I recently saw the Circle of Friends symbol casually referred to as “the spinning logo”. It doesn't actually spin, but that got me wondering: If it did, what might it look like? Its shape does seem to suggest motion.
What first came to mind could be described as having the mechanics of a loosely coupled ratchet. I imagine the ring kicking the dots, which recoil and then spring back into position:
Watching this reminds me of trying—and failing—to pull-start an engine. If I encountered it in a user interface I'd probably interpret its stuttering motion as a sign of uncertainty, of perhaps the success of an attempted error recovery or response from an unreliable service.
The mood noticeably improves when I model the system after a torque limiting clutch. In order to move the dots, the ring must initially accelerate slowly. Then, just as wheel acceleration is modulated for effect in a hurdy gurdy, this system performs the tablecloth trick by skipping a segment before allowing the dots to return to position:
In a user interface, I'd expect this to signify that work is being done. The dots' oscillation suggests that progress is still piecemeal, but the way the ring so gracefully catches them just as they rebound speaks to the work's stability.
To more strongly communicate a message of progress, I inverted roles of the dots and ring, allowing a transfer of momentum from the former to latter, which in turn rotates the whole system. The resulting dynamic resembles an overbalanced wheel in perpetual motion:
I find the continual forward motion of this pleasing. Presented with this as an activity indicator in a user interface, I would feel reassured that progress is being made.
There is, however, a somewhat gruesome problem I've overlooked throughout this exploration. The mere act of separating dot from ring segment is, upon reevaluation of their original symbolism, a tragic beheading of our three friends.