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symmetry
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.

$F$ has rotational symmetry;

$F$ has point symmetry;

$F$ has symmetry about a point;

$F$ is symmetric about a point.
If $V=\mathbb{R}^{2}$, then the last two terms may be used to indicate the specific case in which $E$ is conjugate to $\displaystyle\left(\begin{array}[]{rr}1&0\\ 0&1\end{array}\right)$, i.e. the angle of rotation is $180^{{\circ}}$.
The following are classic examples of rotational symmetry in $\mathbb{R}^{2}$:

Regular polygons: A regular $n$gon is symmetric about its center with valid angles of rotation $\displaystyle\theta=\left(\frac{360k}{n}\right)^{{\circ}}$ for any positive integer $k<n$.
As another example, let $\displaystyle F=\bigcup_{{k=1}}^{4}P_{k}$, where each $P_{k}$ is defined thus:
$\displaystyle\displaystyle P_{1}$  $\displaystyle=$  $\displaystyle\left\{(x,y):0\leq x\leq\frac{4}{1+\sqrt{3}}\text{ and }(2\sqrt{% 3})x\leq y\leq x\right\},$  
$\displaystyle\displaystyle P_{2}$  $\displaystyle=$  $\displaystyle\left\{(x,y):\frac{4}{1+\sqrt{3}}\leq x\leq 2\text{ and }x\leq y% \leq(2+\sqrt{3})x4\right\},$  
$\displaystyle\displaystyle P_{3}$  $\displaystyle=$  $\displaystyle\left\{(x,y):2\leq x\leq\frac{4\sqrt{3}}{1+\sqrt{3}}\text{ and }(% 2+\sqrt{3})x+84\sqrt{3}\leq y\leq(2\sqrt{3})x+4+4\sqrt{3}\right\},$  
$\displaystyle\displaystyle P_{4}$  $\displaystyle=$  $\displaystyle\left\{(x,y):\frac{4\sqrt{3}}{1+\sqrt{3}}\leq x\leq 4\text{ and }% (2+\sqrt{3})x+84\sqrt{3}\leq y\leqx+4\right\}.$ 
Then $F$ 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.
As a final example, the figure
$\{(x,y):3\leq x\leq1\text{ and }(x+1)^{2}+y^{2}\leq 4\}\cup\big([1,1]\times% [2,2]\big)\cup\{(x,y):1\leq x\leq 3\text{ and }(x1)^{2}+y^{2}\leq 4\}$ is symmetric about the origin. The boundary of this figure and the point $(0,0)$ are shown in the following picture.
If $E(F)=F$ and $E$ is a reflection, then $F$ has reflectional symmetry. In the special case that $V=\mathbb{R}^{2}$, the following terms are used:

$F$ has line symmetry;

$F$ has symmetry about a line;

$F$ is symmetric about a line.
The following are classic examples of line symmetry in $\mathbb{R}^{2}$:

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.

Circles: A circle is symmetric about any line passing through its center.
As another example, the isosceles trapezoid defined by
$T=\{(x,y):0\leq x\leq 6\text{ and }0\leq y\leq\min\{x,2,x+6\}\}$ 
is symmetric about $x=3$.
In the picture above, the boundary of $T$ is drawn in black, and the line $x=3$ is drawn in cyan.
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