Difference between revisions of "Definition of Polygons"

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===Number of sides===
 
===Number of sides===
 
Polygons are primarily classified by the number of sides. See the table below.
 
Polygons are primarily classified by the number of sides. See the table below.
 +
 +
===Convexity and non-convexity===
 +
Polygons may be characterized by their convexity or type of non-convexity:
 +
 +
*Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. 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.
 +
*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 term ''complex'' is sometimes used in contrast to ''simple'', but this usage risks confusion with the idea of a ''complex polygon'' as one which exists in the complex Hilbert plane consisting of two complex dimensions.
 +
*Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.
 +
 +
===Equality and symmetry===
 +
*Equiangular: all corner angles are equal.
 +
*Equilateral: all edges are of the same length.
 +
*Regular: both equilateral and equiangular.
 +
*Cyclic: all corners lie on a single circle, called the circumcircle.
 +
*Tangential: all sides are tangent to an inscribed circle.
 +
*Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
 +
*Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential.
 +
 +
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 a ''regular star 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 line ''L'': every line orthogonal to L intersects the polygon not more than twice.
 +
 +
== Naming ==
 +
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.[16]
 +
 +
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.
 +
 +
<div style="text-align: center;"> '''Polygon names and miscellaneous properties''' </div>
  
 
{| class="wikitable"
 
{| class="wikitable"
Line 101: Line 136:
 
|-
 
|-
 
| id=20 | icosagon  
 
| id=20 | icosagon  
 +
| 20
 +
|
 +
|-
 +
| id=21 | icositetragon (or icosikaitetragon)
 +
| 24
 +
|
 +
|-
 +
| id=22 | triacontagon
 +
| 30
 +
|
 +
|-
 +
| id=23 | tetracontagon (or tessaracontagon)
 +
| 40
 +
|
 +
|-
 +
| id=24 | pentacontagon (or pentecontagon)
 +
| 50
 +
|
 +
|-
 +
| id=25 | hexacontagon (or hexecontagon)
 +
| 60
 +
|
 +
|-
 +
| id=26 | heptacontagon (or hebdomecontagon)
 +
| 70
 +
|
 +
|-
 +
| id=27 | octacontagon (or ogdoëcontagon)
 +
| 80
 +
|
 +
|-
 +
| id=28 | enneacontagon (or enenecontagon)
 +
| 90
 +
|
 +
|-
 +
| id=29 | hectogon (or hecatontagon)
 +
| 100
 +
|
 +
|-
 +
| id=30 | 257-gon
 +
| 257
 +
| Constructible polygon
 +
|-
 +
| id=31 | chiliagon
 +
| 1000
 +
| Philosophers including René Descartes, Immanuel Kant, David Hume, have used the chiliagon as an example in discussions.
 +
|-
 +
| id=32 | myriagon
 +
| 10,000
 +
| Used as an example in some philosophical discussions, for example in Descartes's ''Meditations on First Philosophy''
 +
|-
 +
| id=33 | 65537-gon
 +
| 65,537
 +
| Constructible polygon
 +
|-
 +
| id=34 | megagon
 +
| 1,000,000
 +
| As 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. The megagon is also used as an illustration of the convergence of regular polygons to a circle.
 +
|-
 +
| id=35 | apeirogon
 +
| ∞
 +
| A degenerate polygon of infinitely many sides.
 
|}
 
|}
  
−
===Convexity and non-convexity===
+
=== Constructing higher names ===
−
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.
+
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by Kepler, and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra.
−
*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 term ''complex'' is sometimes used in contrast to ''simple'', but this usage risks confusion with the idea of a ''complex polygon'' as one which exists in the complex Hilbert plane consisting of two complex dimensions.
 
−
*Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.
 
  
−
===Equality and symmetry===
+
{| class="wikitable"
−
*Equiangular: all corner angles are equal.
+
! rowspan="2" colspan="2" style="text-align: center;" | Tens
−
*Equilateral: all edges are of the same length.
+
! and
−
*Regular: both equilateral and equiangular.
+
! colspan="2" style="text-align: center;" | Ones
−
*Cyclic: all corners lie on a single circle, called the circumcircle.
+
! final suffix
−
*Tangential: all sides are tangent to an inscribed circle.
+
|-
−
*Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
+
| id=1 rowspan=9 | -kai-
−
*Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential.
+
| 1
−
 
+
| -hena-
−
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 a ''regular star polygon''.
+
| rowspan=9 | -gon
−
 
+
|-
−
=== Miscellaneous ===
+
| id=2 | 20
−
* Rectilinear: the polygon's sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.
+
| icosi- (icosa- when alone)
−
* Monotone with respect to a given line ''L'': every line orthogonal to L intersects the polygon not more than twice.
+
| 2
 +
| -di-
 +
|-
 +
| id=3 | 30
 +
| triaconta- (or triconta-)
 +
| 3
 +
| -tri-
 +
|-
 +
| id=4 | 40
 +
| tetraconta- (or tessaraconta-)
 +
| 4
 +
| -tetra-
 +
|-
 +
| id=5 | 50
 +
| pentaconta- (or penteconta-)
 +
| 5
 +
| -penta-
 +
|-
 +
| id=6 | 60
 +
| hexaconta- (or hexeconta-)
 +
| 6
 +
| -hexa-
 +
|-
 +
| id=7 | 70
 +
| heptaconta- (or hebdomeconta-)
 +
| 7
 +
| -hepta-
 +
|-
 +
| id=8 | 80
 +
| octaconta- (or ogdoëconta-)
 +
| 8
 +
| -octa-
 +
|-
 +
| id=9 | 90
 +
| enneaconta- (or eneneconta-)
 +
| 9
 +
| -ennea-
 +
|}
  
 
== Licensing ==  
 
== Licensing ==  
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [https://en.wikipedia.org/wiki/Polygon Polygon, Wikipedia] under a CC BY-SA license
 
* [https://en.wikipedia.org/wiki/Polygon Polygon, Wikipedia] under a CC BY-SA license

Latest revision as of 21:36, 11 December 2021

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

In geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain (or polygonal circuit). The bounded plane region, the bounding circuit, or the two together, may be called a polygon.

The segments of a polygonal circuit are called its edges or sides. The points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon.

A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons.

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

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.

Classification

Some different types of polygon

Some different types of polygon

Number of sides

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

Convexity and non-convexity

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

  • Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. 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.
  • 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 term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions.
  • Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.

Equality and symmetry

  • Equiangular: all corner angles are equal.
  • Equilateral: all edges are of the same length.
  • Regular: both equilateral and equiangular.
  • Cyclic: all corners lie on a single circle, called the circumcircle.
  • Tangential: all sides are tangent to an inscribed circle.
  • Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
  • Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential.

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 a regular star 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 line L: every line orthogonal to L intersects the polygon not more than twice.

Naming

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.[16]

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 Sides Properties
monogon 1 Not generally recognised as a polygon, although some disciplines such as graph theory sometimes use the term.
digon 2 Not generally recognised as a polygon in the Euclidean plane, although it can exist as a spherical polygon.
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 The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle.
hexagon 6 Can tile the plane.
heptagon (or septagon) 7 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
nonagon (or enneagon) 9 "Nonagon" mixes Latin [novem = 9] with Greek; "enneagon" is pure Greek.
decagon 10
hendecagon (or undecagon) 11 The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and angle trisector.
dodecagon (or duodecagon) 12
tridecagon (or triskaidecagon) 13
tetradecagon (or tetrakaidecagon) 14
pentadecagon (or pentakaidecagon) 15
hexadecagon (or hexakaidecagon) 16
heptadecagon (or heptakaidecagon) 17 Constructible polygon
octadecagon (or octakaidecagon) 18
enneadecagon (or enneakaidecagon) 19
icosagon 20
icositetragon (or icosikaitetragon) 24
triacontagon 30
tetracontagon (or tessaracontagon) 40
pentacontagon (or pentecontagon) 50
hexacontagon (or hexecontagon) 60
heptacontagon (or hebdomecontagon) 70
octacontagon (or ogdoëcontagon) 80
enneacontagon (or enenecontagon) 90
hectogon (or hecatontagon) 100
257-gon 257 Constructible polygon
chiliagon 1000 Philosophers including René Descartes, Immanuel Kant, David Hume, have used the chiliagon as an example in discussions.
myriagon 10,000 Used as an example in some philosophical discussions, for example in Descartes's Meditations on First Philosophy
65537-gon 65,537 Constructible polygon
megagon 1,000,000 As 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. The megagon is also used as an illustration of the convergence of regular polygons to a circle.
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. The "kai" term applies to 13-gons and higher and was used by Kepler, and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra.

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-

Licensing

Content obtained and/or adapted from:

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