Phi line

A phi line is either of two patterns constructed by Brett Berger on October 2, 2022. The first produces a single glider stream with an irrational bit density of 1/φ. The second emits gliders in a line whose slope is φ (the golden ratio). Both patterns incorporate parts of an extensible 8-engine Cordership discovered earlier that same day.

x = 235, y = 185, rule = B3/S23 129bo$128b3o$128bo2bo$128b3o$128b3o14b2o$127bob2obo12b2o$128bo2b3o$ 128b4ob2o$129bob3o$131b3o10bo$132b2o10bo$145bo$143bobo7b2o$128b2o16bo 6b2o$128b2obo$128b4o12bobo$130bo14b2o2$144bo$143bo$142b3obo14b2o$142b 2o2b2o13b2o$142b2o2b2o2$133b2o7b2o6b2o$133b2o7bo4bo2b2o$142bo4bo$105bo 37bobo$101b4ob2o5b3o53b2o$100b2ob3ob2o4b3o53b2o$101bob3obo7b2o37b2o$ 102bo3b4o4b2o37bo$105b2ob3o42bo2bo$106b5o13b2o25b2ob4o$107bo16b2o13b2o 9bo20b3o$139b2o9bo3b3o2bo$150bo2bo2bob3o3bobo10b2o$152b2o4b3o4bo11b2o$ 154b2obobo$134b2o19b2obo14bo$134b2o22bo12b4o$170bo3bo$120b3ob3o43bo2bo $112bo6b4obo2bo43b3o6bo4bo$109b2o4bo2bobo58bob2o2bo$104b2o5b2o3bo10bo 53b3o3bo$104b2o7bob2o3b3o55b3o2b2o2b2o$121b2o2b2o55b2o2bob2o$185bo2bo$ 182b2ob3o$124b2o8b3o46b4o$124b2o7bo26b2o$133bo3bo22b2o11bobo$112b2o17b 2o3b2o35bo2bo$112b2o16bo2bobo2bo34bo2bo$130bo2bobo2b2o34b2o$132b2ob2o 2bo$133bo4bo60b2o$136b2ob2o57bobo$135b4o29b2o29bo$136b2o13b2o15b2o11bo $120b2o29b2o28bo$120b2o59bo2$136bo$137bo$136bo2$159b2o$29bo98b2o29b2o$ 28b3o97b2o11b2o$28bo2bo102bo5bo2bo$28b3o103bo5bo2bo$28b3o14b2o87bo4b2o b2o8b3o$27bob2obo12b2o93b2o13bo$28bo2b3o118bo2bo$28b4ob2o118b2o$29bob 3o102b2o$31b3o10bo91b2o8bo$32b2o10bo99bobo$45bo97bo2bo$43bobo7b2o89b2o 14b3o$28b2o16bo6b2o89b2obobo$28b2obo113b3ob2o$28b4o12bobo99bo3bo$30bo 14b2o96b2o3b3o$147bob2o$44bo100b2o2bo$43bo102b3o$42b3obo14b2o84bo$42b 2o2b2o13b2o$42b2o2b2o2$33b2o7b2o6b2o$33b2o7bo4bo2b2o$42bo4bo$5bo37bobo $b4ob2o5b3o53b2o$2ob3ob2o4b3o53b2o87bo$bob3obo7b2o37b2o101bobo$2bo3b4o 4b2o37bo103b2o$5b2ob3o42bo2bo$6b5o13b2o25b2ob4o$7bo16b2o13b2o9bo20b3o$ 39b2o9bo3b3o2bo$50bo2bo2bob3o3bobo10b2o$52b2o4b3o4bo11b2o$54b2obobo$ 34b2o19b2obo14bo$34b2o22bo12b4o$70bo3bo$20b3ob3o43bo2bo$12bo6b4obo2bo 43b3o6bo4bo$9b2o4bo2bobo58bob2o2bo$4b2o5b2o3bo10bo53b3o3bo$4b2o7bob2o 3b3o55b3o2b2o2b2o10bo$21b2o2b2o55b2o2bob2o7bobo$85bo2bo9b2o$82b2ob3o$ 24b2o8b3o46b4o$24b2o7bo26b2o$33bo3bo22b2o11bobo$12b2o17b2o3b2o35bo2bo$ 12b2o16bo2bobo2bo34bo2bo83b2obo6bo$30bo2bobo2b2o34b2o84b2ob3obobobo$ 32b2ob2o2bo116b2o8b2obo2bo10b2o$33bo4bo60b2o55bo3b5o3bobobo10bobo31b2o $36b2ob2o57bobo57bo2bo3b2obob2o11bo17b2o5b3o6bo$35b4o29b2o29bo57b3o4bo 2bo33b2o5bobo4bobo$36b2o13b2o15b2o11bo63b2o9bo6b2o2bo39bo3b2o2b2o5b2o$ 20b2o29b2o28bo64bo9b2o5b2o8b2o32bo3b2o9bobo$20b2o59bo64bobo17bob5obo 32bo4bo10b3o$147b2o16b7o33bob2o2bobo6b3ob2o3bo$36bo130b4o4bo24b2o2bobo 2b5o6b2ob2o3bobo$37bo69b2o42bo3b3o17bo22b2obo2bo5bo3bo4b2o2b2o2bo3bo$ 36bo55b3o11bo43b7obo15bo23bo2b2o17bobo5bo3bo$92bobo15bo37b2obobob2obo 16b3o20bo3b4o6bo8bo7bo3bo$59b2o31bobo15bo23b2o7bo4b2o3bo2b3o18bo21bo5b 2o6b2o15bo3bo$28b2o29b2o31bo4b2o9bo2bo30bobo3b3ob2obob2o41bo2b3obo4bob 2o8bo6bobo$28b2o11b2o48b3o3b2o10b3o20bo3bo6bo6bobo2bob2o14bo27b4o6b2ob 2o7bobo6bo$34bo5bo2bo53bo12b2o20bo4bo14bo2bo2bo12bobo16bo11bob2o2bo3bo bo9bo$34bo5bo2bo49b4o13bo23bobobo6bo8b2o2bo13b2o16b3o10bo9bo$34bo4b2ob 2o8b3o39b2o12bob2o23bobobo3b3o10b2o35bo10b4o$40b2o13bo38bo13b4o24bo4bo 14bo8b2o25bo12bo$52bo2bo52bobobo6bo17bo3bo25bo24bo4b4o$53b2o40bo13b2ob 2o4bobo46bo10bo17b7o16b2o$36b2o55bobo12bo3bo6b2o17b2o24b3o11b2o13bob5o bo17bobo$36b2o8bo45b4o13b3o19bo12b2o31bobo13b2o8b2o5b2o9bo$44bobo45b3o 15bo18bobo12b2o5b2o47bo2b2o6bo9b2o$43bo2bo47bo35b2o18bobo12b2o33bo2bo 4b3o$44b2o14b3o87bo14b2o29b2obob2o3bo2bo$44b2obobo99b2o44bobobo3b5o3bo $45b3ob2o144bo2bob2o8b2o$46bo3bo56b3o86bobobob3ob2o$43b2o3b3o57bobo86b o6bob2o$47bob2o56bo2bo78bo$45b2o2bo57bo2bo76bobo$46b3o48b2o8bobo4b2o 72b2o$47bo49b2o8bo2bo3b2o$108b2o$70bo38bo$66b5o7b2o$66bo3bobo4b3o$66b 3o3b3o3b3o$72b2o2b3o$72bo$69b3o16b2o$58bo10b2o17b2o13b2o$57bobo43b2o$ 57b2o3$98b2o$98b2o49b2o$149b2o2b2o$66bo86bobo$65bo18bob4o64bo$65bo3bo 3b3obo6b2o4bo$70bo3bobobo5bo3bob2o59b2o4b2o$67b2ob7ob2o5b3obo62bo4b2ob o$69b3ob2o3bo70b3o9bo$75b3o71bo8bo$76bo82bob2o$88b2o71b2o$88b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 800 HEIGHT 700 ]]

Pattern that produces a southeast glider stream with a 1/φ density (click above to open LifeViewer) RLE: here Plaintext: here

x = 320, y = 261, rule = B3/S23 277b3o$275b7o6bo$274b9o4bo$273bo7b3o2b2o$274b3o6bo3bo2bo$275b2o11b3o$ 278bo10bo$279bo18b2o$280bo17b2o7$298b3o5b2o$295b2obobo5b2o$295bo2bobo$ 284bob2o5b3o$278b2o4bobo3b2ob2o$278b2o6b2ob3o2bo2$271bo9bo2bo$280bo3bo $267bo7bo8bo$267bo8bo2bo3b2o$266bo4b2o3bo3b2obo$267bobo6bo$267bob2o2$ 272bo9$278b2o$278b2o$262b3o$262bobo$262bobo$262bo4b2o$261b3o3b2o$267bo $263b4o19b2o$264b2o20b2o$172b3o89bo2$170bo5bo88bo$170b7o12bo73bobo$ 172bo17b2o70b4o$172bo3bo13b2o70b3o$171bo17b2obo71bo$172bo18b3o$173bo4b o11bob2o$173bo4bo11b2o60b3o6bo$172b2obobo12bo6b2o54b2o3b2obo$171bobo 22bo2bo52b2o5bobo$172bo24b2o58b3obo$260b2o2$230bo$230bo7bo$230bo5b3o$ 205b2o22b2o3bobo$187b3o14bo2bo27bo2bo$189bo15b2o27bo2b2o38bo$188b3o58b 2o24b2o$249b2o25b2o3$182bo65bo10b2o$174b3o3b4o29b2o33bo3bobo$174bobo3b 2ob2o27bo2bo30b2obob2o2bo$174b3o2bo33b2o32b6o2bo$179b2o3bo63bo5b2o$ 180b4o67bobo$181b3o54b2o3b2o6bo$152bo79bo3b4o3bobo$152bo5b3o71bob2o2bo 2bo3bo$151bobo3b2o2bo59b2o9bob2o4bo3b2o7b2o$152bo3bo5bo57bo2bo12bo3bo 12b2o$152bo3bo64b2o18bo11b2o$157bo2b2o$158b2o11b2o76b2o2bo$171b2o77b3o $251bo$237b2o$174bo54b2o6b2o7bobo$173b5o50bo2bo17bo$173bo2b3o50b2o18bo $168b2o8b2o69bo$175bo2bo67b2ob2o$165bo9b3o69bobo$155b2o9b2o7b2o71bo$ 154bob3o8b2o$154bo3bo10bo67b2o$155b2obo5b5o6b2o59bo2bo$156bo10bo6bo2bo 59b2o22bo$173b2ob2o81b2o$172bobo85b2o$171b2o$171bo3bo$172bo2bo$159b2o 11b3o70b2o$159b2o83bo2bo$171bo73b2o$171bo$168b2o$168b2o$168bo2bo$169b 3o$170bo82b2o$252bo2bo$160bo92b2o$159bobo$159bobo$159bo2bo$160bobo92b 2o$163bo91bobo$162b2o92b2o3b2o5b2o5b2o$162bo13b2o82bo2bo4b2o5bo$176b2o 83b2o10bobo$273b2o$269b2o$160bo108bobo$159bobo108bo$161bo$161bo73b3o 35b3o$161bo22b2o51bo35b3o$184b2o50bo36b3o$162bo113b3o$158b2ob3o112b3o$ 160b3ob2o110b3o$163bo$302b2o$286b2o14bo$153bo19b2o111b2o12bobo$152bobo 16bo128b2o5b2o$151b2ob2o15bo4bo130b2o$152bo2bo23bo107bo$154b3o13bo8bo 106bo26b3o$154bobo13b2obobobobo107bo6b2o11b2o3bo$155b2o15bo121b2o10bob o3bo3bo$155b2o46b3o71b3o14bo10b2o5bo2bobo$205bo71bo16b3o17bobo2bo$204b o73bo16bobo3b2o12bo3bo$295bo2bo4bo5bo9bo$152bobo141bobo4bo4bobo5b3o$ 152bo2bo119bo20b2o2bo2bo5bo$151b2ob2o119b3o16b3o3bob2o$134bo15b5o123bo 15b3o2b2o2bo$131b4o16b3o123b2o15b2o4b3o$152bo148bo$279b3o5bo16b2o$278b o3b5o2bo14bobo$134bobo141bob6o3bo5b2o9bo$116b3o6bo9bo13b2o128b4o4bo8bo 9b2o$115bo4bo3bobo6b2o13bo2bo131b3o2bo4b3o$115bo4b2o3bo7bo2bo11bo132b 2obob2o3bo2bo$115bo6bo9b2ob2o10b3o21b3o106bobobo3b5o3bo$116b3obo2bobo 21b2o24bo106bo2bob2o8b2o$120b5o25b2o20bo108bobobob3ob2o$125bo23bo2bo 129bo6bob2o$94bo7bo$93b3o5bobo43b2o$92b2ob4o2bo2bo42bo2bo$93b3o2bo49b 3o40b2o$94bo2b2o2bo88bobo$98b2obobo88bo$102bo10b2o18b2o$113b2o18b2o3$ 114b3o$114b3o$120b3o$118bo3bo$116b2obob2o18b2o$98bo17bo24b2o$97b4o6b2o 8b2o$78bo17bo4bo4b3ob2o$64b2o7b2o2bobo15b2o10b2ob3o4bo$64b3o6b4o3bo15b obob2o4b3ob2o3b4o40b2o$64b2obobo7bo19b3o6b2o7b2o2bo38bobo$67b2o10bob2o 15bo15bo4bo40bo$79bobo31b2obob2o$80bo8b2o21b4o$89b2o21b4o$115b2o$101b 2o13b2o$101b2o12bo2$111b3o$89b2o$87b3ob2o4b2o10bo4bo$87bobob2o4b2o10bo $85b2obo24b2o$78b2o3bo26b2ob2o$69b2o5bo3bobo29bo14b2o$69b2o6b2obob3o 41bobo$77bo3b4o43bo$75bo$74bobo$73bo$57bobo6bobo4bo$45b2o9bo9bobo4bobo $57bo2bo6bo6bo$44bo3bo10b3o$49bo$42bo7bo$41bo4bo$42bobobo4bo$42bo8bo$ 43bo4b2obo$44bob2ob3o43b2o$45b3ob2o43bobo$96bo4$24bobo17bobo$24b3o16b 5o$24b2o15b7o$39bo2b3o$39bobo$27bo11bobob3o$25bobo4b2o6b3ob2o$3bo21b3o 16b2o$4o24bo11b2o2bo$24b5o14b2o$17b3o20b3o20b2o$40b2o20bobo$3bobo12b2o 44bo$4bo13b2o$2b2o13bo2bo$2bo2bo11bo28b2o7bo$b2ob2o10b3o27b2o7bo$16b2o 37b2obo$19b2o37bo$18bo2bo31b2obo2bo$53b8o$16b2o36b4o3bo$16bo2bo39bobo$ 17b3o40bo4$6b2o$6b2o7$14b2o$14b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 800 HEIGHT 700 ZOOM 2 ]]

Pattern that emits a southwest glider stream in a line with slope φ (click above to open LifeViewer) RLE: here Plaintext: here

Puffer version

In 1999, Dean Hickerson constructed a pattern that produces a glider stream with density 1/(φ + 1) relative to a complete p120 glider stream, as shown below. Two puffers move east, producing a westward MWSS and a boat every 60 ticks. Two stationary guns produce a glider every 60 ticks and an eastward MWSS every 120 ticks. Half the gliders delete the MWSSs while the other half escape. But when a westward MWSS reaches the guns, it deletes 2 consecutive gliders, allowing one eastward MWSS to escape and suppressing one output glider. When an eastward MWSS hits a boat, it becomes a northeastward glider which deletes one westward MWSS. Thus, each westward MWSS that reaches the guns causes the deletion of a later westward MWSS. If the westward MWSSs are numbered starting at 3, then, for each integer n ≥ 2, MWSS number ⌊(n(φ))⌋ causes the deletion of MWSS number ⌊(n (φ +1))⌋. Also, a glider escapes from the bottom of the pattern about generation 120 ⌊(n (φ + 1))⌋. Relative to a complete p120 glider stream, the resulting stream of gliders has density 1/(φ + 1).

#N p60 version of Irrational 5 #C Two puffers move east, producing a westward MWSS and a boat #C every 60 gens. Two stationary guns produce a glider every 60 #C gens and an eastward MWSS every 120 gens; half of the gliders #C delete the MWSSs while the other half escape. But when a #C westward MWSS reaches the guns, it deletes 2 consecutive gliders, #C allowing one eastward MWSS to escape and suppressing one output #C glider. When an eastward MWSS hits a boat, it becomes a NEward #C glider which deletes one westward MWSS. Thus, each westward #C MWSS that reaches the guns causes the deletion of a later #C westward MWSS. Specifically, if the westward MWSSs are #C numbered starting at 3, then, for each integer K >= 2, MWSS #C number floor(K phi) causes the deletion of MWSS number #C floor(K (phi+1)), where phi = (1 + sqrt(5))/2. Also, a glider #C escapes from the bottom of the pattern about generation #C 120 floor(K (phi+1)). Relative to a complete p120 glider stream, #C the resulting stream of gliders has density 1/(phi+1) = #C (3 - sqrt(5))/2 ~ 0.381966. #O Dean Hickerson, 1/2/1999 x = 151, y = 91, rule = B3/S23 105b2o$103b2ob2o13b2o$103b4o13b4o$104b2o14b2ob2o$122b2o4$119b2o$113b3o 3bobo$92bo17b3o6bob2o$33bo59b2o15bo9b2o$32bobo57b2o16b3o7bo$15b2o15b2o bo$15bobo14b2ob2o3b2o39b2o$6bo11bo13b2obo5bo36b3ob2o37b2o$6b2o7bo2bo 13bobo43b5o29b4o4b4o7b2o$18bo14bo45b3o25bo3bo3bo4b2ob2o4b2ob2o13b2o$ 15bobo90bo6bo6b2o5b4o13b4o$15b2o87bo3bo5bo15b2o14b2ob2o$105b4o39b2o$ 100b3o$98b2ob2o$100b3o18bo$34bo70b4o13b2o$35bo41bo26bo3bo12b2o23b2o$ 33b3o42b2o28bo38b2o$12bobo62b2o28bo38b2o$10bo3bo101bo29bo$3b2o5bo15b2o 89b2o$3bo5bo16bo2bo86b2o$10bo19bo116b2o$10bo3bo15bo107b4o4b4o$12bobo 15bo99bo6bo3bo4b2ob2o$26bo2bo6b2o93bo9bo6b2o$26b2o8bobo88bo3bo8bo$38bo 89b4o$38b2o83b3o$30b2o68b3o18b2ob2o$30bo2bo65bobob2o18b3o$16bobo15bo6b 2o56bo3bobo22b4o$14bo3bo15bo7bo56bobo3bo21bo3bo$14bo19bo65b2obobo25bo$ 7bo5bo16bo2bo68b3o25bo$7b2o5bo15b2o22b4o$14bo3bo34b6o$16bobo34b4ob2o$ 57b2o57bo$117bo$51bo14b2o45bo3bo$49b3o10b4ob2o15b4o26b4o$48bo13b6o15bo 3bo21b3o$48b2o13b4o20bo19b2ob2o$86bo22b3o13b2o8bo$114b4o6b4o8bo$113bo 3bo6b2ob2o3bo3bo$68b2o28b2o17bo8b2o5b4o$68bobo27bobo15bo$9b2o23b2o33bo 29bo$9bobo22bo$2o10bo14bo4bobo70b2o17b2o$o8bo2bo13bobo3b2o70bobo16b2o 7b3o$12bo13b2obo76bo17bo2bo6bo$9bobo14b2ob2o94bo6bobo$9b2o15b2obo97b2o 3b2o$26bobo81b2o$27bo19b2o60bobo$47bo63bo23bo$136bo$132bo3bo$48bo67b4o 13b4o$17b2o27bob2o65bo3bo$17bobo26bo72bo$8b2o10bo14bo10bo71bo$8bo8bo2b o13bobo9bo2bo$20bo13b2obo9b2o$17bobo14b2ob2o4bo$17b2o15b2obo4b2o$34bob o$35bo31b2o9b2o$66b2o9bobo$61bo6bo7b3o$38bo21bobo12b3o$37bobo18b2o3bo 12b3o$37b2obo8bobo6b2o3bo13bobo5b2o$25b2o10b2ob2o6bo2bo6b2o3bo14b2o5bo bo$25bo11b2obo6b2o6b2o3bobo24bo$37bobo5b2o3bo3bo2bo3bo25b2o$38bo8b2o6b 2o$48bo2bo$49bobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 1000 HEIGHT 700 ]]

Pattern that produces a southeast glider stream with with density 1/(φ + 1) (click above to open LifeViewer) RLE: here Plaintext: here

Gallery

• 

  Line with slope φ after approximately 60 million ticks

See also

• Phi calculator