A few years ago I came across this video, showing a complex machine built entirely in Conway’s Game of Life:

The purpose of the machine is to emulate a single Life pixel. With a big enough matrix of these “metapixels”, you can simulate a meta-version of Life on a massive scale. From there you could create a meta-metapixel out of metapixels and so on….

At the time I was reading the book The Recursive Universe by William Poundstone, which gives a detailed breakdown of John Conway’s 1982 proof of self-replicating objects in Life. Since its inception in 1970, Conway’s Game of Life has developed a cult following of researchers, engineers, and hobbyists, pushing each other to construct increasingly elaborate “machines” from pixels on a screen. These investigations have resulted in a lengthy taxonomy of motifs, reactions, and mechanisms, as well as engineering principals and design abstractions. These days you can spend a long time on YouTube exploring the seemingly impossible things people are designing in the “simple” Life universe.

This post is (mostly) some notes I took back in 2015 while trying to understand how this metapixel was designed. Unfortunately, John Conway passed away recently, which got me thinking about Life again (not to sound too sappy). Now that I find myself with a lot of time on my hands, I figured I’d clean up these notes and finally kick this blog off (inaugural post!).

Here we go….

Game of Life

Conway’s Game of Life is a 2D cellular automaton - a simulated world set on a grid of pixels (cells). Though the rules that govern Life are very simple, the results can become quite complex. In the Game of Life, a cell’s behavior is dictated by its current state (alive or dead) and the state of its eight nearest neighbors. The rules of Life are loosely based on population dynamics (copied from wikipedia below):

• Any live cell with fewer than two live neighbors dies, as if caused by under-population.
• Any live cell with two or three live neighbors lives on to the next generation.
• Any live cell with more than three live neighbors dies, as if by overcrowding.
• Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

One of the most interesting things about Life is the variety of patterns that can be constructed within it. Some patterns are readily observed in the wild – spontaneously emerging from a random soup of living and dead cells. Other patterns were meticulously designed by people, often built from smaller subunits with well-characterized behavior. In general, patterns in Life can be broken into a few broad categories, described below. (image source wikipedia)

Stable configurations, called “still lifes”, do not change over time:

block

loaf

boat

beehive

“Oscillators” are dynamic patterns that repeat themselves after a certain number of time steps:

blinker (period 2)