I have always been fascinated by flexagons and have made several over the years, some more successful than others.
My first crochet hexaflexagon was about twenty years ago and, amazingly, it is still surviving. It isn’t the most comfortable cushion in the world but it is great fun. I can absolutely predict what will happen when it is presented to a group of kids (or adults!). First, it is used as a frisbee (It is quite aerodynamic.) One person will put it on their head and it isn’t long before someone discovers that a slight tug will pull it over the head and it will sit snuggly round the neck, like a ruff. Eventually they investigate its strange properties and discover that this there are six possible arrangements.
I made it for my own use but lots of people asked for a pattern. My crochet skills at that time, were fairly limited so the pattern was very basic. Perhaps that was a good thing as hundreds of people have used it since. It was in a book called Twists & Turns but it soon took on a life of its own and became a standalone pattern.
Over the years the names of these things seem to have changed. The flexagons here are now usually referred to as trihexaflexagons. They are made from nine triangles.
I was asked to make a more elaborate version, for an exhibition, so made a fluffy knitted one. (There have never been pattern instructions for this one.) It is basically the same as the crochet version but looks different because there is no ‘background colour’ in the triangles.
Eventually I made a knitted version quite similar to the crochet one – and added the instructions to the original pattern.
Another crochet variation (also included in the pattern) is the hexaflexacube. This isn’t a real mathematical thing. The construction is the same as the others but the colouring makes it look like a drawing of a cube. The crochet technique is a little more sophisticated than the earlier crochet version.
A few months ago I got ambitious when I came across a book, written by someone we worked with way back in MathsYear2000. Kjartan Poskitt published an ebook called Flexomania. It includes several different flexagons and he also has models to download, cut and fold yourself, and videos.
I have a book about polyhedra, including flexagons, that I bought about 25 years ago. I made some of them in paper but never thought about making the more complicated hexaflexagons in anything else. Kjartan’s video inspired me to think again. I think I always knew that adding extra thicknesses of knitting, crochet, and wadding, was not going to work but decided to try anyway. I learned a lot on the way that might come in useful in the future.
The flexagon, that now tends to be called a hexaflexagon, is made from 18 triangles which means that several thicknesses have to pile up on top of each other to make the hexagonal shape. (In some places you will find this one called hexahexaflexagon.)
I originally thought about making two – one knit, the other crochet – but that would have meant an awful lot of work. Plan B was to make half the triangles in knitting and half in crochet. They would have to be exactly the same size to fit together. The other complication was that they had to have a directional design because sometimes the triangles point one way, sometimes they point another, and it is important to be able to tell the difference.
The triangles need enough space between them that they can twist and turn. That part was relatively easy but it does mean that the flexagon sometimes turns more than you want it to. The main difficulty is that the 18 triangles are sometimes (at their most extreme) three piles of five and three piles of one which makes a very uneven cushion.
The cushion is difficult to fold – even for a person with big hands,
Recently I had reason to go back and look at the old Twists & Turns book. There were four more flexagon patterns languishing there so I decided to resurrect them. Calling them cushions is not really accurate. They are foldable, padded things that are fun to play with. They are not particularly comfortable and some of them are infuriating – hence the title of the free ebook is Frustrating Flexagon Cushions.
They are easy to make but time-consuming. They are all made from squares, rectangles, or triangles and can be knit, crochet, or fabric. Once you have made them, and start to fold, it often seems impossible to get back to where you started. Sometimes you know there is another coloured face inside there somewhere but you just can’t find it.
After playing with these ‘cushions’ again, I now think they should all be a bit smaller than the sizes in the pattern. They might not collapse quite as much. The ebook contains all four patterns plus paper versions to cut out and fold.
Face to Face is the easiest. It has fewest pieces so there is less to go wrong. It is made from 6 double-sided squares which are joined in such a way that the cushion has three different sides.
Box and Cox is made from 12 double-sided rectangles. It can be folded to reveal four different coloured faces, and also folded in half.
Flying Colours has 16 double-sided triangles. It can be folded into many different shapes and colour combinations. The photos show just a few of them.
Square-bashing is the most annoying of all. It looks as though it should be easy to manipulate but it certainly isn’t. It is made from 12 double-sided squares.
It starts with 12 squares arranged in a square and ends with a block of four squares.
Ben made another puppet that needed glasses. I know a little more about 3D printing than I did a month ago so I was able to adjust the settings to get a better print.
These are puppet versions of the presenters of the TMRO Space webcast
I doubt if anyone else would want to make the glasses but, if you do need a toy-size pair of glasses, the file is available on Pinshape. It is a .stl file and needs some adjusting, such as rotating through 90 degrees, before it can be printed.
I knitted yet another To Bias Or Not To Bias shawl. I published the pattern about two years ago and I think this is probably the sixth I have knitted for myself. I love this pattern because it is so versatile and works with any yarn.
It is an asymmetric shawl pretending it is symmetric. The rows of holes run parallel to the long edge but the rows of knitting do not.
I was given lots of this particular yarn about twenty years ago and always thought it wasn’t really my kind of thing because it was a bit sparkly. It is very dark, possibly black, with a slightly sparkly blue thread running through it.
I used the yarn recently to make some slippers and realised that when you stand a little way back it doesn’t really look sparkly at all … so I decided to use it for the shawl. It is DK thickness and I used 5.5 mm needles. I wished afterwards that I had used one size bigger to make it even softer and more cuddly.
In recent years the way of knitting shawls and scarves has changed. They used to be simple affairs in basic triangular, round, or rectangular, shapes. Nowadays the shapes are much more varied. To Bias Or Not To Bias concentrates on traditional right-angled isosceles triangles.
Triangles are often started at the bottom point with just a few stitches. The rows of knitting are parallel to the long edge. Increase at both sides and it gets wider and wider as you move up. Stop knitting when the shawl is wide enough.
Some triangles start in the middle of the back of the neck. The increases are usually done in the centre, and at the outer edges, to form the shape.
You can keep knitting until the shawl is big enough but it can be tricky to see the exact size because the knitting has to bend round the point at the bottom and can be difficult to lay flat until the stitches are off the needle.
In this style the rows of knitting are at right angles to the long edge. The triangle starts at one of the side points. Increases are done on one edge while the other edge is knitted straight. Keep knitting until the shawl is half the length you want it to be then start to decrease on the edge where you were increasing before, still keeping the other edge straight, until all the stitches are worked off.
The biased shawl starts at a point with increases at one side and a straight edge at the other side. It begins in exactly the same way as the shawl above. (The rows of knitting are shown reversed in the diagram.)
Continue until the shawl is the size you want. The sloping edge becomes the long edge of the shawl whereas the sloping edges in the previous example are the short sides..
This is the shape you get when you start knitting
… and this is what it looks like at the point of the shawl
The shawl doesn’t need to have holes … and it doesn’t need to be a shawl. The pattern includes instructions for several variations.
I printed a small toy with rotating wheels and was intrigued when I found it is possible to print something all in one go and for the wheels to be able to move. I didn’t need any wheels, and I didn’t want to use anyone else’s designs for wheels, so I experimented just to find out how it is done.
My first problem was that the bottom of the top wheel in this photo is an unsupported surface. 3D printers cannot print in mid air. They need something to fix to. The axle only supports the centre of the wheel. The rest needs to be sitting on a cone which slopes gently outwards from the axle.
The casing for the wheel has to fit round the cone but leave a gap so the wheels can turn. The gap makes a funnel shape. A similar shape, in reverse, is needed for the other wheel. The cones and funnels are at the same angle. The angle and spaces have to be arranged so that the printer can print without filling the gaps.
The first one was too small. The pieces fused together and I couldn’t see exactly where they were fused.
I made changes to the angle and spacing then printed it at a bigger size. It was still wrong but I could begin to see where it was starting to fuse together.
More work on the angles and spacing. I printed an even bigger model and the wheels do turn. It works but doesn’t look good.
I made more small alterations and it’s definitely getting better. I decided to stop at this point.
I moved on to thinking about wheels that were not encased and would rotate just by rolling on their own axles.
I designed a wheel and an axle that should fit inside it. The axle wouldn’t fit.
Version 2: Bigger wheel and axle. The holes in the wheel were also bigger, in comparison, to the axle. I also put a groove round the end of the axle and made a clip to fit in that groove to hold the axle in place in a model.
Version 2: The pieces need cleaning but it is obvious that the axle still won’t fit in the wheel.
Version 3: I made the holes in the wheel even bigger and made a sturdier clip.
Version 3: Once the axle would fit inside it became obvious that it was too short.
Version 4: A more robust clip, a longer axle and very slightly bigger holes in the wheel.
I am not really a sock knitter but I recently wrote a pattern for mathematical slippers and made several pairs in a few weeks. The front, sides and sole of the slippers are knitted flat; the cuffs can be knitted flat or in the round.
Things with a small circumference, such as socks, can be knitted in the round, using four (or more) double-pointed needles, or with a long circular needle, using a method known as ‘magic loop’. Magic loop involves folding the work in half, pulling the ends of the needle through and knitting as though you were knitting flat then repeating this process after every half round. I decided I didn’t want to do either of these. I wanted to just knit round and round without having to stop.
I have many sets of interchangeable circular needles. The tips of the needles can be swapped from one length of cable to another depending on what you want to knit. The shortest cables in most sets of interchangeables make a length (including the needle tips) of about 40 cm (16″) which is far too long for a slipper. It is possible to buy very short fixed circulars but they are quite expensive and I needed various thicknesses. The photo shows 4.5 mm needles from all the sets I have. As you can see two are shorter than the rest but not short enough.
After searching on the internet I came across this tutorial for how to use a set of Boye needles and add your own cables. It just so happened that the shorter pink needles, in the photo, are from a set of Boye needles I have never used. I bought them when I was in US about 12 years ago – because they were pretty. I didn’t like them because the cables are too stiff and they have a very strange bend in them. The hack in the tutorial was perfect – and cheap.
We followed the instructions almost as they are given. WeedEater Line is called strimmer cord in UK. A long roll costs less than £2. The US size for the scews is given as 2-56, which is also known as M2. We took the needle tips to our local family-run ironmongery shop, where they hadn’t heard of these sizes. Their guess was that we needed 2 mm screws. They tried some in the needles and they seemed to fit. They weren’t as long as we wanted but were the only option. The shop owner chopped the heads off the screws so we wouldn’t have to do that at home. The screws cost 5p each so it wasn’t going to be a big loss if they didn’t work.
The most tricky bit was finding narrow tube to fit over the screws and cord. We tried various things we had lying around and discovered that the tube in a pump make-up dispenser was just right. We followed the instructions more-or-less as given, making the cords as short as we possibly could and still allow them to make a circle. We finished off by binding the joins with plumber’s tape.
Knitting the first couple of rounds wasn’t easy but, once the stitches started to spread out a bit, they worked well. I wouldn’t recommend them for fine yarns as the joins are not as slick as other needles but they did the job without too much frustration.
In the meantime we had bought the 3D printer and decided to experiment with more strimmer cord to make fixed circulars with a very small circumference. Each needle tip was basically made from a cylinder with a tall pointed cone on each end. The tip was made in two halves, the cord was glued in the channel, the two halves were glued together and the joint wrapped with plumber’s tape.
In principle these needles work well although we didn’t go on to polish them properly because the tips we made were too short. They were about 6 cm long. The needle was about 24 cm from end to end. I have problems with my hands and the tips were really not long enough for me to manage. Anyone more dexterous would not have the same problem They would need to be about 1 cm longer for me to manage which would still make the whole needle considerably shorter than the hacked Boye needles. We didn’t make any more but probably will in the future.
It is very easy to change the length and diameter of the needles from the original 3D printing files.
Pros of 3D printed needles
If you have strimmer cord you can make them whenever you need them
Tips can be any length
Tips can be any diameter (provided the cord will fit inside)
Cables can be any length
Cons of 3D printed needles
Need polishing to be smooth enough to use
Points need smoothing so they are not too sharp
The sizing might not be as accurate as shop-bought needles
Yesterday I came across this video, on YouTube, of someone making Steve’s Flying Dragon Illusion on a shawl. It is in French so I am not sure what it all says but it is very detailed and I can tell that she explains a lot about the pattern and the yarns she used.
The pattern has instructions for using the dragon on a banner, shawl or square wall-hanging. Our original version was a banner which was made using approximately 270 metres (300 yards) of DK yarn in each colour. It is about 60 cm (24”) wide and 71 cm (28”) high.
Several years ago, Colin Wright (mathematician/magician/juggler extraordinaire) showed me a Magician’s Chain. This is a toy made from metal rings where the rings cascade from the top of the chain. It was fascinating, fun, and quite intriguing. I have seen Colin’s chain, and others, many times since. Then, a few months ago, I was at a maths event with Colin and was idly playing with the chain. I suddenly thought it needed colour to be able to see what was really going on.
My first plan was to buy some split rings and cover them in embroidery silk in shades of blue so they might look like water falling down. I have a tin full of silks from the days when I used to do embroidery so I didn’t have to buy those. There weren’t many shades of blue but I decided that some bright colours would be just as good.
I was thwarted at every step. I ordered 100 metal rings from Ebay. They didn’t arrive when they should have but, before I got round to complaining, I got a message from Ebay to say that my order had been cancelled and my money refunded. The seller mysteriously disappeared.
I tried again and ordered some more from another seller. They didn’t arrive. This was in the run-up to Christmas so this seller asked me to wait until two weeks had elapsed. I did that then emailed him to say they had not turned up. Mysteriously, they dropped through our letterbox late that same afternoon. I have no idea where they materialised from as the normal post had already been delivered earlier in the day. Fortunately, the seller hadn’t sent any more.
Once we had the rings we joined them into a chain, following the instructions in this video. I originally thought that I could crochet round the rings but that didn’t work because it made the rings too bulky and stopped them moving properly. Another problem was that the rings could only be covered when they were already part of the chain because the split would be covered and it would be impossible to get them on or off.
Plan B was to wrap the silk round and round each ring. It was tricky, because a tiny ball of silk had to be passed through the ring so many times – and it didn’t work! We had more ideas for what might work, such as binding the joint with plumber’s tape first or filling the gap with Fimo. Nothing worked.
3D printer to the rescue. We could print the rings except that we only had the two colours of filament that came with the printer and wanted lots of colours so had to buy more. It is possible to buy small amounts of filament but 500 grams costs almost the same as 1 kg. We now have 7 kg. Making the rings took a while. We used 28 for the long chains and printed six at a time although more would probably have fitted on the printer bed.
The rings are approximately 30 mm in diameter and have a gap of 2 degrees in the circumference.
As you can see in the video we made a silver and black chain, then the multi-coloured version. (We also tried a chain with square ‘rings’ just to see what would happen but they hardly moved without shaking the chain repeatedly.)
Whatever your eyes may be telling you, there are only two positions for the rings in the chain.