Turing Bench

t 0 Growing dt ok Clean run

Brush 6 px Speed 0 steps/s Grid 128² Steps 0

Cross-section

ActivatorInhibitorrow 64

Hover the field with Probe to move the cut. Click to pin it. Each curve is scaled to its own range.

01ModelWhich chemistry runs on the substrate▶

Reaction Preset

02KineticsLocal production and decay▶

Changes apply live. Theory in section 05 recomputes as you drag.

03TransportDiffusion ranges and the integrator▶

Activator diffusion D_u

Inhibitor diffusion D_v

Range ratio d = D_v/D_u

–

Inhibitor reach over activator reach

Stability limit dt_max

–

Explicit Euler, this stencil

Time step dt

Steps per frame

Anisotropy D_x/D_y1.00

Boundary Grid

Push dt past the limit to watch the integrator fail. Anisotropy above 1 lets morphogens spread faster along x, which aligns stripes.

04Initial stateSeed, noise and spatial gradients▶

Seed pattern Seed number

Initial noise amplitude

Live noise (breaks parity)

Parameter gradient across x on

Gradient strength (± of base value)

Symmetric seeds are mirrored exactly, so parity holds until live noise or a brush stroke breaks it. Live noise is drawn from the seeded generator, so even noisy runs are reproducible.

05Linear theoryWhat the equations predict before anything grows▶

Uniform steady state analyzed

Jacobian J at that state

State values

Dispersion relation

growth σ(k)measured power

Predicted wavelength

–

2π / k* of fastest mode

Measured wavelength

–

Peak of field's power spectrum

Fastest growth σ_max

–

e-fold time

Prediction / measurement

–

1.00 = theory nailed it

Rounding floor δ* = ulp / (2σ_max·dt)

–

In float32, a smooth disturbance smaller than this changes each cell by less than half a ULP per step, so rounding erases it and the field stays flat even though theory says it should grow. Smaller dt raises the floor.

Turing map

pattern-forming

Shaded where the uniform state is unstable to some wavelength. The dot is your current setting. Click the map to jump there.

06MorphometricsIs the pattern even? Did it stop?▶

Gap disparity

–

Spread of empty-space radii between structures (CV). 0 means every gap is the same size.

Territory disparity

–

CV of each piece's share of substrate

Clark–Evans R

–

1 random · 2.15 perfect hexagon

Pieces

–

Connected structures

Tips · forks

–

Skeleton endpoints · junctions

Box-count dimension

–

Of the skeleton. 1 line · 2 fills plane

Coverage

–

Share of substrate claimed

Parity error

–

Mirror asymmetry about x = center

Change rate

–

Field motion per unit time, normalized

Gap disparity

Coverage

Parity error (log₁₀)

Change rate (log₁₀)

Measure continuously

07Protocols & paritySet-piece experiments and reproducible run records▶

Checkpoint hashes

every 1000 steps

Step	t	This run	Compared

A clean run (no brush strokes, no slider moves after seeding) is fully determined by its record. Paste a record from another browser, machine or the Node runner to check bit-for-bit parity.

Compare against a run record