Pulling the wool over your eyes

Pat Ashforth and Steve Plummer have been knitting and crocheting mathematical images and objects for over twenty years. They’ve given demonstrations at science festivals, schools and craft exhibitions to ‘teach maths in an unconventional way’.

Some of the pieces have taken many thousands of hours to create and some have been bought by the Science Museum. Anything that engages young people with maths or science is laudable but is this anything new or interesting or artistic, worthy of a place in the Science Museum? Why is the Guardian getting so excited about them?

Firstly, does making these pieces out of yarn add anything to them? I spoke to Elaine Jones, a professional crafter, who said:

Most of their blanket designs would be equally interesting (and maybe clearer) as designs on paper. The craft aspect adds nothing, whereas hyperbolic crochet, for example, is a really good way of creating 3D shapes that are not easily made in other ways.

Hyperbolic crochet is something I did and wrote about here quite some time ago.

It started as a teaching aid and has developed from there. There has been a travelling exhibition of coral reefs made from hyperbolic crochet. They are both beautiful and have a scientific purpose – two, in fact. One is solving a problem of three-dimensional mathematical representation, the other is ecological. This is something that would be very hard to do in any other media.

For many craftspeople, maths is a tool, not an end in itself. Elaine continues:

It also made me consider how much maths/geometry there is in traditional pattern design anyway. The article totally overlooks this, making it sound as though using mathematical sequences to create patterns is 'amazing', instead of being something that craftspeople have always done. The difference is that craftspeople are concerned about aesthetics, whereas these people are not (as evidenced by some of their minging colour choices!)

This really is not special at all:

Elaine then says
I have a bit of a bee in my bonnet about science in art/craft. Some of it is great, but some of it is not. Despite that, if someone creates work based on something 'scientific' it immediately attracts more kudos than purely aesthetic work, regardless of its merit. It's an interesting idea, but to be really good, the execution and visual appeal should be good too. Otherwise, it's just mathematicians knitting stuff.

She is absolutely right. Part of the problem for me is that people who don't craft are impressed by any old bit of hand-made crap (which works in your favour if it's a birthday gift). Etsy is full of lumpy ugly stuff that people are charging money for. The fact these pieces are hand-made doesn’t make them ipso facto special. People have been making complex carpets and wall hangings for over a thousand years by instinctively using maths – and many of them were innumerate.

Anyone who has designed a crocheted blanket has spent time thinking about proportion, the relation of one section to another, fractions, how shapes intersect and so on. One of Ashforth and Plummer’s pieces is basically just a bunch of knitted mitred squares with a mathematical gloss.

I did a cushion using the same technique without even realising I was doing maths.

Lots of people make Fibonacci scarves. Craft web site Ravelry has over a hundred designs using the Fibonacci sequence, from scarves to stuffed animals. They also have hyperbolic crochet, pseudospheres, algebraic socks and a Lorenz manifold, which is pretty damn cool.

Many quilters use geometry in their designs. The Canadian quilter Libs Elliott goes further and uses coding to generate random formations of geometric quilts.

And of course, in the Arab world, geometric patterns have been used for around a thousand years in carpets, tiles and hangings.

Fractals are another maths thing widely used in art and craft, like this fractal necklace by Marc Newson.

I could go on but you get the picture. Or the quilt. Or the scarf.

How interested are kids in something crafted anyway? Does a six year old really care about afghan technique or how many hours went into making it?

The couple run school workshops teaching kids to knit just enough to collaborate on their own craft project for a wall hanging in the school, with a bit of added maths. This is the core of their teaching work and, judging by their website, seems to be successful in helping kids to work together and learn social skills. Making something yourself is always satisfying, especially for children. But there’s no metric for gauging how much maths the children learn or whether the workshops benefit them in their studies. Judging from this teacher’s report, they learnt far more about knitting and teamwork than maths.

The pieces the couple produce themselves are nothing special as craft or as mathematically produced objects. They sell patterns for many of their pieces and also other things like knitted toilet roll covers, dolls and puppets which are, frankly, the kind of thing that’s left over after a village hall bazaar.

They are not some amazing novelty act, they are a very small part of a long and varied tradition of using maths to create a wide range of arts and crafts. Treating them as something special ignores this tradition and the people, amateur and professional, who do it every day, as well as the work of professional maths communicators.

So, to Cast Off, putting a maths or science gloss on a piece of craft does not make it unique or wonderful. If you can find a simpler way to make maths accessible, don’t spend thousands of hours on it. Crafters have been using maths for generations. There are far better teaching aids and examples of maths-based crafts available. The Science Museum is surprisingly easy to impress. It's all a bit Emperor's New Clothes.

Hyperbolic Crochet

In 1997 a mathematician at Cornell figured out how to make a model of hyperbolic space that her students could handle so they could understand the concept better. Dr Daina Taimina got the idea from William Thurston but his paper models were too fragile to be of practical use. Crochet to the rescue.

With hyperbolic space, as you move away from a point, the space around it expands exponentially. While all spheres have the same form, hyperbolic surfaces can differ substantially. This quality is characterized by the radius of the plane; the more crenellated the surface, the smaller its radius and the flatter the surface, the greater its radius. A regular Euclidean plane can be thought of as a hyperbolic plane of infinite radius.

To show you what this means in practice, here are my own attempts. This is what one looks like after just two rows of crochet:

And after four rows:

And six:


And eight (at which point, I ran out of yarn):


The width is 9cm and the edge measures over 2.5m. Dr Tamina's biggest one had a width of 4 inches, a perimeter length of 369 inches and weighed nearly a pound.

The photos are not great quality, but you get the idea. There are some better quality ones here. For mathematicians, an increase is made every Nth stitch so the number of stitches from one row to the next is in the constant ratio N to N+1. For crocheters, I used a 3.5mm needle, double acrylic yarn and started with a ring of 6 stitches then increased in every stitch, working in double (or, if you're American, single).

There are examples of hyperbolic geometry in nature, like lettuce, kelp, sea slugs, flatworms and nudibranches. Margaret and Christine Wertheim of the Institute for Figuring have made a crocheted coral reef with kelps, anemones, sea slugs and corals by using variations and permutations of the basic algorithm. Crocheters all over the world have contributed and there are now many sub-reefs, like the Bleached Reef:



After which, my own little sea creature looks a bit lonely:



It's really really easy to do. Who says maths can't be fun?

Next time, the science of kittens and pink things.


Thanks to my crochet guru for this one.