Math for doilies and shawls, part 2

Let's compare the estimates from part 1 to a stitch-by-stitch calculation of a simple hypothetical doily having 16 total rounds. This doily starts with 8 stitches in round 1, increases to 16 stitches in round 3, 24 stitches in round 5, and so on (8 stitches increased every other round, a "standard" increase rate that produces a flat doily). There are 64 stitches in each of final rounds 15 and 16.

The total number of stitches in this doily is (8+8+16+16+24+24+32+32+40+40+48+48+56+56+64+64) = 576

After round 8 of 16, the total number of stitches is (8+8+16+16+24+24+32+32) = 160

160 stitches/576 stitches = .277 (i.e., 28%) In other words, after round 8, about 28% of the stitches have been completed, which more or less corresponds to the "25%" quick estimate from above.

Also, 70% of the total number of rounds (16) is about 11 rounds. After round 11, there are 288 stitches. 288 stitches/576 stitches = .5 (i.e., 50%) In other words, half of the stitches have been knitted, just as predicted by the estimate from part 1.

Of course, increasing eight stitches every other round can produce a square or a circular shape depending on where the increases are placed in the pattern. As far as estimating the number of stitches left to knit, it doesn't matter what shape the pattern is, so long as it is basically flat.

What about triangular shawls, the kind knitted back and forth from a few stitches cast on at the center of the neck with rows getting longer and longer? Like the Kiri shawl by Polly at alltangledup or the Flower Basket shawl by Evelyn Clark? Well, they are essentially half of a square knitted from the center out, where 4 stitches increased every other round is a good rule of thumb for producing a flat triangular shape. The percentages are the same for a triangle as if you were working all the way around a square—when you have knitted half of the total number of rows, you have completed about 25% of the shawl, and when you have knitted 70% of the total number of rows, you have completed about 50% of the shawl (not counting any additional knitting like a sideways-knitted lace edging).