Math198 - Hypergraphics

# 3D Life

## Abstract

The goal of this project is to create a 3 dimensional cellular automaton that would follow rules similar to Conway's Game of Life, except with the addition of a neighbor in both the near and far directions, and then look for any patterns that may emerge.

## Brief Overview

• Suharsh Sivakumar's program was modifiyed to follow the rules of the Game of Life with random starting conditions. (Click here to see his program.)
• Explore some of the special patterns of the Game of Life and functions.
• The lattice program to become a 3 dimensional cellular automata
• Explore and observe how the 3D cellular automata behaves

## Introduction on the Game of Life

Conway's Game of life is a 2 dimensional cellular automaton that evolves into many interesting configurations based on certain initial conditions. The Game of Life consists of a grid of cells each of which is in one of 2 states: dead or alive. Each cell also belongs to a neighborhood. There are many ways to define a neighborhood, in the Game of Life however the neighborhood is defined to be the 8 cells adjacent to the cell in question. The state of each cell in the next generation is determined by the state of its neighborhood, these are the rules for the automaton. In the Game of Life the rules are: A dead cell with exactly 3 live neighbors becomes alive, a live cell with either 2 or 3 live neighbors continues to be alive, and in all other cases the cell is dead in the next generation. The rules of a cellular automaton are applied to every cell simultaneously.

Most initial starting conditions in the game of life will stabilize over time. There are three kinds of stabilizations that have been identified, they are still lives, oscillators, and spaceships. Still lives reach a certain state and then no longer change from generation to generation. Oscillators will oscillate between certain states as the generations go on. Spaceships are patterns that oscillate between certain states but will also fly off in a certain direction indefinitely. Examples of each type of stable state are shown below. (Pictures courtesy of http://en.wikipedia.org/wiki/Conway's_Game_of_Life)

A still life that is called a "boat"

A still life called a "beehive"

An oscillators called a "pulsar"

A spaceship called a "glider"

However the more interesting patterns are the ones that grow indefinately. The first example of a pattern like this is Gosper's glider gun, which will produce a glider every thirty generations.

The gun in it's initail position.

The glider gun in action.

However there exist patterns that exhibit even more interesting behaviors such as universal constructors and patterns that simulate the activities of a computer. Where a universal contructor is a patteren that can create copies of itself (example below).

Each one of the stuctures in the above picture creates a new, identical structure above itself