# 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 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 $\mathbb{R}^{2}$:

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 $\mathbb{R}^{2}$:

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

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.

 Title symmetry Canonical name Symmetry Date of creation 2013-03-22 17:12:29 Last modified on 2013-03-22 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