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| \PMlinkescapetext{This entry is not yet complete.} |
\PMlinkescapetext{This entry is not yet complete.} |
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| Let $V$ be a Euclidean vector space, $F \subseteq V$, and $E \colon V \to V$ be a Euclidean transformation that is not the identity map. |
Let $V$ be a Euclidean vector space, $F \subseteq V$, and $E \colon V \to V$ be a Euclidean transformation that is not the identity map. |
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| The following terms are used to indicate that $E(F)=F$ if $E$ is a rotation: |
The following terms are used to indicate that $E(F)=F$ if $E$ is a rotation: |
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| \begin{itemize} |
\begin{itemize} |
| \item $F$ has \emph{rotational symmetry}; |
\item $F$ has \emph{rotational symmetry}; |
| \item $F$ has \emph{point symmetry}; |
\item $F$ has \emph{point symmetry}; |
| \item $F$ has \emph{symmetry about a point}; |
\item $F$ has \emph{symmetry about a point}; |
| \item $F$ is \emph{symmetric about a point}. |
\item $F$ is \emph{symmetric about a point}. |
| \end{itemize} |
\end{itemize} |
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| 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} |
If $V=\mathbb{R}^2$, then the last two terms may be used to indicate the specific case in which $E$ is conjugate to $\displaystyle \left( \begin{array}{rr} |
| -1 & 0 \\ |
-1 & 0 \\ |
| 0 & -1 \end{array} \right)$, \PMlinkname{i.e.}{Ie} the angle of rotation is $180^{\circ}$. |
0 & -1 \end{array} \right)$, \PMlinkname{i.e.}{Ie} the angle of rotation is $180^{\circ}$. |
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| The following are classic examples of rotational symmetry in $\mathbb{R}^2$: |
For example, let $\displaystyle F=\bigcup_{k=1}^4 P_4$, where $\displaystyle P_1=\left\{ (x,y) : 0 \le x \le \frac{4}{1+\sqrt{3}} \text{ and } (2-\sqrt{3})x \le y \le x \right\}$, $\displaystyle P_2=\left\{ (x,y) : \frac{4}{1+\sqrt{3}} \le x \le 2 \text{ and } x \le y \le (2+\sqrt{3})x-4 \right\}$, $\displaystyle P_3=\left\{ (x,y) : 2 \le x \le \frac{4 \sqrt{3}}{1+\sqrt{3}} \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le (-2-\sqrt{3})x+4+4\sqrt{3} \right\}$, and $\displaystyle P_4=\left\{ (x,y) : \frac{4 \sqrt{3}}{1+\sqrt{3}} \le x \le 4 \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le -x+4 \right\}$ has point symmetry with respect to the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$. The valid angles of rotation for $F$ are $120^{\circ}$ and $240^{\circ}$. The boundary of $F$ and the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$ are shown in the following picture. |
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| \begin{itemize} |
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| \item Regular polygons: A regular $n$-gon is symmetric about its \PMlinkname{center}{Center9} with valid angles of rotation $\displaystyle \theta=\left( \frac{360k}{n} \right)^{\circ}$ for any positive integer $k<n$. |
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| \item Circles: A circle is symmetric about its \PMlinkname{center}{Center8} with uncountably many valid angles of rotation. |
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| \end{itemize} |
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| As another example, let $\displaystyle F=\bigcup_{k=1}^4 P_4$, where $\displaystyle P_1=\left\{ (x,y) : 0 \le x \le \frac{4}{1+\sqrt{3}} \text{ and } (2-\sqrt{3})x \le y \le x \right\}$, $\displaystyle P_2=\left\{ (x,y) : \frac{4}{1+\sqrt{3}} \le x \le 2 \text{ and } x \le y \le (2+\sqrt{3})x-4 \right\}$, $\displaystyle P_3=\left\{ (x,y) : 2 \le x \le \frac{4 \sqrt{3}}{1+\sqrt{3}} \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le (-2-\sqrt{3})x+4+4\sqrt{3} \right\}$, and $\displaystyle P_4=\left\{ (x,y) : \frac{4 \sqrt{3}}{1+\sqrt{3}} \le x \le 4 \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le -x+4 \right\}$ has point symmetry with respect to the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$. The valid angles of rotation for $F$ are $120^{\circ}$ and $240^{\circ}$. The boundary of $F$ and the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$ are shown in the following picture. |
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\begin{center} |
| \begin{pspicture}(0,0)(4,3.5) |
\begin{pspicture}(0,0)(4,3.5) |
| \pspolygon(0,0)(2,0.536)(4,0)(2.5359,1.4641)(2,3.4641)(1.4641,1.4641) |
\pspolygon(0,0)(2,0.536)(4,0)(2.5359,1.4641)(2,3.4641)(1.4641,1.4641) |
| \psdot(2,1.1547) |
\psdot(2,1.1547) |
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\end{pspicture} |
| \end{center} |
\end{center} |
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| As a final example, the figure |
As another example, the figure $\{ (x,y) : -3 \le x \le -1 \text{ and } (x+1)^2+y^2 \le 4 \} \cup \big( [-1,1] \times [-2,2] \big) \cup \{ (x,y) : 1 \le x \le 3 \text{ and } (x-1)^2+y^2 \le 4 \}$ is symmetric about the origin. The boundary of this figure and the point $(0,0)$ are shown in the following picture. |
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| \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 origin. The boundary of this figure and the point $(0,0)$ are shown in the following picture. |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-3,-2)(3,2) |
\begin{pspicture}(-3,-2)(3,2) |
| \psarc(-1,0){2}{180}{270} |
\psarc(-1,0){2}{180}{270} |
| \psline(-1,-2)(1,-2)(1,0)(3,0) |
\psline(-1,-2)(1,-2)(1,0)(3,0) |
| \psarc(1,0){2}{0}{90} |
\psarc(1,0){2}{0}{90} |
| \psline(1,2)(-1,2)(-1,0)(-3,0) |
\psline(1,2)(-1,2)(-1,0)(-3,0) |
| \psdot(0,0) |
\psdot(0,0) |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| \PMlinkescapetext{How to determine $E$ for a specific point} $p \in \mathbb{R}^2$ \PMlinkescapetext{will be added.} |
\PMlinkescapetext{How to determine $E$ for a specific point} $p \in \mathbb{R}^2$ \PMlinkescapetext{will be added.} |
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| If $E(F)=F$ and $E$ is a reflection, then $F$ has \emph{reflectional symmetry}. In the special case that $V=\mathbb{R}^2$, the following terms are used: |
If $E(F)=F$ and $E$ is a reflection, then $F$ has \emph{reflectional symmetry}. In the special case that $V=\mathbb{R}^2$, the following terms are used: |
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| \begin{itemize} |
\begin{itemize} |
| \item $F$ has \emph{line symmetry}; |
\item $F$ has \emph{line symmetry}; |
| \item $F$ has \emph{symmetry about a line}; |
\item $F$ has \emph{symmetry about a line}; |
| \item $F$ is \emph{symmetric about a line}. |
\item $F$ is \emph{symmetric about a line}. |
| \end{itemize} |
\end{itemize} |
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| The following are classic examples of line symmetry in $\mathbb{R}^2$: |
\PMlinkescapetext{Picture in the case of} $V=\mathbb{R}^2$ \PMlinkescapetext{will be added.} |
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| \begin{itemize} |
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| \item Regular polygons: There are $n$ lines of symmetry of a regular $n$-gon. Each of these pass through its center and at least one of its vertices. |
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| \item Circles: A circle is symmetric about any line passing through its center. |
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| \end{itemize} |
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| As another example, the isosceles trapezoid $T=\{ (x,y) : 0 \le x \le 6 \text{ and } 0 \le y \le \min\{x,2,-x+6\} \}$ is symmetric about $x=3$. In the following picture, the boundary of $T$ is drawn in black, and the line $x=3$ is drawn in cyan. |
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| \pspolygon(0,0)(6,0)(4,2)(2,2) |
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| \psline[linecolor=cyan]{<->}(3,-0.5)(3,2.5) |
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| \end{pspicture} |
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| \PMlinkescapetext{How to determine $E$ for a specific line} $\ell \subseteq \mathbb{R}^2$ \PMlinkescapetext{will be added.} |
\PMlinkescapetext{How to determine $E$ for a specific line} $\ell \subseteq \mathbb{R}^2$ \PMlinkescapetext{will be added.} |
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| \PMlinkescapetext{Feel free to add these derivations of $E$ before I do if you wish!} |
\PMlinkescapetext{Feel free to add these before I do if you wish!} |
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| \PMlinkescapetext{Feel free to add to the list of classic examples too!} |
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