Cultivating the Garden of Eden

TL;DR

This paper systematically studies Garden of Eden configurations in Conway's Game of Life, mapping their distribution, characteristics, and structural constraints to deepen understanding of initial state uniqueness.

Contribution

It provides a comprehensive map of GOE and nonGOE configurations up to 11x11 grids, analyzes their density-dependent structure, and explores the network of constraints defining these states.

Findings

Mapped GOE and nonGOE configurations in density/size space.

Characterized how precursor structures vary with grid size.

Cataloged all known GOEs up to 11x11.

Abstract

Garden of Eden (GOE) states in cellular automata are grid configurations which have no precursors, that is, they can only occur as initial conditions. Finding individual configurations that minimize or maximize some criterion of interest (e.g., grid size, density, etc.) has been a popular sport in recreational mathematics, but systematic studies of the set of GOEs for a cellular automaton have been rare. This paper presents the current results of an ongoing computational study of GOE configurations in Conway's Game of Life (GoL) cellular automaton. Specifically, we describe the current status of a map of the layout of GOEs and nonGOEs in density/size space, characterize how the density-dependent structure of the number of precursors varies with increasing grid size as we approach the point where GOEs begin to occur, provide a catalog of all known GOE configurations up to a grid size of 11x11, and initiate a study of the structure of the network of constraints that characterize GOE vs. nonGOE configurations.