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# 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 $ab$*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 $a_{1}a_{2}\ldots a_{n}$*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. 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. 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 $P$.

3.

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.

## Mathematics Subject Classification

51-00*no label found*51G05

*no label found*

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## Comments

## another polygon entry?

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.

## Re: another polygon entry?

You have a very good thought -- please write a understandable definition of polygon!

## Re: another polygon entry?

While I think it's a good idea for there to be another entry on polygon which anyone with a high school education can understand, it wouldn't hurt for this entry to have a diagram or two. In fact, it would be nice for the two entries to have the same pictures.

As for the canonical name, you could do what Anton did the simple entry about length.

## Re: another polygon entry?

Sorry for taking so long to get back to this, but I plan on finally adding a polygon entry as described in previous posts some time this weekend. Of course, as I want it to be an entry that virtually anyone can read, there will be pictures (pretty pictures I hope!).