9.06 Symmetry

In our study of transformations we have looked at

• Reflections: flipping a shape over a line.
• Rotations: spinning a shape about a point.
• Translations: moving objects left or right, and up or down.

A shape has symmetry if it looks the same before and after a transformation. Symmetries relate to self-similarity of things, and this is very common in nature as well as many areas of mathematics.

The two main types of symmetry we will explore are line symmetry (or reflection symmetry) and rotational symmetry.

Line symmetry

The applet below shows a square reflected across a line. When the line has a certain slope, we can position it so that the reflected square overlaps the original square perfectly. A line that leads to this type of reflection is a line of symmetry (or axis of symmetry) for the square.

A square has four lines of symmetry, all passing though its center. Any other line different from these four will not produce a reflected square that overlaps the original square perfectly.

Think about the other regular polygons, such as the triangle, the pentagon, and the hexagon. How many lines of symmetry do each of these shapes have? How many lines of symmetry would there be for a regular polygon with $100$100 sides?

Building symmetric shapes from reflections

Instead of starting with a shape and finding its lines of symmetry, we can make a shape that has line symmetry using the reflection transformation.

For example, when an obtuse isosceles triangle is reflected across its longest side, the shape created by combining the original triangle and the reflected triangle is a rhombus. Since we built this rhombus using a reflection, we know that the rhombus is symmetric and has a line of reflection along its longest diagonal.

Shapes with no line symmetry

A shape that has no lines of symmetry is called asymmetric. Many asymmetric shapes have very irregular features, but we can also create asymmetry using only minor changes to symmetric shapes.

Worked example

Which of these capital letters has line symmetry along their vertical axis? Which have line symmetry along their horizontal axis?

Think: Vertical line symmetry means we can draw a vertical line through the center of the letter, reflect the letter across this line, and have the reflected letter overlap the original letter.

Do: Firstly we can reflect each letter across a vertical line. Each letter and its reflection is shown side-by-side for clarity.

This shows that the letter "A" and the letter "H" both have reflection symmetry across a vertical line. Next let's compare each letter to a copy reflected across a horizontal line.

Now we can see that the letter "B" and the letter "H" have reflection symmetry across a horizontal line. Notice that the letter "R" does not have line symmetry in either case.

Rotational symmetry

To rotate a shape, we specify the amount of rotation and the point about which we are rotating. Clearly, a rotation by $360^\circ$360° will create a shape that will look the same after the transformation as it did before, no matter where the point of rotation is located. So rotations by $360^\circ$360° are not very interesting.

But what about rotations less than $360^\circ$360°? If a rotated shape perfectly overlaps the original shape after a rotation that is less than $360^\circ$360°, then the original shape has rotational symmetry. The point about which this rotation happens is called the center of rotation.

In the applet above, we can see that it takes only $120^\circ$120° of rotation for the triangle to overlap with its original position. In fact, within a full $360^\circ$360° rotation, the triangle overlaps its original position three times. We can say that the order of rotational symmetry for this triangle is $3$3.

Similarly, each rotation by $90^\circ$90° returns the square to its original position. And we can do this four times in a full $360^\circ$360° rotation, so the order of rotational symmetry for a square is $4$4.

In contrast, the only time the irregular polygon overlaps its original position is after a full $360^\circ$360° rotation. This means that this shape does not have rotational symmetry.

Building symmetric shapes from rotations

Similar to the case for line symmetry, we can make a shape that has rotational symmetry using the rotation transformation.

For example, we can start with a shape that has no rotational symmetry and make copies by rotating about one of its vertices. If we can fit a whole number of copies in a full rotation, then the shape that is made of all the copies will have rotational symmetry.

 

Practice questions

Question 1

Question 2

Question 3