[HN Gopher] Conway's Game of Life is omniperiodic
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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.
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(page generated 2023-12-07 23:00 UTC)