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Symmetry

An object is said to have symmetry, if it retains the original shape even in case of a reflection, rotation or translation. Since symmetry for such objects is possible by any of these operations, the types of symmetries are classified accordingly. Some objects may have more than one type of symmetry. The symmetrical property can also be inferred from algebraic functional descriptions of the shapes, in some cases. The concept of symmetry is helpful to study transformations of shapes.

There are mainly three types of symmetries as follows:

  1. Reflection symmetry
  2. Rotational symmetry
  3. Point symmetry

 

Reflection Symmetry

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If one half of a shape is an exact mirror image of the other half of the same shape when reflected across a line, then the shape is said to have a reflection symmetry.

The line across which, it is reflected is called as the line of symmetry or axis of symmetry.

Reflection Symmetry

Some shapes have more than one axis of symmetry. Regular polygons have number of lines of symmetries equal to the number of sides.

Rotational Symmetry

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When some shapes are rotated around the center by some angle, the shape appears to be original after that rotation. Such shapes have a symmetry called rotational symmetries. A rotational symmetry is expressed in terms of its order number.
 
The order number of an object is the number of times the object exhibits the original shape in one complete rotation.
 
When rectangle rotated by 180o, it comes back to the original shape and again it does after a rotation of 360o. Thus, the rectangle has a rotational symmetry of order 2.
In case of the equilateral triangle, the rotational symmetry is of order 3.
In case of parabola, the shape comes back to the original position only after a complete rotation.
But, the order of rotational symmetry here is NOT referred as 1. It is because all the shapes in the world do come back to the initial position, even if they don’t at in between positions. Hence, such shapes are said to have no rotational symmetry, rather than, describing as having a rotational symmetry of order 1.

Rectangle and equilateral triangle have both the reflection and rotational symmetries, whereas the parabola has only reflection symmetry.

Point Symmetry

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Point Symmetry

This is a bit strange type of symmetry. In some shapes, we can find the set of points that are in opposite directions are at equidistant from a particular point. Such shapes are symmetrical about the same point and the symmetry is called as point symmetry.

The graph shown in the first diagram shows the points A and A’ are at equidistant from origin 0. So, the shape of the graph is symmetrical about origin in this case. It is a special case of point symmetry called as origin symmetry.

The functions whose graphs possess origin symmetry are called as odd functions. Algebraically, an odd function can be identified by checking if f(x) = -f(x).

The second diagram also shows a shape having a point symmetry. But, the point is not origin in this case. But, it is another point P. While it satisfies the condition of a point symmetry, the function represented by the shape cannot be an odd function. It may be interesting to note that capital English letters like X, N, I, H, Z and S have point symmetries.  

More topics in  Symmetry
Reflection Symmetry Rotational Symmetry
Point Symmetry Lines of Symmetry
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