Modular Knitting 3

How to use modular garter stitch shapes with 45, 90 ad 135 degree angles.

Previous posts about making the shapes
1. Knitting at MathsJam
2. More Modular Knitting

Now you know how to make the shapes, what can you do with them? You could make a small cushion or go on, as many people have, to make a blanket.

Making a garment is easy. The most important thing is to decide the overall size you need, before you begin.

It is possible to make a sweater, jacket, hat, etc. without any extra shaping. It can be a very simple shape. My (very old) cardigan has tapered sleeves.

I made the cardigan entirely from scraps of leftover yarn. The second photo shows all the shapes, before the edgings were added.

To make something like this

  • Decide how wide you want the back to be and divide the measurement by six.
  • Draw a square with sides of the size you calculated.
  • Knit a diagonal square to fit the drawing.
  • Make a note of the number of stitches on the diagonal.

The photos below show the whole of the back and the first seven shapes that became the top corner of the back.

Shape 1
Width of back will be 54 cm (21″)
Divide by 6.
Square size = 7 cm (3.5″)

Knit the diagonal square.

The square in this example has 20 stitches on the diagonal.

Make a note of the number of stitches on the diagonal of your square.


Shape 2
Divide the number of stitches on the diagonal by 1.4. I will now call this number x

Example: x = 20 / 1.4 = 14

Pick up x stitches along the edge of the square.
Knit straight on wrong side rows.
Knit two together at the end of every right side row until one remains. Fasten off.

The triangle is the same height as the original square.

Shape 3
Pick up x stitches along the edge of the starting square and another x stitches along the edge of the triangle.

Knit two together at the end of every wrong side row.
Knit into the front and back of the last stitch of every right side row.

Continue until you have x garter ridges. Keep the stitches on a stitch holder if you think you might use them again. (Or cast off instead of working the last row.)

Shape 4
Shape 3 is the same height as the starting square so Its sloping side is exactly the same length as the diagonal of the square.

Pick up the same number of stitches you had on the diagonal.

Knit straight on wrong side rows.
Knit two together at the end of every right side row until one remains. Fasten off.


Shape 5
Pick up x stitches along the edge of the square.

Increase in the last stitch of every row.

Continue until you have x garter ridges.

Either cast off on the last row, or keep the stitches on a holder, in case you need them later.

It is easiest to stitch the join between shapes 4 and 5, as the pieces will not match ridge for ridge.

Shape 6
Pick up x stitches along the edge of the square.

Knit straight on wrong side rows.
increase at the end of every right side row.

Knit the same number of ridges as Shape 5. Do not cast off.

The ridges in this shape exactly match the ridges in Shape 5. As an alternative to your normal increase you could pick up one stitch from each ridge end of Shape 5.

Shape 7
Continue on the same stitches, with a different colour.

Decrease at the end of every row. Continue until one remains. Fasten off.

Continue adding shapes in any direction until the width of piece is equivalent to six starting squares. The length can be anything you choose.

The letters in this diagram show you how many stitches to pick up from the exposed edges.

x = the number of stitches you calculated.
d = The number of stitches on the diagonal of the square,

Add shapes in any direction. You can even use more than one colour in a shape. The first photo below includes shapes 8 to 13. The other photo is of an almost complete square.

These shapes are included in a pattern called Cushion Conglomeration Caper. There are two sets of instructions. The short version describes shapes for you to think about, and make, yourself. The long version gives you detailed help and, sometimes, alternative ways of making a shape.

History
CCC was published in 1998, which explains why the photos are so old.

It was originally a project for, and by, the members of the Woolly Thoughts Yahoo group, as an introduction to using Woolly Thoughts methods. The plan was for a different member of the group to add each new shape so that there was no predetermined pattern. Some might refer to this as freeform but this was all using very precise geometric shapes. The only angles allowed in the shapes were 45, 90 and 135 degrees.

The original plan broke down as some members wanted to work more quickly than others. I added the remaining shapes myself so that everyone could work at their own speed. The basic instructions for a new shape were revealed every two days but all help and advice files had to be accessed separately so as not to spoil the fun for those working more slowly.

This blanket was made by two people who started with the Cushion Conglomeration pattern then took it in turns to add new shapes without the knowledge of the other person.

Download the pattern



More Modular Knitting

In an earlier post I described what happens at a typical Woolly Thoughts workshop about modular knitting and why these foolproof methods work.

Begin with a garter stitch square, knit diagonally:

  • Make a slip knot
  • Increase by knitting into the front and back of the loop
  • Increase at the end of every row until you have the size you want
  • It is very important to remember the number of stitches at the widest point
  • Decrease at the end of every row by knitting two together, until all stitches are worked off

To add shapes to the square
Use a calculator to divide the number of stitches at the widest point by 1.4. Round to a whole number. This tells you how many stitches you need to pick up along the edge of the square.

The simplest form of modular knitting is in garter stitch and creates angles of 45, 90 or 135 degrees. There are only a few different things you can do to add shapes onto the square. Each side of the shape can lean to the left, go straight up, or lean to the right. The simplest is to just keep knitting straight but it is not very exciting. The other possibilities are:

Decrease at the end of every right side row

Knit straight on wrong side rows

Knit straight on right side rows

Decrease at the end of every wrong side row

Decrease at the end of every right side row

Increase at the end of every wrong side row

Increase at the end of every right side row

Decrease at the end of every wrong side row

Knit straight on right side rows

Increase at the end of every wrong side row

Increase at the end of every right side row

Knit straight on wrong side rows

Increase at the end of every row

Decrease at the end of every row


Some shapes work off all the stitches and come to a point, others could continue as far as you like. You can even swap between the different angles and end up with a zigzag or other eccentric shape.

If you plan to add more shapes, you need to think carefully about how tall the shapes need to be so that they can fit together. It is often best to make them the same height as your starting square, or half that height.

To be continued …

Continued in Modular Knitting 3

Two Dolphins

New year – new pattern.

Steve has now designed more than a dozen illusion blankets. You can see them all here. This new one is amongst my favourites.

There are many reasons why I particularly like it

  • Dolphins appeal to people of all ages so it isn’t just a baby blanket
  • It is very easy
  • The dolphins are on a garter stitch background (Many of the blankets are alternate ridges of garter and stocking stitch.)
  • The outlines of the dolphins are very streamlined so it is easy to check that no stitches are out of place
  • It reminds me of yin yang

The blanket in the photo is about 80 cm x 100 cm (32” x 40”). It could be made much bigger with a wider border all round.

Buy the pattern

When you look directly at illusion knitting you only see narrow stripes. The picture appears when you look from an angle

Knitting at MathsJam

Last week I gave a talk at MathsJam Annual Gathering. This is an amazing annual event where ‘mathematicians’ from around the world get together. It isn’t the boring number stuff you might imagine. When 200 eccentrics (of the nicest possible kind) get together it leads to mind-bending fun, puzzles, games, performances, cake baking, singing and many other things that would soon dispel the idea of maths being boring.

Each speaker is only allowed a five-minute slot (with plenty of discussion time afterwards). I originally intended to talk about mathematical Barbie clothes, which you can find in other posts on this blog. At the last minute I changed my mind and decided to talk instead about the most basic ideas behind Woolly Thoughts mathematical knitting.

We have been presenting workshops for 25 years and work on the premise that a good many participants will say ‘I can’t do maths’. We aim to convince them that they can. The methods shown here are a ‘foot in the door’ to those people and we know, from experience, that some wonderful things can follow.

What happens at a workshop

Knitters are asked to bring their own choice of needles and yarn, to prove that the methods work with anything. The same thickness of yarn should be used throughout.

They are given a square with a diagonal line marked on it.

The instructions are

  • Work in garter stitch (Knit every row)
  • Start with a slip-knot
  • Knit into the front and back of the loop
  • On every row, knit into the front and back of the last stitch until you have a triangle wide enough to fit up to the diagonal line
  • Make a note of the number of stitches
  • On every row, knit to the last two stitches. Knit two together
  • Fasten off when one stitch remains

It may not be a perfect square but will become square when other shapes are added to it.

The next part involves some elaborate work with a very large calculator where we take each person’s remembered number of stitches, divide it by 1.4 and round to the nearest whole number. We haven’t mentioned maths yet. Calculators aren’t real maths.

The next stage is to pick up this number of stitches along one edge of the square.

This must be with the same needles and the same thickness yarn but can be a different colour. The new triangle is made in exactly the same way as the second half of the square.

It is guaranteed that at this point someone will say ‘I’ve made a house’ … and people will put them together and say ‘We’ve made a row of houses’. All the houses are the same size with different numbers of stitches.

We still haven’t mentioned maths.


Use the number of stitches picked up before and divide by 1.4 to make the small triangles. The sharp-eyed might notice that this is half the number of stitches on the diagonal of the square so now we could start dropping in some maths.

  • Squares, triangles, rectangles
  • Mathematically there is no such thing as a diamond
  • Triangular numbers
  • Consecutive triangular numbers to make a square number
  • Halving and doubling
  • Area
  • and, maybe, Pythagoras Theorem

Why does it work?

Garter stitch forms square stitches. Two rows are the same height as the width of one stitch. This property means it makes right-angled triangles so normal geometry can be used.

The square root of two is really 1.4142… Knitting stretches so the geometry doesn’t need to be quite as accurate as in normal maths. It is near enough to say that the diagonal of the square is 1.4 times the length of the side of the square.

Some knitters don’t care how it works and are happy to just believe that dividing the stitches on the diagonal by 1.4 will give the right answer. Whether they care or not they can still go on to use the basic concept in many different ways.

The ‘houses’ can be arranged in various ways to make patterns. This could be an introduction to tessellations and looking at different shapes. What shape are the houses? What shapes can you see in the arrangements below? What do you notice about the squares? etc.

More shapes can be added, in any direction. The arrangement below is particularly useful. The four yellow corners can be folded in to cover the green square. This makes it very obvious that the yellow area is the same size as the green area. The overall shape is twice the size of the original.

This can be very helpful to knitters. The four yellow triangles will need exactly the same amount of yarn as was used for the green square.

Using these methods it is possible to go on and add shapes in any direction. The angles of the shapes will always be 45, 90 or 135 degrees. The number of stitches needed can be found by doubling and halving. Stitches in the same direction as those on the diagonal of the square will always be a multiple, or fraction of those stitches. Stitches parallel to the sides of the square will be a multiple, or fraction of the number of stitches on the side of the square.

To be continued …

Continued in More Modular Knitting and Modular Knitting 3

Borromean Baubles

These simple mathematical baubles can be made in lots of different ways and either hung as a decoration, or used to hold a small gift.

The Borromeo family originated in Rome – that’s where the Rome bit of their name came from. In the twelfth century they set up a bank in Milan and became very wealthy and influential. For hundreds of years they controlled a large area around Milan. The descendants still own some of the land today and their gardens are a popular tourist attraction.

Mathematicians are more interested in their coat of arms than the history of the family. It contains three interlinked rings that are now known as Borromean Rings. They form just a small part of the crest and are thought to represent the links between three families.

The reason these rings are special is that you cannot remove one and leave the others connected. Technically, they are known as Brunnian links.

It might not be immediately obvious how the baubles are related to rings – but they are. They are basically cubes made from three strips of knitting, crochet, paper, or anything else you can think of. The strips are just wide rings.

These are all knit and crochet.

The video explains how to make a cube. There is almost no stitching to be done. You can put a gift inside and remove it without damaging the cube at all. It also shows how the rings become a cube.

These are some of the cubes from the video

Add extra sparkle and beads to make them more Christmassy.

Use some sparkly yarns
Add beads
Embroider letters
Fancy yarns make the cube look like a ball
Cut shapes from sequin waste and stitch them on
Six are needed if you want one on each face
Four letter words fit nicely round the cube
Think carefully about the placement and orientation of the letters

Small cubes keep their shape very well, even when you hang them or put things inside. It is possible to make larger ‘boxes’ though it probably isn’t worth the effort. There is a lot of knitting involved and fairly accurate measurements need to be taken.


Baubles don’t have to be knit and crochet. They can be made from some quite surprising things that you might have lying around. My favourite are these made from plastic folders. In their natural state the folders are fairly uninteresting. They are usually available in red, blue, green and yellow but when the colours are overlapped you can create a myriad of different shades and the translucent shapes look more like glass or ceramic.

These are quick, cheap and easy to make but not quite so good for putting a gift inside. You need to unfasten the cube to remove the gift but you can fasten it up again afterwards. They need to be measured accurately and folded sharply to make a good cube shape.

Let your imagination run riot with coloured card, holographic foil, etc. You can even mix materials in the same cube.

Download Borromean Baubles pattern
Download Cube Thing pattern

Barbie Spirals

Barbie has lots of new knit and crochet spiral dresses.

In maths there are very specific definitions for some kinds of spirals. You can find lots of information about mathematical spirals on the internet. Some of them refer to ‘circling round a centre point’ but you will also find square, hexagonal and other shaped spirals. In everyday language we refer to anything that goes round and round as a spiral – even though it may not be round.

Barbie’s dresses are different kinds of ’round spirals’. They fall into two main categories – those that are flat and spread outwards to form a shape very like a circle, and those that go upwards to form a cylinder shape. Barbie dresses are small so they are perfect for experimenting with new ideas that could then be used on bigger things like hats, bags, cushions, sweaters, etc.

Flat spirals

2 colours
3 colours
4 colours
6 colours

These dresses are worked in treble crochet. When you work in trebles you need to add 12 trebles in each round to keep the shape flat. If you want to make a hat, coaster, or something similar, you would begin with 12 stitches worked into a ring then keep adding 12 stitches on every round. Barbie’s spirals begin at her waist with 24 stitches. It is best to use 2, 3, 4 or 6 colours because the stitches can be divided easily into these numbers of sections.

Many people think you need to add more and more stitches as the circle gets bigger. This is not true. You only ever need 12 extra stitches even if you are making something as big as a blanket. To make a large circle look ’rounder’, stagger the extra stitches so they don’t lie in lines above each other. Barbie’s dress is not big enough to worry about this.

Treble stitches are quite tall so, as each colour is introduced, the heights of the stitches need to be gradually increased so they can wrap over each other. Start with a few double crochet, then some half-trebles and, eventually, the trebles.

To make Barbie’s skirts softer and more floppy, use a bigger hook but don’t change the stitches.

Spirals don’t have to be in trebles. If you use other stitches the number to add on each round is different:
Double crochet – add 6 stitches on each round
Half trebles – add 8 stitches on each round
Double trebles – add 18 stitches on each round.

Multi-colour cylinder spirals

2-colour treble crochet
3-colour treble crochet
2-colour double crochet
2-colour rib knit

Making a spiral cylinder isn’t much different from making a plain cylinder. The number of stitches for the cylinder has to be a multiple of the number of colours so the stitches can be divided evenly. The 2-colour spiral begins with 11 stitches of each colour (22 in total). The 3-colour spiral begins with 7 in each colour (21 in total). However many colours you use you need to begin with small stitches and gradually change to taller stitches, similar to the beginning of the flat spirals. This would also apply if you were only using one colour. Once the stitches are the right height, just keep going round and round, always working into stitches of the previous colour.

The 2-colour dress worked in double crochet has less obvious spirals because the stitches are smaller. The knitted dress is even less obvious.

Single colour spirals

All knitting in the round is technically a spiral. If you are going round and round with no breaks you are making a spiral. In most cases this is not noticeable. If you want the spiral to stand out it has to have some kind of texture. All four dresses below use the same basic rib pattern but it moves across by one stitch on every round.

The two on the left spiral in one direction, the two on the right go the opposite way. You could describe these as clockwise and anti-clockwise. I prefer to call them forwards and backwards because, as you are knitting they are moving in those directions.

All dresses are worked in a 3 x 3 rib. A cylinder would have to be a multiple of six stitches. Surprisingly the dresses on the left (orange and yellow) have two fewer stitches than those on the right (green and brown). The ‘forwards’ dresses have 23 stitches. Because the spiral is always moving forward, the 24th stitch needed to complete a round is the first stitch of the new round. Conversely the ‘backwards’ spirals would be one stitch short at the end of the round so need to have 25 stitches.

Many people think that if you look at a spiral from the other end it turns the opposite way. When there are problems with yarn, I have often heard people suggest using the ball from the other end because the twist is different. This is not true. Turn Barbie upside down and you can see the spiral is the same. This is also what happens to the yarn. When Barbie’s dress is turned inside out the spiral is reversed.

Two alike
Turn one upside down. The spirals are the same.
Turn one inside out. The spirals go opposite ways.

Toilet roll spirals

The centre of a toilet roll is a flat piece of card wrapped round to make a tube. It is very easy to get confused about the direction to knit to make a similar spiral to wrap round Barbie.

When you unwrap a toilet roll centre you get a long thin parallelogram. Wrapping it round in this orientation would make a cylinder but it would not be as tall as the original.
Turn the parallelogram round and it becomes the right height. The angles of the parallelogram could be changed but the height and width (circumference) of the cylinder must be correct.

The angles of the knitted spirals are not the same as the card. When the colour is moved across by one stitch on each garter ridge it automatically creates a 45 degree angle. When it is moved across by one stitch on alternate ridges it makes a much steeper angle, approximately 63 degrees.

2 colours. 45 degrees
3 colours. 45 degrees
2 colours. 63 degrees

Download the free pattern

You will find many other spirals at woollythoughts.com Here are some of them:

Hyperbolic Barbie

Hyperbolic crochet has become very popular in recent years. It has been used world-wide in various coral reef projects. The world expert in the mathematics of this art is Daina Taimina.

Our latest mathematical pattern for Barbie and friends is Hyperbolic Barbie.

Hyperbolic geometry is quite complicated but you don’t need to understand it to make these simple dresses. The pattern includes three crochet dresses, three knitted dresses and two scarves. It is intended to show what happens when you repeat the same pattern of stitches over and over again, and how making one small change quickly has a dramatic effect.

These two knitted dresses are basically the same. The frills are different. The pink increases in every stitch. The blue increases in alternate stitches.
  • The pink frill begins with 26 stitches
  • Row 2 has 52 stitches
  • Row 3 has 104 stitches
  • Row 4 has 208 stitches
  • If you could add more it would increase to 416.

  • The blue frill begins with 26 stitches
  • Row 2 has 39 stitches
  • Row 3 has 58 stitches (approx)
  • Row 4 has 87 stitches (approx)
  • The next row would have 130 (approx)
These two crochet dresses are basically the same. The dark blue increases in every stitch. The cream increases in alternate stitches.
  • The dark blue frill begins with 30 stitches
  • Round 2 has 60 stitches
  • Round 3 has 120 stitches
  • Round 4 has 240 stitches
  • The next round would have 480

  • The cream frill begins with 30 stitches
  • Round 2 has 45 stitches
  • Round 3 has 67 stitches (approx)
  • Round 4 has 100 stitches (approx)
  • The next round would have 150 stitches (approx)

It is clear to see that the numbers diverge rapidly when the rate of increase is changed.

It is much easier to make hyperbolic frills in crochet than it is in knitting. When you crochet you only have one stitch on the hook. You can work into it as many times as want and can twist and turn the dress to make the stitches. Knitting is in a straight line and all the extra stitches have to get forced into that line. It soon becomes physically impossible to add any more.

Download the free pattern

If you would like to experiment with hyperbolic crochet but don’t want to make Barbie dresses take a look at Squiggly Things.

Carnival of Mathematics 175 – Sum Square Over The Rainbow

You might be wondering what a couple of knitters are doing hosting the Carnival of Mathematics. If you are not familiar with the methods we use to teach maths you might want to check out our main Woolly Thoughts site. It has got rather rambling over the last 20+ years. The Afghans page is a good place to start.

First things first – the number 175.

Steve and I often talk about ‘nice numbers’. That usually means numbers that can be split up nicely when designing things – numbers with lots of factors (or divisors or whatever else you want to call them) or other interesting properties.

At first glance 175 looks like a nice number. It’s a number you come across quite often. It’s a multiple of 25. That has to make it a nice useful number.

Look at it more closely. Is it really much use? I don’t think so. It’s 5 x 5 x 7. Fives are OK but I don’t like Sevens. The only time we have used seven in designs is for rainbows.


I do quite like that 11 + 72 + 53 = 175. Wikipedia tells me that 135, 518, 598, and 1306 also have this property where raising the digits to powers of successive integers, and adding, equals itself.

Fives makes me think of pentominoes. I wondered whether there was some connection between the number 175 and the five squares that make up a pentomino. There isn’t but 5 x (5 + 7) does give me the total number of squares in a set of pentominoes.

Wikipedia also tells me
175 is an odd number, a composite number, and a deficient number. It is a decagonal number, a 19-gonal number, and a centered 29-gonal number.

175 is an Ulam number, and a Zuckerman number. It is the magic constant of the n×n normal magic square and n-Queens Problem for n = 7.

… and now for the Carnival

  • Barney Maunder-Taylor posted a link, on Twitter, to The Car Wheel Games at House of Maths.
    He said I’ve become distracted by car wheels! Here are five games for youngsters (and mathematically inclined adults) to play while out for a walk around town.

  • Lewis Baxter sent a link to Seventeen or Bust : Progress of the search This is part of the Prime Grid project.
    Lewis said This 17 year distributed computing project (which I believe is the longest) was resumed after a pause of about 2.5 years to double check some results. Now the search continues for 5 remaining numbers, with more than 9 million digits, that is conjectured to verify that 78557 is the smallest Sierpinski number.

  • Autograph Maths made an announcement on October 12
    We are now able to bring Autograph to everyone for free, forever.

  • Ari Rubin submitted An Introduction to State Space.
    An introduction to converting equations of motion into their state space representation which is the foundation of many modern numerical methods.

  • Brent Yorgey sent a link to his new post Order of operations considered harmful.
    He warns that this is a somewhat intentionally inflammatory piece on why the way we usually teach and define the “order of operations” is wrong.

  • Colin Beveridge has an interest in ancestry and posted a long thread about calculating the number of ancestors a person might have. He links it to the Birthday Problem. When I asked if it was available elsewhere online it wasn’t but it is now – An Ancestry Thread.
    One of the replies to Colin leads to an older post, on a similar theme, by Tim Urban – Your Family: Past, Present, and Future

  • Katie sent a link to a Duplo game posted on Twitter by Sam Blatherwick.

  • If paper folding is your thing you might like these geometric tessellations by Arseni Kazhamiakin. 

  • Do you prefer facts about heights and lengths and such things? UK Railway Station Trivia has recently been updated.

  • Why mathematicians just can’t quit their blackboards. A Guardian article about a project called Do Not Erase, which is to be published as a book in 2020.

  • Katie sent a link to a blog post that is a snapshot of what being a maths teacher involves – Classroom Reflections

  • Another thing Katie spotted was a post on John D. Cook’s blog – Exact values of sine and cosine

  • If you prefer listening to reading check out The Art of Innovation. This is A partnership between Radio 4 and the Science Museum exploring art and science
    (Use the Episodes tab to find the programmes.)

  • The Aperiodical website has a review of Kit Yates’ new book The Maths of Life and Death

  • To end the month Ed Southall started a Twitter thread about Vampire Numbers and evil numbers, and magic numbers and lots of other Halloween related numbers.

… and finally

Take a fence

The next Carnival of Mathematics (#176) will be hosted by Mehak at Tangent Math. Submit ideas here.

Tilting at Windmills

Before you scroll down, I want to ask you a question – Is this a pattern?

You may think this is a totally pointless question. Try showing it to other people and see if their answer is the same as yours. You might be surprised.

It doesn’t matter where we take this, or what the age or understanding of a group might be, we can ask that one simple question and arguments begin. There will always be many who land firmly on one side of the fence and can’t understand why there should be anyone on the other side. The discussions can get quite animated.

Tilting at Windmills is made entirely from squares that are navy on one side and some shade of pink/purple on the other. They had to be in the correct orientation but the choice of squares was mostly random. However, in a few areas, the colours were carefully chosen to give a hint of a repeated pattern. This had to be enough for the brain to think it was looking for a pattern but quickly be able to dispel the idea. It was deliberately confusing.

If the photo is recoloured I’m sure everyone would agree that it is a pattern. Before you scroll down look at the photo below, or the one at the top, and think about what you can see.

Did you see windmills, or pinwheels? Many people do.
If you didn’t see them before, can you see them now? Are they light or dark? Which way are they turning?

You might have seen, butterflies, bow ties, ‘diamonds’, lozenges, small squares, large squares. or many other different things. (One ten-year old once told us there were diaboloes.) The more you look the more you will see.

The first time we saw groups arguing it came as a bit of a shock. It became apparent that those who declared that it was a pattern were not taking notice of the colours at all. The shapes made a pattern and that was all that mattered. On the other side were those who insisted that if the colours were wrong it couldn’t possibly be a pattern. It certainly justified our arguments for not using colours in our Woolly Thoughts book though we had not been prepared for the extreme reaction whenever a group of people look at the hanging.

This has implications for normal classroom teaching. Almost every teacher provides pupils with coloured shapes for particular tasks but we rarely make it clear whether the colours are strictly relevant to the task. Perhaps those pupils who struggle are not seeing the task as we intended it to be.

Before you scroll down – can you see any tunnels?

We haven’t put this in front of a group for a long time but last week we had a visitor who came to look at what we do so we asked her the usual questions.

I have been looking at this for over 20 years now and I saw something I had never seen before. It might have been the effect of the light at that moment but I suddenly saw three tunnels. The really strange thing is that their outlines do not follow the lines in the divisions of the squares.

It troubled Steve that the inside/outside parts of the tunnels were ‘wrong’. He couldn’t accept them as tunnels but to me they very definitely were. I can see another tunnel above these and one below but they are not as well-defined. There is always something new to be discovered.

Just in case you were struggling to see windmills going in opposite directions, this photo might make them more obvious.

If you want to make your own you can buy the pattern.

Warrington’s Geometric Car Park

This week we decided to go to Warrington – and this was the view from inside the car park. I love all these different size hexagons.

This is the outside of the car park. Unfortunately, it was in shadow. There is a lot of building going on in the town centre.

We really went to Warrington to go to the museum and art gallery which are in an interesting building. We didn’t realise, until we got there, that there was a temporary exhibition Let’s Get Stuck in Traffic! which included knitting.

The artist Marie Jones had filmed herself chatting to people in her car. The videos were projected on to pieces of knitting and she had taken a quote from each person and knitted them into panels.

In another room there was an enormous knitted landscape.

I was also impressed by the children’s play area. I have never seen circular snakes and ladders before.