Hyperbolic Crochet Without Maths!

    Crochet, it always seems to me, is a self directed study. We learn at our own pace and in a unique order. Which must be why it has taken me so long to get round to writing about Hyperbolic crochet.

       Daina Taimina's book Crocheting Adventures with Hyperbolic Planes was first published in 2009 and then a second edition was produced in 2018, so I am rather late joining the party. It's not surprising really, as I am not now and have never been a mathematician. Up until this moment I have got along fine with just the maths I learnt in the first decade of life. I have never needed trigonometry, or algebra or any of the other things that I forgot about a long time ago. Now I wish I could remember all that stuff!

    I told you, only recently, that I always read my crochet books from cover to cover. I tried to do that in this case, I really did. Fortunately the book is packed full of really interesting illustration and the chapter on the history of crochet was surprising and very well researched. If you are a crocheting mathematician you will love this book, but for the rest of us, here is my non mathematical rendering of hyperbolic planes!

    So, the first question has to be, what on earth is a hyperbolic plane? It's basically our understanding of a curved surface. The maths we use for flat triangles and circles, or 'Euclidean' planes is absolutely useless when we step into three dimensions. Let me illustrate.

If I lay hexagonal tiles they form a nice flat surface.

But if I lay 5 hexagons around a pentagon they immediately curve. Of course you will recognise this as a football. (Soccer ball-USA!) This curve is called positive, because the surface joins up with itself. And so we can only add a finite number of tiles.

Let's swap that pentagon for a heptagon. Now something else is happening, this is a negative curve. What we have created is a Hyperbolic plane. We can continue this pattern for ever and it will never join up with itself.

You will remember, I am sure, that in order to find the circumference of a circle you need to use the formulae Circumference=𝜋Radius² but this doesn't work with a sphere because the circumference is smaller in relation to it's larger radius.

    This is what all the fuss is about. But it also turns out that this form of maths is really important in lots of ways. For instance without hyperbolic planes you wouldn't even be able to read this post on the internet. In fact it is helping within all of the scientific disciplines including medicine, architecture and computer sciences.

     So to make a truly mathematical hyperbolic surface we have to use this form of maths. At this point I was throwing my hands up in despair as there was no way I could understand the formulae without going back to school. Fortunately the Institute For Figuring (IFF) have taken pity on us. The IFF are interested in the 'poetic and aesthetic dimensions of science, mathematics and engineering' and are the people behind the crochet coral reefs. This link will take you to The Institute. This link will take you to a PDF they have created with instructions for different crochet forms. They even encourage us to be irregular!

    Let's look at some of those forms.

    The first is a hyperbolic plane.

We start with a simple short row of chain stitches and work in rows of double crochet (Sc-US). You choose how often you want to add increases. The more increases the quicker your work will grow out of all proportion. 

I began with 20 chain plus one turning chain. I chose to increase on every 4th stitch. For the odd stitches left at the end of rows I simply worked them without increases. 

To start with I have a straight line, 

as I add increases my line begins to bend into a circle 

and then crinkle, the more it grows, the more it creates whorls. 

For all the examples in this post I used DMC Natura Just Cotton and a 3mm hook. This hyperbolic plane used about 145 metres of yarn.

    The second is a pseudosphere, basically a trumpet. 

This time we begin with a ring of stitches. I started with four double crochet (Sc-US) in a magic ring. Working in a spiral I worked a few rounds without increase and then began to increase in every 4th stitch. I choose to keep to the same increase ratio as before so that we could compare the two results.

What I've made is a tube that quickly opens out into a trumpet shape before it begins to buckle and soon creates the same whorls that we saw before.  

Just for fun I added a final round in a contrast yarn, this I worked without increase. It has had the effect of opening out the whorls so that they have a looser structure. It used about 167 metres. If I had increased more slowly it would have had a bigger trumpet and might have looked more like a trumpet flower.

    A double Hyperbolic plane is just a row of chain where stitches are worked on both sides. 

Once again I started with 20 chain and one turning chain. I placed one stitch into each chain but three stitches into the end chain. Then working back down the other side, I repeat one stitch in each chain and three into the end chain. From now on I worked in a continuous spiral placing two increase to every three stitches. At first the whole thing starts to form into a spiral and then it creates a rippling surface on either side of the spine. I quickly realised that if I continued increasing at this frequency each round would grow to an unmanageable size, so I changed to increasing in every third stitch.

At this point I also felt inspired to make it look a bit like a Polypore fungus and added extra colour.

    While I would dearly love to understand the Mathematics behind Escher's tiled designs, because we are not being scientific I feel I can break the rules. And because I just can't help asking why? and what if? So why not find out what happens if I work around any shape? So, I have to try growing my hyperbolic sphere out of the top of a sphere and for some mad reason decided to make a frilly egg. 

It reminds me of something, what I am not sure! Not all my experiments have a pleasing outcome. I worked into a ring of front loop stitches and now it occurs to me that I could have made it spiral around the egg shape. So of course I had to try that. 

    This time I made an urchin shape, a flattened sphere. I worked all the Dc (Sc-US) into back loops only, leaving the front loops free for the hyperbolic crochet. Starting at the top, I worked down in a spiral but stopped part way down. I worked backwards and forwards up and down the spiral reducing the rate of increases in each row.

I am very happy with this solid form.

 

    The IFF suggest we experiment with other sizes of stitch, hooks and types of yarn. I could get lost here for a very long time! So, I better sign off before you too get swept up into the vortex and sucked down the rabbit hole along with me.

But before I do, hyperbolic surfaces are very soothing and make great fiddle balls so if even the idea of numbers raises your pulse, well, you know what to crochet next!

Fastening off...