A couple of folks have asked me for some more detail on building this little paper cone so I thought I would create a diagram to show how each row should be pinned. This diagram is as if the project were flat. There are four triangles to each row and the top point of each triangle should be lined up with the seam between triangles of the row above it. Pin only the bottom two points and leave the top point free to form the "spikes" of the cone.
I hope this helps to clarify a bit more. It took me a couple of tries to figure out how to make these ornaments.