Timeline for Astonishing affinity of Wolfram's rule 110 to the numbers 2 and 7
Current License: CC BY-SA 4.0
27 events
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| Aug 30, 2022 at 17:51 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 30, 2022 at 17:50 | comment | added | Hans-Peter Stricker | @MattF. I'll remove 3 and 5 from the title. | |
| Aug 30, 2022 at 17:18 | comment | added | user44143 | The smallest ten numbers are all products (including the empty product) of 2, 3, 5 and 7, and the fractions with smallest numerators and denominators therefore are all products and quotients of 2, 3, 5, and 7 — and in the analysis of this cellular automaton, small numbers occur more frequently than large numbers. The numbers answering particular questions can still be interesting, but the appearance of this group of numbers is not what I would call astonishing. | |
| Aug 30, 2022 at 14:31 | comment | added | Hans-Peter Stricker | Obviously, $\kappa = 4 = 2\cdot 2$ is also missing. | |
| Aug 30, 2022 at 14:17 | comment | added | Hans-Peter Stricker | In turn, $\kappa(N) = 36 = 2\cdot 2\cdot 3 \cdot 3$ is frequent. | |
| Aug 30, 2022 at 14:13 | comment | added | Hans-Peter Stricker | One may ask, why $\kappa(N) = 2\cdot 3, 3 \cdot 3$ and so on are rare or even missing. | |
| Aug 30, 2022 at 14:11 | comment | added | Hans-Peter Stricker | @MattF.: I found it in the many $N$ with $\kappa(N) = 2, 2\cdot 5, 3\cdot 5, 2\cdot 3 \cdot 5$ and the many $N$ with $\kappa(N) = \frac{3}{2}, \frac{3 \cdot 5}{2 \cdot 2}, \frac{3\cdot 5}{2}$. | |
| Aug 30, 2022 at 12:22 | comment | added | user44143 | Where is the affinity suggested in the title for 3 and 5? | |
| Aug 30, 2022 at 10:29 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 30, 2022 at 10:19 | comment | added | Hans-Peter Stricker | @IlkkaTörmä: I added some spacetime diagrams that lead to limit cycles of length $7$. | |
| Aug 30, 2022 at 10:17 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 30, 2022 at 7:57 | comment | added | Hans-Peter Stricker | Let us continue this discussion in chat. | |
| Aug 30, 2022 at 7:57 | comment | added | Hans-Peter Stricker | @VilleSalo: What I have learnt from your comments: I have to think of gliders with their own periods, and the overall period is the least common multiple. So part of the question is: how many gliders do emerge? BTW: I find it hard to see the examples for $\lambda =7$ as gliders. Can you? | |
| Aug 30, 2022 at 7:54 | comment | added | Ville Salo | Just run it for ~4000 steps and you'll see it immediately. | |
| Aug 30, 2022 at 7:50 | comment | added | Hans-Peter Stricker | @VilleSalo: Where does the number $N=3000$ come from? Why does infinite spacetime start to be settled at this size? Why not at $N=1000$ or $N=10000$ oŕ any other finite number? | |
| Aug 30, 2022 at 6:52 | comment | added | Ville Salo | "Huge" because you have a glider gun with a period of over 200 feeding into one border, and a glider with period 17 or something on the other, and the cycle size determines how they collide. Some of the collisions may lead into new case analyses. Things tend to settle down in 110 (in my very limited experience), so I would not be surprised if the case analysis is finite, but you have to do it, and it's not so easy to computer-assist it. | |
| Aug 30, 2022 at 6:45 | comment | added | Ville Salo | It might feasible to analyze the behavior on large enough cycles. Specifically, large enough means far beyond 3000, which is about when the infinite spacetime settles. But that would be a huge undertaking and in principle at any point you could run into something that's impossible to analyze. Possibly very quickly, I just looked at one random situation. | |
| Aug 30, 2022 at 6:41 | comment | added | Hans-Peter Stricker | @VilleSalo: Don't you think the observed spectra (with $N \leq 250$) will continue to look essentially the same for $N \leq 3000$ and beyond? But I believe it's not feasible to calculate the exact cycle lengths for arbitrarily large $N$ in reasonable time. Or do you know a trick to circumvent brut force calculation? | |
| Aug 30, 2022 at 6:34 | comment | added | Ville Salo | In particular, I don't think looking at cycle lengths below 3000 will tell you much about the eventual behavior. | |
| Aug 30, 2022 at 6:29 | comment | added | Ville Salo | There is an obvious structure to the non-cyclic configuration with a single initial dot -- spacetime is semilinear, there's just a few gliders going around. If we want to know what happens with large sizes, one could restrict to cycles where you reach this eventual behavior before you reach the end of the cycle. I looked at one of the collisions that can happen, and it seems to stabilize again. So I wouldn't be surprised if this can be fully analyzed (and whatever you are seeing is just an initial phenomenon). But there are hundreds of cases to consider (I didn't even fully consider one case). | |
| Aug 30, 2022 at 6:25 | comment | added | Hans-Peter Stricker | @IlkkaTörmä: Is your question about transients? I'll add this information. Maybe in a scatterplot that correlates cycle and transient length. | |
| Aug 30, 2022 at 6:21 | comment | added | Ilkka Törmä | What do the spacetime diagrams look like that lead to short limit cycles? Is there some obvious structure that you could extrapolate? | |
| Aug 29, 2022 at 19:21 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 29, 2022 at 18:39 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 29, 2022 at 18:14 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 29, 2022 at 17:32 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
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| Aug 29, 2022 at 17:18 | history | asked | Hans-Peter Stricker | CC BY-SA 4.0 |