More circles

Here is the second part of my previous post on circles.

Method 2, even more so that the method 1, which I have described in that post, uses the elasticity of knitted fabric to make an even circle out of something which is not very even. It is a bit more versatile in the sense that you can change a lot more things and still get a good enough result. The idea is, as I explained before, that if you do increase rounds well separated and with many increase stitches, your eyes won't be able to distinguish a spiral. Furthermore, you can increase this effect by not doing equally spaced increase rows. The result is a more even circle (in the sense that it won't have a prominent spiral), but your eyes will probably be able to distinguish some subtle concentric circles formed by every increase row. There is no way to avoid this. To put it plain, you should follow this method, if you prefer concentric circles to multi-armed spirals.

As I mentioned above, method 2 is more versatile, in the sense that you can make many adaptations to it and get similar (but not identical) results. To exemplify this I give two variants of this method. The 'powers of two' variant can run into problems if you're attempting a very large circle. The reason is that the increase rows become too spaced out. But it works in general. The Fibonacci variant can theoretically run into the same problem, but the increase is smoother so I don't think this will ever happen in practice. This variant is a bit of a joke, in the sense that it complicates things unnecessarily. It is definitely a method for math lovers, because the calculations involved are a bit more complex. I invented it myself, meaning I haven't seen it mentioned anywhere, but I'm sure I'm not the first one to come up with this idea.

Method 2 (powers of two)

1. Cast-on 4 stitches and join in the round.
2. round 1: kbf4 [8st]
3. round 2: knit straight
4. round 3: kbf8 [16st]
5. rounds 4 to 6: knit straight
6. round 7: kfb16 [32st]
7. rounds 8 to 14: knit straight
8. round 15: kfb32 [64st]
9. rounds 16 to 30: knit straight
10. round 31: kfb64 [128st]
11. continue to knit straight every round except for ones with a number which is half the number of stitches in your needles minus 1 (this is the hard way) or, if you prefer, a power of 2 minus 1 (this is easier...), or, to be precise, rounds 63, 127, 255, 511, etc, (even easier) where you double the number of stitches by doing a kfb in every stitch
12. when the circle is about the desired size, to avoid curling, do a few extra rounds in garter stitch and bind-off.
13. Don't forget to block it.

Method 2 (Fibonacci)

1. Cast-on 4 stitches and join in the round (surprise!).
2. round 1: kfb4 [8 st]
3. round 2: k1 kfb1, 4 times [12st]
4. round 3: k2 kfb1, 4 times [16st]
5. round 4: knit straight
6. round 5: k1 kfb1, 8 times [24st]
7. rounds 6 and 7: knit straight
8. round 8: k1 kfb1, 12 times [4x9=36st]
9. rounds 9 to 12: knit straight
10. round 13: (kfb2 k1)x2 k1, 5 times and knit last stitch [4x14=56st]
11. rounds 14 to 20: knit straight
12. round 21: (kfb2 k1)x2 k1, 8 times [4x22=88st]
13. continue knitting every round straight except for rows corresponding to a Fibonacci number which are increase rounds. Every increase round should have evenly spaced increases with a total number of stitches that is equal to the number of the round plus 1 times 4.
14. when the circle is about the desired size, to avoid curling, do a few extra rounds in garter stitch and bind-off.
15. Don't forget to block it.

A word of caution. These patterns, like the other one in my previous post, have been tested with worsted weight yarn, 5mm needles, by myself. Another person, who will apply a different tension, or another type of yarn or needles, may get different results. The circle may start to bulk and turn into something like a kippah. I therefore recommend the usual: start with a swatch before attempting a large circle. You may need to adapt this pattern, by casting on a different number of stitches or removing some straight knitted rounds in the first rounds. Note that problems usually arise only at the centre of the circle where stitches have less space to shrink or stretch. That is a good thing, because you'll be able to notice your circle is not staying flat before wasting too much time.



Math addendum:
How to compute Fibonacci numbers. Start with 0 and 1. Compute the next number by adding the 2 previous ones. This way you get the following sequence
0
1
0+1=1
1+1=2
1+2=3
2+3=5
3+5=8
5+8=13
8+13=21
13+21=34
21+34=55
34+55=89
55+89=144
and so on.


Now how many increases do we need to do in each increase round? OK, I'm going to try to put it in plain English. This should be the number of the current round minus the number of the previous increase round times 4. This is always the number of the previous to the previous round times 4 (with the exception of the first). Here is the list (this is just the above list times 4 except for the first number): 4, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220, and so on. As mentioned above, you should do these increases as evenly spaced as possible.

For completeness, what about the final number of stitches? As mentioned above, it is the number of the increase round plus 1 times 4. That is 8, 12, 16, 24, 36, 56, 88, 140, 224, 360, 580 and so on. I hope I didn't do any mistakes.