If you've been reading this blog for any length of time, then you surely know I'm a huge fan of hand-knit socks. I almost always have a sock WIP on the go, and I have knit enough of them in my knitting career that I could probably knit one in my sleep.
After knitting hundreds of socks, however, I find myself perpetually curious in new ways to knit them (at least within my usual MO of cuff down). I've tried different methods of doing the gusset decreases, doing a simultaneous heel flap and gusset, and of course many different heels. While knitting my daughter's last couple of pairs of socks, I thought about whether it would be possible to create what is essentially a gusset for the heel and have the snug fit one gets with a traditional flap and gusset decreases. The only way to know for sure was to try, and while it took more than one attempt to get the fit just right, I'm happy to report that it worked as well as I'd hoped.
Non-Euclidian is a twist on the traditional stockinette sock. Rather than working back and forth to make a heel flap, increases create a pocket of fabric to hug the heel. Short rows are then worked under the heel to decrease the extra stitches. The result is two triangular areas of fabric covering a three-dimensional part of the foot -- hence the name.
I've written up the pattern in the form of a stockinette sock, but the beauty of the heel is that, much like many other heels, it can be substituted for any other heel. So you can plop it into any other sock pattern or use it in conjunction with your favorite stitch pattern.
The pattern has been graded to five sizes -- to fit a foot circumference of 6, 7 (8, 9, 10) in./15, 18 (20.5, 23, 25.5) cm -- to fit a wide range of child and adult feet. As you can see, it works well with self-striping yarn, but really any kind of sock yarn is going to place nice. I've suggested a dense gauge of 9 stitches per inch for durability (because the heel has no reinforcement like the slip-stitch method usually used in heel flaps), but if you prefer to work to a different gauge, it's easy to adapt the method to your stitch count.