Small models of nature

How colors mix, how waves interfere, cellular automata, how an illness spreads. They look unrelated, but all four rest on the same idea: copy the most important part of a complicated phenomenon with a few very simple rules. That is what we call a “model.”

All four works on this site are simplified teaching models. Real phenomena contain many things the models leave out.

Color Mixing (the site’s “Color Mixing”)

Light on a screen makes colors by adding red, green and blue. All three together give white. Paints and dyes work the other way: the more cyan, magenta and yellow you mix, the more light is taken away, so the result gets darker. The same word “mixing” means addition for light and subtraction for pigment.

The site’s pigments follow a simple model of ideal absorption only. Real paints change with how transparent the pigment is, and often do not follow this model closely.

The first color photograph

Maxwell proposed the three-color method in an 1855 paper on color vision. In 1861 Thomas Sutton took photographs by that method: three black-and-white pictures made through red, green and blue filters, then projected on top of one another with three projectors. The subject was a striped ribbon.

The photographic plates of the time barely responded to red light, so the result is described as inadequate, and Maxwell himself called it so. Later study suggests the three images may really have been recorded with ultraviolet, blue-green and blue light rather than true red, green and blue.

Wave Interference (the site’s “Wave Interference”)

With two sources on water, the waves overlap: in some places they grow, in others they cancel. When both sources start in step, a difference in path length of a whole number of wavelengths reinforces, and a half-wavelength difference cancels. The site’s calculation is checked by tests for exactly this.

Young and the wave nature of light

In the early 1800s Thomas Young argued that light shows interference just as waves do. He presented a paper to the Royal Society in 1801, and in 1803 he is described as demonstrating interference with sunlight and small holes. A double-slit type of experiment was described in that discussion. Bands of bright and dark are explained well by overlapping waves, less well by a stream of particles, so they became evidence for the wave theory of light. Whether Young actually carried out a double-slit experiment has been questioned, though.

Later it was found that light also behaves like particles, and light came to be seen as having both wave and particle sides. The wave model captures one of them.

The site’s model just adds waves of equal strength that start in step. It leaves out water depth, reflections and fading with distance.

Cellular Automata (the site’s “Cellular Automata”)

Each cell of a grid looks at its neighbours and decides its next state. Repeat that one rule again and again. The idea is said to have been conceived in the 1940s by Ulam and von Neumann.

1D: 256 rules

In one dimension, the left, centre and right cells decide the next centre. There are 8 possible triples, so there are 256 rules in all. A rule number writes the eight answers as a binary number. The site’s program checks all 256 rules.

Stephen Wolfram published his first paper on these one-dimensional rules (Rule 30 in particular) in 1983. Rule 110 is very simple, yet Matthew Cook showed it can in principle compute anything (it is “universal”). The full proof was published in 2004.

2D: a survival game

In two dimensions, a cell is born when exactly three live cells surround it, and a live cell survives with two or three. The mathematician John Conway devised it, and Martin Gardner’s column in the October 1970 issue of a magazine made it widely known. The “glider,” which creeps diagonally, is famous. The site checks that a glider moves one cell diagonally every four generations.

In the site’s model, everything beyond the edge of the board counts as empty.

Epidemic Spread (the site’s “Spread Model”)

Sort people into “susceptible,” “infectious” and “immune,” then count how they move between groups each day. This is called the SIR model. It grew out of a theory published by Kermack and McKendrick in 1927.

The model assumes simple things: everyone is equally likely to meet everyone else, and recovery brings lasting immunity. Even so, it captures the “hill shape” in which the number of infectious people rises and then falls. It is said to fit well only for some diseases.

The site checks that the total number of people stays the same each day, that nothing spreads without contact, and that nothing spreads if everyone starts immune. Each trial uses random numbers, so the same settings give slightly different results in another trial.

The site’s epidemic is a simplified model of a fictional illness, not medical advice. Please do not use it to judge real diseases or precautions.

How to look at a model

• Ask what was left out. A model keeps only the most important part of reality. Thinking about what is missing shows its limits.
• Change just one rule. Move the spacing, the rule number or the contacts per day on its own, and you see what really matters.
• Tell where it fits from where it does not. Every model explains well within some range.