Smallest Spaceships Supporting Specific Speeds (5s) Project

[77topaz]

Code: Select all

x = 27, y = 1, rule = W104
2obob2o8b2obob2obob2o!

Code: Select all

x = 15, y = 1, rule = W104
2obobobobobob2o!

Code: Select all

x = 55, y = 1, rule = W104
2obobobobobobobobobobobobobobobobobobobobobobobobobob2o!

what is 7c/8 spaceship, 8c/9, etc?

Eh, did you intend to post three Wolfram Rule 104 patterns that don't have anything to do with spaceships?

[wildmyron]

Slightly smaller 3c/4 orthogonal (again), 6 cells:

Code: Select all

x = 3, y = 7, rule = B2aci3ay4t6a/S03j4it
2bo2$2bo$obo$2bo2$2bo!

30c/30 orthogonal, 7 cells:

Code: Select all

x = 7, y = 5, rule = B2acn4cin/S
obo$6bo$2bo3bo2$obo!

what is 7c/8 spaceship, 8c/9, etc?

We're working on it! As you can see from this thread. To my knowledge there are no ships of those speeds known in any isotropic 2-state rule on the Moore neighbourhood. If you have any hints or clues in the direction of such ships, then please post them! As for the patterns you posted, I can only echo what @77topaz said.

@AforAmpere: nice work with the reductions and new ships. I think the 9c/11 frontend is more promising than you believe. Here are a few puffers which might provide inspiration:

9c/11 (p11) dot puffer / rake

Code: Select all

x = 49, y = 43, rule = B2ac3e4aer5eiknq6cek8/S02ai3ain4aekrtw5cikr6ae
32b2o$32bobo$32bo$34bo4$2bo8bo8bo8bo2$bo2bo5bo2bo5bo2bo5bo2bo5bo2$27bo
13b2o$26bobo6bo6bo$30bo11bo$3bo8bo3bo4bo6bo2bo2bobo7bo$14bo3bo4bo8bobo
bo6bo$2bo8bo2bo3bo11bo3b3o3bo5bo$14bo15bo5bobo5b4o$15bo19bobo3bo2bo3bo
$4bo13bo13bo11b3obo$bo4bo2bo25bo8bobo$o2bobo2bobo3bo3bobo2bo2bo2bo7bob
o7b2o$bo4bo2bo25bo8bobo$4bo13bo13bo11b3obo$15bo19bobo3bo2bo3bo$14bo15b
o5bobo5b4o$2bo8bo2bo3bo11bo3b3o3bo5bo$14bo3bo4bo8bobobo6bo$3bo8bo3bo4b
o6bo2bo2bobo7bo$30bo11bo$26bobo6bo6bo$27bo13b2o2$bo2bo5bo2bo5bo2bo5bo
2bo5bo2$2bo8bo8bo8bo4$34bo$32bo$32bobo$32b2o!

9c/11 (p44) dot puffer:

Code: Select all

x = 26, y = 25, rule = B2ac3e4aer5eiknq6cek8/S02ai3ain4aekrtw5cikr6ae
11bo$10bo2bo$obobo8bo$2bo12bobo$2bo2bo11bo$2b2o4bo9b4o$obo15b3obo$bo
17bo$bo2bo10bo5bo$o2bo2bo9bo$bo2bo11bo7b2o$6bo18bo$3b2o2bo5bobo4bo3bo$
6bo18bo$bo2bo11bo7b2o$o2bo2bo9bo$bo2bo10bo5bo$bo17bo$obo15b3obo$2b2o4b
o9b4o$2bo2bo11bo$2bo12bobo$obobo8bo$10bo2bo$11bo!

9c/11 (p11) dot + backward photon puffer:

Code: Select all

x = 14, y = 21, rule = B2ac3e4aer5eiknq6k/S02ai3ain4aert5ir
3bo$4bo$2b2obo$5bo2$6bo2b3o$o4bo2bob3o$6bo3bo2bo$2bo6b3obo$8bo2bo$obob
o4b2ob2o$8bo2bo$2bo6b3obo$6bo3bo2bo$o4bo2bob3o$6bo2b3o2$5bo$2b2obo$4bo
$3bo!

[AforAmpere]

I think the 9c/11 frontend is more promising than you believe.

I suppose so, here's a ship:

Code: Select all

x = 11, y = 13, rule = B2ac3e4akqr5einq6cei/S02ai3ain4ackrtyz5cij6i
5bo$o3bobo$7bo2$9b2o$10bo$3bobob3o$10bo$9b2o2$7bo$o3bobo$5bo!

Does anyone have a 5c/7 partial?

[praosylen]

Code: Select all

x = 27, y = 1, rule = W104
2obob2o8b2obob2obob2o!

Code: Select all

x = 15, y = 1, rule = W104
2obobobobobob2o!

Code: Select all

x = 55, y = 1, rule = W104
2obobobobobobobobobobobobobobobobobobobobobobobobobob2o!

what is 7c/8 spaceship, 8c/9, etc?

Eh, did you intend to post three Wolfram Rule 104 patterns that don't have anything to do with spaceships? :?

They do, in fact -- B2ac rules emulate W104 on a strip travelling at lightspeed at the edge of any pattern.

Compare:

Code: Select all

x = 27, y = 1, rule = B2ac/S
2obob2o8b2obob2obob2o!

Code: Select all

x = 15, y = 1, rule = B2ac/S
2obobobobobob2o!

Code: Select all

x = 55, y = 1, rule = B2ac/S
2obobobobobobobobobobobobobobobobobobobobobobobobobob2o!

@PHPBB12345 I suspect that there aren't any spaceships faster than 6c/7, or perhaps 13c/15 or 7c/8. The issue is that some part of the frontend, once it has partially decayed, must continue to travel at lightspeed at least one cell "behind" the rest of the front end, and somehow the rest of the front end must be able to send the signal to the decayed parts to "restart" the reaction. So far, 6c/7 and 12c/14 are the fastest I can find engines for, and judging from how difficult it is to construct something that fast, I doubt there are any engines that are much faster.

EDIT: 5c/7, 12 cells:

Code: Select all

x = 9, y = 9, rule = B2aci3cjq4aekqr/S03r4ce6i
b2obob2o2$o7bo$4bo3$2bobobo2$4bo!

The 3c transition is unnecessary to improve the minimum cell count, but the ship has a larger minimum bounding box without it to suppress a large spark.

[AforAmpere]

I suspect that there aren't any spaceships faster than 6c/7, or perhaps 13c/15 or 7c/8.

I disagree, I believe there are ships of higher speeds than 6c/7, based on the random 9c/11 I found today. It seems possible that a 7c/8 or 8c/9 exists, but I agree ships like a 20c/21 probably do not. Nice find with the 5c/7, any other interesting frontends?

11-cell variant:

Code: Select all

x = 5, y = 9, rule = B2aci3cj4akrz/S02n3ry6i
o$4bo$obobo2$4bo2$obobo$4bo$o!

8-cell 10c/14:

Code: Select all

x = 8, y = 5, rule = B2-ek3q4-cijqt/S03ry4ik5e6i
7bo$5bobo$obo$5bobo$7bo!

EDIT, 10 cell 5c/7:

Code: Select all

x = 7, y = 7, rule = B2-ek3cjq4akrz/S03r4c5c6i
6bo$2bobobo2$o5bo2$2bobobo$6bo!

EDIT 2, 9 cell:

Code: Select all

x = 7, y = 5, rule = B2-ek3q4aknrz6i/S03r4i
4bobo$4b2o$o$4b2o$4bobo!

[BlinkerSpawn]

I suspect that there aren't any spaceships faster than 6c/7, or perhaps 13c/15 or 7c/8.

I disagree, I believe there are ships of higher speeds than 6c/7, based on the random 9c/11 I found today.

9c/11 is slower than 5c/6.
I agree with AforAwesome that there must be some limit somewhere in that neighborhood because getting speeds closer and closer to c requires frontends that fall back less and less often, which requires them to somehow take longer and longer before reaching the handful of neighborhoods that cause it to fall back.
The problem of keeping the front end away from the "trigger handful" for long stretches of time bears some similarity of the firing squad syunchronization problem, in a way.

[wildmyron]

@AforAmpere: Nice work finding the 9c/11 ship.

@Others: Thank you for the explanations about B2ac frontends. I can see now how it starts to get harder and harder for patterns which support speeds that are closer to c to exist in a 2-state Moore neighbourhood rule. My apologies to @PHPBB12345.

Here are a few further improvements on the 5c/7 ships.

Smaller 5c/7 orthogonal, 6 cells:

Code: Select all

x = 3, y = 5, rule = B2aci3j4anrz/S03r
2bo$obo2$obo$2bo!

15c/21 orthogonal, 6 cells:

Code: Select all

x = 2, y = 5, rule = B2-ek3cq4ajknqrz5j6i/S03r4it6i
bo$bo$2o$bo$bo!

Here are some more photons. Several of these ships are adjustable by placing the domino replicator further out from the ship which allows multiplying the period of the ship by 2. I've tried to find the smallest examples for each of the periods posted and refrained from adjusting the period of any of the ships to beyond p50.

Smaller 16c/16, 7 cells (p8 adjustable)

Code: Select all

x = 21, y = 6, rule = B2ak3y/S
o19bo$20bo$o$2bo$2bo$o!

Smaller 24c/24, 7 cells (p12 adjustable)

Code: Select all

x = 11, y = 5, rule = B2ak3nry4y5e/S
o9bo$10bo$o$2bo$obo!

32c/32, 7 cells (p8 adjustable)

Code: Select all

x = 37, y = 6, rule = B2ak3y/S
o35bo$36bo$o$2bo$2bo$o!

36c/36, 24 cells

Code: Select all

x = 33, y = 5, rule = B2ak3cnry4ny5e/S
o3bobobo13bo3bobo3bo$o5bobo23bo$o3bo17bo3bo3bo$2bo3bobo19bo$6bobo17bo!

There's also this slightly smaller 36c/36 at 17 cells, but it contains separate p4 and p9 parts which I guess make it a trivial example of a p36 photon. I don't think it counts for this project, and as such I've added an extra dot spark so it doesn't get picked up by my update script.

Code: Select all

x = 17, y = 8, rule = B2ak3cnr4ny5e/S1c3i
bo2bo11bo$6bo$bobo3b2o3bo$8bo3bo$7b2o$obo3bo$2bo$o!

40c/40, 21 cells

Code: Select all

x = 21, y = 7, rule = B2ak3nry4cntyz5e/S
o5bo5bobo$6bo5bobobo$o13bobo$obo11bobobo$obo17bo$20bo$18bo!

48c/48, 7 cells (p12 adjustable)

Code: Select all

x = 25, y = 5, rule = B2ak3nry4y/S
o23bo$24bo$o$2bo$obo!

96c/96, 21 cells (potentially superseded by a p96 example of the p12 adjustable ship). I'm a bit dubious about this one as there's not a lot of interaction between the two separate ships, but I think this one's OK. Any other thoughts?

Code: Select all

x = 29, y = 9, rule = B2akn3nry4ny5e/S
o3bo$4bo$2bo$4bo$o3bo15bo3bo$20bo5bo$18bobo3bo3bo$24bobobo$18bo5bobo!

144c/144, 16 cells. This ones definitely OK, the front ship toggles the rear ship from a p9 to a p6 and eventually back to a p9.

Code: Select all

x = 37, y = 4, rule = B2ak3cnr4cntyz5e/S
26bo5bobobo$o23bo9bobo$obo21bo7bo$obo21bo5bo!

[BlinkerSpawn]

16c/16 and 32c/32 trivially reducible by one cell:

Code: Select all

x = 21, y = 5, rule = B2ak3y/S
o19bo$20bo$o$2bo$2bo!

Code: Select all

x = 37, y = 5, rule = B2ak3y/S
o35bo$36bo$o$2bo$2bo!

[77topaz]

Hmm... it would be interesting to have a version of this project that's restricted to rules that aren't explosive, since many of these rules with photons are explosive.

[AforAmpere]

Hmm... it would be interesting to have a version of this project that's restricted to rules that aren't explosive, since many of these rules with photons are explosive.

Wouldn't reducing the number of rules make the project less interesting, as there would be less ships? Why would collecting spaceships be less interesting with explosive rules?

[77topaz]

Wouldn't reducing the number of rules make the project less interesting, as there would be less ships? Why would collecting spaceships be less interesting with explosive rules?

Well, for one thing, explosive rules can't be searched by algorithms like apgsearch. Secondly, non-explosive rules generally have more options for other types of engineering.

[BlinkerSpawn]

Wouldn't reducing the number of rules make the project less interesting, as there would be less ships? Why would collecting spaceships be less interesting with explosive rules?

Well, for one thing, explosive rules can't be searched by algorithms like apgsearch. Secondly, non-explosive rules generally have more options for other types of engineering.

While true, that's not really the point of this particular thread, as it deals with a specific field of engineered object.

[AforAmpere]

While true, that's not really the point of this particular thread, as it deals with a specific field of engineered object.

Exactly, and non-explosive rules are not banned, so they can still be explored. There has been a lot of investigation into exploding rules with B2a, because they have unusually fast ships sometimes, filling in speed gaps.

[Majestas32]

For speeds that exist in non-explosive rules, but for which the minimal cell count rule is explosive, maybe there can be an extra cell count listed for the minimal non explosive rule.

[AforAmpere]

For speeds that exist in non-explosive rules, but for which the minimal cell count rule is explosive, maybe there can be an extra cell count listed for the minimal non explosive rule.

No, that would be both extremely hard to verify, as there are over 25,000 known speeds to check, and any speed above C/2 is almost necessarily explosive.

[77topaz]

While true, that's not really the point of this particular thread, as it deals with a specific field of engineered object.

I just thought it might be useful to make this distinction, f.e. if someone wanted to use this database for engineering larger patterns at specific speeds, or to find out how frequently these spaceships occur naturally.

For speeds that exist in non-explosive rules, but for which the minimal cell count rule is explosive, maybe there can be an extra cell count listed for the minimal non explosive rule.

That sounds like a good idea to me.

[AforAmpere]

For speeds that exist in non-explosive rules, but for which the minimal cell count rule is explosive, maybe there can be an extra cell count listed for the minimal non explosive rule.

That sounds like a good idea to me.

As I just said, it would be unfeasible to check all of the speeds in the database for explosiveness, and then add a second ship that is in a non-explosive rule. A large percentage of speeds would not exist in non-explosive rules as well. In any case, as with above, engineering ships, in a lot of cases, deals, and works well with modifying explosive rules, you just have to know where to look. Also, other ships, like the ones found by muzik (and modified by a couple of others), are non-explosive, and in the database, and are very engineerable:

Code: Select all

x = 17, y = 3, rule = B2c3aj4a6ack7/S1e2-an3ejnr4i6k7e
b15o$7bo5bo2bo$16o!

In any case, if you want to collect spaceships in non-explosive rules, that is fine, but the goal of this project is to find the smallest of each speed in isotropic non-totalistic rules without B0, no other exceptions.

[Majestas32]

Yeah. It seems there's only a few cases (and anything above c/2o to c/3d is immediately known to be explosive) that are like that anyway.

[AforAmpere]

3-cell 10c/18:

Code: Select all

x = 2, y = 3, rule = B2acn3aek4ijrw5ciq6e/S1c2i3ijky4ajqy5y6ci
bo$o$bo!

EDIT, 7c/13:

Code: Select all

x = 2, y = 3, rule = B2acn3aek4ijr5aiqy6e/S1c2i3ijky4a5y6c
bo$o$bo!

8c/15:

Code: Select all

x = 2, y = 3, rule = B2acn3aek4ekw5iqy/S1c2in3ij4aij5y6c
bo$o$bo!

9c/17:

Code: Select all

x = 2, y = 3, rule = B2acn3aeky4ajqy5iknq/S1c2i3ij4acw5y6c
bo$o$bo!

11c/20:

Code: Select all

x = 2, y = 3, rule = B2acn3aeky4ijrt5aiq6i7e/S1c2ei3ij4aky5ny6c
bo$o$bo!

10c/19:

Code: Select all

x = 2, y = 3, rule = B2acn3aeky4eijqr5aiy6k/S1c2i3aijq4ajw5ny6c7c
bo$o$bo!

9c/15:

Code: Select all

x = 2, y = 3, rule = B2acn3aek4aiqtz5iq/S1c2i3ijy4aew5ey6ci
bo$o$bo!

9c/16:

Code: Select all

x = 2, y = 3, rule = B2acn3aek4eij5ci6i7e/S1c2i3-ekry4aejkq5y6c
bo$o$bo!

Better 8c/14:

Code: Select all

x = 2, y = 3, rule = B2acn3aceky4i5inqy/S1c2i3ij4aekn5y6c
bo$o$bo!

[Macbi]

The following were found with my new program, Logic Life Search.

(2,1)c/3, 5 cells:

Code: Select all

x = 5, y = 6, rule = B2-ei3cjkr4cektyz5-cnr6-ik78/S01e2-ae3cnqry4cqrtwyz5-ain6ekn7e
4bo2$4bo2$2b2o$o!

(2,1)c/4, 4 cells:

Code: Select all

x = 5, y = 4, rule = B2-ei4ciknwyz5jkqr678/S1e2en3-aery4-acert5-acei6ekn78
4bo2$4bo$obo!

3c/4 orthogonal, 4 cells:

Code: Select all

x = 2, y = 5, rule = B2-k3-aijq4-aijky5-inry6-a78/S2-ek3ijry4cinwyz5ekqry6-ak8
o$bo2$bo$bo!

(3,1)c/4, 4 cells:

Code: Select all

x = 3, y = 3, rule = B2ac3-aeij4-ikt5-acin6-a78/S1e2ekn3eknr4aceyz5-aery678
2bo2$3o!

(3,1)c/5, 3 cells:

Code: Select all

x = 3, y = 5, rule = B2-ek3cry4ceknqwz5-acin6-k7c8/S12kn3-ejkr4-ajknr5-acin6-c7e8
o2$2bo2$2bo!

(3,2)c/5, 4 cells:

Code: Select all

x = 3, y = 3, rule = B2ace3cr4ceinwyz56-k7c8/S1e2cen3ac4cejktwy5-qr678
obo$2bo$2bo!

4c/5 orthogonal, 5 cells:

Code: Select all

x = 2, y = 5, rule = B2-ik3ak4-eir5-y678/S2-ce3-jry4-ackwy5-i6-a78
bo$bo$o$bo$bo!

(4,1)c/5, 5 cells:

Code: Select all

x = 3, y = 5, rule = B2-en3ajr4cenw5aeiky6ci8/S1e3aeky4-ek5-ejkq6-ak7c8
2bo2$2bo$2bo$obo!

(3,1)c/6, 3 cells:

Code: Select all

x = 2, y = 5, rule = B2ack3aq4ijknrz5-einq6-in7c8/S03aeijk4crwyz5-cejy6-ck7c8
o2$o2$bo!

(3,2)c/6, 3 cells:

Code: Select all

x = 2, y = 5, rule = B2-e3ekq4cknqwy5-cn6-ae78/S01e2ekn3ekqy4-kyz5eiry6-ac78
o2$o2$bo!

(4,1)c/6, 3 cells:

Code: Select all

x = 2, y = 5, rule = B2-ei3a4-aetyz5-einq6cn78/S02cei3aijqr4cikry5aciqy6-ei7c8
bo2$bo2$o!

5c/6 orthogonal, 5 cells:

Code: Select all

x = 2, y = 5, rule = B2ace3cejr4cjnqwyz5-cy6-ci8/S2-ce3cjknq4kqrw5aeky6-ik78
bo$bo$o$bo$bo!

(5,1)c/6, 5 cells:

Code: Select all

x = 2, y = 6, rule = B2-kn4aijkrz5aejry6cik7/S12in3ckr4acejkt5-acek6-en
bo2$2o2$bo$bo!

(Rules are not minimized. What's the easy way to do that?)

[AforAmpere]

Amazing! I can't even express how amazed I am. My 7c/11 pales in comparison:

Code: Select all

x = 2, y = 3, rule = B2acn3aekn4cir5j/S1c2i3ij4t
bo$o$bo!

And 8c/13:

Code: Select all

x = 2, y = 3, rule = B2acn3aek4eiw5iy/S1c2i3ijq5iy6c
bo$o$bo!

8c/12:

Code: Select all

x = 2, y = 3, rule = B2acn3acekn4et7e/S1c2i3cir4ei
bo$o$bo!

9c/14:

Code: Select all

x = 2, y = 3, rule = B2acn3acek5ci/S1c2i3ijy4ajrt5eiky6ck
bo$o$bo!

9c/16:

Code: Select all

x = 2, y = 3, rule = B2acn3aekn4ceijr5iy6eik7e/S1c2i3ijn4i5ck
bo$o$bo!

10c/15:

Code: Select all

x = 2, y = 3, rule = B2acn3aeky4acjkrw6k7e/S1c2ei3iy4ijknrty5i
bo$o$bo!

[praosylen]

Smaller (in bounding box) (2,1)c/3:

Code: Select all

x = 4, y = 4, rule = B2-e3cjkq4-ijktw5-cn678/S01c2-c3-akn4-enw5-cijn6ikn8
3bo$3bo$3bo$obo!

EDIT: (2,1)c/4, same:

Code: Select all

x = 4, y = 4, rule = B2-ik3c4-aejkt5-cnq678/S01e2-ak3cknr4-ajry5-ijr6-a78
2bo$3bo$o$3bo!

[AforAmpere]

Can you search for a (4,2)c/6, or send the compiled glucose file? I can't seem to compile it.

[praosylen]

Here's the file for Mac:

glucose-syrup.zip

(76.43 KiB) Downloaded 195 times

If you have a different platform, I can't really help you.

EDIT: (4,2)c/6, 4 cells:

Code: Select all

x = 3, y = 4, rule = B2aci3aq4acikwyz5-cqry6n8/S1e2n3aeijr4acjkrwz5-cjnq6ikn7
2bo2$2bo$obo!

EDIT 2: (4,3)c/7, 4 cells:

Code: Select all

x = 3, y = 4, rule = B2-ek3an4cijnqy5-inry6-c7e/S01e2ac3ak4ejkrwz5aejnr6-k7c8
2bo2$2bo$obo!

(5,2)c/7, 4 cells:

Code: Select all

x = 3, y = 4, rule = B2aci3acn4ajnwz5-qy6akn78/S01e2ai3ajq4-nrt5ciq6-e78
2bo2$obo$2bo!
!

(3,2)c/7, 3 cells:

Code: Select all

x = 3, y = 5, rule = B2ain3-eijr4cijy5cjkry6ai8/S01e2ekn3acjr4cij5ainq6n78
o2$2bo2$2bo!

(4,2)c/7, 3 cells:

Code: Select all

x = 2, y = 4, rule = B2-i3aknq4ajkqryz5ckqr6ak7e/S02ae3ijnq4eijknrw5-ceqr6in8
o$bo2$o!

(4,1)c/7, 3 cells:

Code: Select all

x = 3, y = 4, rule = B2ack3acknq4aciqwy5-aijy6c7e8/S02ac3-kq4ceknqt5aceky6ack
obo3$2bo!

(5,1)c/7, 3 cells:

Code: Select all

x = 3, y = 5, rule = B2-k4acijnz5iy6k7e8/S12e3cein4ejnqty5-ajkq6aen7
o2$2bo2$2bo!

(6,1)c/7, 13 cells:

Code: Select all

x = 3, y = 8, rule = B2-kn3ceky4aceity5-jk6-kn7/S2-ei3cejn4aeiqryz5cey6-a7e8
3o$2bo$b2o$2bo$o$2bo$b2o$obo!

This time, I would assume that's the slope limit for (M,N)c/2(M+N) ships.

[AforAmpere]

I have Windows, running Cygwin.

EDIT, that means all ships up to p6 are found!