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Point Incidence Geometry Identity Map Integer Euclidean Transformation Circle Terms From Foreign Languages Used In Mathematics Polygon Positive Boundary In Topology Origin Regular Polygon Euclidean Vector Space Isosceles Trapezoid

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Symmetry

Let 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 if is a rotation:

has rotational symmetry;
has point symmetry;
has symmetry about a point;
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 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 -gon is symmetric about its center with valid angles of rotation $\displaystyle \theta=\left( \frac{360k}{n} \right)^{\circ}$ for any positive integer .
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 is defined thus:

$\displaystyle \displaystyle P_1$	$\displaystyle =$	$\displaystyle \left\{ (x,y) : 0 \le x \le \frac{4}{1+\sqrt{3}} \text{ and } (2-\sqrt{3})x \le y \le x \right\},$
$\displaystyle \displaystyle P_2$	$\displaystyle =$	$\displaystyle \left\{ (x,y) : \frac{4}{1+\sqrt{3}} \le x \le 2 \text{ and } x \le y \le (2+\sqrt{3})x-4 \right\},$
$\displaystyle \displaystyle P_3$	$\displaystyle =$	$\displaystyle \left\{ (x,y) : 2 \le x \le \frac{4 \sqrt{3}}{1+\sqrt{3}} \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le (-2-\sqrt{3})x+4+4\sqrt{3} \right\},$
$\displaystyle \displaystyle P_4$	$\displaystyle =$	$\displaystyle \left\{ (x,y) : \frac{4 \sqrt{3}}{1+\sqrt{3}} \le x \le 4 \text{ and } (-2+\sqrt{3})x+8-4\sqrt{3} \le y \le -x+4 \right\}.$

Then

has point symmetry with respect to the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$ . The valid angles of rotation for

are $120^{\circ}$ and $240^{\circ}$ . The boundary of

and the point $\displaystyle \left( 2, \frac{2}{\sqrt{3}} \right)$ are shown in the following picture.

$\begin{pspicture}(0,0)(4,3.5) \pspolygon(0,0)(2,0.536)(4,0)(2.5359,1.4641)(2,3.4641)(1.4641,1.4641) \psdot(2,1.1547) \end{pspicture}$

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 are shown in the following picture.

$\begin{pspicture}(-3,-2)(3,2) \psarc(-1,0){2}{180}{270} \psline(-1,-2)(1,-2)(1,0... ...\psarc(1,0){2}{0}{90} \psline(1,2)(-1,2)(-1,0)(-3,0) \psdot(0,0) \end{pspicture}$

If and is a reflection, then has reflectional symmetry. In the special case that $V=\mathbb{R}^2$ , the following terms are used:

has line symmetry;
has symmetry about a line;
is symmetric about a line.

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

Regular polygons: There are lines of symmetry of a regular -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

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

is symmetric about

$\begin{pspicture}(0,-1)(6,3) \pspolygon(0,0)(6,0)(4,2)(2,2) \psline[linecolor=cyan]{<->}(3,-0.5)(3,2.5) \end{pspicture}$

In the picture above, the boundary of is drawn in black, and the line is drawn in cyan.

1	Polygon
1	Circle
1	Incidence Geometry
1	Point
1	Euclidean Transformation
1	Positive
1	Terms From Foreign Languages Used In Mathematics
1	Identity Map
1	Integer
2	Origin
2	Boundary In Topology
3	Regular Polygon
3	Euclidean Vector Space
7	Isosceles Trapezoid