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.

The following terms are used to indicate that $E(F)=F$ if $E$ is a rotation:

• $F$ 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 $k<n$ .
• Circles: A circle is symmetric about its center with uncountably many valid angles of rotation.

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 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

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

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 $$T=\{ (x,y) : 0 \le x \le 6 \text{ and } 0 \le y \le \min\{x,2,-x+6\} \}$$ 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.