Definition. Isometry (Motion).
Let $P$ denote the set of all points in
the plane, and $M$ the set of all
permutations of $P$ that preserve
distance between points. The permutations
in $M$ are called
motions or isometries of the plane.
Definition. Rotations.
If $p$ is a fixed point in $P,$
then any rotation of the plane
about $p$ is a motion of the plane.
Rotations were discussed in
Definition. Reflections.
The reflection of the plane $P$
through a line $L$ in $P$, the
mapping that sends each point $p$
in $P$ to the point $q$ such that
$L$ is the perpendicular bisector
of the segment $pq,$ is a motion
of the plane.
Definition. Translations.
A translation of the plane $P$, a mapping
that sends all points the same distance
in the same direction, for instance,
the translation sending $p_1$ to $q_1$
in Figure 8.2 would send $p_2$ to $q_2$
and $p_3$ to $q_3,$ is a motion of the plane.
Note.
See Section 54 for
another type of motion of a plane,
glide reflection.
Theorem 8.1.
The set $M$ of all motions (isometries)
of a plane $P$ is a subgroup of
$\Sym(P).$
Note.
For $p,q\in P$ let $d(p,q)$
denote the distance between $p$
and $q.$ With this notation,
if $\alpha\in\Sym(P)$ then
\(
\alpha\in M
\)
iff
\(
d(\alpha(p),\alpha(q))=d(p,q).
\)
Definition. Symmetry Group.
If $T$ is a set of points in a plane,
then $M_{(T)},$ the group of all motions
leaving $T$ invariant, is called
the group of symmetries
(or symmetry group) of $T.$
Example 8.1.
Symmetry Groups.
Symmetries of the square, rectangle,
and parallelogram are examples of
symmetry groups, discussed in greater
detail next.
Example 8.1.1.
If the vertices of a square are given by
$T=\{a,b,c,d\},$ then $M_{(T)},$ the symmetry
group of $T,$ is called the
group of symmetries (symmetry group) of
the square.
Example 8.1.2.
If the vertices of a rectangle are
given by $T=\{a,b,c,d\},$ then $M_{(T)},$
the symmetry group of $T,$ is called the
group of symmetries (symmetry group) of
the rectangle.
Example 8.1.3.
If the vertices of a parallelogram
are given by $T=\{a,b,c,d\},$ then $M_{(T)},$
the symmetry group of $T,$ is called the
group of symmetries (symmetry group) of
the parallelogram.
Theorem.
Let $T=\{a,b,c,d\},$ be the vertices of a
square, $H$ and $V$ be its horizonal and
vertical bisectors, and $D_1,D_2$ its
diagonals. Then the elements of $M_{(T)},$
the
symmetry group of the square,
are:
\[
\begin{align*}
\mu_1 &=\text{ identity permutation}\\
\mu_2 &=\text{ rotation } 90^\circ
\text{clockwise around } p\\
\mu_3 &=\text{ rotation } 180^\circ
\text{clockwise around } p\\
\mu_4 &=\text{ rotation } 270^\circ
\text{clockwise around } p\\
\mu_5 &=\text{ reflection through }H\\
\mu_6 &=\text{ reflection through }V\\
\mu_7 &=\text{ reflection through }D_1\\
\mu_8 &=\text{ reflection through }D_2
\end{align*}
\]
Furthermore, $M_{(T)}$ is a subgroup of $M$
(
Theorem.
Let $T=\{a,b,c,d\},$ be the vertices of a
rectangle, and $H$ and $V$ be its
horizonal and vertical bisectors.
Then the elements of $M_{(T)},$
the
symmetry group of the rectangle,
are:
\[
\begin{align*}
\mu_1 &=\text{ identity permutation}\\
\mu_3 &=\text{ rotation } 180^\circ
\text{clockwise around } p\\
\mu_5 &=\text{ reflection through }H\\
\mu_6 &=\text{ reflection through }V\\
\end{align*}
\]
Furthermore, $M_{(T)}$ is a subgroup of
the symmetry group of the square,
and $\abs{M_{(T)}}=4.$
Theorem.
Let $T=\{a,b,c,d\},$ be the vertices of a
parallelogram.
Then the elements of $M_{(T)},$ the
symmetry group of the parallelogram,
are:
\[
\begin{align*}
\mu_1 &=\text{ identity permutation}\\
\mu_3 &=\text{ rotation } 180^\circ
\text{clockwise around } p\\
\end{align*}
\]
Furthermore, $M_{(T)}$ is a subgroup of
the symmetry group of the rectangle,
and $\abs{M_{(T)}}=2.$
Note.
Any motion of one of the figures will
permute the vertices of the figure among
themselves and the sides of the figures
among themselves. Moreover, any motion
will be completely determined by the way
it permutes the vertices. Therefore,
there is a natural correspondence between
the symmetry group of the figure and the
group $\Sym(T)=\Sym\{a,b,c,d\}.$
That is to say, there is a natural
correspondence between the symmetry
group of the square (rectangle,
parallelogram) and the symmetric group
on its (respectively) vertices.
This correspondence allows us to
conclude that the order of the symmetry
group of the figure is at most
the same as the order of
$\Sym(T),$ or $4!=24.$
In fact, the order must be less, because
some permutations of the vertices
clearly cannot arise from motions of
the plane.
In fact, the orders are $8,4$ and $2$
respectively. Notice the more symmetric
the figure, the larger its group of
symmetries.
Guevara Note.
Look ahead to
Guevara Note.
Although we reuse $T=\{a,b,c,d\}$
for the vertices of each of the three
figures, respectively, possible because
they are all quadrilaterals, it is
important to realize we are not assuming
these points to be the same set of
vertices between figures. Even if
two figures shared some vertices, they
could not all coincide, since the very
definition of each figure dictates
that the relative position
of their vertices differ.
Therefore, with the understanding that
$T$ is a different set of vertices for each
figure, it follows, in particular, that
$M_{(T)}$ is a different group of symmetries
for each figure. Comparing the listings of
their members, and their orders, reflects
that.
Table 8.1.
Cayley table for Symmetries of Square
| $\circ$ |
$\mu_1$ |
$\mu_2$ |
$\mu_3$ |
$\mu_4$ |
$\mu_5$ |
$\mu_6$ |
$\mu_7$ |
$\mu_8$ |
| $\mu_1$ |
$\mu_1$ |
$\mu_2$ |
$\mu_3$ |
$\mu_4$ |
$\mu_5$ |
$\mu_6$ |
$\mu_7$ |
$\mu_8$ |
| $\mu_2$ |
$\mu_2$ |
$\mu_3$ |
$\mu_4$ |
$\mu_1$ |
$\mu_7$ |
$\mu_8$ |
$\mu_6$ |
$\mu_5$ |
| $\mu_3$ |
$\mu_3$ |
$\mu_4$ |
$\mu_1$ |
$\mu_2$ |
$\mu_6$ |
$\mu_5$ |
$\mu_8$ |
$\mu_7$ |
| $\mu_4$ |
$\mu_4$ |
$\mu_1$ |
$\mu_2$ |
$\mu_3$ |
$\mu_8$ |
$\mu_7$ |
$\mu_5$ |
$\mu_6$ |
| $\mu_5$ |
$\mu_5$ |
$\mu_8$ |
$\mu_6$ |
$\mu_7$ |
$\mu_1$ |
$\mu_3$ |
$\mu_4$ |
$\mu_2$ |
| $\mu_6$ |
$\mu_6$ |
$\mu_7$ |
$\mu_5$ |
$\mu_8$ |
$\mu_3$ |
$\mu_1$ |
$\mu_2$ |
$\mu_4$ |
| $\mu_7$ |
$\mu_7$ |
$\mu_5$ |
$\mu_8$ |
$\mu_6$ |
$\mu_2$ |
$\mu_4$ |
$\mu_1$ |
$\mu_3$ |
| $\mu_8$ |
$\mu_8$ |
$\mu_6$ |
$\mu_7$ |
$\mu_5$ |
$\mu_4$ |
$\mu_2$ |
$\mu_3$ |
$\mu_1$ |
Alex Notes