On the page on Group Orbit of Wolfram MathWorld the following example is posted:
with the caption
For example, consider the action by the circle group $\mathbb S^1$ on the sphere $\mathbb S^2$ by rotations along its axis. Then the north pole is an orbit, as is the south pole. The equator is a one-dimensional orbit, as is a general orbit, corresponding to a line of latitude.
Is it possible to intuitively think of each color dot as a multiplicative cyclic group with a particular n-th roots of unity generator, $\langle \zeta=e^{\frac{2\pi}{n}i}\rangle,$ one at each latitude? Seemingly the angular distances are preserved between dots...
I am not clear as to whether each dot represents a different group, and I miss intuition as to the effect on the sphere as a set (are the elements of the set $\mathbb S^2$ the different latitudes?) being acted upon (by each dot? All at once?).
Adding interest to the topic, I wonder if it can be connected to the opening statement on the page:
In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit.
Can any parallelism be insinuated between this group action, its orbits, and a planetary system?
If possible, avoiding Lie groups would be appreciated.