What you need:
Worsted weight cotton yarn in colors that you like. I used Lily Sugar 'n Cream in Hot Pink and Lion Brand Lion Cotton in Orchid.
Round six-pack rings, trimmed apart into circles. They will not be perfectly circle and will have minor imperfections and stretched-out areas. See above photo on the left for what mine looked like. Don't worry about it too much because you won't be able to tell later. Just make sure the edges are smooth with no points.
Size H (5.0 mm) crochet hook
Begin by attaching your yarn with a slip stitch to a ring. Chain 3 and work 44 double crochet into the ring. Slip stitch to third chain of your beginning ch 3 to join. (See above and below. By the way, I took excruciatingly detailed pictures of pretty much every step, but I have come to the conclusion that I went overboard and they wouldn't be necessary. If you need any help with any of this, let me know and I can make the pictures available.)
If 45 dc seem to be too many or too few to fill up your ring, you can adjust accordingly! Weave in ends.
When I had made my six circles, I ironed them to get rid of a few bumps in the plastic I felt might show through. I used my iron's setting for polyester, and pressed each side of the circle for a short time. This isn't totally necessary, but I do think it helped.
Now lay them out like this with each ring overlapping the next:
You're ready to make the center ring. I found the magic number for my beginning chain to be 45 (plus 3 for turning chain) again but before you commit to that length, weave the chain through the circles as shown below and remove chain stitches if necessary. A tiny bit of overlap is a good thing. Leave long tails for finishing off. So:
Chain 48, double crochet into the third chain from your hook. Double crochet in each remaining stitch.
Weave your center ring through your circles as shown above. Adjust the ring to evenly show through each section. You should have a slight overlap as shown below.
Fasten off behind a plastic ring, sewing the inner ring into a continuous loop. I did not attach it to anything but itself. Now you have a cluster of 7 interlocking rings. You need to crochet the border around the outer rings to hold everything together securely. The appropriate number of double crochets needed in each ring seems to be roughly a third of the number of double crochets you did into each individual ring.
As pictured above, attach your yarn with a slip stitch to the first loop that shows just to the left of the next ring.
Chain 3, double crochet in each double crochet up to the next circle. I used 14 in each circle. Fasten off, and you're done! Unless you want to add a decorative stitch to the edge, of course! I pressed it again with my iron on the polyester setting. Let me know if you make this. I'd love to see pictures!