Saturday, January 06, 2007

Fibonacci 1

In my last post I mentioned my experiments using a fibonacci sequence to make a hat.
This is the simplest of patterns, it uses a fibonacci sequence for both the intervals between increases, and for the actual numbers of stitches to increase.
The resulting hat looks like a hat, (albeit a stylized one) and is a reasonably good fit (it is a large hat, but then, I have a large head, (as do my children) so a large hat is fine.


Lets start with the key numbers, in sequence:
(0), 1, 1, 2, 3, 5, 8 ,13, 21, 34, 55, 89.… the series continues, but this is enough for our purposes.

If you are unfamiliar with a fibonacci sequence, it is formed by added the last 2 numbers together to get the next number.
.....Start with 1 (proceeding it, is, of course, 0(zero)
......So 1 + 0= 1
..Then 1 +1 =2
...Next 2 +1=3
And so on.

This is how I translate this to a knitting pattern.

I decided, the hat should be knit in the round, but the first few ‘rounds’ have so few stitches they were worked as I-cord on 2 DPN’s, and knit more like Rows.
To make this easy to understand, I use the designation R (not a row, but not really a conventional round either!)

Cast on --not R 1, but R 0 (well it is in most patterns, since row/round 1 always starts after the cast on. The cast on generally not considered row 1)

R0, (Cast on) 1
R1 Knit 1 1
R1 (again!)Make 1 (2 stitches on needle)

Ok i could have proceeded like that, but instead, I used a long tail cast on, and a twisted loop, (not a slip knot) to start.

A long tail cast on is a simple cast on combined with a stitch (R1 in effect)

So a loop (not a slip knot) and 1 stitch cast on gives 1 stitch --that hasn‘t been knit (just cast on) and 1 stitch --that has been in effect, knit.
Now on to R2--in R2, 3 stitches are need-so there is a need for an increase:
R2- Knit 1, Make 1, Knit 1 (3 stitches on the needles)
By R3, 5 stitches are needed
R3 -K1, M1, K2, M1 (5 stitches on the needles)
As so begins the ‘pattern’.

Now a proper fibonacci curve progresses at a ratio of 1:0.68

And this can be done in knitting. But this pattern is simplified (and that is precisely what I dislike about it, and what will be resolved in F2!)and it doesn’t really ‘curve’.

The pattern proceeds straight forward: at fixed intervals, a fixed number of increases. The increases, and intervals between the increases, will be based on Fibonocci sequences.
The next increase is at R5, and the increase needs to follow the sequences so the number of stitches will increase from 5 to 8
So, R4 is knit every stitch (no increases or decreases)
and R5 has 3 increases (to see the pattern clearer, the M1's are color coded)
R5--K1, M1, K2, M1, K1, M1, K1 (8 stitches)

From this point forward only rounds with increases will be noted, all other rounds are knit with out any increases.
And at this point, I began working in rounds (8 is the minimum number of stitches needed to work comfortably in the round, in my opinion)
I used a magic loop, but DPN’s could be used, though eventually you’ll need to move to a circular needle.
Round 8: Increase 5 times to bring stitch count to 13:
K1, M1, K2, M1, K1, M1, K2, M1, K1, M1, K1
Round 13: Increase 8 times, to bring stitch count to 21:
*K1, M1, K2, M1, repeat from * across round till you have increased 8 times
Note: round will end with an additional K1
Round 21: Increase 13 times, to bring stitch count to 34:
*K1, M1, K2, M1, repeat from * across round till you have increased 13 times
Note: round will end with K2
Round 34: Increase 21 times, to bring stitch count to 55:
*K1, M1, K2, M1, repeat from * across round till you have increased 21times
Note: round will end with K3.
Round 55: Increase 34 times, to bring stitch count to 89:
*K1, M1, K2, M1, repeat from * across round till you have increased 34 times
Note: round will end with K5.
Work even for 34 rounds.

Hat is complete. Well-- basically complete.

What I found interesting is how the ‘pattern’ of increases (K1, M1, K2, M1) resulted in a remainder (K X at the end of round) that also followed a Fibonacci sequence!

What remained was a bind off.
I decided I didn’t want to break the count with more rows/rounds of ribbing, so on went an I cord.

I also wanted something a bit more substantial than the standard I cord. Partly because I didn’t think a 3 or 4 stitch I-Cord would be enough of an edge to prevent rolling, and partly for style.
I consulted Nicky Epstien’s book, Knitting on the Edge, and found a 6 stitch, cabled I-cord..
Her design is for an sewn on/applied I-cord, I modified this to a Knit-In-Place I-cord bind off.
Cabled I-cord Bind off
With the same yarn (or with a contrasting color) cast on 6
Slip them to the left needle, (unknit)
R1 -K5, SSK (1 stitch bound off), return stitches to left needle
R2--K1, slip 2 onto cable needle, hold at front, K2, then k2 from cable needle, SSK,return stitches to left needle
R3--K5, SSK (1 stitch bound off), return stitches to left needle
R4--K5, SSK (1 stitch bound off), return stitches to left needle
Repeat these 4 rows till all stitches have been bound off.

After a few rows, to make it easier, I turned the last on the left needle, before the 6 I-cord stitches were returned. AND I turned the first stitch of the icord as I returned the stitches to the left needle.
This resulted in the last stitch of the icord and the next stitch to be bound off being ‘twisted’. These 2 stitches were then knit through the back look (un twisting them) which resulted in to decrease that was identical, in effect, to a SSK.


After the last stitch is bound off, graft the last row of the I-cord to the cast on row, making the edging look seamless. (this grafted row, plus the cast on row, corrected the bind off to 92 rows, so the cables pattern was even!)



The finishing touch, (after weaving in all the ends), was a tassel. I have been using pom pom’s a lot of late, and wanted something different.. Besides a tassel is more in keeping with the shape of the hat.
NOTE: as you have surely noticed, I didn’t include yarn/needle size/gauge in this ‘pattern’. Obviously, your choices will make a difference to the finished size of the hat.
My hat fits me. The final 34 rounds (a huge chunk of the hat) are knit straight(no increases)
There are 89 stitches in these 34 rounds. You can swatch and find the right yarn/needle/gauge to make any size hat you desire!

F2 will be a while before its knit.. other projects require my attention!

3 comments:

Anonymous said...

The hat is so pretty. I love how the striping curves around the hat.

Thanks for commenting on my blog about color work. I have and am currently working with stripes. The slip stitch idea sounds like a good next step.

Kay

sulu-design said...

My students will love to hear that this actually turned out to be a hat! They're so skeptical that math is actually used in the real world.

sulu-design said...

I'm returning to post another comment... I thought you might be interested in this blog, since you're into the mathy knitty thing:
http://runawaypenguin.blogspot.com/
Check out this particular post:
http://runawaypenguin.blogspot.com/2007/01/mathematical-knitting.html