My 100th free pattern on Ravelry! This reversible doily from 1902 Butterick Publishing’s Fancy and Practical Knitting, pp. 61-62, has 4 diamond shapes in k2, p2 ribbing edged with yarn overs on a garter stitch background. The doily would be circular if not for 2 rounds near the end of the instructions! These 2 rounds work 3 yarn-overs over 3 stitches in 8 places around the doily to make the corners.
The first 2 rows of the doily are knitted back and forth to have enough stitches to start working in the round comfortably. Any small gap left by this technique can be closed when weaving in the beginning thread end. If preferred, work in the round from the start or use any technique comfortable for you to get started. Some find a crochet “magic loop” and a few rounds of single crochet very helpful for this.
Skills needed: long-tail cast on, knit, purl, yarn over, double yarn over, kf&b (knit front and back of 1 stitch), SKP (slip 1 stitch, knit 1 stitch, pass slipped stitch over), k2tog (knit 2 stitches together) and bind off in knit stitch.
For optional crochet edging: ch (chain), sl st (slip stitch), sc (single crochet) and dc (double crochet).

Gauge is not important in this pattern. Any block-able yarn/thread will work. Make a swatch to check size and appearance.

240 yards of #10 crochet cotton thread will make a 9” doily with crocheted edging in the stated gauge.