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:

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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 $\displaystyle\left(\begin{array}[]{rr}-1&0\\ 0&-1\end{array}\right)$ , i.e. (http://planetmath.org/Ie) the angle of rotation is $180^{\circ}$ .

The following are classic examples of rotational symmetry in $\mathbb{R}^{2}$ :

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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 $k<n$ .
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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 $\displaystyle F=\bigcup_{k=1}^{4}P_{k}$ , 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:

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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

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

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