Talk:Garden of Eden
Theorem
The last sentence in the section contradicts everything else, unless I'm mistaken: "However, surjective cellular automata do not need to be injective." implies that {surjective} > {injective} (and {injective} is contained by {surjective}), while previously it says stuff like "the class of surjective cellular automata and those which are injective over finite configurations coincide." which implies {surjective} = {injective}, which is further supported by "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." and stuff. Although even if that last statement is right,the theorem still proves that Life has Gardens of Eden, so I'm not really sure whether it is or not. Elithrion 18:35, 14 February 2009 (UTC)
- The important distinction is between simply injective and injective over finite patterns. The theorem says that injective over finite patterns iff surjective. Thus, surjective implies injective over finite patterns, but not injective overall (that is, there may be two *infinite* patterns that are mapped to by the same infinite pattern). This could perhaps be made more clear in the article. Anyway, yes, the "picture" is {surjective} = {injective over finite patterns} > {injective}. Nathaniel 19:07, 14 February 2009 (UTC)
- Ah, thanks. I guess I overlooked that distinction. Elithrion 20:42, 14 February 2009 (UTC)
I managed to find a paper documenting the Garden of Eden theorem: [[1]]. (Mutually erasable is equivalent to being finitely non-injective.) Now that I understand the situation better, I'll try to detail a proof that finitely non-injective implies non-surjective, due to a proof in Conway's Winning Ways. FractalFusion 08:23, 25 March 2009 (UTC)
Records
What do you think about insertion of following-like table? --Mtve (talk) 10:20, 7 April 2016 (UTC)
| Year | Author | Box-min | Box-max | Box | Orphan | On-Cells | Density | Symmetry | Note |
|---|---|---|---|---|---|---|---|---|---|
| 1971 | Roger Banks et al. | 9 | 33 | 297 | 297 | 226 | ? | - | - |
| 1973 | Jean Hardouin-Duparc | 6 | 122 | 732 | ? | ? | - | - | |
| 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, but only one parent | |
| 2009 | Nicolay Beluchenko | 11 | 11 | 121 | 109 | 69 | 0,6330 | D4 | Flower_of_Eden (GoE#5) |
| 2009 | Nicolay Beluchenko | 11 | 11 | 121 | 113 | 59 | 0,5221 | - | - |
| 2009 | Nicolay Beluchenko | 11 | 11 | 121 | 110 | 49 | 0,4455 | - | - |
| 2009 | Nicolay Beluchenko | 11 | 13 | 143 | 139 | 58 | 0,4173 | D2 | - |
| 2009 | Nicolay Beluchenko | 11 | 11 | 121 | 115 | 47 | 0,4087 | - | - |
| 2009 | Nicolay Beluchenko | 11 | 11 | 121 | 107 | 51 | 0,4766 | C2dia | - |
| 2009 | Nicolay Beluchenko | 11 | 11 | 121 | 113 | 45 | 0,3982 | C2dia | - |
| 2009 | Nicolay Beluchenko | 12 | 12 | 144 | 129 | 50 | 0,3876 | - | - |
| 2011 | Marijn Heule et al. | 10 | 10 | 100 | 92 | 56 | 0,6087 | D4 | GoE#6 |
| 2012 | Marijn Heule et al. | 13 | 13 | 169 | 153 | 49 | 0,3203 | - | - |
| 2015 | Steven Eker | 9 | 11 | 99 | 99 | 66 | 0,6667 | - | - |
| 2016 | Steven Eker | 8 | 12 | 96 | 96 | 57 | 0,5938 | - | - |
Hello, Mtve. I would just insert it into the article without attempting to discuss it. From my experience, nobody bothers to discuss anything anymore on this wiki. Posting it on talk pages only increases the chance that the information will never make it to the main page.
I would avoid using things like "C2dia" and "D4" as symmetries, because those labels might be ambiguous. FractalFusion (talk) 18:22, 8 April 2016 (UTC)