B0167/S14

[ebcube]

I've been doing a bit of research in this rule. According to Eppstein's gliders database, there are six known gliders in this rule, all of them orthogonal:

Code: Select all

x = 104, y = 24, rule = B0167/S14
6$6b2o11b2o32b2o9b2o12bobobo$5b4o55b2o14bo14bobo$18bo2bo30bo2bo8b2o12b
o2bo11bo2bo2bo$19b2o32b2o26bo12bo3bo$4bob2obo10bo42bo17bo12bo3bo$6b2o
14bo40b2o14bobo11bo2bobo$8b2o13bo55bo2bo16bo$9bo12bo54bobo$9bo12bo$7bo
bobo9bobo54bo$9bo68bobo$77bo2bo$78bobobo$78b3o$78bo!

The tail of the two c/2 spaceships can be easily changed, I've found six different ends so far (including the two in Eppstein's DB):

Code: Select all

x = 28, y = 126, rule = B0167/S14
2$5b2o11b2o$4b4o$17bo2bo$18b2o$3bob2obo10bo$5b2o14bo$7b2o13bo$8bo12bo$
8bo12bo$6bobobo9bobo$8bo10$6b2o10b2o$5b4o$17bo2bo$18b2o$4bob2obo9bo$6b
2o13bo$8b2o12bo$9bo11bo$9bo11bo$8bobo8bobobo$21bo9$6b2o10b2o$5b4o$17bo
2bo$18b2o$4bob2obo9bo$6b2o13bo$8b2o12bo$9bo11bo$9bo9bobo$7bobo11bo$9bo
12bo$10bo11$7b2o10b2o$6b4o$18bo2bo$19b2o$5bob2obo9bo$7b2o13bo$9b2o12bo
$10bo11bo$10bo11bo$10bobo9bobo$8bobo9bobo$8b4o8b4o11$7b2o10b2o$6b4o$
18bo2bo$19b2o$5bob2obo9bo$7b2o13bo$9b2o12bo$10bo11bo$10bo11bo$10bo11bo
$8bo3bo7bo3bo$10bo11bo11$8b2o10b2o$7b4o$19bo2bo$20b2o$6bob2obo9bo$8b2o
13bo$10b2o12bo$11bo11bo$11bo11bo$9bo11bo$11bo11bo$9b3o9b3o$9b3o9b3o$
10bo11bo$8b2o10b2o!

I also found a way to turn the width-1 line of the tail into a width-3 line at c/2, so width-3 c/2 tails can be used to. Unfortunately I've only found one so far. It creates a puffer:

Code: Select all

x = 9, y = 28, rule = B0167/S14
2$3b2o$2b4o3$bob2obo$3b2o$5b2o$6bo$6bo$6bo$4bobo2$4b3o$4b3o$4b3o$2bo2b
2o$4b3o$3b4o$3bob2o$2bo2b2o$4b3o$4b3o$4b3o!

However, there are a lot of possible width-3 tails at speed c, because there is a replicator at speed c inside width-3 lines. It can be used to create oscillators on arbitrarily high periods: for example, this oscillator of period 8188:

Code: Select all

x = 13, y = 64, rule = B0167/S14
4$4b5o$4b5o$4b5o$4b5o$4b5o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b
3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5bobo2$5bobo$5b3o$5b3o$5b3o$5b3o$5b3o
$5bobo2$5bobo$5b3o$5b3o$5b3o$5b3o$5b3o$5bobo2$5bobo$5b3o$5bobo2$5bobo$
5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$5b3o$4b5o$4b5o$4b5o$4b5o$4b5o!

As I said, there are a lot of possible width-3 tails at speed c because of this replicator. But no heads on that speed yet.

Code: Select all

x = 176, y = 116, rule = B0167/S14
3$163bo$36bo129bo$34b2o4bo11bo2bo106b2o2b2o$35b3o4bob7o2bo4bo2b3ob91ob
o4b2o2b2o$34b3o4bo6bo4bo4bo3bo3b90o5bo4bo$35b2o3bo2bo3bo2bobo4b3ob3ob
91ob5obo$33b3obo4b2o$34bo7b2o$32bo9b2o$42b2o8$164bo$167bo$40bo2bobo3bo
113b2o2b2o$41b2o4b2o4bo2b3ob97obo4b2o2b2o$40b4obob3o3bo3bo3b96o5bo4bo$
44b2o2b2o3b2ob3ob97ob5obo$47bo10$164bo$167bo$55bo107b2o2b2o$46bob2o2b
2o8b95obo4b2o2b2o$46bob2o5bo2bo2bo2b93o5bo4bo$49bo4b3ob3ob95ob5obo12$
164bo$6bobo158bo$9bo3b2obo146b2o2b2o$8bobo14bo2b3ob125obo4b2o2b2o$16bo
bobob2o5bo3b124o5bo4bo$16bobobob2ob2ob3ob125ob5obo$8bobo$9bo3b2obo$6bo
bo11$165bo$168bo$164b2o2b2o$45b113obo4b2o2b2o$44bo2b111o5bo4bo$45b113o
b5obo13$166bo$169bo$34bo130b2o2b2o$36b2o3b118obo4b2o2b2o$35bo2b2o3b
116o5bo4bo$35b5ob118ob5obo11$167bo$170bo$166b2o2b2o$40b4ob115obo4b2o2b
2o$38bo5bo2b113o5bo4bo$40b4ob115ob5obo!

There's also a width-2 c/2 head and a 2c/5 tail, a width-2 c/4 head and two width-4 tails at c/2 and c/4. If I found a way to turn a width-2 line into a width-4 at c/4...

Code: Select all

x = 50, y = 35, rule = B0167/S14
3$4bo3bo$15bo$5bo2b5o2b2o$5bo2b5o2b2o$15bo$4bo3bo7$14bo$10b4o2b2o$10b
4o2b2o$14bo8$10b35o$10b34o$8b36o$9b36o!

EDIT: Nevermind. The width-2 c/4 head is actually c/6, and the width-4 c/4 tail is actually c/8. There is, however, a way to turn a width-2 line into a width-4 at c/4, which proves to be completely useless for this.

[apg]

Can you make a XOR-extensible spaceship in this rule? I tried to find one in HighLife, but it is likely to be billions of cells long due to the expected period (p462, or depth-42).

[ssaamm]

I found an interesting little thingy here:

Code: Select all

x = 3, y = 2, rule = B0167/S14
obo$obo!

If it is interrupted by some things, it can make some things naturally:

Code: Select all

x = 11, y = 2, rule = B0167/S14
obo7bo$obo7bo!

Code: Select all

x = 9, y = 3, rule = B0167/S14
8bo$obo$obo4bo!

Code: Select all

x = 11, y = 2, rule = B0167/S14
obo7bo$obo5bo!

Which reminds me, here is a high-heat oscillator created by the last one of those:

Code: Select all

x = 4, y = 5, rule = B0167/S14
b2o$3bo2$o$b2o!

[ebcube]

Which reminds me, here is a high-heat oscillator created by the last one of those:

Code: Select all

x = 4, y = 5, rule = B0167/S14
b2o$3bo2$o$b2o!

Interesting. I've only found two natural oscillators with a period higher than 4. That's one of them. The other one is this:

Code: Select all

x = 4, y = 4, rule = B0167/S14
2o$o$3bo$2b2o!

[ebcube]

Can you make a XOR-extensible spaceship in this rule? I tried to find one in HighLife, but it is likely to be billions of cells long due to the expected period (p462, or depth-42).

What do you mean by "XOR-extensible spaceship"? It is certainly possible to make a extensible spaceship.

[apg]

What do you mean by "XOR-extensible spaceship"? It is certainly possible to make a extensible spaceship.

A XOR-extensible spaceship is a miraculous application of the parity-based replicator (e.g. the one in HighLife). Specifically, if you have a reaction where a line of replicators push some junk by a certain distance (a multiple of the spatial period of the replicator), and another reaction where another line of replicators pull some other junk by the same distance, then there is the possibility of engineering an arbitrarily slow spaceship.

As an example of a pushing reaction in HighLife, here is one by Dean Hickerson, repeated multiple times for clarity:

Code: Select all

x = 1061, y = 1061, rule = B36/S23
6b3o$5bobbo$4bo3bo$4bobbo$4b3o$$o$o$o13b3o$13bobbo$12bo3bo$12bobbo$12b
3o4$22b3o$3o18bobbo$20bo3bo$20bobbo$bboo16b3o$3boo$oboo$3o$bo5bo$6b3o$
5boobo$4boo$5boo3boo22b3o$11boo20bobbo$8boboo20bo3bo$8b3o21bobbo$9bo5b
o16b3o3b3o$14b3o20bobbo$13boobo19bo3bo$12boo22bobbo$13boo3boo16b3o$19b
oo$16boboo$16b3o$17bo5bo$22b3o$21boobo$20boo$21boo27b3o$49bobbo$48bo3b
o$48bobbo$30boo16b3o3b3o$31boo20bobbo$28boboo20bo3bo$28b3o21bobbo$29bo
5bo16b3o$34b3o$33boobo$32boo$33boo3boo22b3o$39boo20bobbo$36boboo20bo3b
o$36b3o21bobbo$37bo22b3o4$47bo$46b3o$45boobo$44boo$45boo4$78b3o$77bobb
o$76bo3bo$76bobbo$58boo16b3o3b3o$59boo20bobbo$56boboo20bo3bo$56b3o21bo
bbo$57bo22b3o3b3o$85bobbo$84bo3bo$84bobbo$67bo16b3o$66b3o$65boobo$64b
oo$65boo3boo$71boo$68boboo$68b3o$69bo28b3o$97bobbo$96bo3bo$96bobbo$79b
o16b3o$78b3o$77boobo$76boo$77boo3boo$83boo$80boboo$80b3o$81bo5bo$86b3o
$85boobo$84boo$85boo4$94boo22b3o$95boo20bobbo$92boboo20bo3bo$92b3o21bo
bbo$93bo22b3o8$130b3o$129bobbo$128bo3bo$128bobbo$128b3o3b3o$133bobbo$
132bo3bo$132bobbo$132b3o8$123bo22b3o$122b3o20bobbo$121boobo19bo3bo$
120boo22bobbo$121boo21b3o4$130boo$131boo$128boboo$128b3o$129bo5bo22b3o
$134b3o20bobbo$133boobo19bo3bo$132boo22bobbo$133boo3boo16b3o3b3o$139b
oo20bobbo$136boboo20bo3bo$136b3o21bobbo$137bo5bo16b3o$142b3o$141boobo$
140boo$141boo4$174b3o$173bobbo$172bo3bo$172bobbo$172b3o3b3o$177bobbo$
176bo3bo$176bobbo$176b3o3b3o$181bobbo$180bo3bo$180bobbo$180b3o8$194b3o
$193bobbo$192bo3bo$192bobbo$192b3o3b3o$197bobbo$196bo3bo$196bobbo$178b
oo16b3o$179boo$176boboo$176b3o$177bo5bo$182b3o$181boobo$180boo$181boo
27b3o$209bobbo$208bo3bo$208bobbo$190boo16b3o$191boo$188boboo$188b3o$
189bo5bo$194b3o$193boobo$192boo$193boo3boo$199boo$196boboo$196b3o$197b
o4$207bo22b3o$206b3o20bobbo$205boobo19bo3bo$204boo22bobbo$205boo3boo
16b3o$211boo$208boboo$208b3o$209bo5bo22b3o$214b3o20bobbo$213boobo19bo
3bo$212boo22bobbo$213boo21b3o4$222boo22b3o$223boo20bobbo$220boboo20bo
3bo$220b3o21bobbo$221bo22b3o4$254b3o$253bobbo$252bo3bo$252bobbo$252b3o
8$243bo$242b3o$241boobo$240boo$241boo3boo22b3o$247boo20bobbo$244boboo
20bo3bo$244b3o21bobbo$245bo5bo16b3o3b3o$250b3o20bobbo$249boobo19bo3bo$
248boo22bobbo$249boo3boo16b3o3b3o$255boo20bobbo$252boboo20bo3bo$252b3o
21bobbo$253bo5bo16b3o$258b3o$257boobo$256boo$257boo3boo$263boo$260bob
oo$260b3o$261bo5bo22b3o$266b3o20bobbo$265boobo19bo3bo$264boo22bobbo$
265boo3boo16b3o$271boo$268boboo$268b3o$269bo5bo$274b3o$273boobo$272boo
$273boo3boo22b3o$279boo20bobbo$276boboo20bo3bo$276b3o21bobbo$277bo22b
3o3b3o$305bobbo$304bo3bo$304bobbo$287bo16b3o$286b3o$285boobo$284boo$
285boo3boo$291boo$288boboo$288b3o$289bo5bo22b3o$294b3o20bobbo$293boobo
19bo3bo$292boo22bobbo$293boo21b3o3b3o$321bobbo$320bo3bo$320bobbo$302b
oo16b3o$303boo$300boboo$300b3o$301bo5bo$306b3o$305boobo$304boo$305boo
3boo$311boo$308boboo$308b3o$309bo5bo$314b3o$313boobo$312boo$313boo3boo
$319boo$316boboo$316b3o$317bo12$358b3o$357bobbo$356bo3bo$356bobbo$339b
o16b3o$338b3o$337boobo$336boo$337boo3boo$343boo$340boboo$340b3o$341bo
28b3o$369bobbo$368bo3bo$368bobbo$351bo16b3o3b3o$350b3o20bobbo$349boobo
19bo3bo$348boo22bobbo$349boo3boo16b3o$355boo$352boboo$352b3o$353bo5bo
22b3o$358b3o20bobbo$357boobo19bo3bo$356boo22bobbo$357boo21b3o4$366boo$
367boo$364boboo$364b3o$365bo5bo$370b3o$369boobo$368boo$369boo3boo22b3o
$375boo20bobbo$372boboo20bo3bo$372b3o21bobbo$373bo22b3o3b3o$401bobbo$
400bo3bo$400bobbo$383bo16b3o3b3o$382b3o20bobbo$381boobo19bo3bo$380boo
22bobbo$381boo3boo16b3o$387boo$384boboo$384b3o$385bo5bo$390b3o$389boob
o$388boo$389boo27b3o$417bobbo$416bo3bo$416bobbo$398boo16b3o3b3o$399boo
20bobbo$396boboo20bo3bo$396b3o21bobbo$397bo22b3o4$430b3o$429bobbo$428b
o3bo$428bobbo$411bo16b3o$410b3o$409boobo$408boo$409boo4$418boo$419boo$
416boboo$416b3o$417bo5bo$422b3o$421boobo$420boo$421boo3boo$427boo$424b
oboo$424b3o$425bo5bo$430b3o$429boobo$428boo$429boo12$470b3o$469bobbo$
468bo3bo$468bobbo$468b3o8$458boo22b3o$459boo20bobbo$456boboo20bo3bo$
456b3o21bobbo$457bo22b3o3b3o$485bobbo$484bo3bo$484bobbo$467bo16b3o$
466b3o$465boobo$464boo$465boo3boo22b3o$471boo20bobbo$468boboo20bo3bo$
468b3o21bobbo$469bo22b3o4$479bo22b3o$478b3o20bobbo$477boobo19bo3bo$
476boo22bobbo$477boo21b3o8$490boo22b3o$491boo20bobbo$488boboo20bo3bo$
488b3o21bobbo$489bo22b3o12$507bo$506b3o$505boobo$504boo$505boo8$542b3o
$541bobbo$540bo3bo$540bobbo$540b3o3b3o$545bobbo$544bo3bo$544bobbo$544b
3o3b3o$549bobbo$548bo3bo$548bobbo$548b3o4$558b3o$557bobbo$556bo3bo$
556bobbo$538boo16b3o$539boo$536boboo$536b3o$537bo28b3o$565bobbo$564bo
3bo$564bobbo$547bo16b3o$546b3o$545boobo$544boo$545boo3boo$551boo$548bo
boo$548b3o$549bo28b3o$577bobbo$576bo3bo$576bobbo$559bo16b3o3b3o$558b3o
20bobbo$557boobo19bo3bo$556boo22bobbo$557boo21b3o8$594b3o$593bobbo$
592bo3bo$592bobbo$592b3o8$606b3o$605bobbo$604bo3bo$604bobbo$586boo16b
3o3b3o$587boo20bobbo$584boboo20bo3bo$584b3o21bobbo$585bo22b3o3b3o$613b
obbo$612bo3bo$612bobbo$595bo16b3o$594b3o$593boobo$592boo$593boo3boo22b
3o$599boo20bobbo$596boboo20bo3bo$596b3o21bobbo$597bo5bo16b3o$602b3o$
601boobo$600boo$601boo3boo$607boo$604boboo$604b3o$605bo5bo$610b3o$609b
oobo$608boo$609boo3boo$615boo$612boboo$612b3o$613bo4$623bo$622b3o$621b
oobo$620boo$621boo3boo$627boo$624boboo$624b3o$625bo5bo$630b3o$629boobo
$628boo$629boo4$638boo$639boo$636boboo$636b3o$637bo5bo$642b3o$641boobo
$640boo$641boo3boo$647boo$644boboo$644b3o$645bo4$655bo$654b3o$653boobo
$652boo$653boo4$686b3o$685bobbo$684bo3bo$684bobbo$684b3o3b3o$689bobbo$
688bo3bo$688bobbo$688b3o12$682boo$683boo$680boboo$680b3o$681bo12$699bo
$698b3o$697boobo$696boo$697boo27b3o$725bobbo$724bo3bo$724bobbo$706boo
16b3o$707boo$704boboo$704b3o$705bo5bo22b3o$710b3o20bobbo$709boobo19bo
3bo$708boo22bobbo$709boo21b3o4$718boo$719boo$716boboo$716b3o$717bo12$
758b3o$757bobbo$756bo3bo$756bobbo$756b3o4$766b3o$765bobbo$764bo3bo$
764bobbo$764b3o4$774b3o$773bobbo$772bo3bo$772bobbo$772b3o4$782b3o$781b
obbo$780bo3bo$780bobbo$763bo16b3o$762b3o$761boobo$760boo$761boo4$770b
oo$771boo$768boboo$768b3o$769bo5bo$774b3o$773boobo$772boo$773boo3boo$
779boo$776boboo$776b3o$777bo5bo$782b3o$781boobo$780boo$781boo4$814b3o$
813bobbo$812bo3bo$812bobbo$794boo16b3o3b3o$795boo20bobbo$792boboo20bo
3bo$792b3o21bobbo$793bo22b3o3b3o$821bobbo$820bo3bo$820bobbo$820b3o4$
830b3o$829bobbo$828bo3bo$828bobbo$828b3o16$827bo$826b3o$825boobo$824b
oo$825boo27b3o$853bobbo$852bo3bo$852bobbo$852b3o4$862b3o$861bobbo$860b
o3bo$860bobbo$842boo16b3o$843boo$840boboo$840b3o$841bo4$851bo$850b3o$
849boobo$848boo$849boo3boo22b3o$855boo20bobbo$852boboo20bo3bo$852b3o
21bobbo$853bo22b3o3b3o$881bobbo$880bo3bo$880bobbo$863bo16b3o$862b3o$
861boobo$860boo$861boo3boo$867boo$864boboo$864b3o$865bo5bo22b3o$870b3o
20bobbo$869boobo19bo3bo$868boo22bobbo$869boo21b3o3b3o$897bobbo$896bo3b
o$896bobbo$878boo16b3o$879boo$876boboo$876b3o$877bo5bo$882b3o$881boobo
$880boo$881boo3boo22b3o$887boo20bobbo$884boboo20bo3bo$884b3o21bobbo$
885bo22b3o3b3o$913bobbo$912bo3bo$912bobbo$895bo16b3o$894b3o$893boobo$
892boo$893boo4$926b3o$925bobbo$924bo3bo$924bobbo$906boo16b3o3b3o$907b
oo20bobbo$904boboo20bo3bo$904b3o21bobbo$905bo22b3o8$942b3o$941bobbo$
940bo3bo$940bobbo$923bo16b3o3b3o$922b3o20bobbo$921boobo19bo3bo$920boo
22bobbo$921boo21b3o3b3o$949bobbo$948bo3bo$948bobbo$930boo16b3o$931boo$
928boboo$928b3o$929bo5bo$934b3o$933boobo$932boo$933boo27b3o$961bobbo$
960bo3bo$960bobbo$942boo16b3o$943boo$940boboo$940b3o$941bo8$955bo$954b
3o$953boobo$952boo$953boo27b3o$981bobbo$980bo3bo$980bobbo$980b3o8$994b
3o$993bobbo$992bo3bo$992bobbo$992b3o4$978boo$979boo$976boboo$976b3o$
977bo5bo22b3o$982b3o20bobbo$981boobo19bo3bo$980boo22bobbo$981boo21b3o
3b3o$1009bobbo$1008bo3bo$1008bobbo$990boo16b3o3b3o$991boo20bobbo$988bo
boo20bo3bo$988b3o21bobbo$989bo22b3o4$1022b3o$1021bobbo$1020bo3bo$1020b
obbo$1020b3o4$1030b3o$1029bobbo$1028bo3bo$1028bobbo$1028b3o4$1038b3o$
1037bobbo$1036bo3bo$1036bobbo$1019bo16b3o$1018b3o$1017boobo$1016boo$
1017boo4$1026boo$1027boo$1024boboo$1024b3o$1025bo5bo22b3o$1030b3o20bo
bbo$1029boobo19bo3bo$1028boo22bobbo$1029boo21b3o3b3o$1057bobbo$1056bo
3bo$1056bobbo$1038boo16b3o$1039boo$1036boboo$1036b3o$1037bo!

So, the front of a XOR-extensible spaceship in HighLife would resemble this. The back would pull a blinker at the same speed, and the entire thing would comprise hundreds of millions of replicators.


Engineering these monstrous spaceships is no mean feat. It requires:

• Discovering a parity-based replicator in the first place;
• Finding a reaction to push junk;
• Finding a reaction to pull junk;
• Enumerating all the possible heads and tails for a certain period;
• Ascertaining the period of the equivalent linear feedback shift register to ensure that it is worthwhile to search;
• Performing a computer search involving billions of comparisons and XOR operations to find a working combination.

And it doesn't even look like your 'replicator' can propagate through vacuum.


As far as I know, there are no non-trivial XOR-extensible spaceships constructed in any rule as of the time of writing.