Symmetry of Plane Figures

Some plane figures also have a property which is called symmetry. Symmetry is also a concept in aesthetics or the science of beauty.

There are two kinds of symmetry – Line Symmetry & Rotational Symmetry.

Line/ Reflection symmetry

Imagine an equilateral triangle. If we draw a line joining any of the vertices to the mid-point of the opposite side, then it divides the triangle into two parts. Both the parts are not only identical but also look like mirror images of each other.

1. If we keep a mirror on this line, we can still see the original triangle. Except that in this triangle one half is the original triangle and the other half is its reflection!

2. If we cut an equilateral triangle out of paper, then the triangle can be folded into half using this line. Both halves would coincide exactly.

Such a line is called a “line of symmetry”.

A little experimentation with a equilateral triangle cut from paper would reveal that it has 3 lines of symmetry!

Students can experiment with different types of triangles and quadrilaterals and see if they have line symmetry and if yes, how many lines of symmetry do they have.

It is obvious that a circle has infinite lines of symmetry. It is the most symmetrical figure.

Line symmetry is also known as Reflection Symmetry.

Vertical Symmetry & Our Mind

The American Psychological Association reported a study "Perception of Symmetry in Infants" showing that "(a) 4-mo-old infants showed no preference for symmetry, but they processed menths, infants preferred vertical symmetry to horizontal symmetry and asymmetry. Thus, preference for symmetry seems to develop late, whereas recognition of vertical symmetry is innate, matures very quickly, or is learned very early."

This preference possibly develops because vertical symmetry is more prevalent than horizontal symmetry in environment.

Rotation Symmetry

Take the equilateral triangle again. If we rotate the triangle about its centre, does it resemble itself before it completes a complete turn? Or can someone rotate the triangle in such a way that one cannot detect the rotation?

A little experimentation would reveal that this would happen when the equilateral triangle is rotated through 120, 240 & 360 degrees! The angles will always be multiples of a particular angle; in this case 120 degrees. We then say that an equilateral triangle has 120-degree symmetry.

It is obvious that any figure will always have a 360-degree symmetry!

Here again, students can experiment with different types of triangles & quadrilaterals and discover if they have a symmetry less than 360 degrees.

Symmetry and Beauty

Humans and many other living beings seem to find symmetrical objects more attractive than asymmetrical objects. Nature is also full of symmetrical structures. Hence, we do not know if human being’s affinity to symmetry is due to evolutionary or environmental factors.

Chapter 23.15 shows the symmetries in some common figures.