Triangle Shawls and fun with algebra

My first Triangle Shawl was Evelyn Clark's Spinner's Shawl pattern, modified to be stockinette rather than garter stitch. It's a bit small, but I did a pretty good job of using most of the yarn I had. I love it and wear it around the house a lot as a little extra layer around the neck. I even got so bold as to wear it in public; even though I am lacking an accessories gene --- wearing anything extra unnecessary to maintaining modesty and a normal temperature usually makes me feel exposed and awkward.

Triangle number two is being knit with Sea Silk. I only have the one hank, I want to use as much of it as possible. This requires a few things easy to find household items; a scale, a calculator and some mathematics.

Pattern is Evelyn Clark's Flower Basket Shawl found in IK. I won't bother to dig out the exact issue, because everyone has already knit this pattern. I chose it because the pattern repeat is only 10 rows and the edging is only 12 rows (10 pattern rows and a bind-off). It will be easier to squeeze in as many pattern repeats as possible instead of a shawl such as the Spinner's Shawl where one must stop with the pattern repeats in time to commit to 40 rows of edging. Admittedly it makes for a more interesting shawl to have a different and complimentary border, but so it goes when the Sea Silk is so dear. Flower Basket it is.

Jessica has a shawl calculator on her blog but alas, it is in excel, something my poor little mac cannot read. So I got some paper and a pencil and worked it out myself. Eventually I did read her spreadsheet on my husband's computer. It is fine, but hides the math. So I am glad to have done this work anyway.

Much calculation, much scratching out, much musing, some greek letters and that Gaussian addition trick and I found a formula to compute how many stitches have been knit given the number of rows knit. Before writing it up, I searched the net a bit more and found that Susan had posted an alternate calculation, much simpler. She didn't simplify into algebra, so I did and it resulted in the same formula I found. Although I'm a little chagrined that I took a more circuitous route, in the end having my formula be the same as hers gives me confidence.

Here it is:
While knitting a triangular shawl where one starts with 7 stitches (ignore the set-up rows which use a trivial amount of yarn in the grand scheme of things) and increases four stitches every right side row, if N is the number of rows that have been knit then the total number of stitches done is N² + 5N. (now that I think about it, I am guessing one could derive this from a simple area of triangle formula.)

How does one use this delightfully compact formula? Well, I have knit 10 pattern repeats so far. With 26 rows to prepare for the main pattern, 10 rows per repeat, I have knit 126 rows. Therefore I have knit 126*126 + 5*126 = 16,506 stitches in total.

My scale says this weighs 55 grams, or about 300 stitches per gram.

Looks like plenty of yarn for another pattern repeat or two. Keeping in mind that the finishing will require 12 rows, how much yarn can I expect to use if I knit a few more repeats?

11 repeats: N = 126 (rows so far) + 10 (11th repeat) + 12 (border/finishing) = 148

148² + 5*148 is 22,644 stitches. At 300 stitches per gram, this will use 76 grams of yarn.

12 repeats: N = 158 Total Stitches = 25,754 or 86 grams

13 repeats: N = 168 Total Stitches = 29,064 or 97 grams of yarn.

Now, to tell the truth, I have 88 grams of yarn. Yes, I started with 100 grams, but we won't talk about how the Flower Basket pattern was not my first attempt, nor what happened to that tangled mess of crap 12 missing grams. So I can knit two more pattern repeats, 12 total, then work the edging. I ought to come very close to using up the 88 grams available, and if I go over by a few yards, well, I might be able to resurrect a few yards from the trashed bit.

Isn't it grand? I just love quadratic expressions, this one is so tidy and clean! And the thought of knitting just two more repeats and the edging sounds infinitely doable, after all, I've already done 10 repeats, what's a few more?

Of course, anyone who has actually knit a triangle like this is smirking at that statement. Looking at it from a more realistic perspective, I have knit 16,506 stitches and I will be knitting a total of 25,754 stitches. Therefore, I am only 64% done.