ROTATIONS

A regular triangle has equal side lengths and equal angles.

What's more, it can rotate ⅓ of the way around a circle and appear unchanged. Had our eyes been closed when it rotated, we would not have noticed a difference.

If the triangle instead rotates by an arbitrary amount, like ¼ of the way around a circle, it will then appear changed, since it is oriented differently.

We can even find ways to color the triangle so that a ⅓ turn still does not change it.

While this will not work for other ways.

Check in: Which of the following colored triangles can be rotated by a ⅓ turn without changing in appearance?

Our triangle can also rotate by more than a ⅓ turn without changing. It can rotate by twice that much - ⅔ of the way around the circle - or by 3 times that much, which is all the way around the circle.

We can keep rotating - by 4 times that much, 5 times that much, 6 times... and keep going. The triangle seems to have an infinite number of rotations! But after 3 they become repetitive.

Check in: How many ways can a square rotate without changing before the ways become repetitive?

The triangle has only 3 unique rotations, so we’ll talk about rotations that are less than a full turn. When we say our triangle ‘has 3 rotations’ we mean it can be rotated by these 3 different turns and appear unchanged.

Other shapes have these same 3 rotations. For this reason, we can say they all share the same symmetry group.

However, their rotations can be removed by adding color.

Now when our shape is rotated, its color shows it.

Now that we can count rotations, we can be more precise when we say a square has more symmetry than a rectangle.

We can also see that a square has more rotational symmetry than a triangle, which in turn has more than a rectangle: A rectangle has only 2 unique rotations, while our triangle has 3, and a square has 4.

We don’t need to stop at 4 rotations. We can find shapes with 5 rotations, 6 rotations, 7, 8, ... and keep going towards infinity.

And these shapes don’t even need to be so simple.