Group Actions and Orbits

Orbits: Symmetries of a Circle

Now we'll take what you understand about the orbits of a square and apply it to a circle. There are infinite ways to reflect or rotate a circle in order to obtain the same circle (same radius, same center location). The group of symmetries of a circle is known as the O2 group; "O" for orthogonal and "2" for 2-dimensional. Under the action of the O2 group, what is the orbit of any one point on the unit circle?

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As you can see, the orbit of any point on the circle is the circle itself. This is because every point on the circle is some reflection or rotation of every other point on the circle. This means that the orbit of the unit circle under the action of the O2 group is the unit circle.