Since shortly after I became a sock knitter, I've been a cuff-down sock knitter almost exclusively. It's not that I had anything against the toe-up sock; I just could never get toe-up socks to fit me as well as cuff-down socks. The heel never seemed to be in the right place, for one, and then there was always the problem of the too-tight bind off. As the years went on, and I became able to knit my standard sock recipe from memory, I pretty much stopped trying toe-up socks, though by then I likely could have cracked the code (so to speak) on getting them to fit my foot. In short, cuff-down socks were easy and didn't require much thinking for me, and as sock knitting has long been my mindless, stress-reducing knitting, that seemed the way to go.
When I released my Non-Euclidian pattern, it got a great response -- including many requests for a toe-up version. At that point, I had a bit of a reckoning with myself. I realized that in many respects the math was already done and that flipping the construction of the sock wouldn't require much effort at all. What did I have to lose by giving it a try? And so I did, starting with a pair of socks for myself to see if it would work before I admitted to myself that I was actually writing a toe-up sock pattern.
Obviously, it worked. And, if I do say so myself, it worked brilliantly. The fit is just as good on me as the cuff-down version, and now I can appreciate one of the big allures of knitting socks toe up: using up as much yarn as possible.
I mentioned in the debut of the cuff-down version that geometry was my favorite math subject in school and that it influenced the name. For the toe-up version, I've once again gone back to geometry for inspiration. My favorite part of geometry was doing proofs, and one shorthand we used in proofs was "CPCTC," or "congruent parts of congruent triangles are congruent." It seemed a fitting name for socks with matching triangles on them.
This version of the sock isn't an exact replica of the original. Rather, it's more of a "flipped" version of the sock. Rather than increasing for the heel, the increases are done on the bottom of the foot, providing space for the usual gusset area. The heel is turned to cup the very back of the heel, and the remaining decreases create a mock heel flap. You can then make your socks as tall as you like, ending with some 2x2 ribbing. To ensure that the top of the sock is comfortable, finish with a stretchy bind off of your choice. I really like one called the Miraculous Elastic Bind Off, which seems to have disappeared from the blog on the Web where I originally found it but is basically the traditional lace bind off worked in rib.
I've now made myself a second pair of toe-up socks using my own patterns, and I'm pretty sure I've been converted (does that make me an ambidextrous sock knitter?). I know it'll be great for making socks out of handspun when my yardage has been a little disappointing and I want to use up as much of the yarn as possible. And it's also pretty great for commercial or indie-dyed yarn when I'm trying to use up as much yarn as possible for Stash Dash purposes. ;-)