I have very little actual information on this sock. It was found somewhere in northern Egypt, and dates from sometime between the 11th and 14th centuries CE. It is made of both natural and dark indigo blue cotton. It was knit in the round in stranded knitting. I don't have the gauge.

My sock came just below my knee. Since it was my first stranded knitting, first sock, and third knitted project ever, I didn't use cotton, as experienced knitters have said it is more difficult than wool to work with. I have used it on historical recreation projects after I had a bit more experience

Also, my toe reconstructs the one on a different Medieval Egyptian sock in the Textile Museum, rather than the sock pictured, as it was difficult to make out the tip of the toe in this photo.

And finally, this was the first time I ever tried to write out knitting directions, so I can't guarantee that they are flawless. My e-mail address is on the bottom of the page. If I've made some egregious errors, please let me know so I can correct them.

I used 5 double pointed needles, size 2 US, 2.75 MM.

I find it easiest to use a set of 5 double pointed needles, rather than 4, and I suggest you give it a try. It makes much more sense to me to knit a tube as a square on 4 needles with a 5th working needle, than to knit a tube as a triangle on 3 needles with a 4th working needle.

You could also use a circular needle of the right size.

WHITE:
Sirdar (brand), "Denim" Tweed DK, 60% acrylic, 25% cotton, 15% wool, machine washable.
This yarn is made of a multitude of textured and uneven strands that are not tightly plied and made working with it a tiny bit tricky. I used a little more than one ball.

BLUE:
Lorna's Laces (brand), "Shepherd Sock", hand-dyed color: "Jeans" , 80% superwash wool, 20% nylon, machine washable.
This is a lovely three-ply yarn, of about sport weight. I used a little more than one skein.

The actual historic socks were of cotton. You could try that if you're a more experienced knitter than I was.

MY GAUGE:
Leg: 7 stitches per inch, 7 rows per inch, using both white and blue yarns. I tried to knit more loosely than I usually do so the leg would be stretchy.
Foot: 8 stitches per inch, 8 rows per inch, using both white and blue yarns. I intentionally knit the foot more tightly.
Heel: 8 stitches per inch, 10 rows per inch, using the blue yarn only. Because the blue was finer than the white and more tightly plied, it naturally knit up more finely.

This historic sock was started from the toe, I don't really know exactly how.

Another Egyptian sock was begun by taking a strand of cotton, forming a circle with it, leaving a longish tail, and casting on the initial stitches over this single strand. When the sock was done, the tail was pulled tight, and the end worked into the sock to keep it tight.

You can start as describe above, casting on 4 stitches in white over the circle. I used a crochet hook to pull up loops, then transferred each one as made to a knitting needle.

Or use your favorite method to cast on 4 sts in white.

Small sock - 4 diamonds on the toe:
1: knit one white, add one blue stitch - on each needle. (8 st total)
2: k 1 white, with blue: k 1, inc 1, k 1 - on each needle. (16 st total)
3: k 1 white, with blue: inc 1, k 1; k 1 white; with blue: k 1, inc 1 - on each needle. (24 st)
4: k 1 white, with blue: inc 1, k 1; k 3 white; with blue: k 1, inc 1 - on each needle. (32 st)

Medium sock - 5 diamonds on the toe:
You can start as describe above, casting on 5 stitches in white over the circle, or use your favorite method to cast on 5 sts.
1: knit one white, add one blue stitch - on each needle. (10 st total)
2: k 1 white, with blue: k 1, inc 1, k 1 - on each needle. (20 st total)
3: k 1 white, with blue: inc 1, k 1; k 1 white; with blue: k 1, inc 1 - on each needle. (30 st)
4: k 1 white, with blue: inc 1, k 1; k 3 white; with blue: k 1, inc 1 - on each needle. (40 st)

In every case:
Continue following the chart for 8 rounds.
On round 9 you no longer increase. Small has 64 sts total; Medium/Large has 80 sts total.

20: Knit all blue.
21: Knit all white.
22: Knit all white.
23: Knit all blue.

Start at the right edge of the chart, knit one blue, etc. and follow the chart for a total of 8 rounds.

33: Knit all blue.
34: Knit all white: k 1, inc 1, knit all except last one, inc 1, k 1 - on each needle. (S: 72 sts; M/L: 90)
35: Knit all white.
36: Knit all blue.

Repeat the pattern as many times as necessary to fit foot. Mine has approximately 2-1/2 repeats, ending on row 5. The original has 3 repeats.

Knit one round all blue.

The heel is formed as a "short row heel", which involves knitting back and forth. You will be knitting on only 2 needles, leaving 2 at rest until the heel is complete.

Knit two rounds all white.
Knit one round all blue.

Work entire chart

Knit one round all blue.
Knit two rounds all white.
Knit one round all blue.

Work entire chart

Knit one round all blue.
Knit two rounds all white.
Knit one round all blue.

First, measure the length of the Allah band and Zig-Zag band including the white and blue band between the them.

Repeat Diapered Diamonds pattern as many times as necessary to fit length of leg, minus the length of Allah and Zig-Zag bands. Mine has approximately 3 repeats. The original has 6-1/2 repeats. Make sure to keep knitting loose enough - after all, it has to stretch over your calves.

Knit one round all blue.
Knit two rounds all white.
Knit one round all blue.

Repeat Zig-Zag chart

Knit one round all blue.
Knit two rounds all white.
Knit one round all blue.

Repeat Allah chart

Knit one round all blue.

How the original sock was actually finished is unclear to me.

Before casting off, you may want to knit another row of blue and cast off with blue. Or you may prefer, as I do, to knit a row of white and cast off with white.

I would say that it is important to cast off in such as way that the top edge isn't unelastic, as it has to stretch over your calf. But it shouldn't be too loose and baggy.

After I cast off, I used the working yarn to make a long chain stitch tie that could wrap around my leg and be tied, to help keep the sock up, as can be seen on a number of modern ethnic Near Eastern and Central Asian socks. I finished the chain with a small tassel of the same yarn. The other option is to secure the sock with a separate band of some sort as a garter.