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[parent] symmetry (Definition)

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$:

As another example, let $ \displaystyle F=\bigcup_{k=1}^4 P_k$, where each $ P_k$ is defined thus:

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

Then $ 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.

\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 $ (0,0)$ 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 $ 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

$\displaystyle T=\{ (x,y) : 0 \le x \le 6$    and $\displaystyle 0 \le y \le \min\{x,2,-x+6\} \}$
is symmetric about $ x=3$.

\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 $ T$ is drawn in black, and the line $ x=3$ is drawn in cyan.



"symmetry" is owned by Wkbj79. [ full author list (2) ]
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See Also: dihedral group, determining rotations and reflections in $\mathbb{R}^2$

Also defines:  symmetry about, symmetric, symmetric about, rotational symmetry, point symmetry, symmetry about a point, symmetric about a point, reflectional symmetry, line symmetry, symmetry about a line, symmetric about a line

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Cross-references: isosceles trapezoid, passing through, vertices, pass through, lines, reflection, origin, boundary, point, circles, integer, positive, regular, regular polygons, angle of rotation, rotation, identity map, Euclidean transformation, Euclidean vector space
There are 154 references to this entry.

This is version 15 of symmetry, born on 2007-06-04, modified 2007-06-14.
Object id is 9530, canonical name is Symmetry2.
Accessed 5251 times total.

Classification:
AMS MSC: 51A15 (Geometry :: Linear incidence geometry :: Structures with parallelism)
 51A10 (Geometry :: Linear incidence geometry :: Homomorphism, automorphism and dualities)
 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
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