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Polygon

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(Redirected fromPolygonal)
Plane figure bounded by line segments
For other uses, seePolygon (disambiguation).

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.

Ingeometry, apolygon (/ˈpɒlɪɡɒn/) is aplanefigure made up ofline segments connected to form aclosed polygonal chain.

The segments of a closed polygonal chain are called itsedges orsides. The points where two edges meet are the polygon'svertices orcorners. Ann-gon is a polygon withn sides; for example, atriangle is a 3-gon.

Asimple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called asolid polygon. The interior of a solid polygon is itsbody, also known as apolygonal region orpolygonal area. In contexts where one is concerned only with simple and solid polygons, apolygon may refer only to a simple polygon or to a solid polygon.

A polygonal chain may cross over itself, creatingstar polygons and otherself-intersecting polygons. Some sources also consider closed polygonal chains inEuclidean space to be a type of polygon (askew polygon), even when the chain does not lie in a single plane.

A polygon is a 2-dimensional example of the more generalpolytope in any number of dimensions. There are many moregeneralizations of polygons defined for different purposes.

Etymology

The wordpolygon derives from theGreek adjective πολύς (polús) 'much', 'many' and γωνία (gōnía) 'corner' or 'angle'. It has been suggested that γόνυ (gónu) 'knee' may be the origin ofgon.[1]

Classification

Some different types of polygon

Number of sides

Polygons are primarily classified by the number of sides.

Convexity and intersection

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. 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. This condition is true for polygons in any geometry, not just Euclidean.[2]
  • 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, and may be convex or concave. All convex polygons are star-shaped.
  • Self-intersecting: the boundary of the polygon crosses itself. The termcomplex is sometimes used in contrast tosimple, but this usage risks confusion with the idea of acomplex polygon as one which exists in the complexHilbert plane consisting of twocomplex dimensions.
  • Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.

Equality and symmetry

The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called aregularstar polygon.

Miscellaneous

  • 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 lineL: every lineorthogonal to L intersects the polygon not more than twice.

Properties and formulas

Partitioning ann-gon inton − 2 triangles

Euclidean geometry is assumed throughout.

Angles

Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:

Area

Coordinates of a non-convex pentagon

In this section, the vertices of the polygon under consideration are taken to be(x0,y0),(x1,y1),…,(xn−1,yn−1){\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})} in order. For convenience in some formulas, the notation(xn,yn) = (x0,y0) will also be used.

Simple polygons

Further information:Shoelace formula

If the polygon is non-self-intersecting (that is,simple), the signedarea is

A=12∑i=0n−1(xiyi+1−xi+1yi)where xn=x0 and yn=y0,{\displaystyle A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0},}

or, usingdeterminants

16A2=∑i=0n−1∑j=0n−1|Qi,jQi,j+1Qi+1,jQi+1,j+1|,{\displaystyle 16A^{2}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}{\begin{vmatrix}Q_{i,j}&Q_{i,j+1}\\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}},}

whereQi,j{\displaystyle Q_{i,j}} is the squared distance between(xi,yi){\displaystyle (x_{i},y_{i})} and(xj,yj).{\displaystyle (x_{j},y_{j}).}[4][5]

The signed area depends on the ordering of the vertices and of theorientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positivex-axis to the positivey-axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct inabsolute value. This is commonly called theshoelace formula orsurveyor's formula.[6]

The areaA of a simple polygon can also be computed if the lengths of the sides,a1,a2, ...,an and theexterior angles,θ1,θ2, ...,θn are known, from:

A=12(a1[a2sin⁡(θ1)+a3sin⁡(θ1+θ2)+⋯+an−1sin⁡(θ1+θ2+⋯+θn−2)]+a2[a3sin⁡(θ2)+a4sin⁡(θ2+θ3)+⋯+an−1sin⁡(θ2+⋯+θn−2)]+⋯+an−2[an−1sin⁡(θn−2)]).{\displaystyle {\begin{aligned}A={\frac {1}{2}}(a_{1}[a_{2}\sin(\theta _{1})+a_{3}\sin(\theta _{1}+\theta _{2})+\cdots +a_{n-1}\sin(\theta _{1}+\theta _{2}+\cdots +\theta _{n-2})]\\{}+a_{2}[a_{3}\sin(\theta _{2})+a_{4}\sin(\theta _{2}+\theta _{3})+\cdots +a_{n-1}\sin(\theta _{2}+\cdots +\theta _{n-2})]\\{}+\cdots +a_{n-2}[a_{n-1}\sin(\theta _{n-2})]).\end{aligned}}}

The formula was described by Lopshits in 1963.[7]

If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points,Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.

In every polygon with perimeterp and areaA, theisoperimetric inequalityp2>4πA{\displaystyle p^{2}>4\pi A} holds.[8]

For any two simple polygons of equal area, theBolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.

The lengths of the sides of a polygon do not in general determine its area.[9] However, if the polygon is simple and cyclic then the sidesdo determine the area.[10] Of alln-gons with given side lengths, the one with the largest area is cyclic. Of alln-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).[11]

Regular polygons

Many specialized formulas apply to the areas ofregular polygons.

The area of a regular polygon is given in terms of the radiusr of itsinscribed circle and its perimeterp by

A=12⋅p⋅r.{\displaystyle A={\tfrac {1}{2}}\cdot p\cdot r.}

This radius is also termed itsapothem and is often represented asa.

The area of a regularn-gon can be expressed in terms of the radiusR of itscircumscribed circle (the unique circle passing through all vertices of the regularn-gon) as follows:[12][13]

A=R2⋅n2⋅sin⁡2πn=R2⋅n⋅sin⁡πn⋅cos⁡πn{\displaystyle A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}}}

Self-intersecting

The area of aself-intersecting polygon can be defined in two different ways, giving different answers:

  • Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call thedensity 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.[14]
  • 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 one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.[citation needed]

Centroid

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are

Cx=16A∑i=0n−1(xi+xi+1)(xiyi+1−xi+1yi),{\displaystyle C_{x}={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}),}
Cy=16A∑i=0n−1(yi+yi+1)(xiyi+1−xi+1yi).{\displaystyle C_{y}={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}).}

In these formulas, the signed value of areaA{\displaystyle A} must be used.

Fortriangles (n = 3), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true forn > 3. Thecentroid of the vertex set of a polygon withn vertices has the coordinates

cx=1n∑i=0n−1xi,{\displaystyle c_{x}={\frac {1}{n}}\sum _{i=0}^{n-1}x_{i},}
cy=1n∑i=0n−1yi.{\displaystyle c_{y}={\frac {1}{n}}\sum _{i=0}^{n-1}y_{i}.}

Generalizations

The idea of a polygon has been generalized in various ways. Some of the more important include:

  • Aspherical polygon is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows thedigon, a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role incartography (map making) and inWythoff's construction of theuniform polyhedra.
  • Askew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. ThePetrie polygons of the regular polytopes are well known examples.
  • Anapeirogon is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions.
  • Askew apeirogon is an infinite sequence of sides and angles that do not lie in a flat plane.
  • Apolygon with holes is an area-connected or multiply-connected planar polygon with one external boundary and one or more interior boundaries (holes).
  • Acomplex polygon is aconfiguration analogous to an ordinary polygon, which exists in thecomplex plane of tworeal and twoimaginary dimensions.
  • Anabstract polygon is an algebraicpartially ordered set representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be arealization of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized.
  • Apolyhedron is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are calledpolytopes.[15] (In other conventions, the wordspolyhedron andpolytope are used in any dimension, with the distinction between the two that a polytope is necessarily bounded.[16])

Naming

The wordpolygon comes fromLate Latinpolygōnum (a noun), fromGreek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining aGreek-derivednumerical prefix with the suffix-gon, e.g.pentagon,dodecagon. Thetriangle,quadrilateral andnonagon are exceptions.

Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.[17]

Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example theregularstarpentagon is also known as thepentagram.

Polygon names and miscellaneous properties
NameSidesProperties
monogon1Not generally recognised as a polygon,[18] although some disciplines such as graph theory sometimes use the term.[19]
digon2Not generally recognised as a polygon in the Euclidean plane, although it can exist as aspherical polygon.[20]
triangle (or trigon)3The simplest polygon which can exist in the Euclidean plane. Cantile the plane.
quadrilateral (or tetragon)4The simplest polygon which can cross itself; the simplest polygon which can be concave; the simplest polygon which can be non-cyclic. Cantile the plane.
pentagon5[21] The simplest polygon which can exist as a regular star. A star pentagon is known as apentagram or pentacle.
hexagon6[21] Cantile the plane.
heptagon (or septagon)7[21] The simplest polygon such that the regular form is notconstructible withcompass and straightedge. However, it can be constructed using aneusis construction.
octagon8[21]
nonagon (or enneagon)9[21]"Nonagon" mixes Latin [novem = 9] with Greek; "enneagon" is pure Greek.
decagon10[21]
hendecagon (or undecagon)11[21] The simplest polygon such that the regular form cannot be constructed with compass, straightedge, andangle trisector. However, it can be constructed with neusis.[22]
dodecagon (or duodecagon)12[21]
tridecagon (or triskaidecagon)13[21]
tetradecagon (or tetrakaidecagon)14[21]
pentadecagon (or pentakaidecagon)15[21]
hexadecagon (or hexakaidecagon)16[21]
heptadecagon (or heptakaidecagon)17Constructible polygon[17]
octadecagon (or octakaidecagon)18[21]
enneadecagon (or enneakaidecagon)19[21]
icosagon20[21]
icositrigon (or icosikaitrigon)23The simplest polygon such that the regular form cannot be constructed withneusis.[23][22]
icositetragon (or icosikaitetragon)24[21]
icosipentagon (or icosikaipentagon)25The simplest polygon such that it is not known if the regular form can be constructed with neusis or not.[23][22]
triacontagon30[21]
tetracontagon (or tessaracontagon)40[21][24]
pentacontagon (or pentecontagon)50[21][24]
hexacontagon (or hexecontagon)60[21][24]
heptacontagon (or hebdomecontagon)70[21][24]
octacontagon (or ogdoëcontagon)80[21][24]
enneacontagon (or enenecontagon)90[21][24]
hectogon (or hecatontagon)[25]100[21]
257-gon257Constructible polygon[17]
chiliagon1000Philosophers includingRené Descartes,[26]Immanuel Kant,[27]David Hume,[28] have used the chiliagon as an example in discussions.
myriagon10,000
65537-gon65,537Constructible polygon[17]
megagon[29][30][31]1,000,000As with René Descartes's example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised.[32][33][34][35][36][37][38] The megagon is also used as an illustration of the convergence ofregular polygons to a circle.[39]
apeirogon∞A degenerate polygon of infinitely many sides.

To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows.[21] The "kai" term applies to 13-gons and higher and was used byKepler, and advocated byJohn H. Conway for clarity of concatenated prefix numbers in the naming ofquasiregular polyhedra,[25] though not all sources use it.

TensandOnesfinal suffix
-kai-1-hena--gon
20icosi- (icosa- when alone)2-di-
30triaconta- (or triconta-)3-tri-
40tetraconta- (or tessaraconta-)4-tetra-
50pentaconta- (or penteconta-)5-penta-
60hexaconta- (or hexeconta-)6-hexa-
70heptaconta- (or hebdomeconta-)7-hepta-
80octaconta- (or ogdoëconta-)8-octa-
90enneaconta- (or eneneconta-)9-ennea-

History

Historical image of polygons (1699)

Polygons have been known since ancient times. Theregular 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 akrater byAristophanes, found atCaere and now in theCapitoline Museum.[40][41]

The first known systematic study of non-convex polygons in general was made byThomas Bradwardine in the 14th century.[42]

In 1952,Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where eachreal dimension is accompanied by animaginary one, to createcomplex polygons.[43]

In nature

TheGiant's Causeway, inNorthern Ireland

Polygons appear in rock formations, most commonly as the flat facets ofcrystals, where the angles between the sides depend on the type of mineral from which the crystal is made.

Regular hexagons can occur when the cooling oflava forms areas of tightly packed columns ofbasalt, which may be seen at theGiant's Causeway inNorthern Ireland, or at theDevil's Postpile inCalifornia.

Inbiology, the surface of the waxhoneycomb made bybees is an array ofhexagons, and the sides and base of each cell are also polygons.

Computer graphics

Main article:Polygon (computer graphics)
This sectionneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources in this section. Unsourced material may be challenged and removed.(October 2018) (Learn how and when to remove this message)

Incomputer graphics, a polygon is aprimitive used in modelling and rendering. They are defined in a database, containing arrays ofvertices (the coordinates of thegeometrical vertices, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, andmaterials.[44][45]

Any surface is modelled as a tessellation calledpolygon mesh. If a square mesh hasn + 1 points (vertices) per side, there aren squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. There are(n + 1)2 / 2(n2) vertices per triangle. Wheren is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).

The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.

In computer graphics andcomputational geometry, it is often necessary to determine whether a given pointP=(x0,y0){\displaystyle P=(x_{0},y_{0})} lies inside a simple polygon given by a sequence of line segments. This is called thepoint in polygon test.[46]

See also

References

Bibliography

  • Coxeter, H.S.M.;Regular Polytopes, Methuen and Co., 1948 (3rd Edition, Dover, 1973).
  • 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

  1. ^Craig, John (1849).A new universal etymological technological, and pronouncing dictionary of the English language. Oxford University. p. 404.Extract of p. 404
  2. ^Magnus, Wilhelm (1974).Noneuclidean tesselations and their groups. Pure and Applied Mathematics. Vol. 61. Academic Press. p. 37.
  3. ^Kappraff, Jay (2002).Beyond measure: a guided tour through nature, myth, and number. World Scientific. p. 258.ISBN 978-981-02-4702-7.
  4. ^B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ.Math. Debrecen 1, 42–50 (1949)
  5. ^Bourke, Paul (July 1988)."Calculating The Area And Centroid Of A Polygon"(PDF). Archived fromthe original(PDF) on 16 September 2012. Retrieved6 Feb 2013.
  6. ^Bart Braden (1986)."The Surveyor's Area Formula"(PDF).The College Mathematics Journal.17 (4):326–337.doi:10.2307/2686282.JSTOR 2686282. Archived fromthe original(PDF) on 2012-11-07.
  7. ^A.M. Lopshits (1963).Computation of areas of oriented figures. translators: J Massalski and C Mills Jr. D C Heath and Company: Boston, MA.
  8. ^"Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality",Forum Mathematicorum 2, 2002, 129–130"(PDF).
  9. ^Robbins, "Polygons inscribed in a circle",American Mathematical Monthly 102, June–July 1995.
  10. ^Pak, Igor (2005). "The area of cyclic polygons: recent progress on Robbins' conjectures".Advances in Applied Mathematics.34 (4):690–696.arXiv:math/0408104.doi:10.1016/j.aam.2004.08.006.MR 2128993.S2CID 6756387.
  11. ^Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 inMathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
  12. ^Area of a regular polygon – derivation from Math Open Reference.
  13. ^A regular polygon with an infinite number of sides is a circle:limn→+∞R2⋅n2⋅sin⁡2πn=π⋅R2{\displaystyle \lim _{n\to +\infty }R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=\pi \cdot R^{2}}.
  14. ^De Villiers, Michael (January 2015)."Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral"(PDF).Learning and Teaching Mathematics.2015 (18):23–28.
  15. ^Coxeter (3rd Ed 1973)
  16. ^Günter Ziegler (1995). "Lectures on Polytopes". SpringerGraduate Texts in Mathematics,ISBN 978-0-387-94365-7. p. 4.
  17. ^abcdMathworld
  18. ^Grunbaum, B.; "Are your polyhedra the same as my polyhedra",Discrete and computational geometry: the Goodman-Pollack Festschrift, Ed. Aronov et al., Springer (2003), p. 464.
  19. ^Hass, Joel; Morgan, Frank (1996). "Geodesic nets on the 2-sphere".Proceedings of the American Mathematical Society.124 (12):3843–3850.doi:10.1090/S0002-9939-96-03492-2.JSTOR 2161556.MR 1343696.
  20. ^Coxeter, H.S.M.;Regular polytopes, Dover Edition (1973), p. 4.
  21. ^abcdefghijklmnopqrstuvwxySalomon, David (2011).The Computer Graphics Manual. Springer Science & Business Media. pp. 88–90.ISBN 978-0-85729-886-7.
  22. ^abcBenjamin, Elliot; Snyder, C (May 2014). "On the construction of the regular hendecagon by marked ruler and compass".Mathematical Proceedings of the Cambridge Philosophical Society.156 (3):409–424.Bibcode:2014MPCPS.156..409B.doi:10.1017/S0305004113000753.
  23. ^abArthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151–164,doi:10.1080/00029890.2002.11919848
  24. ^abcdefThe New Elements of Mathematics: Algebra and Geometry byCharles Sanders Peirce (1976), p.298
  25. ^ab"Naming Polygons and Polyhedra".Ask Dr. Math. The Math Forum – Drexel University. Retrieved3 May 2015.
  26. ^Sepkoski, David (2005)."Nominalism and constructivism in seventeenth-century mathematical philosophy".Historia Mathematica.32:33–59.doi:10.1016/j.hm.2003.09.002.
  27. ^Gottfried Martin (1955),Kant's Metaphysics and Theory of Science, Manchester University Press,p. 22.
  28. ^David Hume,The Philosophical Works of David Hume, Volume 1, Black and Tait, 1826,p. 101.
  29. ^Gibilisco, Stan (2003).Geometry demystified (Online-Ausg. ed.). New York: McGraw-Hill.ISBN 978-0-07-141650-4.
  30. ^Darling, David J.,The universal book of mathematics: from Abracadabra to Zeno's paradoxes, John Wiley & Sons, 2004. p. 249.ISBN 0-471-27047-4.
  31. ^Dugopolski, Mark,College Algebra and Trigonometry, 2nd ed, Addison-Wesley, 1999. p. 505.ISBN 0-201-34712-1.
  32. ^McCormick, John Francis,Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
  33. ^Merrill, John Calhoun and Odell, S. Jack,Philosophy and Journalism, Longman, 1983, p. 47,ISBN 0-582-28157-1.
  34. ^Hospers, John,An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56,ISBN 0-415-15792-7.
  35. ^Mandik, Pete,Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26,ISBN 1-84706-349-7.
  36. ^Kenny, Anthony,The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124,ISBN 0-19-875277-6.
  37. ^Balmes, James,Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
  38. ^Potter, Vincent G.,On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86,ISBN 0-8232-1486-9.
  39. ^Russell, Bertrand,History of Western Philosophy, reprint edition, Routledge, 2004, p. 202,ISBN 0-415-32505-6.
  40. ^Heath, Sir Thomas Little (1981).A History of Greek Mathematics, Volume 1. Courier Dover Publications. p. 162.ISBN 978-0-486-24073-2. Reprint of original 1921 publication with corrected errata. Heath uses the Latinized spelling "Aristophonus" for the vase painter's name.
  41. ^Cratere with the blinding of Polyphemus and a naval battleArchived 2013-11-12 at theWayback Machine, Castellani Halls, Capitoline Museum, accessed 2013-11-11. Two pentagrams are visible near the center of the image,
  42. ^Coxeter, H.S.M.;Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p. 114
  43. ^Shephard, G.C.; "Regular complex polytopes",Proc. London Math. Soc. Series 3 Volume 2, 1952, pp 82–97
  44. ^"opengl vertex specification".
  45. ^"direct3d rendering, based on vertices & triangles". 6 January 2021.
  46. ^Schirra, Stefan (2008). "How Reliable Are Practical Point-in-Polygon Strategies?". In Halperin, Dan; Mehlhorn, Kurt (eds.).Algorithms - ESA 2008: 16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008, Proceedings. Lecture Notes in Computer Science. Vol. 5193. Springer. pp. 744–755.doi:10.1007/978-3-540-87744-8_62.ISBN 978-3-540-87743-1.

External links

Look uppolygon in Wiktionary, the free dictionary.
Wikimedia Commons has media related toPolygons.
Triangles
Quadrilaterals
By number
of sides
1–10 sides
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Star polygons
Classes
Fundamental convexregular anduniform polytopes in dimensions 2–10
FamilyAnBnI2(p) /DnE6 /E7 /E8 /F4 /G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron •CubeDemicubeDodecahedron •Icosahedron
Uniform polychoronPentachoron16-cell •TesseractDemitesseract24-cell120-cell •600-cell
Uniform 5-polytope5-simplex5-orthoplex •5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex •6-cube6-demicube122 •221
Uniform 7-polytope7-simplex7-orthoplex •7-cube7-demicube132 •231 •321
Uniform 8-polytope8-simplex8-orthoplex •8-cube8-demicube142 •241 •421
Uniform 9-polytope9-simplex9-orthoplex •9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex •10-cube10-demicube
Uniformn-polytopen-simplexn-orthoplex •n-cuben-demicube1k2 •2k1 •k21n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds
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