The picture on the right shows the actual effect of reflection along the line La. All points
on the line La remain fixed. But points not on the line exchange places with their
respective counterparts in the other half. Thus the vertex A remains fixed while the
vertices B and C get exchanged.
Note that to find the counterpart of a point P, we draw a perpendicular line from the point
P to the line La and extend it to an equal distance on the other side. The point where the
line ends now, say P’, is the counterpart of the point P. Thus a `reflection along line La’
is `a function’, which takes P to P’ and vice-versa. See the picture below illustrating this.
figure 10

From this example we can see that reflection takes place along a line.
All points
on that
line of reflection
remain
fixed
. In
non-identity
rotations
, as seen earlier, there is
only
one fixed point
namely the
point
through which the
axis of rotation
passes. This fixed
point is called the
rotocenter
. This is the main difference between a reflection and a
rotation for finite figures lying in a plane.
Are there any other reflection or mirror symmetries of an equilateral triangle
? The
answer is
yes
. The pictures below show two more reflection symmetries that are possible.
The effect of the reflection on the vertices is shown in each of the pictures on the right.
figure 11
Let us give names to the reflections we have seen above. Let us call reflection along line
La as
ϕ
a
, the reflection along line Lb as
ϕ
b
, and the reflection along line Lc as
ϕ
c
. The
table below gives the effect that these reflection symmetries have on the vertices of the
equilateral triangle. What happens if you apply a reflection twice? (
Q2
.) As we did in the
case of the rotations, can we define the order of a reflection? (
Q3
.)
Table 2
From Table 2, it is clear that each reflection fixes exactly one vertex, namely the vertex
through which the line of reflection passes and just interchanges the other two vertices.
The lines La, Lb and Lc are also called
lines of symmetry
or
axes of symmetry
for the
equilateral triangle ABC. The three reflections described above are the only ones possible
for an equilateral triangle. So we have described up to now
six
different symmetries of an
equilateral triangle. These are the
three
rotations
I,
ψ
, and
ψ
2
and the
three reflections
ϕ
a
,
ϕ
b
, and
ϕ
c
. From the lesson on permutations you will learn that there are only 6
Reflection
Vertices
A
→
A
B
→
C
ϕ
a
C
→
B
A
→
C
B
→
B
ϕ
b
C
→
A
A
→
B
B
→
A
ϕ
c
C
→
C

permutations possible of the letters A, B, C. Since any symmetry permutes the vertices it
has to be one of the six possible permutations and so there can only be at most six
symmetries of an equilateral triangle. But we have produced six symmetries already.

#### You've reached the end of your free preview.

Want to read all 53 pages?

- Fall '09
- Math, Symmetry group