Isotropic non-totalistic hexagonal rules with spaceships
Posted: December 22nd, 2025, 9:09 am
All things considered, the space of possible range-1 isotropic hexagonal rules isn't that much bigger than the space of outer-totalistic rules. Perhaps it might be worth trying to document as many spaceships as we can that only work in non-totalistic rules and don't in outer-totalistic rules.
Here's what I've found from existing threads so far - there must be others I've missed:
2c/4 orthogonal:
4c/8 orthogonal:
2c/6 orthogonal:
c/3 diagonal:
2c/6 diagonal:
4c/12 diagonal:
6c/18 diagonal:
12c/36 diagonal:
18c/42 diagonal:
36c/84 diagonal:
May be corderizable:
It'd be interesting to see if more oblique slopes are easier to find in non-totalistic rules, since in outer-totalistic rules only (2,1,-3) is known.
Here's what I've found from existing threads so far - there must be others I've missed:
2c/4 orthogonal:
Code: Select all
x = 2, y = 5, rule = B2o4p/S2mH
o$bo$bo$bo$bo!Code: Select all
x = 4, y = 5, rule = B2o/S2mH
2bo$3bo$o2bo$3bo$3bo!Code: Select all
x = 4, y = 5, rule = B2o3m4p/S2mH
2bo$bobo$o2bo$3bo$3bo!Code: Select all
x = 5, y = 5, rule = B2o3m4-m56/S2m3-m4-o56H
bobo$obobo$bob2o$3b2o$4bo!Code: Select all
x = 5, y = 7, rule = B2o3-o4-m56/S2m3o4-o56H
3bo$2b2o$bo2bo$obobo$bo2bo$4bo$4bo!Code: Select all
x = 7, y = 9, rule = B2o3m4-m56/S2m3-m4-o56H
4bo$3bobo$2bobobo$bobo2bo$obo3bo$bo4bo$6bo$6bo$6bo!Code: Select all
x = 2, y = 6, rule = B2-m4-m56/S2m34m56H
o$bo$bo$bo$bo$bo!Code: Select all
x = 5, y = 5, rule = B2/S2-mH
obobo2$3bo$3b2o$4bo!Code: Select all
x = 6, y = 4, rule = B2o3m/S2o4oH
2bo$o4bo$bobobo$3b2o!Code: Select all
x = 6, y = 5, rule = B2o4-m5/S2o4-m5H
obobo$ob2obo$2b3o$3b2o$4bo!Code: Select all
x = 8, y = 6, rule = B2/S2-mH
2bo2$bo3bo$o6bo$2bo3bo$2b2o2b2o!Code: Select all
x = 12, y = 6, rule = B2/S2-mH
2bo3bo2$bo7bo$o10bo$2bo3bo3bo$2b2o2b2o2b2o!Code: Select all
x = 16, y = 6, rule = B2/S2-mH
2bo3bo3bo2$bo11bo$o14bo$2bo3bo3bo3bo$2b2o2b2o2b2o2b2o!Code: Select all
x = 6, y = 6, rule = B2o4-m5/S2o45H
bo$o$ob2obo$2b3o$3b2o$4bo!Code: Select all
x = 6, y = 6, rule = B2o45/S2o45H
b2o$obobo$ob2obo$2b3o$3b2o$4bo!Code: Select all
x = 7, y = 9, rule = B2o3p45/S2o3p45H
2o$o$ob2o$3b2o$ob3obo$2b4o2$3b4o$4b3o!Code: Select all
x = 24, y = 24, rule = B2/S2-mH
21bo$17bo3bo$5bo16b2o$7bo10bo$6bob2o$8b2o10bo$3bo6bo11bo$2bo6bo$3bobo
4b2o$3bo$3b3obo2bo2bo$2bo3bo5bo$2o13bo$2o11bobobo$16bob2o$18b2o$13bo6b
o$12bo6bo$13bobo5bo$13bo$13b3obo$12bo3bo$10b2o$10b2o!Code: Select all
x = 272, y = 188, rule = B2o3m56/S2-p4oH
257bo$258bo$255bo2bo10b2o$258bo12bo$258bo5$226bo$227bo$225bo2bo31b2o$
229bo32bo$229bo4$247b2o$197bo51bobo5bo2bo$198bo52bo4bo3bo$195bo2bo51b
o2bo7bo$198bo51b2obobo2bo$198bo52bo3bo4bo$252bo4b3o$257bobo2$238b2o20b
obo$166bo73bo$167bo92bo$165bo2bo90bo$169bo$169bo3$251bo5bo2bo$229b2o20b
o4bo3bo$137bo93bo18bo2bo7bo$138bo111b2obobo2bo$135bo2bo112bo3bo4bo$138b
o113bo4b3o$138bo118bobo2$260bo$261bo$220b2o38bo$106bo115bo36bo$107bo$
105bo2bo$109bo$109bo2$252bo$252bo$211b2o40bo$77bo135bo$78bo$75bo2bo$78b
o$78bo2$211bo15bo15bo$72bo129bo8bo6bo8bo6bo8bo$72bo129bobo7bo5bobo7bo
5bobo7bo$73bo2bo123b3obo11b3obo11b3obo$74bo126b2o14b2o14b2o$60bo14bo126b
o15bo15bo$61b2o4$85bo$47bo37bo$48b2o35bo2bo$56bo2bo25bo$51b2o4bo2bo25b
o$50b2o2bo5bo$51b2obo4bo$52bobo4bobo139bo$53bo4bo142bo$58b2o36bo105bo
$38bo19bo37bo$39b2o16b3obo35bo2bo$60bobo35bo$58b2obo37bo3$176bo15bo$159b
2o15bo15bo$109bo51bo15bo15bo$29bo26bo2bo49bo53bo15bo$30b2o19b2o4bo2bo
48bo2bo50bob4o10bob4o$50b2o2bo5bo48bo53b3o3bo9b3o3bo$51b2obo4bo50bo52b
o4bo10bo4bo$52bobo4bobo101b2obobo10b2obobo$53bo4bo104bobobob3o7bobobo
b3o$58b2o105b2o14b2o$58bo91b2o17bobobo11bobobo$57b3obo58bo31bo17bo2bo
12bo2bo$20bo39bo59bo51bo15bo$21b2o35b2obo59bo2bo61bo$122bo63bo$123bo63b
o3$139bo$141b2o$51bo81bo2bo2b2o2bo$11bo40bo80bobo2bo$12b2o38bo80bobo6b
o34bo$133bo9bo33bo$134bo43bo$139b3o$138bo$139b2obo$140b3o$o9bo5bo9bo5b
o9bo98bo$3bo7bo7bo7bo7bo7bo$3bo7bo7bo7bo7bo7bo80bo43bo$bobo13bobo13bo
bo88bo43bo$b2o14b2o14b2o90bo43bo$139bo$141b2o$133bo2bo2b2o2bo$133bobo
2bo$135bo6bo$143bo$99bo15bo43bo$82b2o15bo15bo23b3o17bo$84bo15bo15bo21b
o21bo$86bo15bo36b2obo$86bob4o10bob4o32b3o$86b3o3bo9b3o3bo32bo$86bo4bo
10bo4bo$86b2obobo10b2obobo$86bobobob3o7bobobob3o$88b2o14b2o44bo$73b2o
17bobobo11bobobo26bo10bo$75bo17bo2bo12bo2bo28b2o8bo$95bo15bo21bo2bo2b
2o2bo$109bo23bobo2bo$109bo25bo6bo$110bo32bo2$139b3o$62bo75bo$64b2o73b
2obo$56bo2bo2b2o2bo73b3o$56bobo2bo79bo$58bo6bo34bo$66bo33bo$101bo$62b
3o$61bo$62b2obo$63b3o$64bo2$91bo$91bo$92bo$62bo$64b2o$56bo2bo2b2o2bo$
56bobo2bo$58bo6bo$66bo$82bo$62b3o17bo$61bo21bo$62b2obo$63b3o$64bo4$73b
o$62bo10bo$64b2o8bo$56bo2bo2b2o2bo$56bobo2bo$58bo6bo$66bo2$62b3o$61bo
$62b2obo$63b3o$64bo!Code: Select all
x = 585, y = 645, rule = B2o3m56/S2-p4oH
390bo$391bo$388bo2bo$391bo$391bo$403bo$404b2o3$359bo$360bo$358bo2bo$362b
o$362bo$394bo$395b2o3$330bo$331bo$328bo2bo$331bo$331bo$385bo$386b2o3$
299bo$300bo$298bo2bo$302bo$302bo$376bo$377b2o3$270bo$271bo$268bo2bo$271b
o$271bo3$362bo2b3o$362bob2o2b2o$239bo122bo2bobo$240bo121bo2bobo$238bo
2bo112bo7bo2bob2obo$242bo112b2o7bo3b2o$242bo122b2o3b2o$366bo3bobo$369b
o4bo3$210bo$211bo$208bo2bo133bo$211bo134b2o$211bo$362bo2b3o$362bob2o2b
2o$362bo2bobo$362bo2bobo$179bo182bo2bob2obo$180bo183bo3b2o$178bo2bo154b
o28b2o3b2o$182bo154b2o27bo3bobo$182bo186bo4bo5$150bo$151bo215bo$148bo
2bo175bo40bo$151bo176b2o38bo$151bo5$119bo$120bo207bo15bo13bo$118bo2bo
196bo15bo15bo8bo$122bo193bo2b2o5bobo3bo2b2o5bobo3bo2b2o6bo$122bo186bo
15bo4b2o9bo4b2o$317bo13bobo13bobo$313bo6bo9b2o4bo9b2o4bo$319bo15bo15b
o$316b2o14b2o14b2o$90bo$91bo225b2o14b2o14b2o$88bo2bo223bo15bo15bo$91b
o224b2ob2o11b2ob2o11b2ob2o$91bo225b2o14b2o14b2o3$69bo$69bo$70bo2bo$71b
o$65bo6bo$66b2o2$315bo$315bo$82bo233bo$82bo$82bo2bo195bo15bo$82bo$83b
o198bo15bo$283bo15bo$282b2obo12b2obo$52b2o2bo224bo2bo3bo8bo2bo3bo$51b
ob2ob2o215b2o9b2o2bo11b2o2bo$54b3obo34bo181bo9b3obo11b3obo$43bo8b4ob4o
32bo188b2o5bo8b2o5bo$44b2o7b3o38bo2bo185bob5o9bob5o$58bo2bo33bo188bo3b
o11bo3bo$59b3o34bo$60b2o223b5o11b5o3$264b2o$106bo159bo$34bo71bo$35b2o
69bo2bo190bo$106bo193bo$107bo193bo$52b2o2bo$51bob2ob2o197bo5bo2bo$54b
3obo196bo4bo3bo$52b4ob4o193bo2bo7bo$53b3o61bo136b2obobo2bo$25bo32bo2b
o55bo137bo3bo4bo$26b2o31b3o56bo2bo134bo4b3o27bo$60b2o57bo141bobo27bo$
120bo171bo$264bobo2$264bo$263bo$56bo73bo$16bo40bo72bo$17b2o38bo72bo2b
o148bo$130bo151bo$131bo123bo5bo2bo18bo$255bo4bo3bo$254bo2bo7bo$254b2o
bobo2bo$255bo3bo4bo$15bo15bo15bo93bo114bo4b3o$5bo10bo4bo10bo4bo10bo92b
o119bobo191bo$4b4o9bo2b4o9bo2b4o8bo93bo2bo127bo181bo$o2bo4bo7bob2o4bo
7bob2o4bo102bo120bobo6bo182bo$o2bo3bobo6bo2bo3bobo6bo2bo3bobo102bo129b
o148bo$o7bo7bo7bo7bo7bo223bo158bo$4bo3bo11bo3bo11bo3bo222bo158bobo$7b
o15bo15bo382bobo$6bo2bobo10bo2bobo10bo2bobo380bo$6b2o14b2o14b2o114bo$
4b2obo4bo7b2obo4bo7b2obo4bo109bo$6b2o14b2o14b2o114bo2bo97bo5bo2bo$154b
o100bo4bo3bo$155bo98bo2bo7bo$254b2obobo2bo$255bo3bo4bo$256bo2$165bo95b
o$165bo271bo$166bo2bo267bo$167bo270bo$168bo94bo99bo$363bo$362bobo$362b
obo$364bo$178bo$178bo73bo175bo$178bo2bo71b2o173bo$178bo80bo169bo$179b
o79bo$260bo2bo$262b2o3$189bo$189bo229bo$190bo2bo225bo$191bo228bo$192b
o110bo$303bo$302bobo$302bobo$304bo$202bo$202bo207bo$202bo2bo204bo$202b
o31bo6bo169bo$203bo29b2o4bobo$237bo4bo2bo$231bo6bo5b2o$231bo2b2ob2o$231b
ob3o$213bo19bobo4bo$213bo19bo4bo162bo$214bo2bo15b2o4bo161bo$215bo17b5o
bo162bo$216bo26bo$237b3o3bo$242bobo$242bobo$227bo16bo$226bobo$226bo2b
o162bo$226bo2bo162bo$226bo2bo163bo$227bo197b2o$427bo$229bo3bo194bo$231b
2obo194bo$215bo10b3obo2bobo191bo$216b2o8bob2o5bo$227bo7b2o146bo$227bo
3bo3bo147bo$233b2o149bo3$232b3o2$204bo11b2o$203b4o11bo$199bo2bo4bo7b4o
bo9bo15bo15bo15bo15bo15bo46bo2bo3bo9bo$199bo2bo3bobo20bobo3bo9bobo3bo
9bobo3bo9bobo3bo9bobo3bo9bobo3bo12b2o14b2o11bobobo3bo8bo$199bo7bo14bo
7bo6b2o7bo6b2o7bo6b2o7bo6b2o7bo6b2o7bo6b2o9bob2ob2o9bob2ob2o8bo3bo2bo
8bo$203bo3bo5bo2bo4bobo5bobobob2o2bo5bobobob2o2bo5bobobob2o2bo5bobobo
b2o2bo5bobobob2o2bo5bobobob2o2bo8bob2o3bo8bob2o3bo10bobo2bo$206bo6bob
o2bo10bob4o10bob4o10bob4o10bob4o10bob4o10bob4o14b2o14b2o14b4o$205bo2b
obo4bo6bo8bo6bo8bo6bo8bo6bo8bo6bo8bo6bo8bo6bo11bobo13bobo14bo213bo$205b
2o16bo15bo15bo15bo15bo15bo15bo9bobobo11bobobo228bo$203b2obo4bo118bo15b
o231bo$205b2o12b3o13b3o13b3o13b3o13b3o13b3o13b3o44bo$218bo15bo15bo15b
o15bo15bo15bo$219b2obo12b2obo12b2obo12b2obo12b2obo12b2obo12b2obo13b3o
13b3o9b4o$220b3o13b3o13b3o13b3o13b3o13b3o13b3o12bo15bo13b2obo$221bo15b
o15bo15bo15bo15bo15bo13b2o14b2o13b2obo$335bo15bo11b4o$364bobo201bo$365b
o203bo$569bo2$367b2o211b3o$369bo213b2o$581b2o3$559bo$560bo$560bo2$358b
2o$360bo4$550bo$551bo$551bo2$349b2o$351bo$519bo15bo$519bo15bo2$509bo11b
obo13bobo$510b2o9bo2bo12bo2bo$337bo11bo168bobo4bo8bobo4bo$337bobo8bob
o167b2obo4bo7b2obo4bo$337bo2bo6bobobo55b2o109bo6b2o7bo6b2o$336bo2bobo
5b4o58bo109bo2bobo10bo2bobo$337b5o4bobo61bo110b4o12b4o30bo$338b2o2bo5b
o62bo160b2o$339b4o8bo57bo112b4o12b4o32bo$340b3o6b2o149bo72bobo$501b2o
71bo3$351bo183bo$536bo$536bo2$490bo8bo$349bo142bo7bo$340bo8bo139b4o7b
o$340bobo3bo3bo$338b3obo2bo2bo141bo5b3o$339b2o8bo141bo4b2obo26bo$340b
o158b2o26bo$344bobob3o176bo$347b2ob2o$349b2o5$517bo$518bo$490bo8bo18b
o$349bo142bo7bo$340bo8bo139b4o7bo$340bobo3bo3bo$338b3obo2bo2bo141bo5b
3o63b3o$339b2o8bo141bo4b2obo65b2o$340bo158b2o62b2o$344bobob3o157bo$347b
2ob2o157bo$349b2o158bo7$490bo8bo$340bo5bo2bo142bo7bo$340bo4bo3bo139b4o
7bo$339bo2bo7bo$339b2obobo2bo142bo5bo$340bo3bo4bo128b2o11bo$341bo4b3o
131bo$346bobo2$349bobo2$349bo$348bo2$467b2o$389b2o72bo3bobo$391bo71bo
b2o3bo$340bo5bo2bo42bo70bobo5bo81bo$340bo4bo3bo43bo69bob2o87b2o$339bo
2bo7bo40bo73bo2bob2o84bo$339b2obobo2bo108b2o11bo85bobo$340bo3bo4bo108b
o7bobo4b2o81bo$341bo4b3o118b3o$346bobo2$349bobo2$349bo$348bo$447b2o$449b
o17b2o$463bo3bobo$463bob2o3bo$340bo5bo2bo113bobo5bo$340bo4bo3bo113bob
2o$339bo2bo7bo114bo2bob2o$339b2obobo2bo121bo$340bo3bo4bo116bobo4b2o$341b
o4b3o89b2o27b3o$346bobo91bo2$349bobo2$349bo$348bo$470bo$470bo$429b2o40b
o$431bo112b3o$340bo5bo2bo197b2o$340bo4bo3bo195b2o$339bo2bo7bo$339b2ob
obo2bo$340bo3bo4bo$341bo4b3o112bo$346bobo112bo$400bo19b2o10bo3b2o10bo
3b2o8bo$349bobo47b2o21bo9b2o4bo9b2o4bo$434bo15bo$349bo83b2o14b2o$348b
o69bo14b2o14b2o$401b2o$403bo2$420b2o14b2o14b2o$340bo5bo2bo67bo2bo12bo
2bo12bo2bo$340bo4bo3bo53bo14bobobo11bobobo11bobobo$339bo2bo7bo53bo15b
o15bo15bo$339b2obobo2bo56bo$340bo3bo4bo$341bo4b3o$346bobo2$349bobo$382b
o$349bo21b2o8b2o$348bo24bo$374bo160bo$375bo160b2o$373bo9b2o153bo$376b
o8bo151bobo$340bo5bo2bo23bo3b2o2bo156bo$340bo4bo3bo23bobo4b4o$339bo2b
o7bo$339b2obobo2bo12bo14bobob2o$340bo3bo4bo12b2o11bo3b3o2bo$341bo4b3o
5bo2bo2b2o2bo10b3o4b3o$346bobo5bobo2bo16b2o2bo$356bo6bo13b3obo$349bob
o12bo$367b2o12b2o$349bo10b3o6b2o$348bo10bo8bo2bo$360b2obo4b3o$361b3o2b
2o$362bo5b3o2$340bo5bo2bo$340bo4bo2bo$339bo2bo$339b2obobo$340bo3bo6bo
$341bo8b4o$350bob2o6bo$346bo3bobobo6bo14bo$352bo8bo13bobo$376bo$377b2o
$348bo30bo146b3o$529b2o$527b2o2$348bo2bo$343b2o4bo2bo$342b2o2bo5bo$343b
2obo4bo$344bobo4bobo$345bo4bo35b2o$384b2o$351bo34b3o$349b2o2$350b2o6$
394bo$393bobo$394bo$395b2o$397bo5$517bo$518b2o$520bo$519bobo$404b2o114b
o$402b2o$404b3o9$412bo$411bobo$412bo$413b2o$415bo9$422b2o$420b2o$422b
3o$508b3o$511b2o$509b2o6$430bo$429bobo$430bo$431b2o$433bo9$440b2o$438b
2o$440b3o6$499bo$500b2o$502bo$448bo52bobo$447bobo52bo$448bo$449b2o$451b
o9$458b2o$456b2o$458b3o9$466bo$465bobo$466bo$467b2o$469bo20b3o$493b2o
$491b2o7$476b2o$474b2o$476b3o9$484bo$483bobo$484bo$485b2o$487bo$481bo
$482bo$482bo3$448b2o14b2o$447bo15bo$449bobo13bobo$448bo2bo12bo2bo$440b
o8bobo2bo10bobo2bo$441b2o9b2obo12b2obo$451bo3bo11bo3bo$450bobo2bo10bo
bo2bo$450b2o2b2o10b2o2b2o$452bobo13bobo$451b6o10b6o3$431bo$432b2o3$466b
o$467bo$467bo2$427bo2bo$422b2o4bo2bo$421b2o2bo5bo$422b2obo4bo$423bobo
4bobo$424bo4bo27bo$429b2o27bo$429bo28bo$428b3obo$431bobo$429b2obo4$448b
o$449bo$427bo2bo18bo$422b2o4bo2bo$421b2o2bo5bo$422b2obo4bo$423bobo4bo
bo$424bo4bo$429b2o$429bo9bo$428b3obo7bo$431bobo6bo$429b2obo6$427bo2bo
$422b2o4bo2bo$421b2o2bo5bo$422b2obo4bo$423bobo4bobo$424bo4bo2$430bo$428b
2o2$429b2o!Code: Select all
x = 4, y = 7, rule = B2/S2-mH
o2$bo2$2bo$2b2o$3bo!