Conway's Game of Life is omniperiodic(arxiv.org)
arxiv.org
Conway's Game of Life is omniperiodic
https://arxiv.org/abs/2312.02799
102 comments
Our family celebrated the event with a custom t-shirt:https://www.teepublic.com/t-shirt/49108604-life-is-omniperio...
Whoa, that catalog of oscillators is beautiful! Thanks for taking the time to arrange it with your son and for sharing it with us. Can definitely see the appeal of having a giant wall tapestry of it. :D
Seeing all 43 oscillators all at once took my breath away because it reminded me so much of some prime-factor flowers I've been playing with for years. Perhaps you or your son might find them interesting: https://twitter.com/elzr/status/1733007772181233681
Seeing all 43 oscillators all at once took my breath away because it reminded me so much of some prime-factor flowers I've been playing with for years. Perhaps you or your son might find them interesting: https://twitter.com/elzr/status/1733007772181233681
I'm not an expert, and I know "Life" in this context may be a proper noun referring the Game of Life, but I would be very hesitant to generalize this to life in the universe, as life is not Conway's Game of Life, and entropy brings elements into play which is not present in Conway's Game of Life.
https://www.youtube.com/watch?v=yKbJ9leUNDE&list=PLoaVOjvkzQ...
https://www.youtube.com/watch?v=yKbJ9leUNDE&list=PLoaVOjvkzQ...
I just bought one. I bet a lot of people reading this thread did.
Backstory please.
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. :)
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. :)
Get a second mug printed from the same printer, and hide it safely away, in case the original ever breaks, and they are out of business.
If it breaks, that might be a good opportunity to practice impermanence.
Well sure, but i imagine you could get the impermanence lesson, and the mug next Christmas.
I love this. Sounds like your kids are lucky to have such supportive parents. I’d love to share that kind of thing with my son. He’s only 2 though so who knows what he’ll be into.
How did he feel about the recent discovery of the aperiodic monotile?
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.
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.
I got 144 wooden spectre tiles from a laser cutter shop to play with. It's surprisingly difficult (for me, without any mathematical ability) to make a proper tiling beyond ~ 30 tiles. I was never able to densely tile the plane with all 144... thus shattering my hopes of getting these in ceramics and tile a floor with them.
Penrose tilings have similar issues. You can’t just combine them any possible way. There only a few valid vertex arrangements (tile positions around a point) to result in infinite aperiodic tiling. If you mess up, the resulting overlap can happen pretty far away.
Exactly! Beyond a certain number of tiles, I'm constantly having to backtrack in order to fix some overlap. And they happen all over the place, in difficult to predict circumstances.
There's this to help create non-overlapping tilings, but understanding it is completely beyond me (or the amount of time I have to invest, I tell myself...)
https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...
There's this to help create non-overlapping tilings, but understanding it is completely beyond me (or the amount of time I have to invest, I tell myself...)
https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...
I've found laser cutters to be more practical than 3d printers for tiles. I made a jigsaw puzzle using spectre, and managed to confuse a bunch of coworkers.
If you can get him into building eurorack synthesizers, he can start turning some of this math into music.
This module, for example https://www.nonlinearcircuits.com/modules/p/cellular-automat... uses networks of XOR gates to create "is a 16 cell gate and pattern generator using cellular automata rules 90 & 150".
Blinky lights in a grid that steps with those rules and can steer other modules.
I love mine.
The designer has many other wild designs based on chaotic dynamics and many other non-linear mathematical concepts. The name is kind of a giveaway.
This module, for example https://www.nonlinearcircuits.com/modules/p/cellular-automat... uses networks of XOR gates to create "is a 16 cell gate and pattern generator using cellular automata rules 90 & 150".
Blinky lights in a grid that steps with those rules and can steer other modules.
I love mine.
The designer has many other wild designs based on chaotic dynamics and many other non-linear mathematical concepts. The name is kind of a giveaway.
You are a lucky parent!
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. He just came up to me yesterday proposing a refinement to the physics. I oughta put that one up on Github. Good times. :)
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. He just came up to me yesterday proposing a refinement to the physics. I oughta put that one up on Github. Good times. :)
Do you have any resources to best optimize (or at least not squander) a young genius's opportunity, especially in math?
My son (just turning 7) is smarter than I am; and possibly more as an artifact of being on the spectrum, is nearly bored after textual representations ( one tredecillion, ten tredecillion...), ~70 digits of pi, latin/greek alphabet, periodic table, planets/cosmos, and every other aspect of rote memorable/science/math things.
I'm not sure if he's bored now, or simply too busy with trying to learn to read, or burnt out, but I've tried to look for more collegiate material that may suit him, besides looking ahead 4-5 school years, and come up rather short. Some private tutoring material has seemed most helpful, but so far, YouTube has the best material, but obviously that has its own issues.
Besides more moderated Youtube/Kids, i fear im missing something obvious
My son (just turning 7) is smarter than I am; and possibly more as an artifact of being on the spectrum, is nearly bored after textual representations ( one tredecillion, ten tredecillion...), ~70 digits of pi, latin/greek alphabet, periodic table, planets/cosmos, and every other aspect of rote memorable/science/math things.
I'm not sure if he's bored now, or simply too busy with trying to learn to read, or burnt out, but I've tried to look for more collegiate material that may suit him, besides looking ahead 4-5 school years, and come up rather short. Some private tutoring material has seemed most helpful, but so far, YouTube has the best material, but obviously that has its own issues.
Besides more moderated Youtube/Kids, i fear im missing something obvious
I (as noted in another comment: Dove's husband and the father of the child in question) homeschool him. My wife and I both have graduate math-related degrees. We let him pretty much run wild with independent math research, while also making a point to probe for weaknesses -- last year I gave him the tests from some MIT Opencourseware calculus courses and about twice per year-of-calculus we ran into something he hadn't learned and spent a couple days working it up. On the future agenda is Gilbert Strang's linear algebra course, since there are full lectures online and everything.
What we did in past years was, largely, giving him unfettered access to math and math-related channels on youtube. Vi Hart, Numberphile, 3blue1brown, stuff like that. When he showed interest in something specific, we'd get him appropriate materials. Since his elementary school was a public montessori school, his teachers encouraged us to send him with appropriate math workbooks for his capability level.
What we did in past years was, largely, giving him unfettered access to math and math-related channels on youtube. Vi Hart, Numberphile, 3blue1brown, stuff like that. When he showed interest in something specific, we'd get him appropriate materials. Since his elementary school was a public montessori school, his teachers encouraged us to send him with appropriate math workbooks for his capability level.
I envy today's generation and their free access to high quality maths resources haha
yes but they're missing the take-the-radio-apart or take-the-computer-apart or fit-whole-instruction-set-in-one's-head advantage that earlier eras were granted.
Maybe. Those things can still be taught.
"What if there is no Internet?" No computer? Etc...
Papa (grandfather now, raising my granddaughter because... not important, lol) tends to impress the young ones with what can be done with what is in one's head.
I am hopeful. But maybe also naive.
The other idea I like to link to these skills is "thought is action" type mastery. And, just to be clear, that is a state of clarity coupled with having grokked[0] something valuable.
Having realized that state many times in my past makes it easier today. I can go there and perform, doing or dealing with whatever it is efficiently and effectively.
It also can mean agency where ones peers may well lack it.
[0] - ...having achieved a state of understanding so complete it is a part of us, who we are, automatic, almost instinct.
"What if there is no Internet?" No computer? Etc...
Papa (grandfather now, raising my granddaughter because... not important, lol) tends to impress the young ones with what can be done with what is in one's head.
I am hopeful. But maybe also naive.
The other idea I like to link to these skills is "thought is action" type mastery. And, just to be clear, that is a state of clarity coupled with having grokked[0] something valuable.
Having realized that state many times in my past makes it easier today. I can go there and perform, doing or dealing with whatever it is efficiently and effectively.
It also can mean agency where ones peers may well lack it.
[0] - ...having achieved a state of understanding so complete it is a part of us, who we are, automatic, almost instinct.
The take things apart has become obfuscated by modern technology design. Most devices these days just drop one of a dozen chips into a circuit board as it's brains. Program it, and then ship it. More or less black boxing the device. It's one of the reasons the supply chain hit a lot of tech hard, those few chips were low cost and plentiful ... Until the knock on effects of the supply chain issues happened and they weren't as profitable to make compared to the high prices chips companies like auto manufacturers were bulk ordering for a premium.
Fascinating few years, a whole generation of designs rendered unusable - complete designs trying to be reworked for the chips they could get a hold of.
Fascinating few years, a whole generation of designs rendered unusable - complete designs trying to be reworked for the chips they could get a hold of.
My state runs a virtual school online that offers a lot of math classes. That's what my son did all through the middle grades since he was above what was normally offered at the school. This allowed him to do the AP classes for calc and such in high school early and then do college levels of math later in high school. You can definitely do more home school or khan academy stuff as an alternative or in conjunction, but the nice thing about the virtual school is it's state run and certified just like any other public school, so the classes have no problem being added to the transcripts with his normal classes.
www.houseofmath.com might be fun
Maybe you should make more.
Kids.
Because this is amazing
Kids.
Because this is amazing
We (I am Dove's husband) have 2 other children. The 7 year old builds electric circuits (takes after his grandpa who worked for NASA and HP before becoming a pastor), and the 2 year old mostly makes engine noises while driving his monster trucks around the living room.
The seven year old can make an electric motor out of spare parts (batteries, magnets, wires), and routinely makes electromechanical Minecraft contraptions that I don't comprehend.
We just hope they choose to use their powers for good.
We just hope they choose to use their powers for good.
My 7 y.o. can explain a motor but we haven't tried building one, I was eyeing him sets online and was between a few.
I built him a loop in MineCraft redstone to power his randomly-timed spooky lights in his "the backrooms" (SCP), but I don't know how to use the newer redstone stuff and haven't found a (educational) use-case to built him anything with the redstone torch/NOT logic gate etc.
My thirteen-year-old got tangled up in his dress shirt the other day.
Mine started with homopolar motors! They're fun and easy and great kid-level physics and engineering.
I can also recommend Snap Circuits for Christmas. :)
I can also recommend Snap Circuits for Christmas. :)
Can I say it one more time? Get him building eurorack modules. It's like legos for sound. So many possible combinations! Everything is voltage controlled, so the output of anything can be inputs for anything.
And you get to build lots of cool circuits. Designing your own isn't hard, you can get really far with just op-amps and resistor/capacitor circuits.
some of the resistors are potometers, of course. So you have some knobs to turn.
But it is usually bog-standard chips. op-amp, simple gates, 8-bit buffers.
And you get to build lots of cool circuits. Designing your own isn't hard, you can get really far with just op-amps and resistor/capacitor circuits.
some of the resistors are potometers, of course. So you have some knobs to turn.
But it is usually bog-standard chips. op-amp, simple gates, 8-bit buffers.
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Love this. This is art and STEM and love.
I'm wondering why this is a surprise, given GoL has been known to be Turing complete for some time, and why one couldn't just bootstrap oscillators of all periods off of some kind of Turing machine construction.
https://hackaday.com/2020/11/21/a-computer-in-the-game-of-li...
Incidentally, the computer in the previous link reminds me a lot of something I've seen implemented in my favorite CA, Wireworld: https://en.wikipedia.org/wiki/Wireworld
https://hackaday.com/2020/11/21/a-computer-in-the-game-of-li...
Incidentally, the computer in the previous link reminds me a lot of something I've seen implemented in my favorite CA, Wireworld: https://en.wikipedia.org/wiki/Wireworld
Note that each single "step" of the Turing Machine may take n steps of advancing the GoL cellular automaton itself. This means that only periods of n*i where i is an integer corresponding to the number of steps made by the TM, are guaranteed to exist. Worse even, I don't think that a proof of Turing completeness requires there to be any time bounds on the machine, so n might not even be a constant... If that's the case, then Turing completeness only proves that there exists an infinite number of periods of various lengths, but nothing about how comprehensively they "cover" all possible periods.
In other words, Turing completeness is enough to simulate GoL within GoL, but says nothing about the existence of the p41 oscillator
I believe Turing completeness means you can simulate all periods divisible by some number(s) n, but the individual steps in the Turing machine construction would take multiple real steps.
Turning complete does not mean it can finish all operations in specific limited time/step. It only means it can finish the computable calculations "eventually".
But wouldn't Turning completeness allow you to construct any computations, including the ones that will finish operations in specific limited steps?
So if you know how to construct the missing oscillator, great, you can simulate it in GoL run on your GoL-TM (just slower than if you do it on hardware). That doesn't help discovering the unknown oscillator.
As an aside, years into the use of computers to search for oscillators, humans were still beating machines to the punch in discovery of some of the unknown period oscillators, as the paper details.
As an aside, years into the use of computers to search for oscillators, humans were still beating machines to the punch in discovery of some of the unknown period oscillators, as the paper details.
No, your computer is Turning completeness. But it does not mean it will be able to encode a giant AV1 video in a night without dedicated hardware. Although it will do it eventually (probably a week or so).
This was my thought. It would imply that a Turing-complete machine should be able to solve NP problems in O(1) time.
Ah yea, good point!
Where does it say it's a surprise?
On the contrary, in summary it says: "The search has finally ended ..."
On the contrary, in summary it says: "The search has finally ended ..."
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.
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]."
I was speculating two oscillators with periods p and q could be composed (as long as there was no way for them to interact) to create an oscillator of period p*q/gcd(p,q) but wondering why large primes wouldn't be a problem.
I guess this is my answer.
I guess this is my answer.
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.
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.
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.
As a fellow numberphile viewer, I accept your authority on the topic, and will probably repeat this guess as fact later.
I don't think that's surprising, because surely there are a lot more possible ways to get larger periods than small periods? The amount of possible interactions / mechanisms in play is more limited with small periods. I think that groups of periods over some theshold numbers would also yield more readily to "constructive" approaches where you can "build up" the period using oscillators of a smaller period as components.
for example, search for constellations that send gliders back and forth through empty space.
If you have one of period p, you can make one of period p + 8n for every n in Ν by moving the guns further away from each other. Find a few guns that take longer to ‘reload’ upon getting ‘hit’.
Search on, and hopefully, you can find the 7 other modulos.
And it need not be glider guns. It could also be constellations that send a ‘ripple’ through occupied space back and forth.
I guess such ones can be made with smaller periods, but may require larger constants than 8 because you typically can’t move them further apart by a single cell.
Edit: another comment mentions https://conwaylife.com/wiki/P43_Snark_loop, a single construct that can be tweaked to have any period ≥ 43.
If you have one of period p, you can make one of period p + 8n for every n in Ν by moving the guns further away from each other. Find a few guns that take longer to ‘reload’ upon getting ‘hit’.
Search on, and hopefully, you can find the 7 other modulos.
And it need not be glider guns. It could also be constellations that send a ‘ripple’ through occupied space back and forth.
I guess such ones can be made with smaller periods, but may require larger constants than 8 because you typically can’t move them further apart by a single cell.
Edit: another comment mentions https://conwaylife.com/wiki/P43_Snark_loop, a single construct that can be tweaked to have any period ≥ 43.
You can build a Turing Machine in the GoL. Then program a loop. The period needed to execute the loop is rather long (TMs in life aren't very fast), but every period beyond that is trivially just a different TM program. Finding oscillators shorter than the shortest looping TM takes is harder.
would have been great if the last one was 42.
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
https://conwaylife.com/wiki/P43_Snark_loop
That doesn't make any sense to me. A 6-cycle oscillator next to a 7-cycle oscillator should form a 42-cycle oscillator, or am I missing something?
This paper is primarily considering nontrivial oscillators, i.e., oscillators in which at least one cell oscillates at the full period, not just some divisor. Still, there are many nontrivial "LCM" oscillators that can be constructed with two lower-period oscillators that just slightly interact. Such an LCM oscillator for p42 ("unix on 44P7.2") could have been constructed for decades; the actual last nontrivial periods to be discovered were p34, p38, p19, and p41.
If you care for 42 to be special, here's video that might be of interest to you.
The Answer to the Ultimate Question of Life, the Universe and Everything :
https://www.youtube.com/watch?v=ZFxT0sObrYQ
The Answer to the Ultimate Question of Life, the Universe and Everything :
https://www.youtube.com/watch?v=ZFxT0sObrYQ
Or if 42 was the only impossible period.
Off by one
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.
People be watching this and still argue "math is invented" smh
Some math is actually invented, like transfinite numbers.
But feel free to empirically construct one of those...
I'm familiar with the underlying concepts in philosophy, but I can't be sure what _you_ mean here. Care to explain?
Math is an empirical science.
If it is a science, it can't have theorems.
Empiricism is often involves in mathematical discoveries but the key difference is math isn't reliant on empiricism and at least in theory the current body of mathematical knowledge can reasoned entirely from simple axioms.
Empiricism is often involves in mathematical discoveries but the key difference is math isn't reliant on empiricism and at least in theory the current body of mathematical knowledge can reasoned entirely from simple axioms.
Why do you say this? What do you mean?
You recognize that "empirical science" has a clear meaning to most scientists, right? This idea is quite different than formal reasoning (e.g. deduction over theorems).
You recognize that "empirical science" has a clear meaning to most scientists, right? This idea is quite different than formal reasoning (e.g. deduction over theorems).
I might be misunderstanding what they accomplished here - doesn't this mean that oscillators needed to be found for all prime number periods?
for me, random coincidence that this popped up today, I just published a little holiday-themed GoL variant last week here: https://52games52weeks.com/gameofchristmas
for me, random coincidence that this popped up today, I just published a little holiday-themed GoL variant last week here: https://52games52weeks.com/gameofchristmas
No, because increasing the width or height of this loop by one cell increases the period by one.
https://conwaylife.com/?rle=36b2o$35bobo$29b2o4bo$27bo2bo2b2...
https://conwaylife.com/?rle=36b2o$35bobo$29b2o4bo$27bo2bo2b2...
From the abstract:
""" ... At the turn of the millennium, only twelve oscillator periods remained ... . The search has finally ended, with ... the final two periods, 19 and 41, ... """
Note that 19 and 41 are prime.
I'm not familiar with this line of research in the GoL (nor most others) but I assume that it was proved all prime periods (above 41, say) have been known since the turn of the century, or thereabouts.
I suspect what happened is that there were "easy" constructions for large periods that could be done. I would think that once you have the freedom of large periods, you can construct large gadgets and thus prove with "relative ease" that you can get any large prime number period.
My bet is that smaller periods are harder because of the smallness restriction.
""" ... At the turn of the millennium, only twelve oscillator periods remained ... . The search has finally ended, with ... the final two periods, 19 and 41, ... """
Note that 19 and 41 are prime.
I'm not familiar with this line of research in the GoL (nor most others) but I assume that it was proved all prime periods (above 41, say) have been known since the turn of the century, or thereabouts.
I suspect what happened is that there were "easy" constructions for large periods that could be done. I would think that once you have the freedom of large periods, you can construct large gadgets and thus prove with "relative ease" that you can get any large prime number period.
My bet is that smaller periods are harder because of the smallness restriction.
I think that beyond a certain period you have so much logic at your disposal that you can construct two organisms sending messages to each other and vary the distance in increments of one to get any period you want.
You're basically correct, though it's usually done with 4 mechanisms making an adjustable glider loop, which can have any period of 43 or more just by increasing the spacing: https://conwaylife.com/wiki/P43_Snark_loop
Right. If you want to have an oscillator with period $pq$ where $p$ and $q$ are primes, then you can just use independent oscillators with period $p$ to one with period $q$. They'd already discovered a construction for $p \geq 61,$ and most of the values below that, so only 19 and 41 were left.
You also need all powers of primes.
I loved reading this. One bug -the pattern that p11 links to is missing two live cells on the bottom still life, thus doesn't oscillate. The image of it in the pdf appears to be he correct pattern though.
https://conwaylife.com/?rle=10b2o$10b2o$5bo10bo$4bobo8bobo$3...
https://conwaylife.com/?rle=10b2o$10b2o$5bo10bo$4bobo8bobo$3...
Can anyone speak to the significance of this?
"Conway's Game Of Life" is a game of cells on a 2d grid that live/die/reproduce based on how many neighbors they have.
As a result of the game rules, you can get simple behaviors like "everything dies" or "everything is stable". But you can also get more complex behaviors, like things growing for thousands of turns before eventually collapsing and then stabilizing into a few fragments here and there. And you can get behaviors that don't stabilize -- like shapes that evolve in a way that after a certain number of turns the whole shape has moved along the grid ("gliders" or "spaceships") or oscillators of various periods.
Quite a while back there was a loop discovered that allowed for a period 43 oscillator, which could be adjusted by just moving things farther apart and therefore allowed every period of 43 or more. And oscillators of most smaller periods had been discovered -- but 19 and 41 were still unknown, up until both were discovered in rapid succession by 2 different people. So now we know how to make an oscillator for any given period within Conway's Game of Life.
As a result of the game rules, you can get simple behaviors like "everything dies" or "everything is stable". But you can also get more complex behaviors, like things growing for thousands of turns before eventually collapsing and then stabilizing into a few fragments here and there. And you can get behaviors that don't stabilize -- like shapes that evolve in a way that after a certain number of turns the whole shape has moved along the grid ("gliders" or "spaceships") or oscillators of various periods.
Quite a while back there was a loop discovered that allowed for a period 43 oscillator, which could be adjusted by just moving things farther apart and therefore allowed every period of 43 or more. And oscillators of most smaller periods had been discovered -- but 19 and 41 were still unknown, up until both were discovered in rapid succession by 2 different people. So now we know how to make an oscillator for any given period within Conway's Game of Life.
And why does that matter? What do we now know as a result? Or what interesting properties follow?
It's quite interesting in its own context, otherwise, you may not be interested.
Yes, yes. I need help appreciating it :)
> ...in the decades since the creation of Life.
Beautifully put.
Beautifully put.
shame john wasn't here to see it.
The last time I saw John, he was talking at a Mathematical Sciences Research Institute donor dinner. I was audience filler, after consulting on "A Beautiful Mind". He was describing games from his book with Berlekamp, and needed some fresh meat to play him in front of everyone. No one budged. The director gave me a certain look, and I volunteered. One of those "sticks on paper" games that never interested me, I was slaughtered. It all went very fast, some in the audience thought I had actually won a game counted against me. I got the biggest laugh when trying to outrun a certian loss, I enlarged the game board.
>> Life ultimately became way too popular for Dr. Conway’s liking. Whenever the subject came up, he would bellow, “I hate Life!” But in his final years he learned to love Life again. He narrated a documentary, with the working title “Thoughts on Life,” by the Brooklyn-based mathematician and filmmaker Will Cavendish, exploring the deterministic Game of Life versus the Free Will Theorem, a result Dr. Conway proved with his Princeton colleague Simon Kochen.
“I used to go around saying, ‘I hate Life,’” Dr. Conway says in the film. “But then I was giving a lecture somewhere, and I was introduced as ‘John Conway, Creator of Life.’ And I thought, ‘Oh, that’s quite a nice way to be known.’ So I stopped saying ‘I hate Life’ after that.” <<
https://www.nytimes.com/2020/12/28/science/math-conway-game-...
“I used to go around saying, ‘I hate Life,’” Dr. Conway says in the film. “But then I was giving a lecture somewhere, and I was introduced as ‘John Conway, Creator of Life.’ And I thought, ‘Oh, that’s quite a nice way to be known.’ So I stopped saying ‘I hate Life’ after that.” <<
https://www.nytimes.com/2020/12/28/science/math-conway-game-...
To help me get a grasp on the meaning and significance of this finding, could anyone provide examples of things outside the domain of cellular automata that are omniperiodic?
for whomever needs to hear it, on linux, xscreensaver has a great variant on conway's game of life built in as an optional package
I'm surprised it took this long. Intuitively, it should be possible to show that the smallest setup that has a period of 19 cannot be larger than some area related to 19, and therefore ought to be long but not intractable to enumerate. Maybe I'm off and it's really too big already.
The p19 in the paper is 31x21, but let it be 19x19.
That's 361 cells, so 4.7x10^108 'soups' to search.
The paper details the evolution of search methods. Brute force-ish methods were responsible for discovery of some lower period oscillators.
That's 361 cells, so 4.7x10^108 'soups' to search.
The paper details the evolution of search methods. Brute force-ish methods were responsible for discovery of some lower period oscillators.
Glad we figured that out. So how’s the search for unified field theory going?
Not so great, all our greatest thinkers are working on advertising.
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