Additional stuff: If you have KPT5, their Frax4D plugin shows best the connection between the base fractal and the quat (it's interactive, so as you change the fractal, the quat changes in near real-time). Here's a screen capture of the two pertinent windows for that plugin. As they would say on a SAT test; circle:sphere as fractal:quaternion The only parameter that can be changed for a circle is it's radius. Thus as you take cross-section slices of a sphere, you see circles of various radii. With a fractal, you have several more parameters that can change, so the quaternion's cross section cuts can change much more dramatically.

RosemaryR
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"This...this is magnificent!"
"Oh, yeah. Ooooo. Aaaaah. That's how it starts.
Then, later, there's ...running. And....screaming."