# 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.

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. (http://planetmath.org/Ie) 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 (http://planetmath.org/Center9) with valid angles of rotation $\displaystyle\theta=\left(\frac{360k}{n}\right)^{\circ}$ for any positive integer $k.

• Circles: A circle is symmetric about its center (http://planetmath.org/Center8) 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:

 $\displaystyle\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle\left\{(x,y):0\leq x\leq\frac{4}{1+\sqrt{3}}\text{ and }(2-\sqrt{% 3})x\leq y\leq x\right\},$ $\displaystyle\displaystyle P_{2}$ $\displaystyle=$ $\displaystyle\left\{(x,y):\frac{4}{1+\sqrt{3}}\leq x\leq 2\text{ and }x\leq y% \leq(2+\sqrt{3})x-4\right\},$ $\displaystyle\displaystyle P_{3}$ $\displaystyle=$ $\displaystyle\left\{(x,y):2\leq x\leq\frac{4\sqrt{3}}{1+\sqrt{3}}\text{ and }(% -2+\sqrt{3})x+8-4\sqrt{3}\leq y\leq(-2-\sqrt{3})x+4+4\sqrt{3}\right\},$ $\displaystyle\displaystyle P_{4}$ $\displaystyle=$ $\displaystyle\left\{(x,y):\frac{4\sqrt{3}}{1+\sqrt{3}}\leq x\leq 4\text{ and }% (-2+\sqrt{3})x+8-4\sqrt{3}\leq y\leq-x+4\right\}.$

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

$\{(x,y):-3\leq x\leq-1\text{ and }(x+1)^{2}+y^{2}\leq 4\}\cup\big{(}[-1,1]% \times[-2,2]\big{)}\cup\{(x,y):1\leq x\leq 3\text{ and }(x-1)^{2}+y^{2}\leq 4\}$ 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.