Line, Rotational, and Point Symmetry

In my last post, I talked about turning, flipping, and sliding a figure over a plane.  Now I want to talk symmetry in a figure (a polygon).  How many lines of symmetry does a square have? Can you turn a rectangle 90 degrees and still have symmetry? What about 180 degrees? These are the types of questions we will practice asking ourselves in this post.

Line symmetry is when a geometric figure has a line symmetry (l) if it is its own image under a reflection in l.  Sometimes a geometric figure will have just one line of symmetry.  Other times, the figure could have several lines of symmetry. 

 For example, a circle has an infinite amount of line symmetries! This is because it can be turned on a fixed point so many times an still be the same.  

The picture below gives some examples of line symmetry.  The dotted line represents the line of symmetry where each side of the geometric shape is equal.  These pictures only show one line of symmetry for each figure, but can you think of more than one line of symmetry?

The picture below gives you more examples of line symmetry of some shapes you would find in everyday life.  The letter "A" has only one line of symmetry, while the pink flower has several lines of symmetry.  You can also see that the square has more than one line of symmetry! In my other example, it only showed one line.  Now you can see that it actually has four lines of symmetry.

Now lets talk about turning the geometric figure and finding symmetry.  A figure has rotational symmetry, or turn symmetry, when the traced figure can be rotated less than 360 degrees about some point so that it matches the original figure.  It's important to remember that it can only be less than 360 degrees because any figure can be turned 360 degrees and still match the original figure.  

In the example below, the hexagon can be turned 60 degrees and still match the original figure.  When it can rotate, we say that the figure has 60 degrees rotational symmetry.

Point symmetry is also similar to rotational symmetry.  It is said that if a figure has point symmetry, then it also has rotational symmetry because the figure can be turned less than 360 degrees.  However, point symmetry is when a figure can be turn 180 degrees, or a half turn, and be the same as the original image.  

While with point symmetry you may have rotational symmetry, it is not the other way around.  If a shape has rotational symmetry, it only has point symmetry sometimes.  This is because a figure can be turned 60 degrees or 120 degrees or another amount other than 180 degrees.  This is not point symmetry, but is still rotational.

Both figures below have point symmetry.  If you turn them 180 degrees, they will look like the original image.