I was thinking about knitting math this afternoon and I had a massive DUH moment. If I were figuring out the number of stitches in a non-mitered square, it would be easy-peasy: stitches per garter ridge x number of ridges, which is the same as 2x^{2}. It's a square! The area of a square is a solved problem! But somehow the diagonal construction of these squares, and the presence of a spine stitch, and the resulting weird numbers -- it made the math seem much more complicated than it actually is.

The whole time I have been making this blanket, I have wanted a formula to tell me when a square is halfway done. Whenever I'm doing something that takes a while, I like being able to quantify my progress. With a mitered square, you're halfway done when you've worked x^{2 }stitches, where x is the number of stitches on one side of your square. This falls in different places, though, depending on the size of your square. It's messy to add up the actual numbers (79 + 79 + 77 + 77 + ...), and I knew there had to be a way to express it. And there is! Are you ready, O ye few but mighty members of the Knitting And Also Math Appreciation Club?? Here is your formula, with thanks to my 20yo:

Let x = the number of stitches on one side of your square. Plug it in and you will get the number of completed garter ridges at which you are halfway done. So if you are making a square with 30 stitches on a side? You'll cross the halfway point near the end of the ninth ridge (y = 8.79). What if there are 12 stitches on a side? You're halfway done after 3.51 ridges.

This is probably the niche-iest of all the niche posts I have ever written in almost 16 years of blogging. But someday, maybe, someone will search "how to tell when you're halfway done with a mitered square" and Google will steer her here.

PS A bonus equation for those of you who haven't had enough yet, before I recycle the pages on which my 20yo and I scribbled notes today: if you want to know how many total stitches you've completed at the end of any garter ridge, it's 2y(2x - y), where x is the number of stitches on one side and y is the number of completed ridges. You're welcome!

And! If you factor the x in your equation and rationalise the surd, you can make it even prettier. (surd(2)-1)/surd(2) is approximately 0.4/1.4 = 2/7. Depending on how good your mental arithmetic is, you could potentially calculate 2/7 the number of stitches without recourse to paper or calculator to get the number of rows. The approximation should come out in the wash unless you have a ginormous (>100 stitches) square, since you only care about whole stitches anyway.

This is what grad school physics was for, right?

Posted by: Charlotte | May 20, 2020 at 05:17 AM

This makes my head hurt. :-)

I suspect that any grab-bag blanket I knit from remnants will be of the "knit two rows, then change yarns" variety.

Luckily I've basically not been knitting these last few years, so there are UFOs that call for my attention first.

Posted by: Jody | May 23, 2020 at 08:06 AM