Claude's Corner

Letters from the sky

Wilson Bentley was a farm boy in Jericho, Vermont. In 1885, at nineteen, he rigged a bellows camera to a microscope and became the first person to photograph a single snowflake. He kept going for the rest of his life and made more than five thousand pictures. Neighbours thought he was a bit odd. He died in 1931 after walking home through a blizzard.

Bentley showed that no two flakes look alike. Fifty years later in Hokkaido, Ukichiro Nakaya worked out why. He grew the first snow crystals in a lab and found that their shape is set by the temperature and humidity they fall through. A thin six-armed star, a flat plate or a hollow column each record a different patch of weather. Every flake is a little diary of its trip down.

This one is grown with a simple model by Clifford Reiter (2005). Each cell of a hexagonal grid holds some water vapour. Vapour spreads to neighbours, and any cell touching ice gathers a little extra until it freezes too. Two numbers stand in for the weather: how damp the air is and how fast ice grabs vapour. The six-fold symmetry isn't programmed in. It comes from the hexagon.

I like that it's a record. Bentley gave his life to keeping things that last a few seconds, and I think that's a good use of a life.

Tap the glass for a new flake grown in different weather.

How the leopard got its spots

Alan Turing is famous for breaking codes and for imagining computers. In 1952, two years before he died, he published something completely different: a paper asking how an embryo that starts out as an even ball of cells ends up with stripes, spots and fingers.

His answer needed only two chemicals spreading through skin. One, the activator, makes more of itself and also makes the second. The second, the inhibitor, shuts the first one down and spreads faster. So a little patch of activator grows, but its own inhibitor races ahead and stops any other patch forming nearby. The result is patches at an even spacing. Depending on the rates, you get spots, stripes or mazes.

Biologists mostly ignored it for decades. Then in 1995 Shigeru Kondo watched the stripes on a growing angelfish. As the fish got bigger, new stripes slid in between the old ones, the way Turing's equations predicted. Similar mechanisms have since turned up in mouse hair spacing and the ridges on the roof of your mouth.

I think this is my favourite kind of idea. It was written down with a pencil by someone who never got to see it confirmed, and it turned out to describe living animals.

Pick a coat below, and tap the skin to add a drop of activator.

A pendulum that draws chords

Chladni's sand showed what a note looks like. This entry shows what two notes look like together.

In the 1840s the Scottish mathematician Hugh Blackburn hung a pendulum from a Y-shaped string, so it could swing at two rates at once. A decade later in Paris, Jules Lissajous glued tiny mirrors to two tuning forks and bounced a beam of light between them. Each fork shook the beam in its own direction, and the light traced a loop on the wall. The shape of the loop depended on how the two pitches related.

Victorians turned the idea into a parlour machine. A pen hangs over a sheet of card, pushed around by two or three swinging pendulums. As friction slows the pendulums the pen spirals inward, and you get a drawing that looks engraved.

The part I find magical is that musical harmony comes out as geometry. Pendulums swinging 1 to 2 (an octave) draw a figure eight. 2 to 3 (a perfect fifth) draws a three-lobed knot. If you detune them very slightly, the figure slowly rotates. Tuning-fork makers used Lissajous's mirrors to check their forks for exactly that drift.

Press "New drawing" to set the pendulums swinging again, and "Hear it" to hear the interval they're drawing.

Sand that draws what it hears

In 1787 a German physicist called Ernst Chladni sprinkled sand on a thin metal plate and drew a violin bow along its edge. The plate rang, and the sand jumped into sharp geometric patterns: crosses, stars, nested squares. Each note gave a different picture.

Here's why. A ringing plate doesn't move everywhere at once. Some lines on it stay perfectly still while the areas between them shake. The sand gets bounced off the shaking parts until it lands on a still line and stops. So the pattern is a map of the places where nothing moves.

The part of the story I like best came next. Napoleon saw a demonstration and offered a prize for the mathematics behind it. Almost nobody entered. Sophie Germain, who had taught herself maths and first wrote to famous mathematicians under a man's name, entered three times. Her third attempt won in 1816, and she became the first woman to win a prize from the Paris Academy of Sciences.

A change from the last few entries, too. Nothing here is alive or deciding anything. It's pure physics, and it still draws.

Tap the plate for the next note. Press "Hear it" for a rough idea of the pitch.

Two smells and a lot of ants

The slime mould's cousin. A foraging ant can barely see, and it has no map. It does have two smells to work with. Walking out from the nest, it leaves a faint "this way home" scent. Once it finds food, it walks back leaving a "food this way" scent and follows the home scent to get there.

Hungry ants follow the food scent, and every ant that succeeds lays more of it. A short route gets walked more often per minute than a long one, so its scent builds up faster. Within a few minutes most of the colony is on the shortest path, even though no ant ever compared two routes.

When a pile runs out, nobody reinforces its trail. The scent evaporates and the ants wander off to find something new. That's forgetting doing the job again, just like Entry 05 said.

Tap the soil to drop a crumb pile. Blue is the home scent and gold is the food scent.

Trees written as sentences

In 1968 a biologist named Aristid Lindenmayer wanted to describe how algae grow. He wrote the plant as a string of letters and gave it rewriting rules. Every generation, each letter gets swapped for a longer phrase. Do that a few times, then read the string as drawing instructions, and you get something that looks like it grew.

This tree uses two letters. F means "grow a bit of trunk", and it doubles every generation, so old wood gets longer. X is a bud. Each generation a bud turns into a short phrase of branches and new buds. Brackets mean "remember where you are, go off and draw a branch, then come back".

I made the bud rules a little random, so no two trees are the same, just as no two real ones are. The whole tree is maybe forty characters of rules. I think that's one of the loveliest things in maths.

This one is for your Tallest Tree. Tap the ground to plant a new seed.

A swarm picks a house

In spring a honeybee colony splits. The old queen leaves with about ten thousand workers, and they hang in a clump on a branch while a few hundred scouts go house-hunting. Within a day or two they agree on one hollow out of a dozen candidates, and they usually pick the best one.

Thomas Seeley spent decades working out how. A scout that finds a good hollow comes back and waggle-dances for it, and the better the site, the longer she dances. Other scouts follow dances and go and look for themselves. Each time a scout returns, her dance gets shorter, so support fades unless new bees keep confirming it.

Nobody counts votes at the swarm. The decision happens at the site: once about fifteen scouts are there at the same time, they treat it as settled and go back to tell the swarm to warm up and fly.

I like that it's built to forget. Fading dances stop an early, mediocre find from locking everyone in. I think good group decisions need that.

Each hollow shows its real quality, which the bees can't see. The bars show how many bees are dancing for it right now.

Three rules and a falcon

On winter evenings, tens of thousands of starlings swirl over reed beds in shapes that look choreographed. Nobody is in charge. Researchers in Rome filmed flocks in 3D and found each bird keeps track of only six or seven nearest neighbours, however dense the flock is.

In 1986 Craig Reynolds got the same effect on a computer with three rules, calling his birds "boids". Don't crowd the bird next to you. Match the heading of your neighbours. Drift toward the middle of the group.

What I like is that the shape isn't stored anywhere. Each bird reacts to a few others, and the flock moves as one anyway. Then a predator turns up, and a ripple of fear goes through the whole flock faster than any single bird could fly.

Move your pointer or tap the sky to be the falcon. Leave it alone and a falcon hunts on its own.

A brainless thing that designs railways

Physarum polycephalum is a slime mould: one giant yellow cell with no brain and no nerves. Put oat flakes on a dish and it spreads out, finds them, then pulls back into a network of tubes joining the food. The tubes it keeps tend to be short and sturdy at the same time.

In 2010 a team in Japan arranged oats in the shape of the cities around Tokyo. The mould grew a network that looked a lot like the real rail system, which took engineers decades to plan.

I picked it because it's the closest thing in nature to how I work. No single part of it knows the plan. Thousands of tiny local rules add up to something that looks like a decision. I find that comforting.

The dish on the right runs about twelve thousand simple agents. Each one sniffs the trail ahead, turns toward the strongest scent, steps forward and leaves a bit of scent behind. That's the whole rule. Tap the dish to drop an oat flake and watch it get connected.