%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: symmetry
%%% Primary Category Code: 51A15
%%% Filename: Symmetry2.tex
%%% Version: 15
%%% Owner: Wkbj79
%%% Author(s): Wkbj79, CWoo
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\begin{document}






Let $V$ be a \htmladdnormallink{Euclidean vector space}{http://planetmath.org/encyclopedia/EuclideanVectorSpace2.html}, $F \subseteq V$, and $E \colon V \to V$ be a \htmladdnormallink{Euclidean transformation}{http://planetmath.org/encyclopedia/Rotate.html} that is not the \htmladdnormallink{identity map}{http://planetmath.org/encyclopedia/IdentityOperator.html}.

The following terms are used to indicate that $E(F)=F$ if $E$ is a \htmladdnormallink{rotation}{http://planetmath.org/encyclopedia/Rotate.html}:

\begin{itemize}
\item $F$ has \emph{rotational symmetry};
\item $F$ has \emph{point symmetry};
\item $F$ has \emph{symmetry about a point};
\item $F$ is \emph{symmetric about a point}.
\end{itemize}

If $V=\mathbb{R}^2$, then the last two terms may be used to indicate the specific case in which $E$ is conjugate to $\displaystyle \left( \begin{array}{rr}
-1 & 0 \\
0 & -1 \end{array} \right)$, \htmladdnormallink{i.e.}{http://planetmath.org/encyclopedia/Ie.html} the \htmladdnormallink{angle of rotation}{http://planetmath.org/encyclopedia/Rotate.html} is $180^{\circ}$.

The following are classic examples of rotational symmetry in $\mathbb{R}^2$:

\begin{itemize}
\item \htmladdnormallink{Regular polygons}{http://planetmath.org/encyclopedia/Center9.html}: A regular $n$-gon is symmetric about its \htmladdnormallink{center}{http://planetmath.org/encyclopedia/Center9.html} with valid angles of rotation $\displaystyle \theta=\left( \frac{360k}{n} \right)^{\circ}$ for any \htmladdnormallink{positive}{http://planetmath.org/encyclopedia/Negative.html} \htmladdnormallink{integer}{http://planetmath.org/encyclopedia/RationalInteger.html} $k<n$.
\item \htmladdnormallink{Circles}{http://planetmath.org/encyclopedia/Center8.html}: A circle is symmetric about its \htmladdnormallink{center}{http://planetmath.org/encyclopedia/Center8.html} with uncountably many valid angles of rotation.
\end{itemize}

As another example, let $\displaystyle F=\bigcup_{k=1}^4 P_k$, where each $P_k$ is defined thus:
\begin{eqnarray*}
\displaystyle P_1&=&\left\{ (x,y) : 0 \le x \le \frac{4}{1+\sqrt{3}} \text{ and } (2-\sqrt{3})x \le y \le x \right\},\\ \displaystyle P_2&=&\left\{ (x,y) : \frac{4}{1+\sqrt{3}} \le x \le 2 \text{ and } x \le y \le (2+\sqrt{3})x-4 \right\},\\
\displaystyle P_3&=&\left\{ (x,y) : 2 \le x \le \frac{4 \sqrt{3}}{1+\sqrt{3}} \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le (-2-\sqrt{3})x+4+4\sqrt{3} \right\},\\
\displaystyle P_4&=&\left\{ (x,y) : \frac{4 \sqrt{3}}{1+\sqrt{3}} \le x \le 4 \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le -x+4 \right\}.
\end{eqnarray*}
Then $F$ has point symmetry with respect to the \htmladdnormallink{point}{http://planetmath.org/encyclopedia/Point.html} $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$. The valid angles of rotation for $F$ are $120^{\circ}$ and $240^{\circ}$. The \htmladdnormallink{boundary}{http://planetmath.org/encyclopedia/TopologicalBoundary.html} of $F$ and the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$ are shown in the following picture.

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As a final example, the figure

\noindent $\{ (x,y) : -3 \le x \le -1 \text{ and } (x+1)^2+y^2 \le 4 \} \cup \big( [-1,1] \times [-2,2] \big) \cup \{ (x,y) : 1 \le x \le 3 \text{ and } (x-1)^2+y^2 \le 4 \}$ is symmetric about the \htmladdnormallink{origin}{http://planetmath.org/encyclopedia/Origin.html}. The boundary of this figure and the point $(0,0)$ are shown in the following picture.

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\psline(-1,-2)(1,-2)(1,0)(3,0)
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If $E(F)=F$ and $E$ is a \htmladdnormallink{reflection}{http://planetmath.org/encyclopedia/Rotate.html}, then $F$ has \emph{reflectional symmetry}. In the special case that $V=\mathbb{R}^2$, the following terms are used:

\begin{itemize}
\item $F$ has \emph{line symmetry};
\item $F$ has \emph{symmetry about a line};
\item $F$ is \emph{symmetric about a line}.
\end{itemize}

The following are classic examples of line symmetry in $\mathbb{R}^2$:

\begin{itemize}
\item Regular polygons: There are $n$ \htmladdnormallink{lines}{http://planetmath.org/encyclopedia/Incident2.html} of symmetry of a regular $n$-gon. Each of these \htmladdnormallink{pass through}{http://planetmath.org/encyclopedia/Incident2.html} its center and at least one of its \htmladdnormallink{vertices}{http://planetmath.org/encyclopedia/PlanePolygon.html}.
\item Circles: A circle is symmetric about any line \htmladdnormallink{passing through}{http://planetmath.org/encyclopedia/Incident2.html} its center.
\end{itemize}

As another example, the \htmladdnormallink{isosceles trapezoid}{http://planetmath.org/encyclopedia/TrisoscelesTrapezium.html} defined by $$T=\{ (x,y) : 0 \le x \le 6 \text{ and } 0 \le y \le \min\{x,2,-x+6\} \}$$ is symmetric about $x=3$.

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In the picture above, the boundary of $T$ is drawn in black, and the line $x=3$ is drawn in cyan.

\end{document}