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{"id":38872,"date":"2024-01-19T21:58:21","date_gmt":"2024-01-19T21:58:21","guid":{"rendered":"https:\/\/scienceandnerds.com\/2024\/01\/19\/maths-game-of-life-reveals-long-sought-repeating-patterns\/"},"modified":"2024-01-19T21:58:22","modified_gmt":"2024-01-19T21:58:22","slug":"maths-game-of-life-reveals-long-sought-repeating-patterns","status":"publish","type":"post","link":"https:\/\/scienceandnerds.com\/2024\/01\/19\/maths-game-of-life-reveals-long-sought-repeating-patterns\/","title":{"rendered":"Math\u2019s \u2018Game of Life\u2019 Reveals Long-Sought Repeating Patterns"},"content":{"rendered":"

Source:https:\/\/www.quantamagazine.org\/maths-game-of-life-reveals-long-sought-repeating-patterns-20240118\/#comments<\/a><\/br>
\nMath\u2019s \u2018Game of Life\u2019 Reveals Long-Sought Repeating Patterns<\/br>
\n2024-01-19 21:58:21<\/br><\/p>\n

\n

Throughout the 1970s, mathematicians and hobbyists filled in the other short periods and found a smattering of longer ones. Eventually, mathematicians discovered a systematic way to build long-period oscillators. But oscillators with periods between 15 and 43 proved tough to find. \u201cPeople have been trying to figure out the middle for years,\u201d said Maia Karpovich<\/a>, a graduate student at the University of Maryland. Filling in the gaps forced researchers to dream up a slew of new techniques that pushed the boundaries of what was thought possible with cellular automata, as mathematicians call evolving grids like Life.<\/p>\n

Now Karpovich and six co-authors have announced in a December preprint<\/a> that they have found the last two missing periods: 19 and 41. With those gaps filled, Life is now known to be \u201comniperiodic\u201d \u2014 name a positive integer, and there exists a pattern that repeats itself after that many steps.<\/p>\n

The burgeoning community devoted to studying Life, which includes many research mathematicians but also many hobbyists, has found not only oscillators but all kinds of new patterns. They have found patterns that travel across the grid, dubbed spaceships, and patterns that build other patterns: guns, constructors and breeders. They found patterns that compute prime numbers, and even patterns that can execute arbitrarily complicated algorithms.<\/p>\n

Oscillators with periods shorter than 15 can be found manually or with rudimentary algorithms that search for oscillators one cell at a time. But as the period gets bigger, so does the complexity, making brute-force searches far less effective. \u201cFor small periods, you can search directly,\u201d said Matthias Merzenich, a co-author of the new paper who discovered the first period-31 oscillator in 2010. \u201cBut you can\u2019t really go beyond that. You can\u2019t just pick a period and search for it.\u201d (Merzenich earned his doctorate in math from Oregon State University in 2021, but currently works on a farm.)<\/p>\n

In 1996, David Buckingham, a Canadian freelance computer consultant and Life enthusiast who had been searching for patterns since the late 1970s, showed that it was possible to construct oscillators of period 61 and higher by sending a pattern around a closed track in an endless loop. By controlling the length of the loop \u2014 and the time it took the pattern to complete one round trip \u2014 Buckingham found that he could make the period as large as he liked. \u201cIt\u2019s chemistry without the funny smells or broken glassware,\u201d he said. \u201cLike building compounds and then exploring the interactions between them.\u201d This meant that, in one fell swoop, he had come up with a way to construct oscillators of arbitrarily long periods, as long as they were longer than 61.<\/p>\n

There was a slew of results in the mid-1990s, when many of the missing oscillators between 15 and 61 were discovered through creative combinations of known oscillators, which had been given a swath of colorful names. Caterers were combined with traffic lights, volcanoes spit out sparks, and eaters ate gliders.<\/p>\n

By the turn of the 21st century, only a dozen periods were still outstanding. \u201cIt seemed very possible to solve this problem,\u201d Merzenich said. In 2013, a new discovery called the Snark loop improved on Buckingham\u2019s 1996 technique and lowered the cutoff above which it was easy to construct oscillators from 61 to 43. This left only five missing periods. One more was discovered in 2019, and two more in 2022, leaving only 19 and 41 \u2014 both prime. \u201cPrimes are harder because you can\u2019t use small-period oscillators to construct them,\u201d Merzenich said.<\/p>\n

Mitchell Riley, a postdoctoral researcher at New York University Abu Dhabi and another co-author of the new paper, has long been intrigued by a type of oscillator called a hassler. \u201cThe way hasslers work is, you\u2019ve got an active pattern in the middle and some stable stuff on the outside that reacts with it,\u201d Riley explained. The stable stuff, called a catalyst, is there to nudge the active pattern back into its original state.<\/p>\n

Designing them is hard. \u201cAll of these patterns are incredibly fragile,\u201d Riley said. \u201cIf you put a single dot out of place, they usually just explode.\u201d<\/p>\n

Riley created a program called Barrister to look for new catalysts. \u201cWhat we\u2019re looking for are still lifes that are robust. The whole point is we want them to interact with what\u2019s happening in the middle and then recover,\u201d Riley said.<\/p>\n

Riley fed catalysts that Barrister found into another search program that paired them with active patterns. This mostly led to failures, he said. \u201cIt\u2019s fairly rare that one of these catalysts survives the interaction. There\u2019s no guarantee of success. You just sort of cross your fingers and hope that you hit the jackpot. It feels a bit like gambling.\u201d<\/p>\n

Eventually, his bet paid off. After a few near misses \u2014 and a modification to the code that expanded the search to include symmetric patterns \u2014 he found a catalyst interaction that could sustain a period-19 oscillator. \u201cPeople had been trying all kinds of really complicated searches with lots of catalysts and lots of rare active things in the middle, but all that was necessary was finding this new chunky catalyst,\u201d Riley said.<\/p>\n

The final missing period, 41, was found by Nicolo Brown, another co-author, who is still an undergraduate math major at the University of California, Santa Cruz. Brown used gliders as catalysts, an idea first proposed by Merzenich.<\/p>\n

\u201cWe have discovered so much deep behavior over the last 10 years,\u201d Karpovich said. \u201cEveryone\u2019s celebrating for a week \u2014 and then moving on to other things. There are so many other problems to solve.\u201d Can oscillators of a given period be made smaller? Can oscillators be found in which every single cell oscillates? Can guns be made with particular periods? Can spaceships be made to travel at particular speeds?<\/p>\n

As Buckingham put it, \u201cIt\u2019s like being a kid in an infinite toy store.\u201d<\/span><\/p>\n<\/div>\n

<\/br><\/br><\/br><\/p>\n

Uncategorized<\/br>
\n<\/br>
\nSource:
https:\/\/www.quantamagazine.org\/maths-game-of-life-reveals-long-sought-repeating-patterns-20240118\/#comments<\/a><\/br><\/br><\/p>\n","protected":false},"excerpt":{"rendered":"

Source:https:\/\/www.quantamagazine.org\/maths-game-of-life-reveals-long-sought-repeating-patterns-20240118\/#comments Math\u2019s \u2018Game of Life\u2019 Reveals Long-Sought Repeating Patterns 2024-01-19 21:58:21 Throughout the 1970s, mathematicians and hobbyists filled in the other short periods and found a smattering of longer ones. Eventually, mathematicians discovered a systematic way to build long-period oscillators. But oscillators with periods between 15 and 43 proved tough to find. \u201cPeople have been […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","om_disable_all_campaigns":false,"pagelayer_contact_templates":[],"_pagelayer_content":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-38872","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"\nMath\u2019s \u2018Game of Life\u2019 Reveals Long-Sought Repeating Patterns - Science and Nerds<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/scienceandnerds.com\/2024\/01\/19\/maths-game-of-life-reveals-long-sought-repeating-patterns\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Math\u2019s \u2018Game of Life\u2019 Reveals Long-Sought Repeating Patterns - Science and Nerds\" \/>\n<meta property=\"og:description\" content=\"Source:https:\/\/www.quantamagazine.org\/maths-game-of-life-reveals-long-sought-repeating-patterns-20240118\/#comments Math\u2019s \u2018Game of Life\u2019 Reveals Long-Sought Repeating Patterns 2024-01-19 21:58:21 Throughout the 1970s, mathematicians and hobbyists filled in the other short periods and found a smattering of longer ones. 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