Teaching objectives
Why does a leopard have spots and a zebra have stripes? It is not random: it is mathematics. This upper-secondary lab explores Turing's reaction-diffusion model (implemented with the Gray-Scott equations) and shows that two substances diffusing and interacting at different speeds are enough to generate, from scratch, the patterns that cover animal skin. Throughout the rooms students learn to create patterns by tuning the physical reaction parameters, and finish by painting the silhouettes of real animals.
What you'll learn
The intuition that simple local rules produce complex global structures, by controlling continuous rather than discrete parameters.
- Block A (rooms 1-5): 1D diffusion is introduced with a slider, then the three qualitative Gray-Scott regimes without diffusion (equilibrium, oscillation, extinction), and finally reaction + diffusion are combined to watch the pattern emerge. The asymmetry room shows that the speed ratio d = D_U/D_V must exceed a critical threshold (~6) for structure to appear.
- Block B (rooms 6-8): students adjust f and k to generate exactly spots (leopard, giraffe) or stripes (zebra, tiger, bee). The anisotropy room adds angle θ to orient the stripes.
- Recap (room 9): combine all controls on the zebra silhouette to generate stripes on it.
- Block D (rooms 10-12): extensions — phyllotaxis (the golden angle 137.5° of auxin), sea shells with the Gierer-Meinhardt model (time axis = growth), and a growing embryo where the domain expands and the pattern reorganizes, leaving defects (stripe branchings and endings).
Key mathematical ideas
- Gray-Scott reaction-diffusion equations: ∂U/∂t = D_U·∇²U − UV² + f(1−U) and symmetrically for V, with feed parameter f and decay parameter k.
- Turing instability: a homogeneous state that is stable under pure reaction becomes unstable when asymmetric diffusion is added; necessary condition D_V > D_U (the inhibitor diffuses faster than the activator).
- Phase space (f, k): regions of the plane corresponding to spots, stripes, labyrinths, oscillations, and extinction — explored visually in the 2D simulation.
- Golden angle: the 137.5° angle maximizes spiral packing in sunflowers, pine cones, and other plants. Auxin diffusion acts as a lateral inhibitor.
- Gierer-Meinhardt model (shells): growth over time produces a row-by-row fossil record of the pattern, reproducing genera such as Conus, Oliva, and others.
- Topological defects: when the domain grows faster than the pattern can reorganize, stripe branchings and terminations appear — the same defects visible in fingerprints.
Room-by-room contents
Room 1 · Diffusion: milk in coffee
Visual intuition of diffusion: the student clicks on the cup to release drops of milk and watches them disperse on their own. A slider controls the diffusion speed. This is the mathematical foundation of the entire laboratory.
Student tasks
- Click on the cup to release drops and observe how they spread.
- Move the speed slider and compare slow vs. fast diffusion.
Room 2 · Reaction without diffusion: 3 outcomes
A single cell with the Gray-Scott equations and no diffusion. Moving parameters <em>f</em> and <em>k</em> reaches three qualitative regimes: stable equilibrium, permanent oscillation or extinction of V.
Student tasks
- Explore the (f, k) plane with the presets and observe each regime.
- Identify in which zone of the plane equilibrium, oscillation and extinction appear.
Room 3 · Reaction + diffusion = pattern
Side-by-side comparison: on the left, diffusion only (everything mixes uniformly); on the right, reaction plus diffusion. The same seeds produce an emergent pattern only when both processes coexist.
Student tasks
- Observe both simulations in parallel and describe the difference.
- Press reset several times and confirm that the pattern always emerges on the right-hand side.
Room 4 · Asymmetry is key
If U and V diffuse at the same speed no pattern appears. The slider controls the ratio d = D_U/D_V; beyond a critical threshold (≈ 6) structure emerges. Releasing the slider re-seeds the simulation.
Student tasks
- Raise d slowly until the first pattern appears.
- Note (or estimate) the threshold value of d at which structure emerges.
Room 5 · Your first Gray-Scott
Free exploration of the full 2D Gray-Scott simulation. The same parameters f and k from the phase plane produce very different landscapes: spots, stripes, bubbles, labyrinth.
Student tasks
- Move f and k freely and record at least three visually distinct patterns.
- Try to find the zone that produces spots and the zone that produces stripes.
Room 6 · Design: spots
Design challenge: find f and k that generate spots. Once the correct pattern is achieved, it is applied to the silhouette gallery (leopard, jaguar, giraffe) with a tile texture at 0.3×, and can be repainted.
Student tasks
- Adjust f and k until classifyPhase detects spots.
- Observe the same pattern projected onto the leopard, jaguar and giraffe.
Room 7 · Design: stripes
Design challenge: find f and k that generate stripes. The gallery shows the achieved pattern on silhouettes of a zebra, a tiger and a bee. The same zone of the (f, k) plane gives the stripes of several animals.
Student tasks
- Adjust f and k until classifyPhase detects stripes.
- Compare the appearance of stripes on the zebra vs. the tiger vs. the bee.
Room 8 · Anisotropy: orienting the stripes
If diffusion is faster in one direction, stripes align. One slider controls the angle θ and another the intensity of the anisotropy; moving them rotates the stripes visibly.
Student tasks
- Move θ from 0° to 180° and observe the rotation of the stripes.
- Find the angle that produces vertical stripes and the one that produces horizontal stripes.
Room 9 · Capstone: paint your zebra
Final challenge: combine f, k, anisotropy and angle θ to obtain stripes that fit the zebra silhouette (scaled ×1.6). Validation is qualitative: are they stripes? Are they oriented? Do they cover the silhouette well?
Student tasks
- Adjust all parameters until realistic stripes are achieved on the zebra.
- Complete the qualitative validation checklist.
Room 10 · Phyllotaxis and the golden angle
Botanical extension: the diffusion of auxin in plants generates the spiral arrangement of leaves and seeds. The golden angle 137.5° gives the densest packing; the student discovers this by varying the angle.
Student tasks
- Move the arrangement angle and observe how the dot pattern changes.
- Confirm that 137.5° maximises coverage and gives the Fibonacci spiral.
Room 11 · Sea shells (reaction-diffusion in the silhouette)
Gray-Scott reaction-diffusion running <em>inside</em> the shell silhouette: the pattern is seeded on the contour and grows inward. Each area is tinted by when the front reached it (cream at the edge, brown in the centre), like real growth lines, with the Turing texture on top. Three presets (f, k): labyrinth (<em>Conus textile</em>), spotted (<em>Conus marmoreus</em>) and rings.
Student tasks
- Try all three presets and relate each texture to a type of shell.
- Adjust f and k to find other textures (spots, labyrinth, rings).
Room 12 · Embryo that grows
If the domain grows while the pattern is forming, stripes reorganise leaving defects (bifurcations and terminations). Pressing "Grow" bilinearly resamples U and V by a factor of 1.18 without altering f or k, mimicking real embryonic development. This is how fingerprints are formed.
Student tasks
- Let a stable pattern form and then press "Grow" several times.
- Observe the defects (bifurcations and stripe endings) that appear with each growth cycle.
Rooms to project
The most striking ones to show and discuss in class.