We follow Forder  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 defined. Rather than write we write .
If and are distinct points, the line is the set of all points such that or or . It can be shown that the line and the line are the same set of points.
If and are distinct points, a ray is the set of all points such that or or .
If and are distinct points, the open interval is the set of points such that . It is denoted by
If and are distinct points, the closed interval is , and denoted by
The way is the finite set of points along with the open intervals . The points 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 are called the side-intervals of the way. The lines are called the side-lines of the way. The way is said to join to . It is assumed that are not collinear.
A way is said to be simple if it does not meet itself. To be precise, (i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals meet in any point.
A simple polygon is polygon for which the way is simple.
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.
A region is convex if for each pair of points the open interval is contained in
Let and be two sets of points. If there is a set of points such that every way joining a point of to a point of meets then is said to separate from .
If is a polygon, then the angles of the polygon are , and so on.
Now assume that all points of the geometry are in one plane. Let be a polygon. ( is called a plane polygon.)
A ray or line which does not go through a vertex of will be called suitable.
An inside point of is one for which a suitable ray from meets an odd number of times. Points that are not on or inside are said to be outside .
Let be a set of polygons. We say that dissect if the following three conditions are satisfied: (i) and do not have a common inside point for , (ii) each inside point of is inside or on some and (iii) each inside point of is inside .
A convex polygon is one whose inside points are all on the same side of any side-line of the polygon.
Assume that all points are in one plane. Let be a polygon.
It can be shown that separates the other points of the plane into at least two regions and that if is simple there are exactly two regions. Moise proves this directly in , pp. 16-18.
It can be shown that can be dissected into triangles such that every vertex of a is a vertex of .
The following theorem of Euler can be shown: Suppose is dissected into polygons and that the total number of vertices of these polygons is , and the number of open intervals which are sides is . Then
A plane simple polygon with sides is called an -gon, although for small there are more traditional names:
|Number of sides||Name of the polygon|
A plane simple polygon is also called a Jordan polygon.
|Date of creation||2013-03-22 12:10:15|
|Last modified on||2013-03-22 12:10:15|
|Last modified by||Mathprof (13753)|
|Defines||angles of a polygon|