Life from a Table of Numbers

One Number

The tank holds a single kind of bead. Between any two beads there is one thing only: a push or a pull along the line joining them, felt within a radius rMax and not at all beyond it.

The shape of the force is fixed. Closer than 0.3 rMax, two beads always push apart, whatever their kind. Further out comes a tent-shaped pull whose height is the number a. A positive a attracts; a negative one repels.

At every step each bead adds up its neighbours’ pushes and pulls, changes its velocity a little by the total, and moves a little. Its speed halves every 0.04 seconds, as if in a thick liquid. Here a = +0.6: the beads gather into round drops and come to rest.

A Table

More kinds, more numbers. With three kinds of bead, each kind has a number for each kind, itself included: nine in all, in three rows and three columns. Row i, column j says how kind i treats kind j.

The knobs on the console are this table. The cup at the head of a row holds the kind that feels; the cup at the head of a column, the kind that is felt. A pointer turned right pulls, turned left pushes; the lamp under it glows warm for a pull and red for a push.

This table is symmetric: red towards teal is the same number as teal towards red. The beads sort themselves into layers, one wrapped round another like liquids that will not mix. Once sorted, they stay still.

The Third Law

Newton’s third law: the forces two bodies exert on each other are equal and opposite. With a symmetric table every pair’s push and pull cancel, the beads’ forces on one another add up to zero, and they cannot change the swarm’s total momentum; only friction can, and friction only drags it towards zero. Such a system has an energy; friction drains it bit by bit until everything stops.

Nothing makes the table symmetric. The bat-handle switch at the top left of the console flips between two tables: the original, and its symmetric part, each pair of numbers averaged. The meter reads what is left when all the beads’ forces on one another are added up, as a share of the sum of their sizes; its scale is logarithmic, each major division ten times the last.

Every few seconds the four kinds in the tank change tables. With the symmetric one they fall still within seconds and the meter reads zero. Back on the original, the pushes no longer cancel; the beads move again, and keep moving.

Chase and Flight

The simplest lopsided table needs only two kinds. Red towards teal is +0.6, teal towards red −0.6: red is drawn to teal, teal is pushed away by red. Red chases, teal flees. A few red beads nestle into a crescent of teal; the teal moves off, the red follow, and the little clump runs away together like a comet with a tail.

Nobody fitted them with motors. They run only because the two beads push each other with different strengths: the total force is not zero, so they speed up until friction holds them back. The total momentum in the readout is no longer zero.

Membrane and Core

Five kinds, a random table, nothing else changed. The tank fills with small units: one kind gathered in the middle, another drawn round it in a thin ring, like a core and a membrane.

They shuffle about and shoulder each other aside; knocked apart, they gather again, while loose beads keep streaming between them. The rule has no cells and no membranes in it; those names are ours. All that can be written down is the five-by-five table.

The Specimen Rack

One rule; change the table and something else grows. Each of the four watch glasses in the rack holds a structure grown from one table, at the same scale as the tank.

The tank brings them out in turn: comets running with a tail, a unit with a membrane, worms drawn out long and writhing, and rings within rings. The names are ours; there is no agreed classification of these things.

Any Table at All

Set a table at random and, more often than not, something moves. The figure below is our own run: a hundred random tables for four kinds, each started from the same scatter of 1,500 beads and run for 20 seconds, with the mean speed taken over the last second. Of the hundred, 98 were still going faster than 2 mm a second.

The same hundred tables, each reduced to its symmetric part and run again: not one went faster than 2 mm a second, and all were winding down. Whatever keeps moving has a lopsided table.

Lopsided in Nature

Between two real particles, action and reaction are always equal. But when the interaction passes through something else, such as a flowing plasma, a flowing liquid, or a chemical that one releases and the other takes up, what each particle feels can differ.

A 2015 study worked out such mixtures and tested them on two-dimensional bilayers of charged microparticles in a plasma: the two kinds settle at different temperatures. In 2020 chemists watched one kind of oil droplet chase another through a surfactant solution while the other fled, driven by a net flow of oil from one droplet to the other.

A 2021 paper brought lopsided interactions into the theory of phase transitions: with them, an ordered state can keep moving for ever, two groups chasing each other round in a circle, for instance. In the tank three kinds chase head to tail, the first the second, the second the third, the third the first. That is not the paper’s model, only the same kind of lopsidedness.

Where It Came From

In the 1987 “boids”, each bird looks only at a few neighbours, matching their heading, closing in and keeping clear, and the flock forms by itself. Particle life is simpler still: no heading, no field of view, only pushes and pulls.

Handing many particles a lopsided table of attractions comes from a program of the 2010s called Clusters, which its author says he first put online around 2016. Many versions followed under the name particle life; the shape of the force and the constants here are taken from one published version.