Shapes: circle (with or without a hole)

On the first two posts of this series, I described the concentric and the swirl methods to create circles. Both these methods use increases in order to create a circle shape from the centre out. Except for that, they seem to be completely different. But they have something else in common. Take a look at the table below that contains the number of stitches in the every increase round on the concentric method and the previous round of the swirl method.

Row	Concentric	Swirl
2, 1	8	8
4, 3	16	16
8, 7	32	32
16, 15	64	64
32, 31	128	128
64, 63	256	256

You see, what both this methods have in common is that on average there are 4 increases per round. On the swirl method these are worked in alternate rounds, on the concentric method they are worked a lot more rows apart. This is the inspiration behind what I'll call the general method, which is not a single recipe, as the former methods, but a set of rules for creating the recipe that best fits your purpose.

The general method

As stated above, this method is not a single recipe, but a set of rules to create a recipe according to your needs. Like the previous methods, this method knits a circle in the round and from the centre outwards. However, inverting the direction of your knitting is trivial: simply work decreases instead of increases.

The rules:

1. Before we begin with the rules, we define a section as a group of rounds that begins with an increase round and ends just before the next increase round. This means that, except for the first round, a section consist of plain rounds (i.e. rounds with no increases). Furthermore, we define the length of a section as the number of rounds in that section.
2. Start with either the pinhole cast-on for a full circle or any other cast-on for a circle with a hole in the centre (for a circular yoke, for example).
3. The main rule: whenever you want to work an increase round, calculate the number of increases by multiplying the length of the previous section by 4. Then work this number of increases spaced as evenly as possible along the round.
4. Avoid working sections with a length that is larger than the number of stitches on your needles divided by 4 (this relates to the next rule). In principle, far enough from the centre, your sections can be as long as you need, but don't push it.
5. Never work more increases than the number of stitches on your needles. If absolutely necessary, you can correct for missing stitches in a later round, but as stated in the previous rule, this should be avoided.

One final comment, in the main rule, evenly may mean two things: one, evenly along the increase round, two, evenly over different increase rounds, i.e. avoiding knitting the increases along specific lines. The former is very important, the latter not so much as long as the number of increases in each increase round is large enough. For example, remember that in the swirl method we increased along 8 lines and thus obtained an octagon shape, which is close enough to a circle, but not exactly? If instead we had increased 16 stitches every forth round, we would have obtained a hexadecagon, which is undistinguishable from a circle in practice. While increasing 32 stitches every eighth round would result in a triacontadigon, try to distinguish that from a circle!

Example : a triacontadigon

1. Use the pinhole cast-on to cast-on 8 stitches and spread them over 2 or more needles, according to whatever method you prefer to join in the round.
2. Round 1: knit all stitches, if you haven't done so already in the previous step.
3. Round 2: * kfb, repeat from * [16sts].
4. All rounds, except the increase rounds below: knit all stitches.
5. Round 4: *kfb, k1, repeat from * [24sts].
6. Round 6: *kfb, k2, repeat from * [32sts].
7. Round 10: *kfb, k1, repeat from * [48sts].
8. Round 14: *kfb, k2, repeat from * [64sts].
9. Round 22: *kfb, k1, repeat from * [96sts].
10. Round 30: *kfb, k2, repeat from * [128sts].
11. Round 38: *kfb, k3, repeat from * [160sts].
12. Round 46: *kfb, k4, repeat from * [192sts].
13. Round 54: *kfb, k5, repeat from * [224sts].
14. Round 62: *kfb, k6, repeat from * [256sts].
15. Continue in the same way, increasing every 8th round with one extra stitch between kfb's every time.
16. If knitting plain stockinette (or another curling stitch), end with at least six rounds of garter stitch (or another non-curling stitch), before binding-off with your favourite elastic bind-off.
17. Block to shape.

Example : a circular yoke

The above rules do not only apply to a circle started from a centre point. If you want to start with a big hole for you head, in other words, if you're working the recipe for a circular yoke, these rules are just as useful. The prime example in this case is the beautiful traditional Nordic sweaters with stranded-knitting colour work in the yoke. Whether you incorporate the increases on your colour pattern or work different patterns in each section, the rules allow a certain freedom to choose the length of each section of the yoke. For example, in the latter case the length of each section is defined by the length in rows of each colour pattern used, because you don't want to disturb the pattern with the increases. While in the former case, the rules give you a rough idea of the amount of increases that you can incorporate in the colour pattern.

Let us work on an explicit example. Suppose you have a series of stranded-knitting colour-work patterns that you'll be using on the circular yoke of a sweater. Pattern A is 8 rows high by 6 stitches wide, pattern B, 16 rows by 10 stitches, and pattern C, 24 rows by 6 stitches.
Your sweater pattern starts at the collar with a cast-on of 118 stitches. We are going to knit a few rounds before starting pattern A. According to our rules, in the third around we should increase 12 stitches evenly. That means we will end up with 130 stitches. However, 130 is not a multiple of 6. What we can do instead is increase 14 stitches evenly (two stitches extra are not going to make any difference). In the following sections we will do the same: not work the exact number of increases stipulated by the main rule, but always a number very close to it and that is more convenient for our purpose.
Here is a possible recipe for this yoke.

1. Using background colour, cast-on 118 stitches and join in the round.
2. Rounds 1-2: knit straight.
3. Rounds 3: * k8, LI, repeat from *, ending with k6. [132 sts].
4. Rounds 4: knit straight.
5. Rounds 5-12: join other colours and work knitting pattern A.
6. Round 13: knit straight with background colour.
7. Round 14: * (LI, k3) 3 times, LI, k2, repeat from * [180 sts].
8. Round 15: knit straight with background colour.
9. Rounds 16-31: join other colours and work knitting pattern B.
10. Round 32: knit straight with background colour.
11. Round 33: * LI, k3, LI, k2, repeat from * [252 sts].
12. Round 34: knit straight with background colour.
13. Rounds 35-58: join other colours and work knitting pattern C.
14. Round 59: knit straight with background colour.
15. Round 60: * LI, k3, LI, k2, repeat from *, ending with k2 [353 sts].
16. Continue knitting your sweater according to the circular yoke recipe. That is, work the desired number of rounds and, before separating the arms and body, work the necessary number of increases to obtain the desired number of stitches.

Abbreviations:
k: knit;
LI: lifted increase (or any other increase of your choice).