[HN Gopher] Conway's Game of Life is omniperiodic ___________________________________________________________________ Conway's Game of Life is omniperiodic Author : sohkamyung Score : 60 points Date : 2023-12-06 12:47 UTC (1 days ago) (HTM) web link (arxiv.org) (TXT) w3m dump (arxiv.org) | pohl wrote: | Of all the natural numbers in N, how did it turn out that the | last two were so small: 41 and 19? I'd have guessed that maybe | some huge number would have been the most difficult. | DemocracyFTW2 wrote: | After years of watching numberphile on Youtube, my guess would | be that there's a cutoff number n0 for which can be inductively | shown that if n0 is a period then n0+1 is also a period; that | would then eliminate almost all natural numbers from the | search. Such a proof might perhaps use combining periodic | patterns to obtain longer periods. It seems to be difficult to | construct a periodic GoL pattern with a given period, so that | then would explain why of all numbers two very small ones held | out over the decades; this is truly astounding, especially | given how incredibly inventive and industrious people have | gotten with the Game of Life, simulating entire computers on | the canvas. | AeiumNE wrote: | As a fellow numberphile viewer, I accept your authority on | the topic, and will probably repeat this guess as fact later. | mhink wrote: | Actually, progress looks like it's been very steady. You're | right regarding inductive-style proofs, but the ones for | _all_ natural numbers greater than a certain value came | surprisingly late. There appear to have been many more | oscillators with periods of x + yn (for constant x and y, any | natural number n). | | In 1996 there was a paper showing that it was possible to use | a particular family of patterns called "Herschel loops" to | create oscillators of any period >= 61. From there, the only | missing oscillators were 17, 19, 22, 23, 27, 31, 33, 34, 37, | 38, 39, and 41, 49, 51, 53, 57, and 59. | | There were gradual discoveries of new oscillators over the | next several years, then in 2013 there was another pattern | discovered which lowered that upper bound to >= 43. | | At this point, there were only five oscillators missing: 23, | 34, 38, 19, and 41. There appear to have been a few years | where progress stalled; 23 was found in 2019, and then the | last four were found over the course of these past couple | years. | Grimblewald wrote: | would have been great if the last one was 42. | Dove wrote: | 42 is special, though, as the longest cycle that requires a | its own solution. 43+ can be made as a Snark Loop. | | https://conwaylife.com/wiki/P43_Snark_loop | readyplayernull wrote: | Off by one | z2trillion wrote: | From page two of the paper: "Low-period oscillators in Life, | roughly p <= 15, can be found by playing with patterns by hand | or using brute force computer searches. In 1996, David | Buckingham demonstrated [6] using his "Herschel conduits" that | one can construct oscillators with p >= 61 by sending signals | around a closed track; the cutoff for systematically | constructing oscillators was later improved to p >= 43, by Mike | Playle's discovery of the Snark [43]." | Dove wrote: | Our family celebrated the event with a custom | t-shirt:https://www.teepublic.com/t-shirt/49108604-life-is- | omniperio... | xpe wrote: | Backstory please. | Dove wrote: | My son (13) is extremely mathematically adept, and current | research on cellular automata generally and Life specifically | is one of his passions. When the final pattern was discovered | earlier this year, he was extremely excited about it, and we | all wanted to celebrate. So he put together on graph paper | what he felt were the smallest/most iconic/best designs for | each period, and I transferred those designs into a Teepublic | appropriate digital format (using my obvious "Engineer Faking | Things" designer skills. LOL.) | | We then went wild buying shirts for the family, a sticker | (stuck currently on the water filter in the living room), a | mug which was supposed to be mine but which my son guards as | a precious possession, and a giant wall tapestry to hang in | my son's room. I actually wasn't sure all of those were going | to come out, but even on the sticker, you can make out the | individual cells in the more complex designs. Anyway - we | enjoy these things on a daily basis. | | We're kinda a bunch of math geeks. My husband and I both have | masters degrees in the field, and our son, I guess, is a | demonstration of what you can accomplish with selective | breeding. ;) We have a lot of mathematical curiosities around | the house, most of them homemade - penrose and hat tile | fridge magnets, klein bottles, constant width solids, | representations of projective tuning space. You know, the | usual. | | Other enthusiasts in the (very niche) space enjoy seeing the | graphic. Since the time of creation, math has advanced, with | these no longer being the smallest or best examples of some | of these loops. This is exciting for all of us - the advance | of mathematics is usually not this accessible. :) | ipnon wrote: | You are a lucky parent! | Dove wrote: | Thank you! We think so! | | This kid's scratch page is wild - | https://scratch.mit.edu/users/noonagon/ | | One of my favorite joint projects with him was a python | implementation of Game of Life in which individual cells | have velocities, and will crash into and react to each | other. I oughta put that one up on Github. Good times. :) | hrnnnnnn wrote: | How did he feel about the recent discovery of the aperiodic | monotile? | Dove wrote: | We have penrose tiles on the fridge that I 3d printed | years ago and glued magnets into. He begged and begged | and begged me to make similar hat tiles. There were two | problems - the 3d printer was currently broken, and work | had slammed me so hard that I had no time for side | projects. Finally I found an afternoon and we made about | 20 hat tiles together out of vinyl and magnet strips. | They currently live on our fridge. | | Of course then they promptly went and discovered spectre | tiles. | | I have since fixed my 3d printer. And it occurs to me | that that does open up a rather obvious option for a | Christmas present. | xpe wrote: | Total aside. I just looked at https://conwaylife.com/wiki/Ocellus | -- there is one main glider. I noticed the two biggish curvy | Golgi apparatus-looking-things on the sides and wondered if they | play a necessary role in preserving the cyclical pattern. I | guessed they might be important to prevent the three squares on | each end from breaking down when the glider collides with them. I | couldn't tell for sure as the animation was too fast. Well, turns | out there is a very nice LifeViewer that allowed me to play with | the sequence! By stepping the simulation I could see that the | curvy structures play a role. ___________________________________________________________________ (page generated 2023-12-07 23:00 UTC)