Shawl Geometry: Pi Circles

06 May 2013

This is the third post in a series about different shawl shapes and how to knit them. All the posts in the series can be found right here.

PI Circle from the center out

The PI Circle falls into the circular shawl shaping camp of “start with a tiny circle, and radiate concentric circles outwards.” The basic idea is that you start with a small circular cast on, then alternate expanses of straight knitting, with dramatic increases. If you work these in accordance with EZ’s concept of PI Circular Shawl Shaping, (which is based on the mathematical relationship between the rate of growth between a circle’s circumference and radius), then you’ll create a circular shawl that lays flat.

The basic concept is, if section A has X rounds and Y stitches, then section B has 2X rounds and 2Y stitches, section C has 4X rounds and 4Y stitches, section D has 8X rounds and 8Y stitches, etc.

Center Out Knitting Instructions

CO 6sts. Join in the round, being careful not to twist.
Section A: knit 1 round.
Inc Rnd: (yo, k1) around. 12sts.
Section B: knit 2 rounds.
Inc Rnd: (yo, k1) around. 24sts.
Section C: knit 4 rounds.
Inc Rnd: (yo, k1) around. 48sts.
Section D: knit 8 rounds.
Inc Rnd: (yo, k1) around. 96sts.
Section E: knit 16 rounds.
Inc Rnd: (yo, k1) around. 192sts.
Section F: knit 32 rounds.
Inc Rnd: (yo, k1) around. 384sts.
Section G: knit 64 rounds.
Inc Rnd: (yo, k1) around. 768sts.
Continue to desired dimensions.
Bind off loosely.

Pi Circles from the hem in

Knitting any shawl from the hem inwards takes a little more planning than working one from the center out. With a shawl knit from the center out you can keep increasing until you reach your desired size, or until you run out of yarn. But knitting a shawl from the hem into the center requires that you know your size information before you begin, so you can cast on an accurate number of stitches.

Calculating your round count

Determine your final gauge and desired radius of your shawl.
The radius is half the width of your final shawl, or the length from the center to the edge.
[Rnd gauge] x [desired radius] = [# of rnds total]
This will be your magic number.

Calculating your shaping using increases

Determine how many stitches you want as the bind off at the center of your shawl.
Next work out shaping for a PI Circle as you would if you were knitting it from the center out. Work out the instructions for your shawl pattern until you reach your shawl’s magic number.
Remember: Your magic number is your TOTAL number of rounds knit, NOT the number of rounds in one section.

Example calculations
We’ll use the same math we did for the center to hem version of this shawl.
So if your magic number (number of rounds total) is 100 and your desired BO number is 6sts, the calculations for your shaping would be as follows:
CO 6sts.
Section A: knit 1 round.
Inc Rnd: (yo, k1) around. 12sts.
Section B: knit 2 rounds.
Inc Rnd: (yo, k1) around. 24sts.
Section C: knit 4 rounds.
Inc Rnd: (yo, k1) around. 48sts.
Section D: knit 8 rounds.
Inc Rnd: (yo, k1) around. 96sts.
Section E: knit 16 rounds.
Inc Rnd: (yo, k1) around. 192sts.
Section F: knit 32 rounds.
Inc Rnd: (yo, k1) around. 384sts.
Section G: knit 31 rounds.

Notice how we stopped in the middle of a section?
If we hadn’t stopped at our magic number of rows (100) we would have knit 64 rounds, not 31, in that last section (section G).
We stopped because, if you add up all of the rows (not counting the CO) you get 100. Which is the magic number for this shawl.

Turning Increases into Decreases

Once you’ve worked out your calculations as though you were knitting from the center out, you knit a shawl from the edge in, by working backwards.
To do this, the number of stitches in your final increase round (when working center out) becomes your cast on number. And rather than working increase rounds that double your stitch count, you work decrease rounds that cut the number of stitches on your needles in half.

Knitting Instructions

CO 384sts. Join in the round, being careful not to twist.
Section A: knit 31 rounds.
Dec Rnd: (k2tog) around. 192sts.
Section B: knit 32 rounds.
Dec Rnd: (k2tog) around. 96sts.
Section C: knit 16 rounds.
Dec Rnd: (k2tog) around. 48sts.
Section D: knit 8 rounds.
Dec Rnd: (k2tog) around. 24sts.
Section E: knit 4 rounds.
Dec Rnd: (k2tog) around. 12sts.
Section F: knit 2 rounds.
Dec Rnd: (k2tog) around. 6sts.
Section G: knit 1 round.
Bind off loosely.
Break yarn leaving a tail, and thread tail through live stitches, pull tight and weave in your end. (The same way you would finish off the crown of a hat.)

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