Polygon

For other uses, see Polygon (disambiguation).
Some polygons of different kinds: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (both), and self-intersecting with varying densities of different regions.

In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.

The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other self-intersecting polygons. 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 may sometimes be allowed. These and other generalizations of polygons are described below.

Etymology

The word "polygon" derives from the Greek adjective πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". It has been suggested that γόνυ (gónu) "knee" may be the origin of “gon”,[1]

Classification

Some different types of polygon

Number of sides

Polygons are primarily classified by the number of sides. See table below.

Convexity and non-convexity

Polygons may be characterized by their convexity or type of non-convexity:

Equality and symmetry

Miscellaneous

Properties and formulas

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 and centroid

Simple polygons

Coordinates of a non-convex pentagon.

For a non-self-intersecting (simple) polygon with n vertices xi, yi ( i = 1 to n), the signed area and the Cartesian coordinates of the centroid are given by:[3]

A = \frac{1}{2}  \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i)  , \,
16 A^{2} = \sum_{i=1}^{n} \sum_{j=1}^{n} \begin{vmatrix}  Q_{i,j} & Q_{i,j+1} , \\
Q_{i+1,j} & Q_{i+1,j+1} \\ \end{vmatrix} , \,

where  Q_{i,j} is the squared distance between (x_i, y_i) and (x_j, y_j); [4] and

C_x = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i), \,
C_y = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i).\,

To close the polygon, the first and last vertices are the same, i.e., xn, yn = x0, y0. The vertices must be ordered according 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 C_x and C_y, the signed value of A (which in this case is negative) should be used. This is commonly called the shoelace formula or Surveyor's formula.[5]

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

\begin{align}A = \frac12 ( 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{align}

The formula was described by Lopshits in 1963.[6]

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 perimeter p and area A , the isoperimetric inequality p^2 > 4\pi A holds.[7]

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

A = \tfrac{1}{2} \cdot p \cdot 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 is

A = \frac{ns}{4} \sqrt{4-s^{2}}.

The area of a regular n-gon in terms of the radius R of its circumscribed circle and its perimeter p is given by

A = \frac {R}{2} \cdot p \cdot \sqrt{1- \tfrac{p^{2}}{4n^{2}R^{2}}}.

The area of a regular n-gon inscribed in a unit-radius circle, with side s and interior angle \alpha, can also be expressed trigonometrically as

A = \frac{ns^{2}}{4}\cot \frac{\pi}{n} = \frac{ns^{2}}{4}\cot\frac{\alpha}{n-2}=n \cdot \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n} = n \cdot \sin \frac{\alpha}{n-2} \cdot \cos \frac{\alpha}{n-2}.

The lengths of the sides of a polygon do not in general determine the area.[8] 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).[9]

Self-intersecting polygons

The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer:

Generalizations of polygons

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

Naming polygons

The word "polygon" comes from Late Latin polygōnum (a noun), from Greek 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 a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions.

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

Exceptions exist for side counts that are more 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 the regular star pentagon is also known as the pentagram.

Polygon names and miscellaneous properties
Name Edges Properties
monogon 1 Not generally recognised as a polygon,[11] although some disciplines such as graph theory sometimes use the term.[12]
digon 2 Not generally recognised as a polygon in the Euclidean plane, although it can exist as a spherical polygon.[13]
triangle (or trigon) 3 The simplest polygon which can exist in the Euclidean plane. Can tile the plane.
quadrilateral (or tetragon) 4 The simplest polygon which can cross itself; the simplest polygon which can be concave; the simplest polygon which can be non-cyclic. Can tile the plane.
pentagon 5 [14] The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle.
hexagon 6 [14] Can tile the plane.
heptagon 7 [14] The simplest polygon such that the regular form is not constructible with compass and straightedge. However, it can be constructed using a Neusis construction.
octagon 8 [14]
nonagon (or enneagon) 9 [14]"Nonagon" mixes Latin [novem = 9] with Greek, "enneagon" is pure Greek.
decagon 10 [14]
hendecagon (or undecagon) 11 [14] The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and angle trisector.
dodecagon (or duodecagon) 12 [14]
tridecagon (or triskaidecagon) 13 [14]
tetradecagon (or tetrakaidecagon) 14 [14]
pentadecagon (or pentakaidecagon) 15 [14]
hexadecagon (or hexakaidecagon) 16 [14]
heptadecagon (or heptakaidecagon) 17 Constructible polygon[10]
octadecagon (or octakaidecagon) 18 [14]
enneadecagon (or enneakaidecagon) 19 [14]
icosagon 20 [14]
icositetragon (or icosikaitetragon) 24 [14]
triacontagon 30 [14]
tetracontagon (or tessaracontagon) 40 [14][15]
pentacontagon (or pentecontagon) 50 [14][15]
hexacontagon (or hexecontagon) 60 [14][15]
heptacontagon (or hebdomecontagon) 70 [14][15]
octacontagon (or ogdoëcontagon) 80 [14][15]
enneacontagon (or enenecontagon) 90 [14][15]
hectogon (or hecatontagon)[16] 100 [14]
  257 Constructible polygon[10]
chiliagon 1000 Philosophers including René Descartes,[17] Immanuel Kant,[18] David Hume,[19] have used the chiliagon as an example in discussions.
myriagon 10,000 Used as an example in some philosophical discussions, for example in Descartes' Meditations on First Philosophy
  65,537 Constructible polygon[10]
megagon[20][21][22] 1,000,000 As with René Descartes' example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised.[23][24][25][26][27][28][29] The megagon is also used as an illustration of the convergence of regular polygons to a circle.[30]
apeirogon A degenerate polygon of infinitely many sides.

Constructing higher names

To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows.[14] The "kai" term applies to 13-gons and higher was used by Kepler, and advocated by John H. Conway for clarity to concatenated prefix numbers in the naming of quasiregular polyhedra.[31]

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

History

Historical image of polygons (1699)

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with the pentagram, a non-convex regular polygon (star polygon), appearing as early as the 7th century B.C. on a krater by Aristonothos, found at Caere and now in the Capitoline Museum.[32][33]

The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century.[34]

In 1952, Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.[35]

Polygons in nature

Polygons appear in rock formations, most commonly as the flat facets of crystals, 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 of lava forms areas of tightly packed columns of basalt, which may be seen at the Giant's Causeway in Northern Ireland, or at the Devil's 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 each cell are also polygons.

Polygons in computer graphics

A polygon in a computer graphics (image generation) system is a two-dimensional shape that is modelled and stored within its database. A polygon can be colored, shaded and textured, and its position in the database is defined by the coordinates of its vertices (corners).

Naming conventions differ from those of mathematicians:

Any surface is modelled as a tessellation called polygon mesh. If a square mesh has n + 1 points (vertices) per side, there are n 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. Where n 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 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.

See also

References

Bibliography

Notes

  1. Craig, John (1849). A new universal etymological technological, and pronouncing dictionary of the English language. Oxford University. p. 404. Extract of page 404
  2. Kappraff, Jay (2002). Beyond measure: a guided tour through nature, myth, and number. World Scientific. p. 258. ISBN 978-981-02-4702-7.
  3. Bourke, Paul (July 1988). "Calculating The Area And Centroid Of A Polygon" (PDF). Retrieved 6 Feb 2013.
  4. B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949)
  5. Bart Braden (1986). "The Surveyor's Area Formula" (PDF). The College Mathematics Journal 17 (4): 326–337. doi:10.2307/2686282.
  6. A.M. Lopshits (1963). Computation of areas of oriented figures. translators: J Massalski and C Mills, Jr. D C Heath and Company: Boston, MA.
  7. Dergiades,Nikolaos, "An elementary proof of the isoperimetric inequality", Forum Mathematicorum 2, 2002, 129–130.
  8. Robbins, "Polygons inscribed in a circle," American Mathematical Monthly 102, June–July 1995.
  9. Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
  10. 1 2 3 4 Mathworld
  11. 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.
  12. 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.
  13. Coxeter, H.S.M.; Regular polytopes, Dover Edition (1973), Page 4.
  14. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Salomon, David (2011). The Computer Graphics Manual. Springer Science & Business Media. pp. 88–90. ISBN 978-0-85729-886-7.
  15. 1 2 3 4 5 6 The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  16. http://mathforum.org/dr.math/faq/faq.polygon.names.html
  17. Sepkoski, David (2005). "Nominalism and constructivism in seventeenth-century mathematical philosophy" (PDF). Historia Mathematica 32: 33–59. doi:10.1016/j.hm.2003.09.002. Retrieved 18 April 2012.
  18. Gottfried Martin (1955), Kant's Metaphysics and Theory of Science, Manchester University Press, p. 22.
  19. David Hume, The Philosophical Works of David Hume, Volume 1, Black and Tait, 1826, p. 101.
  20. Gibilisco, Stan (2003). Geometry demystified (Online-Ausg. ed.). New York: McGraw-Hill. ISBN 978-0-07-141650-4.
  21. 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.
  22. Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.
  23. McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
  24. Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0-582-28157-1.
  25. Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.
  26. Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.
  27. Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.
  28. Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
  29. Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9.
  30. Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.
  31. "Naming Polygons and Polyhedra". Ask Dr. Math. The Math Forum – Drexel University. Retrieved 3 May 2015.
  32. Heath, Sir Thomas Little (1981), A History of Greek Mathematics, Volume 1, Courier Dover Publications, p. 162, ISBN 9780486240732. Reprint of original 1921 publication with corrected errata. Heath uses the spelling "Aristonophus" for the vase painter's name.
  33. 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,
  34. Coxeter, H.S.M.; Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p.114
  35. Shephard, G.C.; "Regular complex polytopes", Proc. London Math. Soc. Series 3 Volume 2, 1952, pp 82-97

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