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polygon (Definition)

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 $ab$ 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 $[oa$ 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 $a_1a_2\ldots a_n$ 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 $a_1a_2, \ldots , a_{n-1}a_n$ are called the side-lines of the way. The way $a_1a_2\ldots a_n$ 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 $\angle a_na_1a_2, \angle a_1a_2a_3$, 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.

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 $P$.
  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
    \begin{displaymath} v-e+f = 1 \end{displaymath}

    .

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.

Bibliography

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.



"polygon" is owned by Mathprof. [ full author list (6) | owner history (4) ]
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See Also: regular polygon, semiperimeter, equilateral polygon, equiangular polygon, pentagon, polygon, hexagon

Also defines:  side, vertex, vertices, simple polygon, side-lines, ray, simple way, way, region, convex region, Jordan polygon, angles of a polygon, plane polygon, broken line

Attachments:
Schwarz-Christoffel transformation (Result) by pahio
coordinate way (Definition) by pahio
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Cross-references: hexagon, pentagon, quadrilateral, number, Euler, triangles, odd number, plane, meets, contained, convex, coplanar, collinear, finite set, line, betweenness relation, points, geometry, interval, term
There are 224 references to this entry.

This is version 39 of polygon, born on 2002-01-05, modified 2007-09-18.
Object id is 1384, canonical name is Polygon.
Accessed 33159 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51G05 (Geometry :: Ordered geometries )
Pending Errata and Addenda
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Discussion
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forum policy
another polygon entry? by Wkbj79 on 2007-09-14 12:18:53
Although Mathprof has adopted and edited this entry so that the concepts are mathematically precise, I am concerned about the accessibility of this entry to non-mathematicians. The term "polygon" is one that people encounter very early on and thus, in my opinion, should have an entry that gives basic information about polygons that are both mathematically precise and accessible to the general population.

Also, I am pretty sure that this entry went up for adoption due to the fact that terms such as "interior angle" and "exterior angle" are difficult to define in a mathematically precise way. Nevertheless, these terms (along with "angle sum") are commonly used and should appear somewhere in PM.

I have been toying with creating another polygon entry which is meant for people who do not have the mathematical background that is necessary to understand the bulk of the content of the current entry. Before doing this, I wanted to get other people's opinions on this matter.
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