A Garden of Eden is a pattern that has no parents and thus can only occur in generation 0. The term was first used in connection with cellular automata by John W. Tukey, many years before Conway's Game of Life was conceived. It was known from the start that Gardens of Eden exist in Life because of a theorem by Edward Moore that guarantees their existence in a wide class of cellular automata.
The Garden of Eden theorem was proved by Edward Moore and John Myhill pre-1970 and shows that a wide class of cellular automata must contain Garden of Eden patterns. Of particular interest is that Conway's Game of Life falls into this class, and thus Gardens of Eden were known to exist right from the day it was conceived.
Statement of the theorem
A finite pattern (or finite configuration) is a pattern with a finite number of cells. A cellular automaton is said to be injective over finite patterns if no two distinct finite patterns map into the same finite pattern. It is said to be surjective if every pattern is mapped to by some other pattern. Thus, by definition a cellular automaton contains Gardens of Eden if and only if it is not surjective.
The Garden of Eden theorem states that the class of surjective cellular automata and those which are injective over finite configurations coincide. In other words, a cellular automaton has a Garden of Eden if and only if it has two different finite configurations that evolve into the same configuration in one step.
As a corollary, every injective cellular automaton (i.e., one with one-to-one global mapping for both finite and infinite patterns) is surjective and hence bijective. However, surjective cellular automata do not need to be injective over infinite patterns (and thus need not be injective in general).
Application to Conway's Game of Life
The theorem applies to Conway's Game of Life because it is easy to find two different finite patterns that are mapped into the same configuration. The configuration in which every cell is dead, and the one in which exactly one cell is alive both lead to the one in which every cell is dead. The Garden of Eden theorem then implies that there must exist a Garden of Eden pattern.
Pre-block and grin are both parents of the block. The Garden of Eden theorem thus says that Gardens of Eden exist in Conway's Game of Life.
Orphans
A related concept to Gardens of Eden is that of orphans, which are finite patterns, including both live and dead cells, that cannot occur as part of the evolution of another pattern. That is, they are Gardens of Eden that can be extended in any way to form other Gardens of Eden. A minimal orphan is an orphan that, if any subset of its cells is removed, is no longer an orphan.[1]
Explicit examples
Several Gardens of Eden and orphans have been constructed, the first by Roger Banks et al. at MIT in 1971. It had a bounding box of size 33 × 9 and had 226 cells. Jean Hardouin-Duparc found the second and third orphans by computer search in 1973, which had bounding boxes of size 122 × 6 and 117 × 6. It was long suspected that no height-5 Gardens of Eden exist, but in April 2016, Steven Eker found a Garden of Eden fitting inside a 5 × 83 bounding box.[2] Eker also proved that any Garden of Eden must have a height greater than 3. On June 11, 2023, Andrew J. Wade proved that there are no height-4 orphans, assuming that all specified cells (living and dead) are counted.[3] It remains an open question whether there exist Gardens of Eden where live cells are restricted to four consecutive rows.[4]
Many smaller Gardens of Eden have been found in more recent years. Garden of Eden 2 was found by Achim Flammenkamp in 1991, contained 143 cells, and had a bounding box of size 14 × 14. Garden of Eden 3 was found by Achim Flammenkamp in 2004, contained 81 cells, and had a bounding box of size 13 × 12. The smallest known Garden of Eden for about five years was Garden of Eden 4,[5] which was also found by Achim Flammenkamp in 2004. It contained 72 cells and had a bounding box of size 12 × 11. Smaller Gardens of Eden were subsequently found, including Garden of Eden 5 and Garden of Eden 6, which contains 56 live cells and a bounding box of 10 × 10 and was found on December 14, 2011. For the smallest Garden of Eden currently known, refer to the table below.
Computer searches have revealed that there are no Gardens of Eden contained within a 7 × 7bounding box,[6] and that there are no reflectionally or rotationally symmetric Gardens of Eden within an 8 × 8 or 9 × 9 bounding box.[7]Randall D. Beer showed that an 8 × 8 Garden of Eden must have a density between 0.15 and 0.8, and so between 8 and 51 on-cells, and gave a heuristic argument suggesting that most likely none exist.
Garden of Eden 11, with the smallest known orphan[18]
↑Orphans with arbitrarily low density can be constructed using two copies of American Dream[14] so the record in this table only applies to small, compact orphans.
↑ 7.07.1Christiaan Hartman, Marijn J. H. Heule, Kees Kwekkeboom, Alain Noels (August 9, 2013). "Symmetry in Gardens of Eden". The Electronic Journal of Combinatorics.
↑ 8.08.1Achim Flammenkamp (December 1, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
↑ 9.09.1Achim Flammenkamp (December 1, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
↑ 10.010.1Achim Flammenkamp (December 21, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
↑ 11.011.1Achim Flammenkamp (December 21, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
↑Achim Flammenkamp (December 17, 2011). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
↑ beebop (February 28, 2012). Patterns of the Year 2011 (discussion thread) at the ConwayLife.com forums
