11.06 Lines and points of symmetry
Introduction
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
Exploration
The applet below shows a square reflected across a line.
Choose an orientation for the line of symmetry by dragging the point.
When the line of reflection goes through the center of the square the reflected square overlaps the original square perfectly.
A line that reflects a shape onto itself is called a line of symmetry (or axis of symmetry).
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 sides?
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.
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.
Neither of these shapes have line symmetry.
Examples
Example 1
How many lines of symmetry does this figure have?
A line that reflects a shape onto itself is called a line of symmetry (or axis of symmetry).
Asymmetric shapes are shapes without lines of symmetry.
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\degree 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\degree are not very interesting.
But what about rotations less than 360\degree? If a rotated shape perfectly overlaps the original shape after a rotation that is less than 360\degree, then the original shape has rotational symmetry. The point about which this rotation happens is called the centre of rotation.
Exploration
Choose a shape and drag the point on the slider.
In the applet above, we can see that it takes only 120\degree of rotation for the triangle to overlap with its original position. In fact, within a full 360\degree rotation, the triangle overlaps its original position three times. We can say that the order of rotational symmetry for this triangle is 3.
Each rotation by 90\degree returns a square to its original position. And we can do this four times in a full 360\degree rotation, so the order of rotational symmetry for a square is 4.
In contrast, the only time an irregular polygon overlaps its original position is after a full 360\degree rotation. This means that this shape does not have rotational symmetry.
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.
Examples
Example 2
Which of the following shapes have rotational symmetry? Select all that apply.
Example 3
For each shape, determine the type or types of symmetry present.
A shape has a rotational symmetry, if the rotated shape perfectly overlaps the original shape after a rotation that is less than 360\degree.
The point about which this rotation happens is called the centre of rotation.
The number of times a shape can rotate onto itself in a full 360\degree rotation, is the order of rotational symmetry.