Reflection Symmetry
If you stand
in front of a plane mirror, you can observe your image behind the mirror. This
is due to the reflection of light rays from the mirror. The image in the mirror is as
far behind the mirror as the object is in front of the mirror. So, a mirror
behaves like a line of symmetry. This phenomenon is called mirror reflection.
Thus, every line of symmetry is considered as a mirror line.
If we cut a shape along the mirror line and place it on the other part, it will coincide exactly with the other part or we can say, one part is the mirror image of the other or the figure is symmetrical about this line.
Rotational Symmetry
A shape is
said to have a rotational symmetry when it looks the same even after a rotation.
Rotation can be clockwise or anti-clockwise.
For example,
when a wind mill or wheel of a vehicle, etc., rotates about its axis, it looks
the same. Thus, these objects have a rotational symmetry.
Order of Rotational Symmetry
The number
of distinct orientations in which the shape looks the same as the original is
called its order of rotational symmetry.
A full turn
means rotation through 360°.
A half turn
means rotation through 180°.
A quarter
turn means rotation through 90°.
A three-fourths
turn means rotation through 270°.
Angle of Rotation
The angle
through which an object is rotated and the object looks like the original, is
called the angle of rotation.
When it completes four quarter turns, the square reaches its original position. But a rotation by 90° also gives a shape exactly as the original one. So, a square also has a rotational symmetry and the angle of rotation is 90°. The order of the rotational symmetry of a square is 4.
If x° is the angle of rotation for a figure, then the order of its rotational symmetry is 360° ÷ x °.
Every object always has a rotational symmetry of order 1 as it comes back to its original position after a rotation of 360°.
Example 1: Find the order of rotational symmetry for
the following objects.
a. An
equilateral triangle
b. A
square
Solution: a. An equilateral triangle acquires its original position when
it is rotated through 120°, 240° and 360°.
Here, angle of rotation = 120°
Therefore, order of rotational symmetry = 360°/120° = 3
Thus, order
of rotational symmetry of an equilateral triangle = 3.
b. A square acquires its original
position when it is rotated through 90°, 180°, 270° and 360°.
Here, angle of rotation = 90°
Therefore, order of rotational symmetry = 360°/90° = 4
Thus, order
of rotational symmetry of a square = 4.
Example 2: When an object is rotated through 60°, it looks the same. Find the
order of rotational symmetry of the object.
Solution: Here, angle of rotation = 60°
Therefore, order of rotational symmetry = 360°/60° = 6
Thus, order
of rotational symmetry of the object = 6.
Example 3: The order of rotational symmetry of a regular
polygon is 8. Name the regular polygon and find its angle of rotation.
Solution: A regular polygon has as many order of
rotational symmetry as it has sides.
Thus, the
regular polygon has 8 sides and hence it is octagon.
Angle of
rotation = 360°/8 = 45°
Translation Symmetry
You can see
that many designs are made up of motifs of the same shape and size. These shapes
are repeatedly arranged with equal intervals.
For example,
floor tiles, design of clothes, etc.
A pattern or
a design is created by a single motif or shape by changing its position
repeatedly. Such designs are said to have translation
symmetry.
