Four rules.
Emergence in Conway's Game of Life
Phil Johnson · Reality after Einstein · 19 October 2026
Open on your phone, we'll use it soon:
philjohnsongh.github.io/life
Act 1 · Four rules
Everything you will see today comes from four rules
Lonely
A live square with 0 or 1 live neighbours dies.
Fine
A live square with 2 or 3 live neighbours stays alive.
Crowded
A live square with 4 or more live neighbours dies.
Birth
An empty square with exactly 3 live neighbours comes alive.
Every square has eight neighbours: sides and corners.
All squares change at the same moment. Count everything first, then update.
Idea 1 · That is the whole universe. There is nothing else.
BY HAND
Your turn: run it by hand
Groups of three. One counts, one writes, one checks.
- Write the neighbour count in every square first, empty ones too.
- Then apply the four rules to get Generation 1.
- Blinker first. Glider if you finish.
Corners count. Nothing changes until every square has been counted.
CHECK
Check your answers
Nobody told it to move. No square moved. The glider did.
ON YOUR PHONE
Open the site
philjohnsongh.github.io/life
1. Tap squares to draw the blinker.
2. Press Step. Was your handout right?
3. Press Play.
Show counts on the Small board shows the neighbour numbers.
Act 2 · Nobody designed the glider
A zoo nobody designed
- Still lifes never change.
- Oscillators repeat.
- Spaceships travel.
- Guns make spaceships, forever.
The rules never mention any of these. People found them, and named them, like animals in a new forest.
ON YOUR PHONE · EVERYONE
Fill the board at random. Raise your hand.
1. Big board. Press Random.
2. Speed 30. Press Play. Hand up.
3. Hand down when it says Settled.
What is left when the boiling stops? Blocks, blinkers, beehives: ash.
So is that all Life can do?
ON YOUR PHONE · 3 MINUTES
Challenge: five squares
1. Clear the board. Draw any five squares.
2. Press Play.
3. Watch the generation counter until it says Settled.
4. Shout out your best number.
Act 2 · Nobody designed the glider
Seven squares. No shortcut.
The acorn: seven squares.
You know every rule. You ran them by hand. And still the only way to find out what seven squares do is to run them.
It is a theorem that some simple yes-or-no questions about Life patterns cannot be answered by any computer, in general.
Idea 2 · The rules don't know what they will do.
Act 2 · Nobody designed the glider
"No shortcut" is a theorem
1931 · Kurt Gödel
Incompleteness
Any system of mathematics rich enough to count has true statements it can never prove. Not "not yet": never.
1936 · Alan Turing
The halting problem
He invented the universal computer on paper, then proved no program can tell, for every program, whether it will ever stop.
1970 · Conway's Life
Undecidable
Life can run any program (Act 3). So "will this pattern ever settle?" has no general answer. Run it, and wait.
Three faces of one limit: some yes-or-no questions have answers that no proof and no computer can reach. Four rules are already enough to hit it.
Act 2 · Nobody designed the glider
A pattern with no past
Life runs forwards only. Many different boards lead to the same next board, so you can never run it backwards.
That means some boards have no yesterday at all. Nothing in the universe can produce them. They can only be a beginning.
Life people call them Gardens of Eden. Proving one exists took a theorem (Moore and Myhill, 1962–63); finding small ones takes a computer search.
Act 2 · Nobody designed the glider
A glider is as real as a traffic jam
No square moves. A square has no idea it is part of a glider. It only lives or dies.
And yet the glider moves. It has a speed. You can count gliders. You can bounce them off each other.
A new level. Real, and never mentioned by the rules.
Cars → the traffic jam
Water molecules → wetness
Squares that live or die → the glider
LIVE
A glider can carry a message
Two guns. The right one is a clock. The left one carries the message: for a 1 its glider goes through, for a 0 we delete it.
Where a message glider meets a clock glider, both die.
What comes out is the opposite of what went in: a NOT gate. Two more kinds of collision give AND, OR and memory.
Act 3 · It goes all the way up
…and from gates, a computer
Nicolas Loizeau, 2016: a programmable 8-bit computer made of glider streams, with a screen. It runs programs. Slowly.
Anything your laptop can compute, this can compute.
At home: Alan Zucconi, Let's build a computer in Conway's Game of Life (23 min).
Act 3 · It goes all the way up
Hands up: what are these tiles made of?
Vote first. Then watch.
Each tile is a Game of Life machine of about four million squares. The tiles together are playing… the Game of Life.
Idea 3 · It goes all the way up.
Video: Phillip Bradbury, "Life in life" (2012), built on Brice Due's OTCA metapixel.
Close
There is no ceiling
1
Four rules. Nothing else.
You ran the whole universe by hand.
2
Nobody designed the glider.
It is still real: a new level the rules never mention. And the rules don't know what they will do.
3
It goes all the way up.
Messages, gates, a computer, Life inside Life. Simple rules put no ceiling on what emerges.
Complex order does not need a complex cause. It needs simple rules, a start, and time.
John Conway, 1937 – 2020
Our universe runs on a few simple rules too.
So which are you: the squares, or the glider?
Keep playing: philjohnsongh.github.io/life · the challenges are on the site.
