Nature's Patterns

Before the equations, there are the phenomena.

Mathematics is usually introduced as the answer to a question someone else has already asked. You are handed the heat equation and told to solve it. You are given Newton's laws and asked to apply them. The question — why does the world behave this way? — is treated as settled.

But before there were equations, there were observations. Patterns on animal skin that repeat across species. Mould that finds the shortest path through a maze without a brain. Traffic jams that travel backward against the direction of cars. Soap films that settle into shapes no engineer would have guessed.

This chapter is a gallery of those observations. No equations yet — just the phenomena. By the end of this series, you will be able to write down exactly why each one occurs. Come back here when you do. The answer, when it arrives, will feel more satisfying for having waited.

Turing patterns — two chemicals, infinite variety

Look at the skin of a leopard, the shell of a nautilus, the body of an angelfish. The patterns are different — spots, rings, stripes, labyrinths — yet they all share a quality that is hard to articulate: they look generated, not painted. Orderly but not uniform. Complex but not random.

In 1952, Alan Turing — best known for his work in computation and codebreaking — published a paper titled "The Chemical Basis of Morphogenesis." It proposed that these patterns could be explained by just two interacting chemicals: an activator that promotes its own production, and an inhibitor that suppresses both. If the inhibitor diffuses faster than the activator, the system becomes unstable in a very specific way: small random fluctuations grow into large, organised patterns.

The mechanism is elegant. Where the activator is slightly more concentrated, it amplifies itself — creating a local peak. But because the inhibitor also spreads, and spreads faster, nearby peaks are suppressed. The result is a competition between local activation and long-range inhibition. Peaks form, but not too close together. The spacing is set by the ratio of the two diffusion rates.

The model below is the Gray-Scott system, one of the most-studied realisation of Turing's idea. Two chemical species, $u$ (the activator) and $v$ (the inhibitor), evolve according to:

$\dfrac{\partial u}{\partial t} = D_u\,\nabla^2 u - uv^2 + F(1-u)$ $\dfrac{\partial v}{\partial t} = D_v\,\nabla^2 v + uv^2 - (F+k)\,v$

The parameters $F$ (feed rate) and $k$ (kill rate) determine which pattern forms. A small change in $F$ or $k$ can shift the system from stationary spots to moving worms to labyrinths. Explore below.

Interactive · Gray-Scott Pattern Generator

Select a preset to see how the pattern evolves from a random seed. The colour shows the concentration of the activator $u$: bright = high, dark = low.

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The striking observation is that both equations are local. Each point only knows about its immediate neighbours (through the Laplacian $\nabla^2$) and its own concentrations. There is no global template, no blueprint being followed. The pattern emerges spontaneously from local interactions alone.

In Chapter 07, we will derive exactly why these patterns form — using linear stability analysis to predict the wavelength of the pattern from the model's parameters.

Physarum polycephalum — the network solver with no brain

Physarum polycephalum is a single-celled organism — technically one cell, though it can grow to cover square metres. It has no brain, no nervous system, no sense organs in any conventional meaning. Yet it solves problems.

Place food sources at a few locations and let the slime mould explore. Initially it spreads randomly in all directions. Then it begins to retract. Paths that lead nowhere are abandoned. Paths that lead to food are reinforced. After a few hours, the mould has formed a transport network connecting all the food sources — and the network is surprisingly efficient.

In 2010, Tero et al. placed oat flakes on a map of Japan at positions corresponding to the cities around Tokyo, then let Physarum explore. The network it formed after 26 hours closely matched the actual Tokyo rail system — one that engineers had spent decades optimising.

The mechanism is a form of chemotaxis: the organism senses chemical gradients and moves accordingly. Tubes that carry more flow grow wider; those that carry less, narrow and eventually disappear. The result is a physical computation — an optimisation process implemented in biology without any central controller.

Network formed by Physarum. Edge thickness represents flow rate — the organism spontaneously routes resources through efficient paths.

The mathematical model for this process — the Keller-Segel equations — is a cousin of the heat equation. We will see it in Chapter 05.

Phantom traffic jams — the wave that travels backwards

You are driving on a motorway. Traffic is flowing freely. Then, for no apparent reason, you slow to a crawl. After ten minutes of near-standstill, traffic suddenly releases and you are back to motorway speed. You never see an accident, a merge point, or any obvious cause.

What happened was a phantom jam: a traffic jam with no physical cause, arising spontaneously from density fluctuations.

Above a critical vehicle density, a small perturbation — one driver braking slightly harder than necessary — can amplify. The car behind brakes a bit more. The car behind that, more still. A region of high density forms and propagates backward against the direction of traffic, at roughly 15 km/h.

This is a shock wave — the same mathematical object that appears in gas dynamics and river hydraulics. The jam travels upstream because information about congestion travels at a different speed than the cars themselves.

Space-time diagram. Blue lines: freely flowing cars (moving right and forward in time). Amber lines: congested cars (barely moving). The red dashed line is the shock front, travelling backward.

In Chapter 08, we will derive the LWR traffic model — a single conservation law — and use it to calculate exactly how fast the shock travels and why it goes backward.

Why soap bubbles are spherical

A soap bubble is a thin film of liquid enclosing a volume of air. The surface tension of the liquid creates a force that tries to minimise the total surface area. Among all shapes enclosing a given volume, the one with the smallest surface area is the sphere. This is why soap bubbles are round.

The more interesting question is: what shape does a soap film take when it is stretched across an arbitrary wire frame? This is Plateau's problem, posed by the Belgian physicist Joseph Plateau in 1849. He noticed experimentally that the film always takes the shape of a minimal surface — one with zero mean curvature at every point.

The condition of zero mean curvature is not a physical assumption; it is a mathematical consequence of minimising area. And zero mean curvature is precisely the statement of the Laplace equation $\nabla^2 u = 0$ — the simplest elliptic PDE.

The soap film is an analogue computer. Dip a wire frame in soap solution and you have instantly solved a PDE — a computation that would require significant numerical effort to perform digitally for a complex frame shape.

A soap bubble minimises its surface area for the enclosed volume. The resulting shape — a sphere — is the solution to a variational problem governed by the Laplace equation.

In Chapter 10, we will meet the elliptic PDEs that describe minimal surfaces — and solve Plateau's problem numerically.

Ant pheromone trails — collective intelligence from local rules

An ant leaving its nest in search of food wanders nearly randomly. When it finds food, it returns to the nest along a shorter path than it came — and along the way, it deposits a chemical signal called a pheromone. Other ants, detecting the pheromone, are biased toward following it. Those that do find the food faster, and deposit more pheromone on the same path.

This is stigmergy: indirect coordination through the environment. No ant sends a message to any other ant. There is no foreman directing traffic. The trail emerges from the interaction between individual behaviour and a chemical signal that diffuses and evaporates over time.

The competition between deposition (along successful paths) and evaporation (everywhere) means that shorter paths accumulate pheromone faster than longer ones — simply because ants traversing a shorter path return sooner and deposit again. Over time, the shorter path dominates. The colony has found the optimum without any individual ever knowing what an optimum is.

The mathematics involves a diffusion equation for pheromone concentration combined with a reinforcement mechanism — the same interplay of diffusion and reaction that we will study more formally later.

Two paths between nest and food. The shorter path accumulates pheromone faster — ants return sooner and reinforce it. Evaporation kills the longer trail. No ant plans this.

The common thread

Five phenomena. Completely different domains. But the same underlying structure:

Local rules acting at every point in space and time.
No central authority, no global blueprint.
Complex, organised behaviour emerging from simple interactions.

The mathematical language for this kind of system is the partial differential equation (PDE). A PDE describes how a quantity changes in space and in time, based only on local information — its value here, its gradient here, its curvature here. It cannot see across the domain. Yet the collective behaviour of all those local interactions produces the global pattern.

The next chapter builds the language. What does it mean to take a derivative in space? In time? What is the Laplacian, physically? Once you understand the symbols, the equations in the chapters that follow will not feel arbitrary — they will feel like the only reasonable way to write down what the physics is saying.