Garden of Eden

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.

Garden of Eden theorem

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 bits. 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 that can not 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.

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.[1] Eker also proved that any Garden of Eden must have a height greater than 3. It is not known if any height-4 Gardens of Eden exist.

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,[2] 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 6×6 bounding box.[3]

Many Gardens of Eden found by Nicolay Beluchenko in 2009
Download RLE: click here

Records

List of records and notable patterns, red are records. Box-height is assumed to be less than or equal to box-width.

Year Author Box-height Box-width Box-cells Orphan On-Cells Density Symmetry Note
1971 Roger Banks et al. 9 33 297 297 226 0.7609 - GoE#1
1973 Jean Hardouin-Duparc 6 122 732 732 576 0.7869 -
1973 Jean Hardouin-Duparc 6 117 702 ? ? -
1991 Achim Flammenkamp 14 14 196 196 143 0.7296 - GoE#2
2004 Achim Flammenkamp 12 13 156 136 81 0.5956 - GoE#3
2004 Achim Flammenkamp 11 12 132 113 72 0.6372 - GoE#4
2004 Achim Flammenkamp 10 13 130 ? ? - not a GoE, one parent
2009 Nicolay Beluchenko 11 11 121 109 69 0.6330 C4 GoE#5
2009 Nicolay Beluchenko 11 11 121 113 59 0.5221 -
2009 Nicolay Beluchenko 11 11 121 110 49 0.4455 D2_x
2009 Nicolay Beluchenko 11 13 143 139 58 0.4173 -
2009 Nicolay Beluchenko 11 11 121 115 47 0.4087 D2_x
2009 Nicolay Beluchenko 11 11 121 107 51 0.4766 D4_x
2009 Nicolay Beluchenko 11 11 121 113 45 0.3982 D4_x
2009 Nicolay Beluchenko 12 12 144 129 50 0.3876 -
2011 Marijn Heule et al. 11 11 121 93 65 0.6989 D8 almost GoE#6
2011 Marijn Heule et al. 11 11 121 119 45 0.3781 D8 almost GoE#6
2011 Marijn Heule et al. 13 13 169 153 49 0.3203 D8
2011 Marijn Heule et al. 10 10 100 92 56 0.6087 C4 GoE#6
2015 Steven Eker 9 11 99 99 66 0.6667 -
2016 Steven Eker 8 12 96 96 57 0.5938 -
2016 Steven Eker 5 83 415 410 284 0.6927 -
2016 Steven Eker 5 45 225 223 139 0.6233 D2_+1
2016 Steven Eker 9 11 99 89 55 0.6180 -
2017 Steven Eker 9 11 99 88 50 0.5682 - GoE#11
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gollark: =tex ha \space I \space have \space cracked \space it!
gollark: =tex \space
gollark: =tex \spave
gollark: =tex AnyoneGotFunIdeasToWorkOn?

See also

References
  1. Achim Flammenkamp (April 14, 2016). "Garden of Eden / Orphan". Retrieved on April 30, 2016.
  2. Achim Flammenkamp (November 7, 2008). "Garden of Eden / Orphan". Retrieved on February 14, 2009.
  3. Adam P. Goucher (January 14, 2012). "Gardens of Eden". Retrieved on June 18, 2012.

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