This method is only suitable for small to medium sized paper pieces (1/2 to 1 inch edges in length; 1 to 2.5 cm). The difference here is that you do NOT stitch through the paper piece at all. This has the advantage that there are no basting stitches that need to be removed and the whole process is a little faster. I have used this method for many years and it produces really nice and neat results!

For Hexagons:

The preparations are the same as usual:

Place the paper shape on the wrong side of the fabric (pin the paper in place if needed). Cut the fabric patch at least ¼ inch (6-7 mm) larger than the paper using good scissors. It is important to have enough fabric excess to easily cover the paper shape all the way around, remember this is your seam allowance, do not make your  life more difficult than necessary! Some quilters recommend to cut squares from the fabric when they are using Hexagons shapes. I have not tried this myself, but I imagine it will work fine.

In this method you start at a corner:

Fold over the seam allowance along the edge of the paper, and at the corner fold over the next fabric edge. Finger press the fabric folds, hold both folds in place and stitch through the meeting point to gather the two folds with a single stitch. Do not stitch through the paper! No stitching should be visible on the front of the paper piece!

Pull the stitch tight and turn the paper piece in your hand to the next corner. Fold over the next fabric edge and again stitch through the folds to gather them with a loop of thread.

Continue in this fashion all the way around the paper shape. Most right handed quilters find is easy to work their way around anti-clock wise.

At the last corner finish off with a second stitch (on top of the first stitch). To secure the end  of the thread weave a few tiny stitches through the seam allowance and the snip off the thread.

Voila! You are done. No basting stitches visible!

This method works just as well for Diamonds or other paper shapes, as long as they are not too large (or too small!).