Symmetry

Symmetry is everywhere around us, and an intuitive concept: different parts of an object look the same in some way. But using transformations, we can give a much more precise, mathematical definition of what symmetry really means:

An object is symmetric if it looks the same, even after applying a certain transformation.

We can reflect this butterfly, and it looks the same afterwards. We say that it has reflectional symmetry.

We can rotate this flower, and it looks the same afterwards. We say that it has rotational symmetry.

Line Symmetry

You know that a polygon is a closed figure made of several line segments.

The polygon made up of the least number of line segments is the triangle. (Can there be a polygon that you can draw with still fewer line segments? Think about it).

A polygon is said to be regular if all its sides are of length and all its angles are of measure.

Thus, an equilateral triangle is a regular polygon of three sides.

Can you name the regular polygon of four sides?

An equilateral triangle is regular because each of its sides has same length and each of its angles measures ° .

A square is also regular because all its sides are of length and each of its angles is a right angle (i.e., °). Its diagonals are seen to be perpendicular bisectors of one another.

If a pentagon is regular, naturally, its sides should have equal length. You will, later on, learn that the measure of each of its angles is 108°.

A regular hexagon has all its sides equal and each of its angles measures °. You will learn more of these figures later.

The regular polygons are symmetrical figures and hence their lines of symmetry are quite interesting,

Each regular polygon has as many lines of symmetry as it has sides. We say, they have multiplesingle lines of symmetry.

In the below given regular polygons, draw their respective axes of symmetry and add in the number of axes possible in the provided blanks

Rotational Symmetry

What do you say when the hands of a clock go round?

You say that they rotate. The hands of a clock rotate in only onetwo direction, about a fixed point, the centre of the clock-face.

Rotation, like movement of the hands of a clock, is called a clockwise rotation; otherwise it is said to be anticlockwise.

What can you say about the rotation of the blades of a ceiling fan? It rotates in a clockwiseanticlockwise direction.

If you spin the wheel of a bicycle, it rotates. It can rotate in either way: both clockwise and anticlockwise.

When an object rotates, its shape and size do not change. The rotation turns an object about a fixed point. This fixed point is the centre of rotation.

The angle of turning during rotation is called the angle of rotation. A full turn, you know, means a rotation of 360°.

A half-turn means rotation by °; a quarter-turn is rotation by °. When it is 12 o’clock, the hands of a clock are together. By 3 o’clock, the minute hand would have made three complete turns; but the hour hand would have made only a quarter-turn. What can you say about their positions at 6 O’clock?

We can rotate this paper wind mill, and it looks the same afterwards. We say that it has rotational symmetry.

Have you ever made a paper windmill? The Paper windmill in the picture looks symmetrical but you do not find any line of symmetry. No folding can help you to have coincident halves. However if you rotate it by 90° about the fixed point, the windmill will look exactly the same. We say the windmill has a rotationalline symmetry.

In a full turn, there are precisely four positions (on rotation through the angles 90°, 180°, 270° and 360°) when the windmill looks exactly the same. Because of this, we say it has a rotational symmetry of order 4. Here is one more example for rotational symmetry.

Consider a square with A,B,C,D as of its corners.

Let us perform quarter-turns about the centre of the square marked x.

Thus a square has a rotational symmetry of order 4 about its centre. Observe that in this case,

(i) The centre of rotation is the centreedge of the square.

(ii) The angle of rotation is °.

(iii) The direction of rotation is clockwiseanticlockwise.

(iv) The order of rotational symmetry is 436.

The point where we have the pin is the centre of rotation. It is the intersecting point of the diagonals in this case.

Every object has a rotational symmetry of order , as it occupies same position after a rotation of 360° (i.e., one complete revolution). Such cases have no interest for us.

You have around you many shapes, which possess rotational symmetry

For example, when you slice certain fruits, the cross-sections are shapes with rotational symmetry. This might surprise you when you notice them

Patterns

In the previous sections, we saw two different kinds of symmetry corresponding to two different transformations: rotations and reflections. But there is also a symmetry for the third kind of rigid transformation: translations.

Translational symmetry does not work for isolated objects like flowers or butterflies, but it does for regular patterns that extend into every direction:

Hexagonal honyecomb

Ceramic wall tiling

In addition to reflectional, rotational and translational symmetry, there even is a fourth kind: glide reflections. This is a combination of a reflection and a translation in the same direction as the axis of reflection.