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| \section{Definitions} |
\section{Definitions} |
| We follow Forder \cite{forder} for most of this entry. |
We follow Forder \cite{forder} for most of this entry. |
| The term polygon can be defined if one has a definition of an interval. For this |
The term polygon can be defined if one has a definition of an interval. For this |
| entry we use betweenness geometry. A betweenness geometry |
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. |
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. |
Rather than write $(a,b,c) \in B$ we write a*b*c. |
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| \begin{enumerate} |
\begin{enumerate} |
| \item If $a$ and $b$ are distinct points, the \emph{line $ab$} is the set of |
\item If $a$ and $b$ are distinct points, the \emph{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 |
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. |
that the line $ab$ and the line $ba$ are the same set of points. |
| \item If $o$ and $a$ are distinct points, a \emph{ray $[oa$} is the set of all points $p$ such that |
\item If $o$ and $a$ are distinct points, a \emph{ray $[oa$} is the set of all points $p$ such that |
| $p=o$ or $o*p*a$ or $o*a*p$. |
$p=o$ or $o*p*a$ or $o*a*p$. |
| \item If $a$ and $b$ are distinct points, the \emph{open interval} is the set of points |
\item If $a$ and $b$ are distinct points, the \emph{open interval} is the set of points |
| $p$ such that $a*p*b$. It is denoted by $(a,b).$ |
$p$ such that $a*p*b$. It is denoted by $(a,b).$ |
| \item If $a$ and $b$ are distinct points, the \emph{closed interval} is |
\item If $a$ and $b$ are distinct points, the \emph{closed interval} is |
| $(a,b) \cup \{a\} \cup \{b\}$, and denoted by $[a,b].$ |
$(a,b) \cup \{a\} \cup \{b\}$, and denoted by $[a,b].$ |
| \item The \emph{way $a_1a_2\ldots a_n$} is the finite set of points $\{a_1, \ldots , a_n\}$ |
\item The \emph{way $a_1a_2\ldots a_n$} is the finite set of points $\{a_1, \ldots , a_n\}$ |
| along with the open intervals $(a_1, a_2), (a_2,a_3), \ldots, (a_{n-1}, a_n)$. |
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 \emph{vertices} of the way, and the |
The points $a_1, \ldots, a_n$ are called the \emph{vertices} of the way, and the |
| open intervals are called the \emph{sides} of the way. |
open intervals are called the \emph{sides} of the way. |
| A way is also called a \emph{broken line}. |
A way is also called a \emph{broken line}. |
| The closed intervals $[a_1,a_2], \ldots, [a_{n-1},a_n]$ are called the \emph{side-intervals} of |
The closed intervals $[a_1,a_2], \ldots, [a_{n-1},a_n]$ are called the \emph{side-intervals} of |
| the way. The lines $a_1a_2, \ldots , a_{n-1}a_n$ are called the \emph{side-lines} |
the way. The lines $a_1a_2, \ldots , a_{n-1}a_n$ are called the \emph{side-lines} |
| of the way. |
of the way. |
| The way $a_1a_2\ldots a_n$ is said to \emph{join} $a_1$ to $a_n$. |
The way $a_1a_2\ldots a_n$ is said to \emph{join} $a_1$ to $a_n$. |
| It is assumed that $a_{i-1}, a_i, a_{i+1}$ are not collinear. |
It is assumed that $a_{i-1}, a_i, a_{i+1}$ are not collinear. |
| \item A way is said to be \emph{simple} if it does not meet itself. To be precise, |
\item A way is said to be \emph{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 |
(i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals |
| meet in any point. |
meet in any point. |
| \item A \emph{polygon} is a way $a_1 a_2 \ldots a_n$ for which $a_1 = a_n$. Notice that there is |
\item A \emph{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. |
no assumption that the points are coplanar. |
| \item A \emph{simple polygon} is polygon for which the way is simple. |
\item A \emph{simple polygon} is polygon for which the way is simple. |
| \item A \emph{region} is a set of points not all collinear, any two of which can be joined by points of a way using |
\item A \emph{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. |
only points of the region. |
| \item A region $R$ is \emph{convex} if for each pair of points $a,b \in R$ the open interval $(a,b)$ is |
\item A region $R$ is \emph{convex} if for each pair of points $a,b \in R$ the open interval $(a,b)$ is |
| contained in $R.$ |
contained in $R.$ |
| \item Let $X$ and $Y$ be two sets of points. If there is a set of points $S$ such that every way |
\item 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 \emph{separate} |
joining a point of $X$ to a point of $Y$ meets $S$ then $S$ is said to \emph{separate} |
| $X$ from $Y$. |
$X$ from $Y$. |
| \item If $a_1 a_2 \ldots a_n$ is a polygon, then the \emph{angles of the polygon} are |
\item If $a_1 a_2 \ldots a_n$ is a polygon, then the \emph{angles of the polygon} are |
| $\angle a_na_1a_2, \angle a_1a_2a_3$, and so on. |
$\angle a_na_1a_2, \angle a_1a_2a_3$, and so on. |
| \end{enumerate} |
\end{enumerate} |
|
|
| Now assume that all points of the geometry are in one plane. Let $P$ be a polygon. ($P$ is called |
Now assume that all points of the geometry are in one plane. Let $P$ be a polygon. ($P$ is called |
| a \emph{plane polygon}.) |
a \emph{plane polygon}.) |
| \begin{enumerate} |
\begin{enumerate} |
| \item A ray or line which does not go through a vertex of $P$ will be called \emph{suitable}. |
\item A ray or line which does not go through a vertex of $P$ will be called \emph{suitable}. |
| \item An \emph{inside point} $a$ of $P$ is one for which a suitable ray from $a$ |
\item An \emph{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 \emph{outside} |
meets $P$ an odd number of times. Points that are not on or inside $P$ are said to be \emph{outside} |
| $P$. |
$P$. |
| \item Let $\{P_i\}$ be a set of polygons. We say that $\{P_i\}$ \emph{dissect} $P$ if the following |
\item Let $\{P_i\}$ be a set of polygons. We say that $\{P_i\}$ \emph{dissect} $P$ if the following |
| three conditions are satisfied: (i) $P_i$ and $P_j$ do not have a common inside point for $i \not = j$, |
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 |
(ii) each inside point of $P$ is inside or on some $P_i$ and (iii) each inside point of $P_i$ is |
| inside $P$. |
inside $P$. |
| \item A \emph{convex polygon} is one whose inside points are all on the same side of any side-line |
\item A \emph{convex polygon} is one whose inside points are all on the same side of any side-line |
| of the polygon. |
of the polygon. |
| \end{enumerate} |
\end{enumerate} |
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| \section{Theorems} |
\section{Theorems} |
| Assume that all points are in one plane. Let $P$ be a polygon. |
Assume that all points are in one plane. Let $P$ be a polygon. |
| \begin{enumerate} |
\begin{enumerate} |
| \item It can be shown that $P$ separates the other points of the plane into at least two regions and that |
\item 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 \cite{moise}, pp. 16-18. |
if $P$ is simple there are exactly two regions. Moise proves this directly in \cite{moise}, pp. 16-18. |
| \item It can be shown that $P$ can be dissected into triangles $\{T_i\}$ such that |
\item 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 $P$. |
every vertex of a $T_i$ is a vertex of $P$. |
| \item The following theorem of Euler can be shown: Suppose $P$ is dissected into $f>1$ polygons |
\item 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 |
and that the total number of vertices of these polygons is $v$, and the number of open intervals |
| which are sides is $e$. Then |
which are sides is $e$. Then |
| $$ |
$$ |
| v-e+f = 1 |
v-e+f = 1 |
| $$. |
$$. |
| \end{enumerate} |
\end{enumerate} |
| |
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| A plane simple polygon with $n$ sides is called an $n$-gon, although for small $n$ |
A plane simple polygon with $n$ sides is called an $n$-gon, although for small $n$ |
| there are more traditional names: |
there are more traditional names: |
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|
| \begin{center} |
\begin{center} |
| \begin{tabular}{||c|c||} \hline |
\begin{tabular}{||c|c||} \hline |
| Number of sides& Name of the polygon \\ \hline |
Number of sides& Name of the polygon \\ \hline |
| 3 & triangle \\ |
3 & triangle \\ |
| 4 & quadrilateral \\ |
4 & quadrilateral \\ |
| 5 & pentagon\\ |
5 & pentagon\\ |
| 6 & hexagon \\ |
6 & hexagon \\ |
| 7 & heptagon\\ |
7 & heptagon\\ |
| 8 & octagon\\ |
8 & octagon\\ |
| 10 & decagon\\ |
10 & decagon\\ |
| \hline |
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| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
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| A plane simple polygon is also called a \emph{Jordan polygon}. |
A plane simple polygon is also called a \emph{Jordan polygon}. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{borsuk-szmielew} |
\bibitem{borsuk-szmielew} |
| K. Borsuk and W. Szmielew, \emph{Foundations of Geometry}, |
K. Borsuk and W. Szmielew, \emph{Foundations of Geometry}, |
| North-Holland Publishing Company, 1960. |
North-Holland Publishing Company, 1960. |
| \bibitem{forder} |
\bibitem{forder} |
| H.G. Forder, \emph{The Foundations of Euclidean Geometry}, |
H.G. Forder, \emph{The Foundations of Euclidean Geometry}, |
| Dover Publications, 1958. |
Dover Publications, 1958. |
| \bibitem{moise} |
\bibitem{moise} |
| E.E. Moise, \emph{Geometric Topology in Dimensions 2 and 3}, |
E.E. Moise, \emph{Geometric Topology in Dimensions 2 and 3}, |
| Springer-Verlag, 1977. |
Springer-Verlag, 1977. |
| |
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| \end{thebibliography} |
\end{thebibliography} |
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| \PMlinkescapeword{segments} |
\PMlinkescapeword{segments} |
| \PMlinkescapeword{maximal} |
\PMlinkescapeword{maximal} |
| \PMlinkescapeword{name} |
\PMlinkescapeword{name} |
| \PMlinkescapeword{names} |
\PMlinkescapeword{names} |
| \PMlinkescapeword{meet} |
\PMlinkescapeword{meet} |
| \PMlinkescapeword{opens} |
\PMlinkescapeword{opens} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
| \PMlinkescapeword{divides} |
\PMlinkescapeword{divides} |
| \PMlinkescapeword{bounded} |
\PMlinkescapeword{bounded} |
| \PMlinkescapeword{unbounded} |
\PMlinkescapeword{unbounded} |
| \PMlinkescapeword{complex} |
\PMlinkescapeword{complex} |
