Over the past year, 242 mathematical granny squares made their way to me from all over the world. They came from mathematicians, fibre artists and hobby crafters from countries including Australia, Denmark, France, Germany, Norway, Scotland and Sweden, and from across the US. Most of these makers had never met or communicated with each other, and did not know what motifs others had chosen. But when everyone’s squares were laid out together, they matched, and became an artwork called Granny Life.
How did I get a matching set of granny squares from makers who never spoke to each other? Well, first of all, I gave everyone detailed instructions for how to crochet the pattern, what yarn to use, what size the squares had to be, and so on. But most importantly, I used mathematics as a design engine, by creating a custom 2D cellular automaton — a simple rule for switching stitches on and off — to produce all of the motifs.
Above are six crocheted granny squares with motifs made from our rule. Each motif is generated from the ones before it, so these squares are part of a family that shares certain resemblances. No two motifs are the same, yet they share shapes and patterns and look good together. Because every contributor’s crocheted motif came from the same rule, the squares combine into a single coherent artwork. About a hundred people contributed squares to the finished Granny Life artwork. This article is about the mathematics that made that possible.
Cellular automata
A 2D cellular automaton is a two-dimensional grid of squares, called cells, each of which is either on or off. Over time the cells switch on and off according to a fixed rule; that’s the ‘automaton’ part. The most famous 2D cellular automaton (CA) is known as Conway’s game of life, named after its inventor, mathematician John Conway.
In Conway’s CA, each cell looks at its eight surrounding neighbours and counts how many of them are on. A cell that is alive (denoted by a white cell) will survive into the next generation if exactly two or three of its neighbours are alive, and switches off otherwise. A cell that is off switches on and is born if exactly three of its neighbours are alive, and stays off otherwise (off or dead cells are denoted by a colour). Any rule of this kind is pinned down by saying when cells are born and when they survive. Conway’s rule can be given in shorthand by B3/S23: born with 3, survives with 2 or 3.
Of course, there are many other CA rules that could have been considered, and each would make different patterns. Some particularly famous ones include HighLife (B36/S23) and Maze (B3/S12345), and each has its own characteristic features that mathematicians and computational hobbyists enjoy exploring. Fun fact: the ‘Maze’ rule really does produce maze-like structures!
Can you find your way to the centre?
When choosing a rule, one could also make different choices about which cells count as neighbours; for example, instead of choosing all eight surrounding cells, choosing only the four directly above, below, left and right. For Granny Life I went the other way and used only the four diagonal neighbours, together with the rule B1/S0: a cell is born if exactly one of its neighbours is on, and survives only if none of them are. There is a standard way of numbering rules of this kind, and in that numbering B1/S0 comes out as number 6. Rule 6 happens to be particularly good at making balanced, matching motifs.
The first three iterations of Conway’s rule starting with a plus. The white squares are alive and the green squares are dead; the number in each square is the number of living neighbours.
The first three iterations of rule 6 starting with a plus.
Stitch neighbourhoods
Cellular automata on standard grids are well studied. But a crochet granny square is not a standard grid. It is built from clusters of stitches, worked in rounds from the centre outward. Granny square designs traditionally have 90-degree rotational symmetries, and in fact the full set of symmetries of a square: four-fold rotational symmetry and also mirror symmetry across the horizontal, vertical and diagonal axes. If we choose a CA rule that depends only on the number of neighbours around each cell (and not their arrangement), then that rule will preserve any symmetries in our starting generation. For example if we start with a ‘plus’ then the mirror and rotational symmetries of that ‘plus’ will be in every generation afterwards. Within each quadrant of a granny square grid, the clusters are staggered like bricks in a wall. Each rectangle in the figure is one cluster of crochet stitches, and the rounds run from the red clusters at the centre to the white border clusters at the edge. Each cluster has four diagonal neighbours. The right-hand diagram shows these neighbourhoods for several different types of cluster. Every cluster on the granny square has exactly four diagonal neighbours, described more precisely below.
The stitches in a granny square/A four-fold centre seed (red) and some stitch neighbourhoods
For the Granny Life project we will start with a small, four-fold centre seed, shown in red above. This is the crochet version of the ‘plus’ seed used earlier on the standard grids. We will choose a CA rule to follow on the stitch neighbourhoods shown in the figure. When we consider stitches close to the edge we need to add an extra rule: wrap around to the other side to find your neighbours. The white border stitches are added around the edge afterwards for aesthetics, and take no part in the rule. The outermost round of clusters, just inside the border, is mirrored across the square: the clusters along the left edge repeat the ones along the right edge, and the top repeats the bottom. This means that a cluster on one edge has a mirror and also neighbours on the opposite edge, as if the square wrapped around.
Crochet stitches
Terminology for crochet stitches is different in the US and the UK. In UK terms the simplest crochet stitches are called double crochet, half-treble crochet, treble crochet and double-treble crochet. In US terms these same stitches are called single crochet, half-double crochet, double crochet and treble crochet. The Granny Life project was made entirely with half-treble (UK) — AKA half-double (US) — crochet stitches. This stitch is made by starting with a crochet hook in a loop of working yarn, wrapping the yarn again around the hook to make a second loop, inserting the hook into the work and pulling up a third loop on the hook, and then finally catching a loop with the hook end and pulling it through all three loops on the hook.
Rule 6
The rule I chose for the Granny Life artwork is the same rule 6 mentioned earlier, except now running on the bricked granny square lattice with its diagonal neighbourhoods.
Rule 6 at a glance: the five cases in which the centre stitch is on in the next generation
In the Granny Life artwork, each crafter could choose their own yarn colour for the off stitches, but everyone used the same ivory yarn for the on stitches.
Rule 6 is a rule for introverts: a stitch that is on survives only if all four of its neighbours are off, and a stitch that is off turns on only if just one of its neighbours is on. Above we see the first eight generations of rule 6, starting from a small plus of four stitches at the centre of the square.
The first eight generations of rule 6, starting with a plus
Starting from the centre plus, rule 6 runs through 1407 different motifs before repeating, so Granny Life contributors had plenty of rule 6 motifs to choose from. Within these motifs, the same small shapes turn up again and again. That shared vocabulary is what lets squares from different makers work well together in the artwork.
The Granny Life project
At grannylifecrochet.com you can see individual photos of all 242 squares that were contributed to the artwork. Each crafter chose their own rule 6 motif using an interactive app on the website, and their own yarn colour. They then used our custom two-colour Pixie Square crochet pattern to create their motif.
A sample of rule 6 motifs
The squares that people contributed to Granny Life were many different colours, and not all of them looked very good next to each other. To solve this problem, I sorted the colours loosely into warm and cool, and gave each its own side of the artwork: Day and Night. Then I created another app, at mathgrrl.com/simultaneous-queens, to arrange each side so that similar colours sat as far apart as possible. More precisely, I divided each side of the work into eleven colour groups each made up of 11 squares, and used our app to find an arrangement in which eight of the groups each formed a set of eleven non-attacking queens on an $11\times 11$ chessboard. Think of placing eleven queens on a chessboard so that no two are in the same row, column, or diagonal; eight of our 11-square groups were placed in this type of arrangement. The remaining three groups followed Latin square rules: no two in the same row or column. The interplay of sameness and difference is at the heart of this artwork, and at every stage mathematics set the terms of that interplay.
Some of the completed squares, sorted in piles according to colour
The squares were joined with an extra border row and a bespoke ‘join-as-you-go’ method. It took more than 10 skeins of yarn just to create the joins and borders.
Joining the granny squares while travelling on the Eurostar train to the exhibition in Paris
Final artwork
The finished Granny Life artwork is made up of 242 squares, a separate $11 \times 11$ grid on each side. The work is nearly six feet wide and six feet tall, and weighs about 11kg. Crochet famously sags, so to keep the piece from stretching I sandwiched a single sheet of curtain fabric between the two sides and sewed each side to it along the border of every square. A loop sewn sturdily into the top of the fabric carries the whole weight for hanging.
Day (left) and Night (right)
Granny Life was made by a community of more than 250 people, about a hundred of whom crocheted and submitted squares for the finished artwork. It was first shown at the ‘Creation, between art and mathematics’ exhibition at Maison Poincaré in Paris, and then travelled to the juried exhibition at the Bridges mathematical art conference in Galway, in Ireland. Some of the contributors were able to visit their own squares in Paris!
From a distance, the work is a field of colour, warm on one side and cool on the other, with no two similar colours lined up in any direction. Step closer and the motifs appear, the same small shapes recurring from square to square, made by people who never met. For months, I strictly enforced the rule that every motif had to be different. Then one of the last squares arrived, and it was an unexpected repeat. I could have left it out of the artwork, but I wanted to respect their work and include them both! So it turns out that Granny Life has one pair of motif-twins. Finding them is left as a puzzle for the reader.
Crocheters with Day…
…and Night
More from Chalkdust
In conversation with Alex Bellos
Ashleigh Ratcliffe and Sophie Maclean talk to this year's Zeeman medal winner about writing, football and donkeys
The domestication of prehistoric wolves
Alice Henman teaches us to sit..., stay..., wait...wait...no, come back! Fenton!
Can you turn a beach ball into a rubber ring?
Alba Sendón Blanco mixes scissors and beach inflatables to great effect.
The paradox of the surprise party
Jeremy Budd gets into party planning
Oπnions: How I became a Fields medal killjoy
Michael J Barany doesn’t want one
Naively adding fractions
Wayne Voo asks: what if ½ + ⅔ was ⅗?