polygon
1 Definitions
We follow Forder [2] for most of this entry. The term polygon^{} can be defined if one has a definition of an interval^{}. For this entry we use betweenness geometry^{}. A betweenness geometry is just one for which there is a set of points and a betweenness relation $B$ defined. Rather than write $(a,b,c)\in B$ we write $a*b*c$.

1.
If $a$ and $b$ are distinct points, the line $a\mathit{}b$ is the set of all points $p$ such that $p*a*b$ or $a*p*b$ or $a*b*p$. It can be shown that the line $ab$ and the line $ba$ are the same set of points.

2.
If $o$ and $a$ are distinct points, a ray $\mathrm{[}oa$ is the set of all points $p$ such that $p=o$ or $o*p*a$ or $o*a*p$.

3.
If $a$ and $b$ are distinct points, the open interval is the set of points $p$ such that $a*p*b$. It is denoted by $(a,b).$

4.
If $a$ and $b$ are distinct points, the closed interval is $(a,b)\cup \{a\}\cup \{b\}$, and denoted by $[a,b].$

5.
The way ${a}_{\mathrm{1}}\mathit{}{a}_{\mathrm{2}}\mathit{}\mathrm{\dots}\mathit{}{a}_{n}$ is the finite set^{} of points $\{{a}_{1},\mathrm{\dots},{a}_{n}\}$ along with the open intervals $({a}_{1},{a}_{2}),({a}_{2},{a}_{3}),\mathrm{\dots},({a}_{n1},{a}_{n})$. The points ${a}_{1},\mathrm{\dots},{a}_{n}$ are called the vertices of the way, and the open intervals are called the sides of the way. A way is also called a broken line. The closed intervals $[{a}_{1},{a}_{2}],\mathrm{\dots},[{a}_{n1},{a}_{n}]$ are called the sideintervals of the way. The lines ${a}_{1}{a}_{2},\mathrm{\dots},{a}_{n1}{a}_{n}$ are called the sidelines of the way. The way ${a}_{1}{a}_{2}\mathrm{\dots}{a}_{n}$ is said to join ${a}_{1}$ to ${a}_{n}$. It is assumed that ${a}_{i1},{a}_{i},{a}_{i+1}$ are not collinear^{}.

6.
A way is said to be simple if it does not meet itself. To be precise, (i) no two sideintervals meet in any point which is not a vertex, and (ii) no three sideintervals meet in any point.

7.
A polygon is a way ${a}_{1}{a}_{2}\mathrm{\dots}{a}_{n}$ for which ${a}_{1}={a}_{n}$. Notice that there is no assumption^{} that the points are coplanar^{}.

8.
A simple polygon is polygon for which the way is simple.

9.
A region is a set of points not all collinear, any two of which can be joined by points of a way using only points of the region.

10.
A region $R$ is convex if for each pair of points $a,b\in R$ the open interval $(a,b)$ is contained in $R.$

11.
Let $X$ and $Y$ be two sets of points. If there is a set of points $S$ such that every way joining a point of $X$ to a point of $Y$ meets $S$ then $S$ is said to separate $X$ from $Y$.

12.
If ${a}_{1}{a}_{2}\mathrm{\dots}{a}_{n}$ is a polygon, then the angles of the polygon are $\mathrm{\angle}{a}_{n}{a}_{1}{a}_{2},\mathrm{\angle}{a}_{1}{a}_{2}{a}_{3}$, and so on.
Now assume that all points of the geometry are in one plane. Let $P$ be a polygon. ($P$ is called a plane polygon.)

1.
A ray or line which does not go through a vertex of $P$ will be called suitable.

2.
An inside point $a$ of $P$ is one for which a suitable ray from $a$ meets $P$ an odd number^{} of times. Points that are not on or inside $P$ are said to be outside $P$.

3.
Let $\{{P}_{i}\}$ be a set of polygons. We say that $\{{P}_{i}\}$ dissect $P$ if the following three conditions are satisfied: (i) ${P}_{i}$ and ${P}_{j}$ do not have a common inside point for $i\ne j$, (ii) each inside point of $P$ is inside or on some ${P}_{i}$ and (iii) each inside point of ${P}_{i}$ is inside $P$.

4.
A convex polygon is one whose inside points are all on the same side of any sideline of the polygon.
2 Theorems
Assume that all points are in one plane. Let $P$ be a polygon.

1.
It can be shown that $P$ separates the other points of the plane into at least two regions and that if $P$ is simple there are exactly two regions. Moise proves this directly in [3], pp. 1618.

2.
It can be shown that $P$ can be dissected into triangles^{} $\{{T}_{i}\}$ such that every vertex of a ${T}_{i}$ is a vertex of $P$.

3.
The following theorem^{} of Euler can be shown: Suppose $P$ is dissected into $f>1$ polygons and that the total number of vertices of these polygons is $v$, and the number of open intervals which are sides is $e$. Then
$$ve+f=1$$ .
A plane simple polygon with $n$ sides is called an $n$gon, although for small $n$ there are more traditional names:
Number of sides  Name of the polygon 

3  triangle 
4  quadrilateral^{} 
5  pentagon^{} 
6  hexagon^{} 
7  heptagon 
8  octagon 
10  decagon 
A plane simple polygon is also called a Jordan polygon.
References
 1 K. Borsuk and W. Szmielew, Foundations of Geometry, NorthHolland Publishing Company, 1960.
 2 H.G. Forder, The Foundations of Euclidean Geometry, Dover Publications, 1958.
 3 E.E. Moise, Geometric Topology in Dimensions 2 and 3, SpringerVerlag, 1977.
Title  polygon 
Canonical name  Polygon 
Date of creation  20130322 12:10:15 
Last modified on  20130322 12:10:15 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  43 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 5100 
Classification  msc 51G05 
Related topic  RegularPolygon 
Related topic  Semiperimeter 
Related topic  EquilateralPolygon 
Related topic  EquiangularPolygon 
Related topic  Pentagon 
Related topic  BasicPolygon 
Related topic  Hexagon 
Related topic  GeneralizedPythagoreanTheorem 
Defines  side 
Defines  vertex 
Defines  vertices 
Defines  simple polygon 
Defines  sidelines 
Defines  ray 
Defines  simple way 
Defines  way 
Defines  region 
Defines  convex region 
Defines  Jordan polygon 
Defines  angles of a polygon 
Defines  plane polygon 
Defines  broken line 