Here's a CP I came up with a while ago. It turns out, it is pretty much Maekawa's Turkey feathers four times over, but I still think it is an interesting application:
(a PDF is available at
http://samplesizeofone.com/share/diamond-base.pdf)
Here's what it looks like folded:
There are a few interesting things to note. First, the elements of the CP can be repeated ad infinitum. Second, the base where the diamond joins with the rest of the paper (which forms a plane beneath it) can be made arbitrarily small given enough time and thin enough paper. Of course, it turns out to be impractical to repeat it more than a few times even using very thin paper in my experience.
The original goal of this design was to tessellate this pattern (which is trivial to do) and fold a crane out of each diamond so its wing touched the plane at a point. I also wished to do one that showed the folding sequence of the crane. Alas, due to its unwieldy nature I doubt I'll ever realize this.
Thoughts are appreciated, as are suggestions as to where I can order some arbitrarily thin paper and a pair of VERY small tweezers