The Pulled Stitch Blog: March 2024

Isn't Crochet great? Just when you think that there is nothing new or unexpected you turn a corner and Bam! It just happened to me. There I was scrolling through the usual suspects on Instagram and got slapped in the face by Topological crochet!

I was already in the middle of writing about Hyperbolic crochet so maybe the Instagram algorithm worked for a change. These two things are not unrelated.

For the mathematicians; these are Seifert surfaces, un-orientable surfaces, Torus knots and rubber-sheet geometry.

For the rest of us; these are complex Möbius strips. Being a crocheter you'll already be aware of Möbius scarves and cowls. A continuous strip with no beginning or end and no back or front and therefore un-orientable.

Just a simple Möbius strip is much more interesting than I realised. Try these experiments.

Take a strip of paper, mark a line down the length, twist the strip once and join the ends. Now cut it down the middle on the marked line. I thought I would end up with two separate loops. Try the same thing again but this time twist the paper twice before joining it.

I went a bit mad with my experiments and cut the paper into three and also four strips.

Of course there are maths involved but as crocheters we don't need them to play. The Instagram post I saw was from MoMath, the National Museum of Mathematics in New York. They run online courses but they also have an Introductory handout which is what I have been using. The author is Shiying Dong, clay_mushi on Instagram. You might find these two videos hosted by GrapefruitGeko on YouTube helpful. The short explainer and the longer tutorial.

The actual crochet for the two models, included in the handout, is simple. But the idea that a Möbius loop has only one surface is so contrary to all the rules of crochet that it takes a while to comprehend it.

The Möbius strip.

I skipped making the Möbius strip to start with and rushed straight into making the Trefoil Knot. Then I realised it wasn't as easy as I thought! We all have very individual brains and mine don't do well with abstract thoughts! Making the band was a good opportunity to practise all the skills before attempting the more complex models. You make a length of chain, twist 180o and join. Then you start to crochet along the edge of the chain, in the normal way, until you reach the join. Suddenly you are crocheting along the bottom of the chain. To give the strip more shape, the final round is crocheted over a length of wire.

The Trefoil Knot.

This is a three armed Möbius strip. It's made up of a length of chain which is joined in a particular way and then crochet added in rounds. The instructions looked so simple that I started off thinking that it would take no time at all to produce the first model. I failed!! I'm not even sure where I went wrong. So I cut three strips of card, joined them at top and bottom with a single twist in each. Now I traced along one edge all the way around the construction and back to the beginning. I proved to myself that there was only one surface.

I began again! This time with more consideration. This time I paid more attention to the pattern and I used a small soft ball so that I could take more control of my chain shape. I failed yet again! Third time lucky? Oh yes indeed!

It is just possible to do this without truly understanding what you are doing, but if you have a brain like mine that really needs to be able to see and understand, then I think I can help! It's ironic, these type of mathematical models are supposed to aid in a greater understanding of the maths involved. But I needed to understand the physical structure to understand how to make it. So let's start again!

Making a card model is an advantage. It gives you a structure to compare to your crochet which will twist around as you work. I made mine using strips of thin card and some double sided tape.
Use a heavier weight yarn that doesn't split easily and can be easily be pulled out when you realise you've made a mistake.
A ball to work around. This ball can be anything that is the right size and that you can stick pins into. Maybe it is tightly wound yarn or a stuffed amigurumi structure.
Stitch markers are vital and they need to be the right kind. I have a selection and I struggled. They need to be substantial enough to hold your structure together and yet thin enough so that you can still get your hook into the stitch with the markers in place.
If you can't get the stitch markers to work for you try using loops of fine yarn instead. (see Borromean Rings, below.)
Work at a table and not in your lap. You need to have complete control of this twisting monster!
Getting the chain structure right is vital. And count your stitches, of course.
This is a lesson in patience and determination, so if at first you don't succeed, try again!

So, let's start with the chain. I used a hook .5mm larger than the main hook, because you are going to be placing at least two stitches into each chain.

We begin with 49 chain and place a stitch marker in the 1st stitch and then every 16th stitch. Use two different colours and place then in the order as shown.

Now we make the structure.

Pin the stitch, with the marker at the tail end, to the top of your ball. Keep the chain straight and bring it down the bottom of the ball. Now we twist the chain 180o. We need to be very deliberate as each twist has to be done identically and in the same direction. It doesn't matter which direction as long as you are consistent. Pin this stitch in place.

Keeping the chain straight, bring it around the ball and back to the top. Twist the chain 180o again and place the stitch maker through the stitch already pinned in place. Remove the first stitch marker and put it through the two stitches but facing the opposite direction. Get a third stitch marker and put it through the two stitches facing the remaining direction.

Take the chain for the last time, keeping it straight bring it back down to the bottom. Twist 180o.

Grab your hook, I'm using the smaller sized hook. The instructions do not clearly explain this part, so this is my interpretation.

Place the hook through the stitch already pinned to the ball, from front to back.

Now place the hook into the marked stitch and carefully remove that last stitch so that you are left with a live loop on the hook. Yarn over the hook and pull through all the loops on the stitch, chain 2. Now place both stitch markers through the base chain.

Before we release the chain structure from the ball let's check what we have.

At the top of the ball we should see three chain branches with the front of the chain face up.

At the bottom of the ball we see three chain branches face down. Check that each turn has been made in the same direction.

Now you can remove the ball and begin to crochet. Remember to go through the chains from front to back, that way you will find your way around the branches in the right order.

Round1

The chain2, we have already made, counts as your first Treble (Dc-US), make a Half Treble (HDc-US) in the next stitch, Dcx13 (Scx13-US), Half Treble (Hdc-US) in the last stitch before the branch point.

This is where it has gone wrong for me before!

Look carefully to see which branch you will be working on next. Remember you are going through the chain from front to back. Make a Treble in the centre chain and remove a single stitch marker.

Repeat the stitch sequence.

Repeat for the third sequence.

You should now have worked through all the chain stitches once and from now on you will be working through the bottom of the stitches repeating the whole sequence until you reach the start. Slip stitch to the top of the chain2. Check your stitch count!

Let's check we have got this right!

All of the branches should twist in the same direction. You can trace the top of the stitches and prove to yourself that this is one continuous surface. If you have it right give yourself a gold star. If not, pull it out back to the chain base and better luck next time!

Round2

From now on it's easy!

Let's look at what we have.

You can see that each branch has an outer and an inner edge. One edge runs on the outside of the twist (OuterSequence), we will increase along this edge. The second edge runs on the inside of the edge (InnerSequence), we will decrease along this edge.

We start with the outer edge sequence. The chart above shows half of each sequence, OuterHalfSequence and InnerHalfSequence.

[UK] Chain2 (replaces first Treble in the following sequence), Trx4, 2Tr in each of the next 2 stitches, 2DTr in the next 4 stitches, 2Tr in next 2, Trx4.

[US] Chain2 (replaces first Dc in the following sequence), Dcx4, 2Dc in each of the next 2 stitches, 2Tr in the next 4 stitches, 2Dc in next 2, Dcx4.

Now we work the inner edge sequence.

[UK] Tr, Htr, Dcx2, Dc2together x4, Dcx2, Htr, Tr.

[US] Dc, Hdc, Scx2, Sc2together x4, Scx2, Hdc, Dc.

Repeat twice more and needle join over the chain2 at the start. Fasten Off.

Round3

Use a second colour and 20 gauge copper wire. I'm using 0.6mm wire that I found in my local craft store. Hold the wire along the top of your stitches and double crochet (Sc-US) over it, working around the edge. You can add increases along the outer edge but I just worked even. The longer of the GrapefruitGecko videos has a good suggestion for how to join your wire ends.

Once you have finished adjust the wired edges to best advantage!

The Borromean Rings.

Are you ready for a real challenge!

In geometry these are two independent rings which are locked together by a third ring. What we are making is a triangular pyramid. It was very hard for me to understand how there could be "three components each consisting of four edges". In the end I tried wrapping strips of coloured paper around a ball so that I could twist them and trace my route.

This is the two dimensional version of what I am trying to produce.

To make this you need four different colours of yarn and three each of four different coloured stitch markers. It is vital for your understanding that you place the stitch markers in the same order as I have. Of course the chances of you having enough stitch markers in the same colours is unlikely, so make a copy of the sketch with your replacement colours.

Alternative to Stitch Markers

I had a great deal of trouble keeping my chain structure secure so I came up with an alternative. I used three pieces of No10 crochet thread per joint. Using a blunt ended needle, I put the threads through the back and front loops of the chain only leaving the third loop free. You will catch this loop in as you make the next row of crochet.

The instructions that follow are slightly different to Shiying Dong's. I nearly drove myself crazy trying to get this to work. The changes are made to hopefully make it easier.

The thread markers hold the structure more securely and also means that your working yarn is not caught in the stitch markers.

Although I have shown 'live loop' in the next diagram, I am advising you to fasten off your chain lengths before joining. I found that having the live loops only complicated matters!

So let's make the chain. For illustrative purposes shown here with two yarn colours for the chain. You will use your main yarn colour. I'm using 12 stitches per 'leg'.

For the long chain, Chain49, fasten off.

For the second, Chain25, fasten off.

Add a stitch marker to the first stitch and then every 12th stitch. You will have 5 stitch markers in the Chain49 and 3 in the Chain25.

Here is the placement to copy.

The Chain structure.

Attach the first stitch of the short chain length to the centre stitch of the long chain length with a stitch marker, all with the chain face up, and pin to the top of your ball. Add two more stitch markers all facing in opposite directions.

Now we join the rest of the chain strips to create a triangular pyramid by joining the colour markers as appropriate. Twist each chain strip 180o before joining to the next marker. Add the additional stitch markers.

Round1

We will crochet with the same stitch pattern we worked for round1 of the Trefoil. There will be a Tr (Dc-US) at each branch point, with a Htr (Hdc-US) on either side, and 9 Dc (Sc-US) in between.

With your main yarn colour, start at the pink marker. I began with a standing stitch but you might prefer to begin with chain2 instead. Create one circle as shown in the 2D diagram. Trace the route in your mind to determine which side of the chain you are working into for each leg. Unlike with the Trefoil you will not always be working into the chain from front to back. Keep in mind not just the next branch point but also the one beyond as well.

Work as follows: from Pink, to Yellow, to Green, to Blue, to Pink. Needle join to the start.

It is also helpful if you work over or weave in your yarn ends as soon as possible.

For your second circle begin at the yellow marker, as follows: from Yellow, to Green, to Pink, to Blue, to Yellow. Needle join to start.

For the third circle begin at the Blue branch point and work as follows: from Blue, to Green, to Pink, to Yellow, to Blue. Needle join to start.

Leave one marker in the blue branch point through to the end of round two, this will help to orientate your model.

Round2

This will be the wired edging.

You are working three separate circles again but this time with three different colours.

Begin at any point on a circle with your selected colour and working over wire, double crochet (Sc-US) around that circle. Needle join to the start and weave away your yarn ends. Repeat for the remaining circles. Shape as desired with the blue branch point at the top of your pyramid.

I wasn't as impressed by the finished product as I'd hoped, not after all the blood sweat and tears, so I made a second using 9 stitches per side as per the original PDF instructions.

Dual Möbius Strip.

Because I always like to give you something more, and if you've got this far you're a super hero, here's something easy to finish off with! I have no idea what it is called, so you can call it what you like. It's basically a four armed Trefoil. Here's what it looks like in card.

I'm using three colours of yarn. One for the chain, Rounds1 & 2 and two for the wired round.

The chain structure.

Let's use 16 stitches per edge and the same stitch pattern as the Trefoil. This time we will begin with Chain65. Fasten off.

Place a stitch marker in the 1st chain and every 16th stitch thereafter. Use two colours and mark in the same way as the Trefoil.

Pin the end of the chain to the top of your ball. Keep the chain straight and bring it down the bottom of the ball. Now we twist the chain 180o. Pin this stitch in place.

Keeping the chain straight, bring it back to the top. Twist the chain 180o again and place the stitch maker through the stitch already pinned in place. Remove the first stitch marker and put it through the two stitches but facing the opposite direction.

Repeat taking the chain length back down and up again.

Check your structure.

At the top you should only see the front face of the chain, at the bottom you should only see the back face of the chain. Check each 'leg' twists in the same direction.

Round1

We will work the same stitch pattern as before. So with the same yarn colour begin in the top branch with a standing Treble (Dc-US) or Chain2, Htr (Hdc-US), Dc13 (Sc13-US), Htr (Hdc-US). Work into the front of the chain stitches all around. Repeat the sequence until you get back to the start. Needle join to start.

Begin again but work into the base of the chain repeating the sequence as before.

Round2

With the same colour, starting at the top branch, work the Outer edge sequence and the inner edge sequence that we used for the Trefoil, twice. Fasten off.

Repeat again for the remaining edges.

Round3