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 $a b$ 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 $a b$ and the line $b a$ are the same set of points.
2.

If $o$ and $a$ are distinct points, a ray $[oa$ 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 $a_{1}a_{2}\ldots a_{n}$ is the finite set of points $\{a_{1},\ldots,a_{n}\}$ along with the open intervals $(a_{1},a_{2}),(a_{2},a_{3}),\ldots,(a_{n-1},a_{n})$ . The points $a_{1},\ldots,a_{n}$ 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 $[a_{1},a_{2}],\ldots,[a_{n-1},a_{n}]$ are called the side-intervals of the way. The lines $a_{1}a_{2},\ldots,a_{n-1}a_{n}$ are called the side-lines of the way. The way $a_{1}a_{2}\ldots a_{n}$ is said to join $a_{1}$ to $a_{n}$ . It is assumed that $a_{i-1},a_{i},a_{i+1}$ are not collinear.
6.

A way is said to be simple if it does not meet itself. To be precise, (i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals meet in any point.
7.

A polygon is a way $a_{1}a_{2}\ldots a_{n}$ 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 $a_{1}a_{2}\ldots a_{n}$ is a polygon, then the angles of the polygon are $\angle a_{n}a_{1}a_{2},\angle a_{1}a_{2}a_{3}$ , 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 $i\not=j$ , (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 side-line 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. 16-18.
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 $P$ .

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

$v-e+f=1$

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, North-Holland 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, Springer-Verlag, 1977.

Title	polygon
Canonical name	Polygon
Date of creation	2013-03-22 12:10:15
Last modified on	2013-03-22 12:10:15
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	43
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 51-00
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	side-lines
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