More than 25 years ago I met John Sharp at a maths conference and was introduced to Sliceforms. John had written a book which was about to be published and is still in print. I don’t know whether John invented the name Sliceforms. I can’t find any evidence of them existing earlier. On the publisher’s page it says:

Sliceform modelling is a technique which lies happily on the borders between art and mathematics.

I have made many sliceforms over the years so decided to 3D print some.


Sliceforms made from paper, or card, fold down flat because the material is thin. 3D printed sliceforms will never go as flat as paper but they are more tactile, won’t crease, and make nice noises.

The cube is the easiest to print because it just needs 18 identical pieces which slot together easily. It takes longest to print because there are no small pieces.

See more photos and download the print files.

The sphere requires a total of 18 pieces, in five different sizes.

See more photos and download the print files.


The cone requires a total of 18 pieces from 10 different print files.

See more photos and download the print files.

The saddle (which is also known as a hyperbolic paraboloid) needs 18 pieces from 10 different print files.

See more photos and download the print files.

The monkey saddle requires 18 pieces from 14 different print files. It is called a monkey saddle because a monkey can sit with a leg at each side and his tail at the back.

See more photos and download the print files.

I experimented with the size of the gaps until the pieces would fit together, and flex easily. I settled on a gap of 1.2 mm.

The photo shows the smallest slice of the sphere. It is difficult to see the gaps in the slice but you can see them where the slice has been removed from the top of the sphere.

It was 25 years ago today …

… when the original Woolly Thoughts was published … and a lot has happened since then.

It is very confusing because everything we do is called Woolly Thoughts but it was originally just this garish pink book.

In 1994 we somehow persuaded Souvenir Press to publish a knitting book that had no colour in it and this was the result.

In 2007 I wrote a book, called Picking Up Threads, about how Woolly Thoughts came about and what happened in the years after its publication.

I have now put the entire document on the Woolly Thoughts web site. You can read it here. Be warned – it is very long.

This is my original description of Picking Up Threads:

It all started with a question on an Internet knitting list: ‘How do you know when you’ve arrived in knitting?’. I couldn’t find a simple answer and my thoughts, which are still inconclusive, led me to write this book.

The three main threads – Mathematics, Knitting and the rise and rise of Information Technology – have combined with the effects of a chronic illness to lead me to where I am now. It is my story – or part of it – dominated by the creations that have evolved on the way and the experiences they have led me into such as using basic Mathematics with small children to mammoth events with 10,000 people in a football stadium, running workshops in an Italian palace with the assistance of a simultaneous translator and my new bathroom featuring on television.

If you thought knitting was boring, think again!

The story ends in 2007 but a great deal more has happened since then. I suspect this may be because I joined Ravelry in 2007 and much of what we have done is documented there. The history of all the afghans we have made, up to the present day, can be found in Afghan Stories.

The original book eventually went out of print. We saw books changing hands for vasts amounts of money. Fortunately, a new edition was published by Dover Publications (who, at that time, mostly published out-of-copyright books and books by dead authors). The advantages of this are that the price is much lower and it will stay in print ‘forever’. The disadvantage is that authors don’t get royalties. We were happy to agree to that as it made the book accessible to anyone.

The only difference in the new version is that it was all shrunk down slightly and the page with the ‘accurate’ squares is no longer accurate.

To read about why we didn’t want to use colour jump to page 6b
To read about the event in the football stadium jump to page 18d
To read about the Italian palace jump to page 25d
To read about how our bathroom came to appear on television jump to page 26a

3D printed Dyson adapter

We have far more vacuum cleaners than anyone can possibly need. We have a big house, on four floors but that’s not a good reason for so many. Part of the problem is that, because we have a lot of space, we tend to keep things, or bits of things, that might ‘come in useful’. At a recent count we had

  • a large upright Dyson, which we have had for a very long time. It is quite heavy and Pat won’t use it because she once got a nasty injury from it
  • Henry, who is great fun and very efficient, but the tools need a certain amount of brute force to attach and remove. They had become increasingly difficult for Pat to attach and remove
  • a Vax Air cordles, which is an easily managed machine for anyone with limited mobility. It can be used like a normal upright vacuum or can be taken apart so you can hold one piece in one hand, the tools in the other, and reach anywhere you want. It has two batteries so is always ready to go. We did have a problem with its wheels some years ago but Vax replaced the entire handle/wheel section immediately.
  • a hand-held Dyson DC 30 (which replaced an earlier model that died after several years). It doesn’t have a long battery life but is excellent for little jobs (and cleaning the stairs).
  • two wet & dry vacuums. One is very old – a relic of when we were renovating the house; the other is newer and can cope with the water that sometimes collects in part of the cellar.
  • a Vax carpet vacuum/shampooer which now blows a fuse each time you turn the brushes on. We have ill-treated it for too long because it was also used for sucking up water in the cellar.
  • a pond vacuum. The houses around here, in this old mill town, have little, or no, garden but we are fortunate to have a small area at the front of the house where we have fish.

We suddenly decided that we didn’t really need them all. The first to go was poor Henry. It was sad to see him go, especially as he lived in the kitchen in a cupboard designated as ‘Henry’s cupboard’, along with his bag of tools and attachments. We sold him to a local person who was a great Henry fan and she was delighted to have him for her daughter who had just moved into a flat of her own.

Amongst Henry’s tools we found a brush from an even older Dyson which had been made to fit Henry with a series of adapters stuck together with gaffer tape. We kept it because, now we have a 3D printer I could make a part that would efficiently join it to the DC30.

It was fairly easy to design. It just needs to be the right size to fit the brush at one side (32 mm) and the Dyson at the other (35 mm). The ridge near the end lets it clip into the Dyson’s fitting.

I believe this would fit several other Dyson models and that attachments from various other vacuums would fit.

Download the file from Pinshape

Square Puzzle

The challenge is to turn a square of fabric into a jacket without cutting off any pieces or adding anything on. It can be for a child, adult, or, in this particular case, Barbie.

You are allowed to make cuts into the fabric, and join it back together as you wish. Nothing can be added or removed.


This is the way clothes would have been made from animal skins to use the maximum amount of skin with the minimum amount of skill and effort.

Does the problem become easier if some cuts are made for you?

Fold in the sides.
Fold down the top.

The underneath edges of the sleeves can be stitched, or laced, together and the rest of the top can be joined to the front panels, to give a recognisable jacket shape.


Somewhere we have a large-person-size version of this jacket made from heavy navy woollen fabric. The slits are created with shiny metal open-ended zips which can put it back together as a square or a jacket. Unfortunately we haven’t been able to find it to photograph. Neither do we have a video of Ben zipping himself into it during a talk at a maths conference some years ago.

Barbie size will have to suffice. The advantage of Barbie is that her neck is so narrow you don’t need to make allowances for it. A human garment would need to have some extra cuts at the neck to let it wrap round.

With knitting, and crochet, you can add the slits as you go. The square can then be folded to the correct shape. This one is fastened with buttons.

Barbie’s pink and grey outfit consists entirely of squares. Her trousers are four identical squares – one for the back, one for the front, and one for each leg. Her top is made from two squares, joined at the sides and shoulders. Her hat is a square folded in half.

The square could be made from four smaller squares. This allows for some extra patterning. These squares are knitted on the diagonal, starting at a point.

The red lines show the positions of the slits, where the squares should not be joined together in the flat view.

The front view and back view create very different effects.

The black and white trousers are made from rectangles knitted on the diagonal. A rectangle is merely an elongated square.

Increase at the end of every row in the same way as for a diagonal square until you reach the width you want (red line). Continue to increase at one edge and decrease at the other. It doesn’t look much like a rectangle until you stand it up. 

When the long side of the shape is the length you want (blue line) start decreasing at both edges. This shaping will always give you 90 degree angles so could be used for any rectangle, such as a for a scarf.

Barbie’s brown outfit is very similar to the others except that it is crochet.

The fluffy outfit uses the same pattern knitted longer and with extended sleeves. The dress is a simple square, seamed at the back.

All outfits, plus a mini-tutorial about squares, are included in the same pattern.

Download the free pattern for Square Barbie

My Face Cloth

Because Pat was making things at small size I decided to have a go at making the head of my Lady with an Ermine using the finest yarn we had in the house.

This is what you see from the side
This is what you see from directly in front

It is very difficult to photograph illusion knitting. It needs to be seen in real-life to appreciate how it changes as you move around. In ideal lighting the image disappears completely in the straight-on version and you only see narrow stripes.

I used DMC Petra 5, which is crochet cotton. Using a hard cotton like this means that the colours don’t blend together quite as much as with other yarns but it shows that the effect still works regardless of the size or type of yarn. The finished piece is about 22 cm square. It was knitted on 2 mm needles. Anyone experienced in miniature knitting would be able to make it much smaller.

My original Lady was based on the portrait by Leonardo da Vinci. Here is the small one pinned to the larger version

… and the full-version of the original

Download the free pattern for the head
Buy the pattern for Lady with an Ermine

Sum Wear

You could be forgiven for thinking I am beginning to lose the plot. My latest design is for Barbie! This is not just any Barbie. This is Fibonacci Barbie.

This is a project that has been languishing for more than 20 years. I started to make some mathematically-inspired clothes for a teddy bear, which I took to a maths and art conference at Oxford University in about 1997. I didn’t make any more because I didn’t see the point in writing patterns that would only fit my bear.

The idea resurfaced seriously last year when I decided to design my own toy that could be dressed. It ground to a halt again because my toy never developed its own character and wasn’t particularly appealing.

Fast forward to May 2019 when I designed an illusion knitting outfit for Barbie. You can read about the reason in an earlier post. This set my mind working again. Barbie is ubiquitous. Most knitters wouldn’t have too much difficulty getting their hands on one (or an ‘imitation’ version. It doesn’t have to be the real thing.)

By now I had realised that making clothes for Barbie wasn’t the important thing. It was just another way of introducing mathematical ideas to people who said they can’t ‘do maths’.

The pattern includes instructions for making the three dresses shown. Two are knit, one is crochet. If you know anything at all about Woolly Thoughts designs you won’t be surprised to know that one is knitted in unusual directions. More importantly, the final section of the pattern is about using the Fibonacci Sequence for other things. The ways in which the sequence can be used are really quite limited but there are a few tricks to help you on your way.

If I can convince just one person that you can’t use the numbers out of order I will regard that as a success.

This is the first of an irregular series. It is about using particular numbers. Others will include interesting geometric constructions and other aspects of maths, They will be collected together under the name Sum Wear.

We even managed to get a ‘shop sign’ for the collection. You can see how it compares to a human-sized sign. We couldn’t call it Barbie’s Sum Wear because the sign only came with one letter B.

All the patterns will use very basic yarns, mostly DK, so they can be made from leftovers. I have various problems with my hands and working with small things is particularly challenging. I would like to use finer yarns and add more detail but that just isn’t possible.

Download the free pattern for Fibonacci Barbie.

Moon Landing

It was in the week I went on my first visit to the school where I was to start teaching, in my first post, in September. Everyone was very excited. Strangely, I don’t remember watching anything at home but I do remember groups of the children huddled round televisions. Televisions in schools were still a fairly rare thing and not everyone had TV at home in those days. My visit couldn’t have been on the day of the landing because that was a Sunday so it is difficult to know which bits are real memory and which bits are the result of seeing the film so many times since.

Almost ten years ago Steve designed an illusion knit of the moon. This was one of his earliest designs. At the time he wrote:

This Moonrise Illusion knit is very much an experiment in using images other than those of people or recognisable characters. Would it be possible to get enough shading into a piece of knitting to indicate areas in a landscape?

This is the picture he started with

When you look directly at illusion knitting you only see narrow stripes. The image is revealed when you look from the side.

I can’t think of the moon landing without remembering two girls I taught some fifteen years later. The father was a massive space fan. He named his first daughter Moonrock, which isn’t so bad. The second girl was called Lunar Landing. Fortunately, that was an easily adapted name and, in later life, she called herself Luna

This is the result.

The Moonrise pattern is available on Ravelry.

Alan Turing

On July 15, Bank of England Governor, Mark Carney, announced that Alan Turing  will be the face on the new £50 note. You may not be surprised to know that we have Turing-related knitting.

In 2014 the film The Imitation Game was released. It is described as a historical drama. Shortly afterwards I came across a page on the web site Information is Beautiful. David McCandless, the owner of the site, had analysed several recent films which claim to be based on true stories. From his data, the least true of them all was The Imitation Game, which tells the story of Alan Turing cracking the Enigma code at Bletchley Park. The way he represented the data made it look so much like a scarf that I felt compelled to knit it.

David compiled a spreadsheet which breaks down the film scene by scene then compares the action with information from other sources. He gives the beginning and ending times of each element, and the duration. Below are the categories he used to indicate the various levels of truth:

  • UNKNOWN (White) We couldn’t verify it or the sources were secret (i.e. personal diaries).
  • FALSE (Red) Out and out didn’t happen, or outrageous dramatic licence taken.
  • FALSE-ISH (Pink) Pretty false but with reasonable/understandable dramatic licence.
  • TRUE-ISH (Light blue) Some tweaks but true in spirit. Or a mix of true and false.
  • TRUE (Blue) Pretty much as it happened,

I omitted some of the film’s credits then took the rest of David’s times and converted them to seconds. I divided by ten and rounded to the nearest ten so that I could use one row for every ten seconds of the film. That gave a scarf with 667 rows.

I had to make the scarf in 4 ply yarn otherwise it would have been much too long and heavy. It was quite difficult to find two close shades of red and two close shades of blue that were different enough to be noticeable. The lighter red is still not showing well in the photos. (Reds are notoriously difficult to photograph.) I used grey instead of white.

I used a very short circular needle and knitted in the round so all the ends of yarn were hidden inside. The finished length is just about six feet.

I gave the scarf to Timandra Harkness, who is a writer, presenter and comedian, with a special interest in data.


The amazing portrait below was made by someone who learned the techniques for designing illusion knits from Steve’s (free) online tutorials.

When you look at illusion knitting from directly in front you only see narrow stripes. The apple and binary code use illusion knitting, the other lettering is intarsia. When you look from an angle you see Turing’s face in the apple and his name in the code.

The creator was a bit disappointed with her work. I think it was a triumph! If you want to read everything she wrote, including the transcription of the code, go to her page on the Woolly Thoughts World of Illusion Knitting web site where she ended by saying:

I’m really glad I made it though – Turing is one of my heroes and I really wanted to make a tribute to him. And I love the way his face turned out. Overall I’m probably pleased enough – I just wanted it to be fantastic!

Two Kinds of Magic

Illusion knitting seems like magic. Things appear and disappear. A Kind of Magic is also the name of the chorus who became National Mixed Chorus Champions 2019 at the British Association of Barbershop Singers Convention. The chorus was an amalgamation of The Cottontown Chorus and Amersham A Cappella. Steve designed a wall-hanging to commemorate this fantastic win.

The video shows how the image in illusion knitting changes as your view changes. Watch the video with your sound turned on as the soundtrack is of the same two choruses singing in eight-part harmony at a rehearsal of It’s a Kind of Magic.

Both choruses are multi-medal winners in their own right. A couple of years ago they sang two songs together in eight-part harmony. For such talented groups that was comparatively easy because they were able to learn the two halves separately with one combined rehearsal at the last minute. The two songs were The Climb and It’s a Kind of Magic.

Their competition pieces were much more difficult. The rules for competition Barbershop are quite strict. Only four parts are allowed. Generally speaking this means that many of the men have to sing lower parts than they usually do and many of the women have to move upwards. It was quite challenging and made even more so because the two groups are 200 miles apart and they only had one real-life rehearsal. (For more info, and to watch the two competition songs, go to this previous post.)

In the bar, in the evening after the win, one of the Amersham ladies said, ‘Are you the person who was knitting at the rehearsal?’ It suddenly seemed a good idea to make something specially for the chorus so, as soon as we got home, Steve set about designing a shawl/wall-hanging, incorporating the Amersham logo and the words A Kind of Magic.

Finding the right yarn was tricky. We wanted purple and white (because these are Amersham’s colours) – with sparkles. We went on a journey and managed to find Woolcraft Diamonds. Amazingly, today, we went into our local branch of Poundstretcher and they had some yarn. There must have been less than 100 balls in total. Two were white and sparkly, two were purple and sparkly! (For those who don’t know, Poundstretcher is a cheap British store that sells a very odd mixture of things.)

Small block letters are quite easy to do in illusion knitting. Fancy writing is more tricky. It has to be big to be able to show the small details of the font. There was just enough room to fit the lettering onto a shawl. Although it can be worn as a shawl, it is much more impressive when it is seen flat. When someone wears the shawl you only see flashes of some of the lettering. It has a channel at the top so it can be hung on a pole.

We do not often add a fringe to a shawl but it seemed a nice finishing touch this time. There are as many pieces of fringe as there are stripes. The strands line up with the stripes.

Whenever Amersham go on stage in a competition they take a Barbie doll dressed to match their stage costume. I don’t usually make doll clothes but thought it was only right that Barbie should have a shawl. She looked a bit strange with just a shawl so she also got a dress, hat and boots.

Barbie is too small to show any kind of illusion but the dress and shawl use illusion techniques to make the purple appear more prominent in the shawl and the white more noticeable in the dress.

Barbie wouldn’t stand up to have her photograph taken. Steve downloaded a file from Thingiverse to 3D print a stand for her. It had a support that went round her waist so it spoiled the effect of the stripes in her dress. He used the base of the original stand with a shorter piece for her to lean against.

The pattern for the Amersham shawl will be of little interest to anyone other than Amersham members but it is available for anyone to download from Ravelry.
The Barbie pattern is also available as a free download.

Flatland Lock

About ten years ago we came across a puzzle called Bank Raid, in Leonardo’s Mirror & Other Puzzles by Ivan Moscovich.

The puzzle itself was inspired by the 2- dimensional world of Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott.

At the time we made a crochet blanket. Obviously it was not possible to physically move the pieces so it was just a reasoning puzzle.

I have now gone back to it to make a version where you can actually move the pieces.

The complete puzzle.

The lid is fixed to the base at the four corners. The sides of the frame are open for the pieces to slide out.

The four pieces of the lock can move up, down, left and right.

How do you remove all the pieces?

One of the photos below shows the complete shapes of the four pieces; the other shows what you see when the lid is not in place.

The four pieces
What you see when the lid is not fixed on

The files for making the puzzle can be downloaded from Thingiverse or Pinshape.

There are six pieces in total
The studs on the back allow the pieces to move along the grooves in the base

The video shows the solution

You can download the afghan pattern from Ravelry