Closed-form heel formula

In my notes on modifying the heel from my Widdershins pattern, I gave a recursive formula to determine the number of stitches needed for the heel flap. In the comments on that entry, Carol posted a closed form:

Embarrassingly, I had to ask my son, who's studying math at university what k was. He told me to think about it. It finally clicked.
I still couldn't figure out how to program the formula but did get it into closed form.
I think it reduces to
h(n) = n – 2 – 2 * int(n/9),
where int(n/9) stands for the integer part of n/9.
btw. I really like the heel.

Of course, I had to try this out. Being a bit to lazy to do a real proof, I just coded it up quickly and compared the two functions for values up to 150. If anyone is planning on knitting a sock over 300 stitches around, I could go higher.

>>> def h1(n):
...   def h(k):
...     if k < n/3:
...       return k
...     else:
...       return 4 + h(k-6)
...   return h(n)
...
>>> h1(33)
25
>>> def h2(n): return n-2-2*(n/9)
...
>>> h2(33)
25
>>> for i in range(150):
...   val1 = h1(i)
...   val2 = h2(i)
...   if val1 != val2:
...     print i, val1, val2
...
8 4 6
17 11 13
26 18 20
35 25 27
44 32 34
53 39 41
62 46 48
71 53 55
80 60 62
89 67 69
98 74 76
107 81 83
116 88 90
125 95 97
134 102 104
143 109 111

For some numbers, my formula suggests one more pair of turning rows than Carol's formula, but given that my original n/3 came from me thinking, "Oh,one-third of the stitches in the center sounds reasonable." I don't think it'll make too much difference unless you are really picky about how pointy you like your sock heels. Thank you, Carol! I was feeling a bit guilty about not providing a closed form, but the guilt didn't motivate me quite enough to get out my college textbooks and remember how to solve it myself. Now I don't have to!
Categories: knitting, socks, Widdershins, python

Notes on Widdershins heel

I promised to make some notes on adapting the reversed round heel shaping from Widdershins for a different number of stitches. (This still counts as the weekend, right?)

Here are my original heel shaping instructions, before they were edited for Knitty. Note that I didn't specify what sort of increases or short row method to use. In my original socks, I actually used paired increased, but thought that might be a bit too picky for something that will usually be hidden inside my shoe. For the short rows, I've been using the yarnover method lately. So, if m1 increases and w&t short rows aren't working for you, feel free to try another technique. The socks don't depend on those particular methods.

Heel Gussets:
On the following rounds, continue to work instep in pattern while increasing on stitch from beginning and end of sole stitches every other round until there are 47 sole stitches.

Turn Heel:
The sole stitches now consist of 14 sts for right gusset, 19 heel sts, 14 sts for left gusset. Work across right gusset. You will now be turning the heel on the 19 heel sts. The heel is shaped with a combination of short rows and increases. It is your choice how to handle the turning stitches.
Row 1: k to 2 sts before end of heel sts, inc1, k1, turn
Row 2: p to 2 sts before end of heel sts, inc1, p1, turn
Row 3: k to 4 sts before gap, inc1, k1, turn
Row 4: p to 4 sts before gap, inc1, p1, turn
Rep rows 3 and 4 until there are 27 heel sts

Work 1 round even to smooth out the short rows, ending in center of heel sts.

Heel Flap:
You will now work in rows across heel sts, joining the gusset sts.

Row 1: k to last heel st, ssk last heel st and 1 st from left gusset
Row 2: sl st, p to last heel st, p2tog last heel st and 1 st from right gusset
Row 3: *sl st, k1* across heel, ssk last heel st and 1 st from left gusset

Rep rows 2 and 3 until left gusset sts are gone. (There will be 1 unworked st left on the right gusset that will be picked up on the next round)

Work across instep sts. K2tog final gusset st and first heel st. Continue instep pattern across heel sts.

So, where did the numbers come from? I based my numbers on a top-down round heel and then reversed the shaping. The number of gusset increases is based on the size of the two gussets plus the heel below the turning. In a top-down heel, you knit a heel flap on half the stitches (usually for about as many rows as stitches), then turn the heel with short rows, working decreases along the way. There is often a line in the instructions like "knit to one stitch before the gap, ssk, k1, turn" and similar purl shaping. After the turning rows, stitches are picked up along the heel flap (usually about one stitch for every two rows) and the gusset stitches are decreased away on subsequent rounds until we are back to the original number of stitches. So, for my 54 stitch sock, I would work the heel over 27 stitches, the flap would be 28 rows long (I need an even number here and it's better to have a slightly deeper heel than one that is too shallow.), the turning rows would decrease from 27 down to 19 stitches, and I would pick up 14 stitches on either side of the heel flap.

For the reversed round heel we have to do a little advance thinking to figure out how many gusset increases we need, work the turning rows with increases as shown above until you have increased the heel stitches to their full width, then decrease away the gussets until you are back you the original number of stitches and work the cuff.

I'm a big geek, so I actually came up with an equation to figure out the approximate number of heel stitches. For a round heel with a flap n stitches wide, the number of stitches below the heel turning is h(n) where h(k) = k for k<n/3 and 4+h(k-6) otherwise. (I also wrote a small Python script to compute that, rather than coming up with a closed form solution. I'm a lazy geek.) Of course, the easiest thing to do is use the numbers from a preexisting top-down round-heeled sock.

I hope this helped. I'm glad so many of you like my pattern and are interested in adapting it.
Categories: knitting, socks, knitty, Widdershins

Widdershins update

I've had multiple people contact me about adapting the heel shaping of Widdershins for a different number of stitches, so I'll write up some general instructions this weekend.
Categories: knitting, socks, Widdershins, pattern, knitty

I'm in Knitty!

My sock pattern is in the summer issue of Knitty! I've reported a couple of corrections they need to make (a few details did not make it through the formatting process unharmed), but at least it's there. Oddly enough, mine is not the only pattern for toe up socks with a heel flap in this issue.

See more progress on: publish my knitting designs

Categories: knitting, knitty, socks, pattern