Our family celebrated the event with a custom t-shirt:https://www.teepublic.com/t-shirt/49108604-life-is-omniperio...
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
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.
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).
Ah yea, good point!
Where does it say it's a surprise?
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.
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.
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.
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.
This would only get you multiples of your "clock rate."
You don't want a clunky TM, in GOL terms just a 'signal conduit' with a reasonably short delay on the reflectors so the periodicity can be reduced.
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.
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 :
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.
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).
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
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...
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.
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...
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.
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.
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-...
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?
I'm not sure the concept is very meaningful outside of CA. There aren't really infinites in the real world, and most domains they'd be possible they are probably trivial. Perhaps outstanding combinatory or graph theory problems are analogous
Thanks, that’s helpful to know!
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.
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.
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
You should add attribution on the poster to GP as it is an original creation of theirs and their kid.
He did - see the replies to the top tweet.
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...
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. :)
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.
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?
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.
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.
You are a lucky parent!
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
Maybe you should make more.
Kids.
Because this is amazing
Love this. This is art and STEM and love.