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Symmetries

We have seen number-theoretical and combinatorial examples of groups. Now we look at some geometrical ones.


\begin{de}
A {\em symmetry} (or {\em isometry}) of the real plane ${\mathbb{R}}^...
...f(X),f(Y))$(where $d(A,B)$\ denotes the distance between $A$\ and $B$).
\end{de}


\begin{exs}
% latex2html id marker 428Translation by a vector (see Figure \ref...
...flection in a line (see Figure \ref{fig23})
are well known symmetries.
\end{exs}

Figure: Translation by the vector $\vec{a}$.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf1.ps}\hspace{1mm}
\end{center}\end{figure}

Figure 2: Rotation about the point $P$ by the angle $\alpha $.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf2.ps}\hspace{1mm}
\end{center}\end{figure}

Figure 3: Reflection in the line $l$.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf3.ps}\hspace{1mm}
\end{center}\end{figure}

Below we list some facts about symmetries.

1)
Every symmetry is a bijection.
2)
The composition of two symmetries is again a symmetry.
3)
The inverse of a symmetry is again a symmetry.
4)
The set of all symmetries is a group under composition of mappings.
5)
A symmetry preserves angles.
6)
Every symmetry is either a translation, or a rotation, or a reflection, or a product of a translation and a reflection (called a glide-reflection).

Now, if we are given a figure $F$ in the plane (i.e. a set of points, like a line, or a triangle or a square, etc.) we can consider those symmetries of the plane which map this figure onto itself. It is easy to see that these symmetries also form a group; this group is called the group of symmetries of $F$. It is worth remarking here that if $F$ is a finite (bounded) figure, then it follows from 6) that every symmetry of $F$ is either a rotation or a reflection.


\begin{ex}
% latex2html id marker 464Consider an equilateral triangle $T$\ (se...
..._b & \sigma_c & \sigma_a & \id & \rho_{120}
\end{array}\end{displaymath}\end{ex}

Figure 4: Symmetries of an equilateral triangle.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf4.ps}\hspace{1mm}
\end{center}\end{figure}


\begin{ex}
% latex2html id marker 497An isosceles triangle (Figure \ref{fig25}...
...
\id & id & \sigma\\
\sigma & \sigma & \id
\end{array}\end{displaymath}\end{ex}

Figure 5: Symmetries of an isosceles triangle.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf5.ps}\hspace{1mm}
\end{center}\end{figure}


\begin{ex}
A scalene triangle has a unique symmetry -- the identity mapping $\id$.
\end{ex}


\begin{ex}
A circle has infinitely many symmetries: all the rotations about the ...
... reflections in all the lines passing through
the centre of the circle.
\end{ex}


\begin{ex}
% latex2html id marker 516A non-square rectangle (Figure \ref{fig26...
...\\
\rho & \rho & \sigma_y & \sigma_x & \id
\end{array}\end{displaymath}\end{ex}

Figure 6: Symmetries of a rectangle.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf6.ps}\hspace{1mm}
\end{center}\end{figure}


\begin{ex}
% latex2html id marker 531A regular $n$-gon (Figure \ref{fig27}, fo...
...\rho^{-1}\rho^{-1}\sigma=\rho^{-1}\sigma=\rho^5\sigma.
\end{displaymath}\end{ex}

Figure 7: Symmetries of a regular hexagon.
\begin{figure}\begin{center}
\hspace{1mm}\epsffile{lnotesf7.ps}\hspace{1mm}
\end{center}\end{figure}

Groups of symmetries of infinite figures are also of interest. Here one often considers a repeating pattern which fills a plane, rather like a wallpaper patterns. It is possible to classify all these groups, and it turns out that there are precisely 17 of them.

One can also consider the symmetries of the 3-dimensional space, rather than the plane, and also symmetries of 3-dimensional figures. Here, the analogue of wallpapers are crystals, and the classification of all possible groups arising here (there $230$ of them) is a significant piece of information in the study of crystals (called crystallography).


next up previous contents
Next: Order of an element Up: MT2002 Algebra Previous: Permutations   Contents

Edmund F Robertson

11 September 2006