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 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.
Orphans
A related concept to Gardens of Eden is that of orphans, which are finite patterns 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. 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,^{[3]} 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.^{[4]}
Records
The following is a list of Gardens of Eden. Numbers marked red are records. Width of the bounding box is more than or equal to height.
Year | Author | Bounding box width × height | Orphan | Population | Density | Symmetry | Pattern | Note |
---|---|---|---|---|---|---|---|---|
1971 | Roger Banks et al. | 33 × 9 = 297 | 226 | 0.7609 | C1 | Garden of Eden 1, which is still a Garden of Eden even after removing the five rightmost columns^{[2]} | ||
1973 | Jean Hardouin-Duparc | 122 × 6 = 732 | size = 732 | 576 | 0.7869 | - | - | not a GoE, one predecessor^{[2]} |
1973 | Jean Hardouin-Duparc | 117 × 6 = 702 | - | - | - | - | - | ^{[2]} |
1991 | Achim Flammenkamp | 14 × 14 = 196 | 143 | 0.7296 | C1 | Garden of Eden 2^{[2]} | ||
2004 | Achim Flammenkamp | 13 × 12 = 156 | 81 | 0.5956 | C1 | Garden of Eden 3^{[2]} | ||
2004 | Achim Flammenkamp | 12 × 11 = 132 | 72 | 0.6372 | C1 | Garden of Eden 4^{[3]} | ||
2004 | Achim Flammenkamp | 13 × 10 = 130 | - | - | - | C1 | - | not a GoE, one unique predecessor^{[3]} |
2009 | Nicolay Beluchenko | 11 × 11 = 121 | 69 | 0.6330 | C4_1 | Flower of Eden, or Garden of Eden 5^{[5]} | ||
2009 | Nicolay Beluchenko | 11 × 11 = 121 | size = 113 | 59 | 0.5221 | C1 | - | ^{[5]} |
2009 | Nicolay Beluchenko | 11 × 11 = 121 | size = 110 | 49 | 0.4455 | D2_x | - | ^{[6]} |
2009 | Nicolay Beluchenko | 13 × 11 = 143 | size = 139 | 58 | 0.4173 | C1 | - | ^{[6]} |
2009 | Nicolay Beluchenko | 11 × 11 = 121 | size = 115 | 47 | 0.4087 | D2_x | - | ^{[7]} |
2009 | Nicolay Beluchenko | 11 × 11 = 121 | size = 107 | 51 | 0.4766 | D2_x | - | ^{[7]} |
2009 | Nicolay Beluchenko | 11 × 11 = 121 | 45 | 0.3982 | D2_x | 45-cell Garden of Eden, the smallest known in terms of population^{[8]} | ||
2009 | Nicolay Beluchenko | 12 × 12 = 144 | size = 129 | 50 | 0.3876 | C1 | - | ^{[8]} |
2011 | Marijn Heule et al. | 11 × 11 = 121 | size = 93 | 65 | 0.6989 | D8_1 | - | almost Garden of Eden 6 |
2011 | Marijn Heule et al. | 11 × 11 = 121 | size = 119 | 45 | 0.3781 | D8_1 | - | almost Garden of Eden 6 |
2011 | Marijn Heule et al. | 10 × 10 = 100 | 56 | 0.6087 | C4_4 | Garden of Eden 6^{[9]}^{[10]} ranked third place in the Pattern of the Year 2011 competition on the ConwayLife.com forums, behind the lobster and the fully universal Turing machine.^{[11]} | ||
2012 | Marijn Heule et al. | 13 × 13 = 169 | size = 153 | 49 | 0.3203 | D8_1 | - | ^{[12]} |
2015 | Steven Eker | 11 × 9 = 99 | size = 99 | 66 | 0.6667 | C1 | - | ^{[13]} |
2016 | Steven Eker | 12 × 8 = 96 | 57 | 0.5938 | C1 | Garden of Eden 8×12, the smallest known in terms of bounding box^{[13]} | ||
2016 | Steven Eker | 83 × 5 = 415 | size = 410 | 284 | 0.6927 | C1 | - | ^{[2]} |
2016 | Steven Eker | 45 × 5 = 225 | 139 | 0.6233 | D2_+1 | Garden of Eden 5×45, the shortest known with the smallest height^{[14]} | ||
2016 | Steven Eker | 9 × 11 = 99 | size = 89 | 55 | 0.6180 | C1 | - | ^{[14]} |
2017 | Steven Eker | 9 × 11 = 99 | 50 | 0.5682 | C1 | Garden of Eden 11, with the smallest known orphan^{[15]} |
See also
References
- ↑ Dave Greene (July 16, 2022). Re: Belated POTY 2012 nomination thread (discussion thread) at the ConwayLife.com forums
- ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} ^{2.4} ^{2.5} ^{2.6} Achim Flammenkamp (January 27, 2017). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ ^{3.0} ^{3.1} ^{3.2} Achim Flammenkamp (December 1, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ Gardens of Eden at Game of Life News. Posted by Adam P. Goucher on January 14, 2012.
- ↑ ^{5.0} ^{5.1} Achim Flammenkamp (December 1, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ ^{6.0} ^{6.1} Achim Flammenkamp (December 1, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ ^{7.0} ^{7.1} Achim Flammenkamp (December 21, 2009). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ ^{8.0} ^{8.1} Achim 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.
- ↑ Marijn J. H. Heule et al. (August 9, 2013). "Symmetry in Gardens of Eden". Electron. J. Comb.. Retrieved on January 5, 2021.
- ↑ beebop (February 28, 2012). Patterns of the Year 2011 (discussion thread) at the ConwayLife.com forums
- ↑ Achim Flammenkamp (February 2, 2012). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ ^{13.0} ^{13.1} Achim Flammenkamp (April 13, 2016). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ ^{14.0} ^{14.1} Achim Flammenkamp (September 18, 2016). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
- ↑ Achim Flammenkamp (January 27, 2017). "Garden of Eden / Orphan". Achim's Game of Life Page. Retrieved on December 24, 2020.
External links
- Garden of Eden at the Life Lexicon
- Garden of Eden (cellular automaton) at Wikipedia
Forum threads
- orphan pattern / garden of eden (discussion thread) at the ConwayLife.com forums
- Conway's Life: Garden of Eden Pattern (discussion thread) at the ConwayLife.com forums
- Finding Gardens of Eden (discussion thread) at the ConwayLife.com forums