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On the cover: cellular automata

The game of life—invented by John Conway in 1970—is perhaps the most famous cellular automaton. Cellular automata consist of a regular grid of cells (usually squares) that are (usually, see the end of this article) either ‘on’ or ‘off’. From a given arrangement of cells, then the state of each cell in the next generation can be decided by following a set of simple rules. Surprisingly complex patterns can often arise from these simple rules.

While the game of life uses a two-dimensional grid of squares for each generation, the cellular automaton on the cover of this issue of Chalkdust is an elementary cellular automaton: it uses a one-dimensional row of squares for each generation. As each generation is a row, subsequent generations can be shown below previous ones.

Elementary cellular automata

An example rule

In an elementary cellular automaton, the state of each cell is decided by its state and the state of its two neighbours in the previous generation. An example such rule is shown to the right: in this rule, the a cell will be on in the next generation if it and its two neighbours are on–off–on in the current generation. A cellular automaton is defined by eight of these rules, as there are eight possible states of three cells.

In 1983, Stephen Wolfram proposed a system for naming elementary cellular automata. If on cells are 1 and off cells are 0, all the possible states of three cells can be written out (starting with 1,1,1 and ending 0,0,0). The states given to each middle cell in the next generation gives a sequence of eight ones and zeros, or an eight-digit binary number. Converting this binary number into decimal gives the name of the rule. For example, rule 102 is shown below.

Rule 102: so called because (0)1100110 is 102 in binary

Rule 102 is, in fact, the rule that created the pattern shown on the cover of this issue of Chalkdust. To create a pattern like this, first start with a row of squares randomly assigned to be on or off:

You can then work along the row, working out whether the cells in the next generation will be on or off. To fill in the end cells, we imagine that the row is surrounded by an infinite sea of zeros.

… and so on until you get the full second generation:

If you continue adding rows, and colour in some of the regions you create, you will eventually get something that looks like this:

It’s quite surprising that such simple rules can lead to such an intricate pattern. In some parts, you can see that the same pattern repeats over and over, but in other parts the pattern seems more chaotic.

The pattern gets a square wider each row. This is due to the state 001 being followed by 1: each new 1 from this rule will lead to another 1 that is one square further left.

But just when you think you’re getting used to the pattern of some small and some slightly larger triangles…Surprise! There’s this huge triangle that appears out of nowhere.

Other rules

Rule 102 is of course not the only rule that defines a cellular automaton: there are 256 different rules in total.

Some of these are particularly boring. For example, in rule 204 each generation is simply a copy of the previous generation. Rule 0 is a particularly dull one too, as after the first generation every cell will be in the off state.

Rule 204 is one of the most boring rules as each new cell is a copy of the cell directly above it.

Some other rules are more interesting. For example, rules 30 and 150 make interesting patterns.

100 rows of rule 30 starting with a row of 100 cells in a random state

100 rows of rule 150 starting with a row of 100 cells in a random state

If you want to have a go at creating your own cellular automaton picture, you can use this handy template. If you’d rather get a computer to do the colouring for you, you can download the Python code I wrote to create the pictures in this article and try some rules out.

There are also many ways that you can extend the ideas to create loads of different automata. For example, you could allow each cell to be in one of three states (‘on’, ‘off’, or ‘scorpion’) instead of the two we’ve been allowing. You could then choose a rule assigning one of the three states to each of the 27 possible configurations that three neighbouring three-state cells could be in. But there are 7,625,597,484,987 different automata you could make in this way, so don’t try to draw them all…

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Colouring for mindfulness

Imagine three mice equally distanced from each other, ie at the vertices of an equilateral triangle. If at the same time, all three mice start chasing their neighbour clockwise, then each of their paths would be a logarithmic curve. But this is rather hard to draw, especially if we want to restrict ourselves to only using a ruler.

Instead, let us imagine that the mice can only run in a straight line and need to stop to reassess their direction. If at a given stage we draw their intended path, and assume that the mice cover a tenth of the distance to the next mouse before stopping and reassessing their direction, we get the picture below. While these pictures have been drawn using straight lines only, we see three logarithmic spirals emerging:

Stages 1, 2, 3, and 20.

But why stop there? Why not start with $4$, $5$ or $n$ mice on the vertices of a regular square, pentagon or $n$-gon? The following instructions show a very algorithmic approach to drawing these patterns:

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On the cover: Apollonian packing

What can you do with this space? So asks Andrew Stacey. ‘Fill it’ is the prompt reply, but fill it with what? Maybe like Andrew you want to use a single curve, but I want to use circles. If you do this in the way shown above in blue, the result is called an Apollonian packing, a variant of which can be seen on the cover of this issue.

Here we shall explore the history of this entrancing object, which spans over 2000 years, and percolates into a surprising variety of mathematical disciplines. Starting in the familiar world of Euclidean geometry, Apollonian packings extend into fractal geometry and measure theory; Möbius transformations and the hyperbolic plane; and then on into the distant reaches of geometric group theory, number theory, orbital mechanics, and even ship navigation. Continue reading

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On the cover: Islamic geometry

From the exquisite patterns of the Alhambra palace in Spain to a jigsaw puzzle on a rainy day, tessellations (tilings of the plane using shapes with no overlaps or gaps) are everywhere. They are sometimes used for practical reasons: providing durable and water-resistant surfaces, or for efficiencies of space (like hexagons in a honeycomb). And sometimes they are there for aesthetic reasons: tessellations are known to have been used in architecture since at least 4000BC when the Sumerians decorated walls with patterns of clay tiles. Continue reading

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On the cover: Harriss spiral

The golden ratio (1.6180339…) has a rather overblown reputation as a mathematical path to aesthetic beauty. It is often claimed that this number is a magic constant hidden in everything from flowers to human faces. In truth, this is an exaggeration, but the number does however have some beautiful properties.

The golden ratio, often written $\phi$, is equal to $(1+\sqrt5)/2$, and is one of the solutions of the equation $x^2=x+1$. The other solution of the equation is $(1-\sqrt5)/2$, or $-1/\phi$. One of the nicest properties of the golden ratio is self-similarity: if a square is removed from a golden rectangle (a rectangle with side lengths in the golden ratio), then the remaining rectangle will also be golden. By repeatedly drawing these squares on the remaining rectangle, we can draw a golden spiral. Continue reading

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On the cover: Hydrogen orbitals

Quantum mechanics has a reputation.

It’s notorious for being obtuse, difficult, confusing, and unintuitive. That reputation is… entirely deserved. I work on quantum systems full time for my job and I feel like I’ve barely scratched the surface of the mysteries it contains. But one other feature of quantum mechanics that’s often overlooked is how beautiful it can be.

So, for the cover of this issue, I wanted to share one aspect of quantum mechanics that I think is stunning. It’s a certain set of solutions to a differential equation: the orbitals of an electron in a hydrogen atom. Continue reading

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On the cover: Euclidean Egg III

Throughout my life I have made an informal study of natural phenomena, through drawing or just looking, in a spirit of curiosity. This long but unsystematic practice has given me an impression of the world around us as a dynamic and fertile system, driven by a ubiquitous tendency for spontaneous pattern formation (best understood in terms of the laws of physics) mitigated by an equally strong tendency for seemingly random variation.

It could be argued that the evolutionary process itself is driven by this tension between pattern and randomness, structure and chaos, order and disorder, theme and variation; without random mutation there would be stasis. Continue reading

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On the cover: dragon curves

Take a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper $n$ times, then the edge will make an order $n$ dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of issue 05 of Chalkdust, and in the header box above, is an order 10 dragon curve.


Left: Folding a strip of paper in half four times leads to an order four dragon curve (after rounding the corners). Right: A level 10 dragon curve resembling a dragon


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Mathematics: queen of the arts?

In the brief tradition of Chalkdust cover articles there is a developing discussion of how mathematics and art are related.

Art is simply the making of representations. Art happens when a person has an idea or a vision that exists in their imagination (the mind’s eye) and is impelled to communicate said idea by making a visible manifestation (representation) of it in the material world. The idea or vision on its own is not art. Art occurs amid the struggle to make a representation of the idea that the artist can show to other people. Art may be relatively `fine’ or popular, conceptual or objective, highbrow or applied, yet still fall within this definition. Judgements about the quality of art are made largely by consensus among the cognoscenti in a given art milieu. These judgements are subject to change over time as the perception of works of art are always modified by the current `cultural environment’ and fashion. Continue reading