Sizing Knitted Pentagons

I thought some people may be interested in how I sized the pentagons in plum blossom. I used a yarn with a slightly different gauge than called for in the pattern and had to make several changes. I started by swatching in both stockinette and 1x1 rib. Then I tried to find the number of stitches to cast on for a pentagon with the correct height (~6.5 inches for my size). Unfortunately, the schematic on page 48 only gave the height of the pentagon and not the length of each side.

I did some websurfing and found that the ratio of a pentagon's chord and it's side is the golden ratio. There are many proofs on the web. Here is one. But the height of a pentagon (which is given in the pattern schematic) is shorter than the chord. (A chord is a line connecting any two non-adjacent vertices of a pentagon.) I worked out a formula for calculating the length of a pentagon side as a function of the height using trigonometry. But, I can't find my calculations and this solution was much more elegant than mine.

If s is the length of a pentagon side and x is it's height, then

s = 0.65 x
x = 1.54 s

Plugging in the numbers from the schematics on page 52 and 64 ,which give the dimensions of both the pentagon heights and sides, showed that the formula is correct.

I wanted a 40" chest pullover which meant I needed a 6.5 inch high pentagon. That translates into a pentagon 4.25 inches long on each side. My swatch suggested that 20 stitches of 1x1 rib using my yarn and size 4 needles would work. I made a test pentagon, washed and dried it, and then measured it. It was just shy of 6.5 inches because the sides of the knitted pentagon are somewhat curved, not straight. Nevertheless, I plowed ahead with the rest of the pentagon yoke.

When I tried on the completed pentagon yoke, it was slightly too tight. I guessed that, once completed, the weight of the sweater body and sleeves would stretch out the yoke. That turned out to be correct. Whew!

Suppose you want to knit a pentagon using another stitch? The ratio of the sides to the height remains the same, whether you make the pentagons in stockinette, rib or garter stitch. The decrease ratio may need to be adjusted for the row gauge. Stockinette and rib have the same row gauge so decreasing one stitch per round (or two stiches every other round) gives pentagons with a slight volcano effect (protrusion in the middle) which blocks out flat. The book's instructions for the garter stitch pentagons in the mosaic pullover also decrease 2 stitches every other round and the pentagons appear flatter.

Still skeptical? Grumperina wanted to make a size petite sweater and used stockinette for her pentagons. She used 15 stitches on each side. If she achieved a perfect 4.5 sts/in gauge, then her pentagon had a side of 3.33 inches and a height of 5.13 inches which is pretty much dead on to the 5.25 height given in the schematic.

keywords: knitting, geometry, knitting nature, norah gaughan, swirled pentagon pullover, knitted pentagon