Dividing surfaces with curves?

 From:  Dreamcube017
Hello. Just checking out the ptrogram and it's really awesome.

The only problem is that I've project some curves onto a sphere and would like to have the surfaces seporated. For example, in the image, there's a polygon projected to a sphere. I'd like the face of the polygon to be separate so that I can do things such as insets and stuff, but I can't seem to figure out what I'm doing wrong. I've tried a few of the booleans which sort of work, but don't give the exact results I am looking for.

Here's the image.