Tropism

A living figure. Every body in a is made by the table in b.

aDish: live simulation

bAffinity aij, row toward column

cForce law

dAggregation and speed, last 5 s

Fig. 1 | Life from a table of affinities. a, Live simulation of 5,000 particles of 6 species on a periodic square of side 18.2 rmax, now at t = 0.0 s. Particles closer than β = 0.30 rmax push apart; farther out, up to rmax, each is drawn toward or pushed from the other by its affinity in b. Velocities decay with a half-life of 40 ms and forces are scaled by k = 10. b, Affinity of species i (row) toward species j (column). Filled discs attract, open rings repel, and area is proportional to |a|. Drag a disc up or right to strengthen attraction, down or left toward repulsion. c, Force on a particle of species i from one of species j at distance r, for the pair under the pointer; at rest, for the most lopsided pair in b. d, Aggregation (neighbours of the same species within rmax, relative to a well-mixed gas; now 1.0×) and mean speed of each species. Numbers with a dotted underline can be dragged.

World code t1

Revision 0, as generated.

Extended Data

Authored worlds, ordered from two species to eight. Each panel is a snapshot taken after the world settled from a uniform gas. Open one in Fig. 1 to watch it assemble from the start.

    Methods

    Model

    The dish holds N particles of S species on a unit square whose edges wrap, so a particle leaving the right side re-enters on the left. Every pair closer than the interaction radius rmax exerts a force along the line between them. The force on a particle of species i from a particle of species j at distance r depends only on that distance and on one number, the affinity aij:

    F(r) = r / β − 1r < β aij (1 − |2r − 1 − β| / (1 − β))β ≤ r < 1 0r ≥ 1

    Here r is measured in units of rmax. The first branch is a short-range repulsion shared by every pair, so particles never sit on top of one another. The second is a tent that peaks at aij halfway between β and rmax: positive affinities pull i toward j, negative ones push it away. Panel c draws this curve for the pair under the pointer, and at rest for the most lopsided pair in the table.

    The table is not symmetric. aij says how i responds to j, and aji how j responds to i; nothing forces the two to agree. That asymmetry breaks Newton's third law on purpose, which is why the dish never settles into stillness: a species drawn toward another that flees it will chase it forever.

    Integration

    Each step advances time by Δt = 0.004 s, and by default the page plays the dish at a tenth of real time, so a second on screen is 0.1 s in the dish. Forces are summed over neighbours, scaled by k rmax, and added to the velocity, which first decays by the factor 0.5Δt / t½ where t½ is the velocity half-life. All velocities are updated from the same snapshot of positions before any particle moves, so the result does not depend on the order particles are visited. Neighbours are found with a uniform grid of cells at least rmax wide, so each step costs time proportional to N rather than N2.

    Reading the table

    A few tendencies make the table readable, though larger tables combine them in ways that are hard to predict:

    • A positive diagonal entry aii makes a species clump; a negative one spreads it into a gas.
    • When aij is positive and aji negative, i chases j and j flees. Pairs like this make bodies that travel.
    • When both are positive, the two species share bodies; when both are negative, they hold each other apart and draw boundaries.
    • A cycle, where i chases j, j chases k, and k chases i, tends to produce chains and rotating rings.

    Statistics

    Panel d plots two quantities per species, sampled ten times a second. Aggregation is the mean number of same-species neighbours within rmax, divided by the number expected if that species were spread uniformly, (Ns − 1) π rmax2. A value of 1 means well mixed; 10 means each particle has ten times as many kin nearby as chance would give it. Speed is the mean particle speed in rmax per second.

    Working in the dish

    Drag in panel a to stir, to seed new particles of a chosen species, to erase, or to pan. Click a particle to track it; its recent path is drawn behind it. Scroll or pinch to zoom, double-click to reset the view. Every change to the table or the parameters is a revision you can undo. Table 2 lists the keyboard shortcuts.

    Reproducibility

    A world code records the table, the number of species and particles, every parameter, and the random seed for the starting positions. Loading a code rebuilds the same starting gas, and the simulation runs identically in the same browser engine until someone touches the dish. Codes use only letters, digits, dots, dashes and underscores, so they survive being pasted into a chat or a link.

    Sound

    Sonify gives each species one voice on a pentatonic scale, lowest for A. A voice grows louder as its species clusters and brighter as it speeds up, so a dish that is condensing swells and a dish that is chasing shimmers. Sound is off until you turn it on.

    Credits

    The model belongs to the family of particle-life simulations; its force law follows the one popularised by Tom Mohr's Particle Life, which descends from Jeffrey Ventrella's Clusters. The species colours follow Okabe and Ito's palette for colour-blind readers, paired with plot markers so no species depends on colour alone. Type is set in Archivo and STIX Two Text. The worlds in Extended Data were chosen by a seeded search and named by hand; they are illustrations, not observations of real organisms.