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.

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.

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. 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. It remains an open question whether there exist Gardens of Eden where live cells are restricted to four consecutive rows.

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, 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.

#N Gardens of Eden #O Nicolay Beluchenko #C Many Gardens of Eden found in 2009. #C www.conwaylife.com/wiki/index.php?title=Garden_of_Eden x = 276, y = 141, rule = b3/s23 o2bob3obobo2b5o12bo2b3ob8obo12b2ob2obob3ob3o2b2o197b$obobo2b2obobobob 2o13b2o2b5ob2ob3obo12b2o2bobob4ob4o198b$2obo2bob2obob6o13b3o2bob6ob3o 11bob2o2b3ob2o3bobo198b$ob2obob2obobo2bo2bo13bo2b2o3bo2bobobo12b2ob2ob ob2ob4obo2bo196b$3obobo3b2o3b2o14bo2bob3ob2o2bob3o11b4obob3o2b2o4b3o 195b$3o2bobobob4o2bo13b2ob3ob2o2bob3obo12bo2bobobob3ob3ob3o195b$bobo2b obob3obobobo11bo2b2o2bobobobobob2o11bobob4obobobob2obo197b$obob2obobob ob3ob2o12bobob2ob3ob3obobo12bobob2obob9o197b$o2b2obob3ob2ob4o12b4ob3ob 4obo2bo11bobob2obobo2b3obo199b12$bobob2o2bob3o2bo13b6obob3o2b2o230b$3o b5obobo2b2o13b3obobobo2b2obob2o228b$bob2obo2bobob2obo13b2obobob5obob2o 229b$ob4obob2ob2obo14bob5ob3obo2b3o228b$2obob5obob4o13b2ob2ob5ob4o230b $obob3ob3obo2bo14b3ob4obobob6o227b$2obobobo2b6o14b4ob3o2b6ob2o227b$ob 3obo2b4obo15bobobobob4o2b3obo227b$b3obob4ob2obo14b2obobobob3ob2obobo 227b$ob4o2b3o2b2o15b11ob6o228b11$b2ob2o2bob2o8b4o2bo4bobo242b$ob3ob4o 10b2o2b2o2bob3o243b$2ob3ob3obo8b3obobo2bo3bo242b$b3ob3ob2o9bob2obob4ob o243b$3o2b5obo9b4obob2obobo242b$ob3obob3o10bobob4obo2bo242b$2obobo2bob 2o8bobobobob5o243b$ob3o2bob3o9bob3obob3o244b$bo2b3obob2o8bob2ob2obobob 2o242b$3ob3o2bobo9bob3obo3b2o243b$3b3o2bobo9bobob3obob4o242b10$b2o2b4o 2bo8b4ob5obo8b2obobob5o8b11obo7bo2bobo2b3o10bobob2obob2o9bob2ob2o2b3o 8b2o2b2o2b3o9b2o2b3o2bobo7b2o2bob3ob2o8bob3obobob4o6b2ob8obo8bo3b2o3bo b3o9bob4ob2ob2o$b2obobo2bobo9bob3ob4o9b5obobo2bo8b2obob5obo8b3obob2obo 10bobobo2bobob2o8b5obob3o8b3obob2ob2o9bob2o3bob4o8b2o2bobobo2bo7bob2ob obob2o9b3obobo2bob2o9bobo2bob3obo8b2obo2bob2o3bo$3bobobo2bo9bobobobo2b o10bobob2ob2obo10b4o2bobo10bobo2bo2b2obo9b3obo2bo2bo8bob2o2bo3b3o7bob 2obob3obo8bobobobo2bobo8bo2b4obob3o8b2obob3obo2bo6bo2bobobob3obo7bobob ob2o2b2o8bob3obobo2bobob$obobobobo2bo8b2obob3ob2o10bobob2ob4o9bob3o2b 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161b$7ob4o9b5o2b4o9bob2ob3ob2obo7b2obob4obob2o7bobob2obob4o9bobobo2bob 2o161b$b3ob3ob4o7b4obobob2o10b2obob3obob2o9bobob2obobo9b5ob3obobo8b3ob 4obo163b$ob4ob4obo8bobo2bob4o9bobob5obobo10bob4obo10b2ob3ob3ob2o181b$b 2obob2obobo14b3obo11b3obobob3o11bo2b2o2bo13bob4o2bo182b$45bo3bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 2 ]]

Many Gardens of Eden found by Nicolay Beluchenko in 2009 (click above to open LifeViewer) RLE: here Plaintext: here

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	size = 297	226	0.7609	C1	x = 33, y = 9, rule = B3/S23 33o$2obob3ob3ob2obobobobobobobobobo$obob3ob3ob4ob3obobobobobobo$5ob3ob 3ob4ob14o$obob2ob3ob3obob3obobobobobobo$4ob3ob3ob5ob2obobobobobobo$b2o b3ob3ob3obobob13o$2ob2ob3ob3ob2ob4obobobobobobo$18ob14o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here Truncated form	Garden of Eden 1, which is still a Garden of Eden even after removing the five rightmost columns
1973	Jean Hardouin-Duparc	122 × 6 = 732	size = 732	576	0.7869	C1	-	not a GoE, one predecessor
1973	Jean Hardouin-Duparc	117 × 6 = 702	-	-	-	-	-
1991	Achim Flammenkamp	14 × 14 = 196	size = 196	143	0.7296	C1	x = 14, y = 14, rule = B3/S23 2obobobob2obo$ob3ob3ob2obo$4ob3ob2obo$3obobobob4o$b3obob3ob2o$7ob4obo$ bobob8o$ob3ob2obobobo$6ob6o$ob2ob5obobo$3ob9o$b3obobobob3o$3obobobob2o bo$ob12o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 2
2004	Achim Flammenkamp	13 × 12 = 156	size = 136 size = 134 (alternative form from Nicolay Beluchenko, 2006)	81	0.5956	C1	x = 13, y = 12, rule = B3/S23 2bob3o$2obob5obo$obob2obobo$b4obob3o$obob2ob3obo$b3ob2obobo$2bo3b3o2b 2o$bob2obobob2o$3ob4obobo$bob4o3bo$bobob2o2bo$b2obo2b2o2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 3
2004	Achim Flammenkamp	12 × 11 = 132	size = 113	72	0.6372	C1	x = 12, y = 11, rule = B3/S23 bob2ob2o2bo$2bob3ob3o$2b2ob3obobo$bob3ob3obo$5o2b4o$b2ob3obo2bo$b3obob o2bo$b2ob3o2b2o$ob3ob3obo$o2b2o2bobobo$10bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 4
2004	Achim Flammenkamp	13 × 10 = 130	predecessor (size = 180) size = 130	90	0.6923	C1	x = 13, y = 10, rule = B3/S23 bob3ob6o$3ob3obobobo$ob3ob3obo$4o2b4ob2o$2ob3obo2bobo$3obobo2b4o$2ob3o2b3obo$b3ob3ob3o$4o2b3ob2o$ob3obob3obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) x = 15, y = 12, rule = B3/S23 o2b3o6b2o$2bo6b2o2b2o$b2o3b2ob2o$bo2b2obo3bo2bo$bo2bo5bo2bo$2bo2b2ob2o3bo$bo3bo3b2ob2o$o2b2o2bo2bo$bo4bo7bo$3bo3bobob4o$obob3o2b2o2bo$o4bobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer)	not a GoE, one unique predecessor
2009	Nicolay Beluchenko	11 × 11 = 121	size = 109	69	0.6330	C4_1	x = 11, y = 11, rule = B3/S23 b3o2b2o$b2obobob3o$b3o2b5o$obobobobobo$4obobobo$4b3o$bobobob4o$obobobo bobo$5o2b3o$3obobob2o$3b2o2b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Flower of Eden, or Garden of Eden 5
2009	Nicolay Beluchenko	11 × 11 = 121	size = 113	59	0.5221	C1	-
2009	Nicolay Beluchenko	11 × 11 = 121	size = 110	49	0.4455	D2_x	-
2009	Nicolay Beluchenko	13 × 11 = 143	size = 139	58	0.4173	C1	-
2009	Nicolay Beluchenko	11 × 11 = 121	size = 115	47	0.4087	D2_x	-
2009	Nicolay Beluchenko	11 × 11 = 121	size = 107	51	0.4766	D2_x	-
2009	Nicolay Beluchenko	11 × 11 = 121	size = 113	45	0.3982	D2_x	x = 11, y = 11, rule = B3/S23 3b2o3bo$2bo2bobobo$bobo2bo3bo$obobo2bobo$o2bobo2bo$bo2b3o2bo$2bo2bobo 2bo$bobo2bobobo$o3bo2bobo$bobobo2bo$2bo3b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	45-cell Garden of Eden, the smallest known in terms of population
2009	Nicolay Beluchenko	12 × 12 = 144	size = 129	50	0.3876	C1	-
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	size = 92	56	0.6087	C4_4	x = 10, y = 10, rule = B3/S23 bob3obo$2bobobo2bo$ob3o2b2o$bob5obo$o2bo2b4o$4o2bo2bo$ob5obo$b2o2b3obo $o2bobobo$2bob3obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 6 ranked third place in the Pattern of the Year 2011 competition on the ConwayLife.com forums, behind the lobster and the fully universal Turing machine.
2012	Marijn Heule et al.	13 × 13 = 169	size = 153	49	0.3203	D8_1	-
2015	Steven Eker	11 × 9 = 99	size = 99	66	0.6667	C1	-
2016	Steven Eker	12 × 8 = 96	size = 96	57	0.5938	C1	x = 12, y = 8, rule = B3/S23 o2b2obo2b3o$b2o2b2obobo$obobo2bobobo$b4o2b2o2bo$o3b3ob4o$2obobo2bob2o$ obo2b3obobo$b4obob4o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 8×12, the smallest known in terms of bounding box
2016	Steven Eker	83 × 5 = 415	size = 410	284	0.6927	C1	-
2016	Steven Eker	45 × 5 = 225	size = 223	139	0.6233	D2_+1	x = 45, y = 5, rule = B3/S23 2obobo2bob3obo2b3ob7o2b6o2bobobobo$b2obob2obobobobobob3obobobobobobob 3obobobo$2b2ob2o2b4o2b6o2b6o2b5ob5obo$b2obob2obobobobobob3obobobobobob ob3obobobo$2obobo2bob3obo2b3ob7o2b6o2bobobobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 WIDTH 800 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 5×45, the shortest known with the smallest height
2016	Steven Eker	9 × 11 = 99	size =89	55	0.6180	C1	-
2017	Steven Eker	9 × 11 = 99	size = 88	50	0.5682	C1	x = 11, y = 9, rule = B3/S23 2b2ob3obo$bo2bobobo$ob3o2b3o$bobobo2bobo$b3o2b2o2bo$2o3b2o2bo$obobob3o $o2bobo2bo$b3o2b4o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL THUMBSIZE 4 ZOOM 16 ]] (click above to open LifeViewer) RLE: here Plaintext: here	Garden of Eden 11, with the smallest known orphan

See also

• Grandfather problem
• Unique father problem

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
• Gardens of Eden (discussion thread) at the ConwayLife.com forums