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The simplest kind of symmetry, and the kind you’re probably most familiar with, is line
symmetry. This is where you can draw a line through a shape, and if you reflect the shape
in that line, you get back the original shape in the same place. You can reflect the shape
across the line without changing the shape.
For example, this shape has two different lines of symmetry. I can reflect it this way,
across a vertical line, or this way, across a horizontal line. We say it has two lines
of symmetry.
This arrow has only one line of symmetry, through the centre.
This cross actually has four lines of symmetry: through each of the bars of the cross, but
you can also reflect it through a horizontal or vertical line as well.
How many lines of symmetry can you find in this star? There are five of them.
Here’s another shape that has only one line of symmetry.
Now this next shape looks kind of symmetric, but it doesn’t have any lines of symmetry.
I cannot draw a line through the shape, and reflect it over that line to get an identical
image. Perhaps I need a better definition, another kind of symmetry.
There is another kind of symmetry, and it’s called rotational symmetry. This is where
a shape can be rotated by an angle smaller than a full turn, and have the image be identical:
the same shape in the same position. The order of rotational symmetry is the number of times
this repeats within a full three hundred and sixty degrees of rotation.
For example, if I rotate that first shape by exactly one hundred and eighty degrees,
I’ll end up with the same shape I started with in the same position. And then of course
if I rotate it another one hundred and eighty degrees, I get back to where I started. We
say this shape has rotational symmetry of order two: there are two different positions
around the full rotation where the shape looks exactly the same.
The arrow cannot be rotated through any angle less than a full turn. Technically you could
say it has rotational symmetry of order one, but since that’s pretty meaningless we usually
just say it has no rotational symmetry.
The thick cross has rotational symmetry of order four. There are four different positions
as you rotate it around where it looks the same as it did at the start. Each position
is ninety degrees apart, three hundred and sixty divided by the order, four.
What’s the rotational symmetry of the star? Rotate it around. It has rotational symmetry
of order five.
Does this teardrop shape have rotational symmetry? No, it doesn’t.
And now we come back to this interesting shape. Although it has no lines of symmetry, look
what happens as we rotate it. It repeats every one hundred and twenty degrees. It has rotational
symmetry of order three.
Let’s try one more. This interesting shape also has no lines of symmetry. But if you
rotate it through one hundred and eighty degrees, look: it’s the same. It has rotational symmetry
of order two.