symmetry
Let $V$ be a Euclidean vector space, $F\subseteq V$, and $E:V\to V$ be a Euclidean transformation that is not the identity map.
The following terms are used to indicate that $E(F)=F$ if $E$ is a rotation:

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$F$ has rotational symmetry^{};

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$F$ has point symmetry;

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$F$ has symmetry about a point;

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$F$ 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 $\left(\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill 1\end{array}\right)$, i.e. (http://planetmath.org/Ie) the angle of rotation is ${180}^{\circ}$.
The following are classic examples of rotational symmetry in ${\mathbb{R}}^{2}$:

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Regular polygons^{}: A regular^{} $n$gon is symmetric about its center (http://planetmath.org/Center9) with valid angles of rotation $\theta ={\left({\displaystyle \frac{360k}{n}}\right)}^{\circ}$ for any positive integer $$.

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Circles: A circle is symmetric about its center (http://planetmath.org/Center8) with uncountably many valid angles of rotation.
As another example, let $F={\displaystyle \bigcup _{k=1}^{4}}{P}_{k}$, where each ${P}_{k}$ is defined thus:
${P}_{1}$  $=$  $\{(x,y):0\le x\le {\displaystyle \frac{4}{1+\sqrt{3}}}\text{and}(2\sqrt{3})x\le y\le x\},$  
${P}_{2}$  $=$  $\{(x,y):{\displaystyle \frac{4}{1+\sqrt{3}}}\le x\le 2\text{and}x\le y\le (2+\sqrt{3})x4\},$  
${P}_{3}$  $=$  $\{(x,y):2\le x\le {\displaystyle \frac{4\sqrt{3}}{1+\sqrt{3}}}\text{and}(2+\sqrt{3})x+84\sqrt{3}\le y\le (2\sqrt{3})x+4+4\sqrt{3}\},$  
${P}_{4}$  $=$  $\{(x,y):{\displaystyle \frac{4\sqrt{3}}{1+\sqrt{3}}}\le x\le 4\text{and}(2+\sqrt{3})x+84\sqrt{3}\le y\le x+4\}.$ 
Then $F$ has point symmetry with respect to the point $(2,{\displaystyle \frac{2}{\sqrt{3}}})$. The valid angles of rotation for $F$ are ${120}^{\circ}$ and ${240}^{\circ}$. The boundary of $F$ and the point $(2,{\displaystyle \frac{2}{\sqrt{3}}})$ are shown in the following picture.
As a final example, the figure
$\{(x,y):3\le x\le 1\text{and}{(x+1)}^{2}+{y}^{2}\le 4\}\cup \left([1,1]\times [2,2]\right)\cup \{(x,y):1\le x\le 3\text{and}{(x1)}^{2}+{y}^{2}\le 4\}$ is symmetric about the origin. The boundary of this figure and the point $(0,0)$ are shown in the following picture.
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:

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$F$ has line symmetry;

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$F$ has symmetry about a line;

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

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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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Circles: A circle is symmetric about any line passing through its center.
As another example, the isosceles trapezoid^{} defined by
$$T=\{(x,y):0\le x\le 6\text{and}0\le y\le \mathrm{min}\{x,2,x+6\}\}$$ 
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.
Title  symmetry 
Canonical name  Symmetry 
Date of creation  20130322 17:12:29 
Last modified on  20130322 17:12:29 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  18 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 51A10 
Classification  msc 15A04 
Classification  msc 51A15 
Related topic  DihedralGroup 
Related topic  DeterminingRotationsAndReflectionsInMathbbR2 
Defines  symmetry about 
Defines  symmetric^{} 
Defines  symmetric about 
Defines  rotational symmetry 
Defines  point symmetry 
Defines  symmetry about a point 
Defines  symmetric about a point 
Defines  reflectional symmetry 
Defines  line symmetry 
Defines  symmetry about a line 
Defines  symmetric about a line 