##
Transformations

A transformation is a one-to-one mapping on a set of points. The most
common transformations map the points of the plane onto themselves, in
a way which keeps all lengths the same. These transformations are called
isometries. Another common sort of transformation which does not preserve
lengths are dilatations.
There are four isometries in the plane: translations,
rotations, reflections,
and glide reflections.

###
Translations

A *translation* slides all the points in the plane the same distance
in the same direction. This has no effect on the sense
of figures in the plane. There are no invariant points (points that map
onto themselves) under a translation.

###
Rotations

A *rotation* turns
all the points in the plane around one point, which is called the center
of rotation. A rotation does not change the sense
of figures in the plane. The center of rotation is the only invariant point
(point that maps onto itself) under a rotation. A rotation of 180 degrees
is called a *half turn*. A rotation of 90 degrees is called a *quarter
turn*.

###
Reflections

A *reflection* flips all the points in the plane over a line, which
is called the mirror. A reflection changes the the sense
of figures in the plane. All the points in the mirror contains all the
invariant points (points that map onto themselves) under a reflection.

###
Glide reflections

A *glide reflection* translates the plane
and then reflects it across a mirror parallel
to the direction of the translations. A glide reflection changes the sense
of figures in the plane. There are no invariant points (points that map
onto themselves) under a glide reflection.

###
The sense of a figure

The *sense* or *handedness*
of a figure refers to the order of points as one goes around the figure.
The three triangles in the figure are all congruent, but triangle ABC has
the same sense as triangle DEF, and the opposite sense to triangle GHI.
A figure has a visible sense if it looks different under a reflection.
For example the letter R, if it is reflected, can not be mapped back onto
itself by any sequence of translations or rotations.
If a figure does not have a visible sense then it is said to have bilateral
symmetry. The letter A for example, is symmetrical
around a vertical line through its center. If it reflected by a vertical
mirror it looks identical. Even if it reflected by a non-vertical mirror,
it can be rotated onto itself.

###
Symmetry

If a figure looks the same under a transformation then it is said to be
symmetrical under that transformation. For example, the letter A is symmetrical
under a reflection around a vertical mirror through its center. This sort
of symmetry is called bilateral symmetry.
The letter N is symmetrical under a half turn around
the midpoint of its diagonal stroke.
There are many interesting symmetrical
patterns which cover the plane, all based on combinations of translations,
rotations, reflections,
and glide reflections. Another interesting set of
symmetrical patterns are frieze patterns.

[Transformations] [Plane
Symmetry] [Geometry] [Resources]
[Copyright]
This page maintained by David A Reid

Email: *david.reid@acadiau.ca*