Conway's Game of Life is omniperiodic
arxiv.org
arxiv.org
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. :)
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. :)
Kids.
Because this is amazing
We just hope they choose to use their powers for good.
I can also recommend Snap Circuits for Christmas. :)
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.
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
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 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.
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.
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.
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...
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.
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
https://www.youtube.com/watch?v=yKbJ9leUNDE&list=PLoaVOjvkzQ...
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
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.
On the contrary, in summary it says: "The search has finally ended ..."
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.
The Answer to the Ultimate Question of Life, the Universe and Everything :
I guess this is my answer.
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.
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.
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).
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
""" ... 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.
https://conwaylife.com/?rle=36b2o$35bobo$29b2o4bo$27bo2bo2b2...
https://conwaylife.com/?rle=10b2o$10b2o$5bo10bo$4bobo8bobo$3...
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.
Beautifully put.
“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-...
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.