This is one of the "hard" blocks in the quilt and I have to admit I had to think about this one a little. There was quite a bit of experimentation so be sure to read the instructions. The photos may be a little confusing if you don't.
Here we go.
Print or trace the block diagram onto foundation paper and, if you're not using freezer paper for foundation piecing, freezer paper as well.
Important! You have to mirror this diagram.
Cut out and number the outside sections as in the photo.
Foundation piece the numbered sections and trim the seam allowances to 1/4".
Cut the four octagons out of freezer paper.
Iron them onto the wrong side of main block fabric and cut out, leaving 1/4" seam allowance all around. (Ignore the stitching in the photo. It was an experiment gone wrong.) Also note that these aren't "real" octagons. Their sides are not the same length.
Here's where it gets a little unconventional.
Press the seam allowances on three of the short sides of each octagon toward the freezer paper and secure by sewing a small seam. The photo only shows two sides, but I later discovered that three is better.
Sew the octagons together in pairs, along the long edges. Press the seams open.
Sew the two octagon sections together as in the photos below. (Note that here 3 of the short sides are folded over.) Press these seams open as well.
Cut away all the excess thread ends.
Cut a piece of background fabric 4" x 4" and lay the octagon unit right side up on top of it. Baste it securely to the background fabric.
Using an applique stitch, applique the octagon unit to the background fabric along all the folded over edges.
Remove the machine stitching holding the folded seams in place.
Trim the block section to 3 5/16" x 3 5/16". I know this is a weird measurement but I measured it more than once.
Sew the smaller edge sections to two sides of the square.
Sew the other two edge sections to the other two sides of the block.