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What is This?

HexaGenesis is an implementation of Conway's Game of Life on a hexagonal grid. Conway's Game of Life is a cellular automaton devised by mathematician John Conway in 1970. It's a zero-player game, meaning its evolution is determined by its initial state, requiring no further input.

In the original Game of Life, cells are arranged on a square grid where each cell has eight neighbors, following the B3/S23 rule: cells are Born with 3 neighbors and Survive with 2 or 3 neighbors. HexaGenesis adapts this classic simulation to a hexagonal grid, where each cell has six neighbors instead of eight, creating unique emergent patterns and behaviors while maintaining the core principles of the original.

Why Conway's Game of Life?

Conway's Game of Life demonstrates the phenomenon of great complexity arising from simplicity - a set of basic rules that define the life and death of a single cell can produce an astonishing variety of behaviors: stable patterns that remain unchanged, oscillators that repeat in cycles, spaceships that travel across the grid, and patterns that grow indefinitely.

The game has captured the imagination of mathematicians, computer scientists, and hobbyists for over 50 years. It showcases concepts central to complexity theory, emergence, and computational universality—some patterns can even function as computers, proving that even simple rules can give rise to computation. The endless variety of patterns, each following the same fundamental rules, makes it both an educational tool and an artistic medium.

HexaGenesis is an adaptation of the Game of Life to a hexagonal grid. Instead of eight neighbors, each cell has only six, six which are laid out in a different geometric pattern than a standard grid. This introduces notable differences in the way patterns emerge, converge, terminate, and cycle. Additionally, this app lets you pick and choose the rules for life and death, allowing you to discover patterns of increasing chaos or stability.

Game Rules

The simulation follows modified Conway's rules adapted for hexagonal grids. The default rules are:

• Cell Death: A cell dies if it has ... neighbors (isolation or overcrowding)
• Cell Birth: An empty cell becomes alive if it has exactly ... neighbors

You can customize these rules using the Advanced Settings to discover more patterns that arise when the rules of life and death are altered.

Getting Started

Head over to the App to start experimenting. Click cells to toggle them on/off, then hit Play to watch the simulation unfold.

Spaceships and Ever-Expanding Patterns

A spaceship is a pattern that re-appears after a certain number of steps in a different position. From a high level view, such a structure might look like an arcade-style spaceship as it propagates through the grid. In HexaGenesis, it is impossible to make a spaceship if the configuration for birth doesn't include any numbers less than three. That is, if it is impossible for an inactive cell to be activated with fewer than three neighbors, a spaceship cannot be formed. This is due to the hexagonal layout of the grid.

To demonstrate: imagine you try to design a spaceship structure yourself. For the structure to move, it must be configured in such a way that cells beyond the structure's perimeter have enough neighboring cells to come alive. On the vertical axis, where cells are stacked, a simple wall wouldn't allow cells beyond it to be formed because each empty cell bordering the wall would only have, at most, two active neighbors. Any attempt to solve this problem by adding additional cells would eventually lead to the same situation. On the horizontal axis, you cannot form a direct wall, but you can form a snake-like pattern. However, this will inevitably shrink to some degree and either terminate entirely or form a stable pattern. Similar to the vertical case, the perimeter will eventually get to a point where cells beyond it will not have enough neighbors to come alive.

It may be possible to form a spaceship if we set the configuration to allow birth for two or one neighbors. However, I've observed that such configurations very easily lead to ever-expanding states, states where the perimiter on at least one side of the pattern expands indefinitely while perimeters on all other sides either expand or remain stable.