Oschene's Octahedral Dib's Box
Here is my solution for constructing Oschene's Octahedral Dib's Box without printing the CP. You will still need to find a way of obtaining a circle of paper though.
See his original CP here: www.flickr.com/photos/oschene/4609233017/
And model here: www.flickr.com/photos/oschene/4609246357/
1. Mark the center.
2. Bring any point from the edge to the center and pinch sides.
3. Bring one pinch to the center and pinch as shown.
4. Pivot around the same pinch to bring another edge-point to the center, and pinch as shown.
5. Make the bold creases, starting with the one that almost goes through the center.
6. Mark the midpoint of this crease as shown.
7. Use the previous mark as reference for these 3 creases.
8. Finish the CP.
To collapse, reference Oschene's original CP for the mountain/valley designations.