I'd heard of using a rectangle to begin a toe up sock but had never seen a
pattern, so I came up with my own method. It uses a set of 4 dpns, but
could easily be adapted to 5 dpn's if needed.
Knit a rectangle 8 sts wide and 6 rows long (stocking stitch)
With ndl 1 k8 (along the 'working' edge of the rectangle), with ndl 2
pickup 4 sts along side of rectangle (ends of the six rows) and 4 more sts
from other end (halfway across the cast on edge), on 3rd ndl pick up 4 sts
from other end (other half of the cast on edge) and 4 more sts from other
side (other end of the 6 rows) - you'll have 8 + 8 + 8 = 24 stitches in
total
Knit one round even
Move first 2 sts from ndl 2 and last 2 sts from ndl 3 to your first
ndl. You'll have 12 on first ndl and 6 on each of the other ndls (with
the picked up sts)
On ndl 1 k1 m1 k10 m1 k1, on ndl 2 k1 m1 k5, on ndl 3 k5 m1 k1
Work one round even
On ndl 1 k1 m1 k12 m1 k1, on ndl 2 k1 m1 k6, on ndl 3 k6 m1 k1
Work one round even
Continue increasing in this manner til you have the number of sts you
need for the foot of the sock
You could change the frequency of the increase rounds, the number of
sts picked up, and / or the size of the rectangle if you like.