Three words:
Cumulative Sum Generator.
This is a device that takes in a stream of gliders emitted at generations 960*a, 960*b, 960*c ... and outputs a stream of gliders emitted 960*a, 960*b, 960*c ... generations apart. The plan was to be able to generate any polynomial sequence P(n) of degree
d, under the requirement that for all
k not greater than
d, the
kth differences of P(n) were positive, a requirement that is fulfilled by all polynomials with positive initial coefficient for sufficiently large input values
n. Such a pattern can be constructed by stacking copies of this filter. I needed this in the construction of a larger project, which I haven't completed (yet).
Here is an example of the pattern with input gliders representing a = 3, b = 4, c = 5, d = 7, e = 9:
Code: Select all
x = 1588, y = 1628, rule = B3/S23
254b2o$254b3o$254b2obo$258bo13bo$250bo3b2o3bo12bo$260bo11bo$257bobo$
257bo$252bo4bobo13bo$177b2o92b2ob2o4b2o$167bo3b2o4b3o76b4o20b2o$167bob
2o2bo6bo75b2o12bobo$167bo2b5o2b4o89bo3bo$171b3o3b3o90b2o2bo9b2o$170b3o
98b3o9bo$170b2o17b2o81bo5bo2bo$170bo18b2o87bo2b5o$278b2ob4o$284b2o$
284b3o$265b3o16b3o$260b2o3bobo6bo$260b2o2bo3bo3b2ob2o$197b2o48bobo14bo
2bobo6bo$197b2o47bo25bo3bo$176bo8b2o2bo57bo2bo15b3o4b3o$175bo2bo5b3o2b
2o58bobo15bo$169b3o2bo3bo6b2o2b3o60bo$169b3o2bo4bo5b2o2b2o60b3o$173bo
3b2o72bo12bo2bo$177b2o54bo15b2o13bo2bo$174b2o57bo17bo12b2o$170bobo62bo
5bo7bo2bo$161bob3o4bobo61bo6bobo5bobo$159b2obo4bo5bo59bo3bo2bo8bo$161b
2o4bo2bo2bo60bo2bobob2o$140bo26bo2bo2bo65bob2o$140bo7bo17bo4bo$140bo5b
3o13bo2bo$139b2o3bobo16b2o$145bo2bo97b2o$144bo2b2o96bo2bo$246b2o4$143b
o$142bobo97b2o3bo$142bobo95b3o4b2o5b2o$143bo91b3obo8b2o3bo2bo$239bobo
2b5o5b2o$240bo3b3o3$127b3o21bo$131bo18bobo74bo3b2o2b2obo$131b2o17bobo
74bo3b2o4bob2o$131bo2bo16bo26bo48bo3b2o2b2ob2o$132bobo44bo51b2o$132bo
44b3o49bo2bo$105bobo23b2o96bobo$104bo25bobo75bo21bo$105bo2bo23b2o75bob
2obobo$107bobo19b2o72b3o3b4o3bo$110bo11b3o6bo77bo2b2ob2o$109b3o96bo$
109bo96b3o$107b2o97b3o$109bo13b3o101b3o$107bo2bo12b3o100bo2bo$107bobo
13b2o101bo$107bo14bo103bo2bo$120b2o2bo101bo2bo$121bo3bo102b3o$121bo2bo
102bobo$122b3o102bo$176b2o51bo$175bobo45b3o4b2o$177bo44bob2o6bo$208bob
2ob2o7bob2o5b2o$207b5o2b2o7b3obo3bo$111b2o94bo2b3obo8b2o5bo$111b2o100b
o12bo2b2o$226b2obo3b2o$181b3o44bo2$58bobo118bo5bo54bo$57bo121b7o12bo
15b2o23bo$58bo2bo119bo17b2o13b2o23b3o$60bobo56b2o60bo3bo13b2o$63bo11b
3o41b2o59bo17b2obo23bo$62b3o116bo18b3o22bo$62bo81b2o36bo4bo11bob2o23bo
$60b2o81bobo36bo4bo11b2o$62bo13b3o66bo35b2obobo12bo6b2o17b2o$60bo2bo
12b3o5b2o94bobo23b2o15bo2bobo$60bobo13b2o6b2o95bo41bo4bo$60bo14bo147bo
$73b2o2bo146b3o$74bo3bo9bo$74bo2bo10bo$75b3o7bo$81b2o6bo$82b4o110b3o
15bo$86bo3bo107bo10bobobob2o$70bo16bo109bo2bo8bob3ob2o$69bobo16bobo84b
o10bobo9b2ob2o6bo5b2o$64b2o2b2ob2o4b3o9bo85bo10bob4o7bo11b4o$64b2o2b5o
4b4o31b2o61bo11b3o2b2ob3obo2bo9b2o$72bo3b2o2b2o29bobo58bob3obo11bobo2b
4o3bo$50b3o17bo6b2obo32bo58bob5o13bobo7bo$54bo15b3o4b3o96bobo13bob4o4b
o$54b2o14b3o5bo71bobo20b2o19b2o2bo3b2o$54bo2bo91bo23b2o16bo2bo5bobo$
55bobo92bo2bo20bo26bo$55bo96bobo19b2o15bo$45bo8b2o11b2obo84bo11b3o5bo
3b2o11b2o$28b3o8bo5bo7bobo12b2o84b3o18bo$27bo3bo6bo7bo8b2o97bo18bob2o$
28b2o9b4o2b2o5b2o98b2o20bo$30b2ob2o9b3o7bo99bo13b3o34b2o$32b2o11bobo
104bo2bo12b3o35b2o$46bo100bo4bobo13b2o35bo$46bo99b2o4bo14bo$80b2o63bob
2o16b2o2bo$79bobo63bo2b2o10bo5bo3bo$38bo42bo61b4ob3o8bobo4bo2bo$37bobo
102bo2bobo2b2o7bobo5b3o$37bobo103bobobob3o8bo$24b3o11bo105b2ob3obo$25b
o97b3o18b4o2bobo$25bo2bo93bo24bo3bo$25bo2bo49bo43bo3bo20bo3bo$26bobo
50bo40b2o3b2o20bo2bo17bo$46bo31bo40bo2bobo2bo39bobo$2bo42bobo71bo2bobo
2b2o38bobo$2bo42bobo73b2ob2o2bo39bo$2bo43bo75bo4bo$3b2o120b2ob2o$3b3o
19b2o97b4o$3b2o120b2o13b2o$18b2o120b2o$18b2o2$23b3o99bo15bo6bo$2bo19bo
2bo100bo14bo2b6o$bobo17bo3bo99bo14bo8bo$bo2bo18bo117b2obob3o$19bo2bo
118b2obob2o$18bo124bo$o2bo126b2o$2ob3o117bo5bo2bo$2b2o2bo11b3o102bo5bo
2bo$3bo2bo11bobo102bo4b2ob2o8b3o$3bobo14b2o107b2o13bo$18bo2bo119bo2bo$
19b2o121b2o$19bo$15bo119bo$16b2o115bobo$14bobo111b2o2bo2bo$6b2o6bo113b
2o3b2o$6b2o4b2o119b2o2bo$13bobo118b2obo$14b2o117b2o3bo$133bo2bobo$133b
o4bo2$135bobo$14b2o119bobo$14b2o15$146b2o$145b2o$147bo59$393bo$393bo2$
393bo$392bobo2$392b3o3$392b3o2$392bobo21b2o$393bo16b2o4b2o$410bo$393bo
8bo5bobo$393bo7b2o5b2o6bo$400b2o13bobo$399b3o12bo3bo$400b2o3b2o7b5o$
390b2o9b2o10b2o3b2o$390bobo9bo11b5o$390bo24b3o$416bo4$417b2o$417b3o5$
407b3o$409bo7b3o$408bo7bo3bo2$415bo5bo$415b2o3b2o2$389b2o9b2o$389bobo
7bobo16bo$384b2o6bo8bo5bo9bobo$380b2obo2bo2bo2bo13bobo8bobo$380b2o2b2o
6bo13b2obo7bo$389bobo14b2ob2o6bo$389b2o15b2obo7bo2bo$406bobo3b2o4b2o$
407bo4bobo$414bo$414b2o130$386b2o$385b2o$387bo238$626b2o$625b2o$627bo
478$1106b2o$1105b2o$1107bo478$1586b2o$1585b2o$1587bo!
I feel that many optimizations could be used, but unfortunately (and inexplicably, considering my eye-catching title
) the thread on which I introduced this pattern has been mostly silent. I also don't know if this pattern is similar to any prior work (other than the sqrtgun, cube root gun, and filters discussed
here).
The pattern has the potential to accept input with a period of 64 instead of 960 if a stable glider duplicator is used in place of the P30 one, but I don't know of any that produce a glider with the appropriate lane and color. Does anyone else know?
(tell me if I actually do have some that I dont remember) BUT, I think this is my best FIND.
