfspaces (it is dened by its sides) while a convex123.2. ELEMENTS 13A 2-dimensional polytope.polytope is the convex hull of a nite number of points (it is dened by its vertices).3.2 ElementsApolytope comprises elements of dierent dimensionality such as vertices, edges, faces, cells and so on. Terminologyfor these is not fully consistent across dierent authors. For example some authors use face to refer to an (n 1)-dimensional element while others use face to denote a 2-face specically. Authors may use j-face or j-facet to indicatean element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an (n 1)-dimensional element.[6]The terms adopted in this article are given in the table below:An n-dimensional polytope is bounded by a number of (n 1)-dimensional facets. These facets are themselvespolytopes, whose facets are (n 2)-dimensional ridges of the original polytope. Every ridge arises as the intersection14 CHAPTER 3. POLYTOPEof two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facetsgive rise to (n 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes maybe referred to as faces, or specically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, andconsists of a single point.A 1-dimensional face is called an edge, and consists of a line segment.A 2-dimensionalface consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.3.3 Important classes of polytope3.3.1 Regular polytopesMain article: Regular polytopeA regular polytope is the most highly symmetrical kind, with the various groups of elements being transitive on thesymmetries of the polytope, such that the polytope is said to be transitive on its ags. Thus, the dual of a regularpolytope is also regular.There are three main classes of regular polytope which occur in any number n of dimensions:Simplices, including the equilateral triangle and the regular tetrahedron.Hypercubes or measure polytopes, including the square and the cube.Orthoplexes or cross polytopes, including the square and regular octahedron.Dimensions two, three and four include regular gures which have vefold symmetries and some of which are non-convex stars, and in two dimensions there are innitely many regular polygons of n-fold symmetry, both convex and(for n 5) star. But in higher dimensions there are no other regular polytopes.[1]In three dimensions the convex Platonic solids include the vefold-symmetric dodecahedron and icosahedron, andthere are also four star Kepler-Poinsot polyhedra with vefold symmetry, bringing the total to nine regular polyhedra.In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two withvefold symmetry. There are ten star Schli-Hess 4-polytopes, all with vefold symmetry, giving in all sixteenregular 4-polytopes.3.3.2 Convex polytopesMain article: Convex polytopeA polytope may be convex. The convex polytopes are the simplest kind of polytopes, and form the basis for severaldierent generalizations of the concept of polytopes. A convex polytope is sometimes dened as the intersection ofa set of half-spaces. This denition allows a polytope to be neither bounded nor nite. Polytopes are dened in thisway, e.g., in linear programming. A polytope is bounded if there is a ball of nite radius that contains it. A polytopeis said to be pointed if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example ofa non-pointed polytope is the set {(x, y) R2| x 0} . A polytope is nite if it is dened in terms of a nitenumber of objects, e.g., as an intersection of a nite number of half-planes.3.3.3 Star polytopesMain article: Star polytopeA non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes. Some regularpolytopes are stars.[1]3.4. GENERALISATIONS OF A POLYTOPE 153.4 Generalisations of a polytope3.4.1 Innite polytopesNot all manifolds are nite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea maybe extended to innite manifolds. plane tilings, space-lling (honeycombs) and hyperbolic tilings are in this sensepolytopes, and are sometimes called apeirotopes because they have innitely many cells.Among these, there are regular forms including the regular skewpolyhedra and the innite series of tilings representedby the regular apeirogon, square tiling, cubic honeycomb, and so on.3.4.2 Abstract polytopesMain article: Abstract polytopeThe theory of abstract polytopes attempts to detach polytopes from the space containing them, considering theirpurely combinatorial properties. This allows the denition of the term to be extended to include objects for which itis dicult to dene an intuitive underlying space, such as the 11-cell.An abstract polytope is a partially ordered set of elements or members, which obeys certain rules. It is a purelyalgebraic structure, and the theory was developed in order to avoid some of the issues which make it dicult toreconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said tobe a realization in some real space of the associated abstract polytope.3.5 DualityEvery n-polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and soon generally interchanging its (j1)-dimensional elements for (nj)-dimensional elements (for j = 1 to n1), whileretaining the connectivity or incidence between elements.For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in the Schli symbols forregular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example {4, 3,3} is dual to {3, 3, 4}.In the case of a geometric polytope, some geometric rule for dualising is necessary, see for example the rules describedfor dual polyhedra. Depending on circumstance, the dual gure may or may not be another geometric polytope.[7]If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.3.5.1 Self-dual polytopesIf a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities,then the dual gure will be identical to the original and the polytope is self-dual.Some common self-dual polytopes include:Every regular n-simplex, in any number of dimensions, with Schlai symbol {3n}, is self-dual. These includethe equilateral triangle {3} and regular tetrahedron {3, 3}.In 2 dimensions, all regular polygons (regular 2-polytopes)In 3 dimensions, the canonical polygonal pyramids and elongated pyramids, also the innite square tiling {4,4}.In 4 dimensions, the 24-cell, with Schlai symbol {3,4,3}.16 CHAPTER 3. POLYTOPEThe 5-cell (4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.3.6 HistoryPolygons and polyhedra have been known since ancient times.An early hint of higher dimensions came in 1827 when Mbius discovered that two mirror-image solids can besuperimposed by rotating one of them through a fourth mathematical dimension. By the 1850s, a handful of othermathematicians such as Cayley and Grassman had considered higher dimensions. Ludwig Schli was the rst of theseto consider analogues of polygons and polyhedra in such higher spaces. In 1852 he described the six convex regular4-polytopes, but his work was not published until 1901, six years after his death. By 1854, Bernhard Riemann'sHabilitationsschrift had rmly established the geometry of higher dimensions, and thus the concept of n-dimensionalpolytopes was made acceptable. Schlis polytopes were rediscovered many times in the following decades, evenduring his lifetime.In 1882 Hoppe, writing in German, coined the word polytop to refer to this more general concept of polygons andpolyhedra. In due course Alicia Boole Stott, the daughter of logician George Boole, introduced polytope into theEnglish language.[1]In 1895, Thorold Gosset not only rediscovered Schlis regular polytopes, but also investigated the ideas of semiregularpolytopes and space-lling tessellations in higher dimensions. Polytopes were also studied in non-Euclidean spacessuch as hyperbolic space.During the early part of the 20th century, higher-dimensional spaces became fashionable, and together with the idea3.7. USES 17of higher polytopes, inspired artists such as Picasso to create the movement known as cubism.An important milestone was reached in 1948 with H. S. M. Coxeter's book Regular Polytopes, summarizing work todate and adding ndings of his own.Meanwhile the topological idea of the piecewise decomposition of a manifold into a CW-complex led to the treatmentof such decompositions as polytopes. Branko Grnbaum published his inuential work on Convex Polytopes in 1967.More recently, the concept of a polytope has been further generalized. In 1952 Shephard developed the idea ofcomplex polytopes in complex space, where each real dimension has an imaginary one associated with it. Coxeterdeveloped the idea further. Complex polytopes do not have closed surfaces in the usual way, and are better understoodas congurations.[8]The conceptual issues raised by complex polytopes, duality and other phenomena led Grnbaum and others to themore general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea wasthat of incidence complexes, which studied the incidence or connection of the various elements with one another.These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of suchelements. McMullen and Schulte published their book Abstract Regular Polytopes in 2002.Enumerating the uniform polytopes, convex and nonconvex, in four or more dimensions remains an outstandingproblem.In modern times, polytopes and related concepts have found many important applications in elds as diverse ascomputer graphics, optimization, search engines, cosmology, quantum mechanics and numerous other elds.3.7 UsesIn the study of optimization, linear programming studies the maxima and minima of linear functions constricted tothe boundary of an n-dimensional polytope.In linear programming, polytopes occur in the use of Generalized barycentric coordinates and Slack variables.3.8 See alsoList of regular polytopesConvex polytopeRegular polytopeSemiregular polytopeUniform polytopeAbstract polytopeBounding volume-Discrete oriented polytopeRegular forms1. Simplex2. hypercube3. Cross-polytopeIntersection of a polyhedron with a lineExtension of a polyhedronCoxeter groupBy dimension:1. 2-polytope or polygon18 CHAPTER 3. POLYTOPE2. 3-polytope or polyhedron3. 4-polytope or polychoron4. 5-polytope5. 6-polytope6. 7-polytope7. 8-polytope8. 9-polytope9. 10-polytopePolyformPolytope de MontralSchli symbolHoneycomb (geometry)Amplituhedron3.9 ReferencesNotes[1] Coxeter (1973)[2] Richeson, S.; Eulers Gem: The Polyhedron Formula and the Birth of Topology, Princeton University, 2008.[3] Grnbaum (2003)[4] Cromwell, P.; Polyhedra, CUP (ppbk 1999) pp 205 .[5] Nemhauser and Wolsey, Integer and Combinatorial Optimization, 1999, ISBN 978-0471359432, Denition 2.2.[6] Regular polytopes, p. 127 The part of the polytope that lies in one of the hyperplanes is called a cell[7] Wenninger, M.; Dual Models, CUP (1983).[8] Coxeter, H.S.M.; Regular Complex Polytopes, 1974SourcesCoxeter, Harold Scott MacDonald (1973), Regular Polytopes, New York:Dover Publications, ISBN 978-0-486-61480-9.Grnbaum, Branko (2003), Kaibel, Volker; Klee, Victor; Ziegler, Gnter M., eds., Convex polytopes (2nd ed.),New York & London: Springer-Verlag, ISBN 0-387-00424-6.Ziegler, Gnter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics 152, Berlin, New York:Springer-Verlag.3.10 External linksWeisstein, Eric W., Polytope, MathWorld.Math will rock your world application of polytopes to a database of articles used to support custom newsfeeds via the Internet (Business Week Online)Regular and semi-regular convex polytopes a short historical overview:3.11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 193.11 Text and image sources, contributors, and licenses3.11.1 Text Base (group theory) Source: https://en.wikipedia.org/wiki/Base_(group_theory)?oldid=631685857 Contributors: Tango, Charles Matthews,Dysprosia, Tobias Bergemann, Jh51681, Andreas Kaufmann, Shanes, Malber, Luke Maurits, CBM, David Eppstein, VolkovBot, JackSchmidt,Addbot, AnomieBOT, Erik9bot, Mark viking and Anonymous: 4 Polygon Source: https://en.wikipedia.org/wiki/Polygon?oldid=667943109 Contributors: Damian Yerrick, AxelBoldt, Lee Daniel Crocker,Mav, Bryan Derksen, Zundark, The Anome, Tarquin, Malcolm Farmer, BenBaker, Youssefsan, XJaM, Rmhermen, Christian List, TheOstrich, Edemaine, Juuitchan, Edward, Ubiquity, Patrick, Michael Hardy, Norm, Dominus, Shyamal, Graue, Eric119, Minesweeper,Ellywa, BigFatBuddha, Panoramix, Mxn, Revolver, Charles Matthews, CecilBlade, Dcoetzee, Andrewman327, Gutza, Wik, Alem-bert~enwiki, Jerzy, Phil Boswell, Rhys~enwiki, Donarreiskoer, Robbot, Fredrik, Altenmann, Mushroom, PrimeFan, Tosha, Giftlite,Dbenbenn, Mikez, Lupin, Herbee, Mark.murphy, Peruvianllama, Curps, Hans-Friedrich Tamke, Eequor, Edcolins, Chowbok, LucasVB,Antandrus, Joseph Myers, Maximaximax, Gauss, Latitude0116, Tomruen, Pmanderson, Icairns, Burschik, Karl Dickman, Qef, Ma'ameMichu, Rich Farmbrough, Guanabot, Cacycle, Paul August, Goochelaar, Nabla, RJHall, Kwamikagami, Mwanner, Susvolans, Mickey-mousechen~enwiki, Fiveless, Vervin, Smalljim, Kevin Lamoreau, Nk, Physicistjedi, Rje, Douglasr007, AzaToth, Krazykillaz, BRW, Cm-prince, HenryLi, Oleg Alexandrov, OwenX, Woohookitty, Georgia guy, LOL, Uncle G, Duncan.france, Schzmo, Ghostofgauss, Paxsim-ius, Graham87, MC MasterChef, Rjwilmsi, Chris Ashley, Anthonymorris, Indiedan, Salix alba, Tawker, Oblivious, The Deviant, FlaBot,Mathbot, Margosbot~enwiki, Harmil, RexNL, Quuxplusone, Chobot, DVdm, Siddhant, YurikBot, Wavelength, Karlscherer3, RussBot,Stephenb, Thane, Gustavb, NawlinWiki, Grafen, Matticus78, JulesH, BOT-Superzerocool, Jemebius, MaxDZ8, Googl, Gulliveig, Ojii-san, JoanneB, Anclation~enwiki, 158-152-12-77, TMott, Robert L, Whitehat101, DVD R W, SmackBot, Reedy, Unyoyega, Pgk, BiT,Alsandro, Freddy S., Gilliam, Ohnoitsjamie, Skizzik, XxAvalanchexX, JAn Dudk, Bluebot, Keegan, TimBentley, MalafayaBot, Octahe-dron80, Gracenotes, Hgrosser, Can't sleep, clown will eat me, Tamfang, Vanished User 0001, Yidisheryid, Gogino, SundarBot, Snapping-Turtle, RedKnight7, Hgilbert, Neshatian, DMacks, Vina-iwbot~enwiki, John Reid, Smremde, Zchenyu, Lambiam, Ninjagecko, Nat2,Bjankuloski06en~enwiki, 16@r, A. Parrot, Sandb, Ryulong, Jasmbspidy, Krispos42, Aktalo, Ziusudra, AbsolutDan, KangKnight, Xcen-taur, Avg, CmdrObot, Enselic, MarsRover, Lentower, WeggeBot, Captmog, Sopoforic, SyntaxError55, Pascal.Tesson, DumbBOT, Clsn,Editor at Large, Riojajar~enwiki, Thijs!bot, Wikid77, Calvinballing, Thescaryworker, Vertium, Escarbot, RetiredUser124642196, Hm-rox, Trlkly, AntiVandalBot, Ben pcc, John.d.page, Mhaitham.shammaa, Marianne-ja, Fireice, Myanw, Steelpillow, Deadbeef, Deective,Xeno, JNW, Think outside the box, CountingPine, David Eppstein, Martynas Patasius, Vssun, HappyUser, Deepsol~enwiki, MartinBot,Shivdas, Tonea, Ron2, WarthogDemon, TomyDuby, Rascus93, Cuno56, Goingstuckey, Lystrablue, Squids and Chips, Idioma-bot, Lights,VolkovBot, Am Fiosaigear~enwiki, TXiKiBoT, A4bot, Qxz, H3xx, Voorlandt, Martin451, Raymondwinn, Maxim, Ian Strachan, Yan-nis1962, Edwin Herdman, All am rejects, IndulgentReader, Pythian Habenero, Neparis, Gustav von Humpelschmumpel, GamesSmash,SieBot, Yea booooooooooiiiii, Argyle Smit, Flyer22, Akshat1992, JuanFox, Trang Oul, Oxymoron83, Lectron6, OKBot, StaticGull, An-chor Link Bot, Denisarona, D cushman, ClueBot, Justin W Smith, Abhinav, Drmies, DragonBot, Excirial, Cat890, Erebus Morgaine,Psinu, SockPuppetForTomruen, Paddo44, Bradv, Ost316, Amy Dolman, Addbot, Download, Possumman, Numbo3-bot, Ehrenkater,Erutuon, Tide rolls, OlEnglish, Romanskolduns, Ghoongta, Gota 93, Killy mcgee, Snaily, Ben Ben, Legobot, Luckas-bot, Yobot, LegobotII, Johnlemartirao, Wonder, AnomieBOT, DemocraticLuntz, Jim1138, LeftClicker, Randomtomato, Materialscientist, Xqbot, Cyn-daquazy, Jsharpminor, Frosted14, ProtectionTaggingBot, RibotBOT, Entropeter, Aaron Kauppi, Geometryfan, FrescoBot, OgreBot,Pinethicket, Fraxtil, Double sharp, TobeBot, Bluest, Duoduoduo, 4, Updatehelper, Ripchip Bot, Beyond My Ken, EmausBot, Robo-tukas11, Thecheesykid, John Cline, Wayne Slam, Amiruchka, Chewings72, NTox, Senator2029, LZ6387, Whoop whoop pull up, ClueBotNG, This lousy T-shirt, O.Koslowski, Helpful Pixie Bot, August132001, Leice17soccer, Titodutta, WNYY98, Anuj.Kumar.Aggarwal,Hz.tiang, Supernerd11, Gallagher783, Chmarkine, Brad7777, Belnapj, Tipu564, Chip123456, BattyBot, Lukas, StarryGrandma, Chris-Gualtieri, Kelvinsong, Lugia2453, Epicjerrit, Mathuncle, Robopiekiller, Adamtenzer, DavidLeighEllis, Rl1610, SyntaxMedium1, The-pro200123, Person3400, Evagavilan, Tmason101, Pour11, Loraof, Oleaster, Joseph2302, Mandalsir94 and Anonymous: 414 Polytope Source: https://en.wikipedia.org/wiki/Polytope?oldid=669670022 Contributors: Zundark, The Anome, Tomo, Charles Matthews,Hyacinth, Phys, Jaredwf, Altenmann, Gandalf61, Henrygb, Giftlite, Fropu, Waltpohl, Leonard G., Mike40033, Eequor, Tagishsimon,Vadmium, Gdr, Joseph Myers, Tomruen, Icairns, Almit39, Cacycle, Zaslav, Pjrich, Keenan Pepper, David Haslam, Ruud Koot, SCE-hardt, OneWeirdDude, Salix alba, Boccobrock, Doc glasgow, Mathbot, Masnevets, Tdoune, Siddhant, Wavelength, Karlscherer3, Zim-bricchio, Jpbowen, Mysid, Tetracube, Tribaal, Deville, Arthur Rubin, NeilN, SmackBot, Eskimbot, Octahedron80, Nbarth, Mhym, UU,16@r, Cbuckley, Lavaka, Ylloh, MotherFunctor, Thijs!bot, Nadav1, Wayiran, Oatmealcookiemon, Steelpillow, Albmont, David Epp-stein, DirkOliverTheis, N4nojohn, C quest000, Chiswick Chap, VolkovBot, Hersfold, JohnBlackburne, Dchmelik, Gentlemath, Jduchi,Hagman, SieBot, YonaBot, Phuggins, Taemyr, Wendy.krieger, Daveagp, PixelBot, Editor2020, Darkicebot, Addbot, Wickey-nl, SpBot,Luckas-bot, Yobot, AnomieBOT, Ciphers, Orange Knight of Passion, Yupimanoob, Twri, Aaron Kauppi, Chriskv, FrescoBot, DenaturedAlcohol, DivineAlpha, ClickRick, Mjs1991, Double sharp, 4, Distortiondude, WikitanvirBot, NotAnonymous0, ZroBot, Cobaltcigs,MarcoMoellerHamburg, Baseball Watcher, Trevor x1968, Frietjes, BG19bot, Brad7777, BeaumontTaz, Nigellwh, Plesantdreams, Crys-tallizedcarbon, MetazoanMarek and Anonymous: 363.11.2 Images File:A_2-dimensional_polytope.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/A_2-dimensional_polytope.svg Li-cense: CC BY-SA 3.0 Contributors: Own work Original artist: Ylloh File:Assorted_polygons.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1f/Assorted_polygons.svg License: Public do-main Contributors: Based on Polygons.png by Guy Inchbald (Public Domain) Original artist: CountingPine File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? 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