Mathematical Knitting with Alan Turing

(Photo by Phil Parker)

Hand crafts are an excellent way to soothe the nerves and get into a meditative state. Alan Turing, like many wonderful people, could get rather anxious at times. Knitting was thus an excellent hobby for him. It also was a great tie-in with the knitting sheep/White Queen of ALICE THROUGH THE LOOKING GLASS, another favorite of Alan’s.

It has been noted that his attempts at making mittens required some intervention on the part of a relative to complete the project. It also has been observed that Alan liked to knit Möebius strips.

Möebius strips are a rather paradoxical shape because they look like they have two sides but if you follow the strip all the way around with your fingers, you will find it only has one side. A related shape is the Klein bottle whose inside and outside are one. instructions to make your own Möebius strip out of paper and then how to knit your own Möebius strip, just like Alan! Like the knitting blogger we’re citing, Alan was left-handed, was Harvey Milk, his lover in FOR THE LOVE OF ALAN TURINGPaper instructions are courtesy oföbius-strip-and-how-can-you-make-one

  1. Cut about a two-inch wide strip of paper from your full sheet and lay it out in front of you so that the long side of the paper is laying horizontally.

2. Write the letter “A” at the top-left corner of this strip, the letter “B” at the bottom-left corner, the letter “C” at the top-right corner, and the letter “D” at the bottom-right corner.

3. Hold the strip of paper in front of you. Now twist it one-half a turn so that the letters “A” and “B” on the left still face you but the letters “C” and “D” on the right now face away from you.

4. Bring the two short edges of your twisted strip together and tape them to make one long twisted loop. Corner “B” should match up with corner “C” and corner “A” should match up with corner “D”. Knitting instructions are courtesy of

Here are two knitting options.

Method 1:

Knit from the inside out, using scrap yarn. You’ll need two different colors of yarn, one for the Möbius band itself and a scrap yarn for casting on. I find plain cotton yarn the easiest to work with in all cases, but especially for the scrap. You’ll also need a sufficiently flexible circular needle. I think I usually use a size 7 or 8 in 24″ length for this, but other sizes should work as well.

Cast on 90 stitches in your scrap yarn. Because you’ll be removing the scrap yarn later, use a cast on method that easily pulls out.

.I create a slipknot loop and, using my fingers, pull another loop through it to start. Then I wrap the free end of the yarn over the needle, and (again using my fingers) pull a loop through the previous loop. Then I repeat.

If that made no sense, try this description instead.) In case you’re wondering why the scrap yarn is necessary at all, it’s to avoid any appearance of a seam in the finished product. This means that your Möbius band will have a central circle of 90 stitches in circumference.

If you’re making a wearable Möbius band rather than just a mathematical manipulative, you will of course want to adjust this number for gauge and fit. Onto the cast-on row, knit one row loosely, using the yarn-for-the-Möbius-band-itself. Leave a bit of a tail when you begin so that you have some to knit in when you’re done. (No knots allowed!) Note: you may want to purl one row instead. I think that whether you should purl or knit depends on the handedness of your cast-on row and the handedness of your knitting; the point is that some ways of knitting will ensnare the scrap yarn so it won’t pull out easily, so be aware of the issue.

Now, if you were doing ordinary circular knitting, you’d continue by stretching the other end of your row to the other tip of the needle, and knitting onto that. However, you’re going to introduce the intrinsic twist by instead knitting into the loops between the stitches of the row you just knit.

In order to do this, don’t stretch the other end of the row. Leave it where it is, and bring the tip of the needle to it. Then rotate the-other-end-of-the-row a bit so that you can access the loops between your stitches. Because these loops are not at the tip of a needle, you cannot do the ordinary (insert needle into loop)-(wrap yarn ’round needle)-(pull new loop through old)-(slide old loop off needle).

Instead, you’ll just do the first three of these operations, leaving your old loops still on the skinny part of the circlar needle… but instead of just knitting or just purling, you must *k1p1*. This has the effect of casting on an additional 89 stitches so that each ‘row’ has 179 stitches. (Note that if you cast on n stitches originally, knitting into the loops between will add n-1 stitches, giving an odd number of stitches total. This ensures that your *k1p1* will become seed stitch rather than ribbing, and that your seam will be invisible.)

Bizarre though this may seem, it’s consistent with the fact that a Möbius Band has only one edge, so it will appear that you’ll be knitting twice ’round the strip to traverse the edge once. Also, this will probably hurt your fingers or at least be rather uncomfortable, so don’t be shocked when that happens. When you’ve finished knitting into the loops-between-the-stitches, your needle will be loop-de-looped. Now you can just do *k1p1* forever, or rather, until you feel like you’re done. For a cute li’l strip, five ‘rows’ should be fine.

For a scarf, you’ll want more like twenty. When you’re done, just cast off as usual. There are three tasks that remain: knit in the end of the yarn, get rid of the scrap yarn, and knit in the beginning of the yarn.

When knitting in the beginning of the yarn, look carefully at your stitches so that you don’t create a hole or piece of seam. This is a modified/expanded version of Maria Iano-Fletcher’s translation of Miles Reid’s pattern.

Method 2:

Knit from the inside out, no scrap yarn needed. Once you’re used to it, this is the fastest method. However, the yarn forming the central circle doesn’t look quite as spiffy as when scrap yarn is used, because this method induces a little bit of additional torque on the yarn. (At least, it does when I do it. This could be a peculiarity of my less-than-orthodox left-handed knitting style.)Also, lots of people seem to have trouble figuring out how to do this, so be forewarned that it does require some thinking.

You’ll need some yarn and a circular needle that can coil twice ’round a circle (the way they’re conventionally used requires that they coil only once ’round a circle). Make a slip knot and slip it over one point of your needle. Grab the point in your right hand and hold the knot so it doesn’t slide away. Bend the needle so that the other point is (a) pointing to the left (b) in front of the point with the slipknot (c) in front of the yarn.

Now, using the cable instead of a piece of scrap yarn, do stranded cast-on. (Online instructions can be found on YouTube by Cat Bordhi—note that to be mathematically correct, you’ll need to omit her last yo in order to have an odd number of stitches—or on Eunny Jang’s blog.

Printed instructions for stranded cast-on can be found in June Hemmons Hiatt’s The Principles of Knitting, p. 138 of the first edition; in Mary Thomas’s Knitting Book, p. 66, it’s called invisible cast-on.) The number of stitches you cast on will be the number in the central circle of the Möbius band, which is roughly half the number of stitches on the boundary of the Möbius band. Pull the yarn to the front, so that it’s coming toward you from behind the cable (not the point of the needle). This will ensure that you end up with an odd number of stitches, which is necessary for the whole seed stitch thing to work out correctly.

What you do next depends on your handedness. If you’re left-handed, like me, you’ll turn the whole business around and knit into the slip knot. If you’re right-handed, you’ll purl into the slip knot.

After that, continue in seed stitch (*k1p1*), always knitting/purling by slipping the needle into the side of the stitch closest to you. I know, that sounds silly to say—that’s how we always knit/purl—but you might feel like you’re putting the needle in from a funny direction or in a funny piece of the stitch.

And you might actually be. Just alter your usual stitch in whatever way you have to, so that none of your stitches twist. (If that confuses you, I recommend reading The Principles of Knitting, pages 23-24 and 32-34 of the first edition, or Anna Zilboorg’s Knitting for Anarchists, pages 14-22.)

Then keep going until you’ve reached the desired width. Bind off in pattern and weave in the end. Now go back, untie the slip knot (which may take some doing) and weave in that end in pattern as well, being careful not to leave a hole. If you don’t remove the slip knot, it will be very easy to see where you started.

Congratulations! You have now been able to share the joy of knitting and math at the same time!


Alan’s Sponge Cake

How Did Alan Turing Break the Enigma?

In the Navy


5 Replies to “Mathematical Knitting with Alan Turing”

Leave a Reply