## B0 hyper-relativistic speeds

For discussion of other cellular automata.
AforAmpere
Posts: 1077
Joined: July 1st, 2016, 3:58 pm

### B0 hyper-relativistic speeds

The purpose of this thread is to find B0 ships that transcend the speeds normally allowed in 2-state rules. One example, and the only one I know of is this 3c/4 diagonal ship:

Code: Select all

x = 3, y = 6, rule = B02357/S23456
b2o$o$obo$b2o$b2o$b2o!  This ship goes faster than anything possible without B0, which raises the question: how many other ships exist? I am unsure if any other ships are known that go faster than (m,n)c/(m+n), like this 3c/4 diagonal, or any that exist. EDIT, a 5c/8 wave, not stablizable: Code: Select all x = 32, y = 31, rule = B023456-a7/S01c23456 2$27b2o$26bo$26bo$24b2o$23bo$23bo$21b2o$20bo$20bo$18b2o$17bo$17bo$15b
2o$14bo$14bo$12b2o$11bo$11bo$9b2o$8bo$8bo$6b2o$5bo$5bo!  It's more of a self-replicating ripple. It's not a wave because it has C2 symmetry. Code: Select all x = 4, y = 3, rule = B3-q4z5y/S234k5j 2b2o$b2o$2o!  LaundryPizza03 at Wikipedia wildmyron Posts: 1343 Joined: August 9th, 2013, 12:45 am ### Re: B0 hyper-relativistic speeds Here's a very similar, but slightly smaller, example which also travels at 3c/4 diagonal Code: Select all x = 4, y = 3, rule = B02ace3r5kq/S2aek3n5a6ak b2o$o2bo$2obo! Edit: Extensible 3c/4 diagonal tagalong. Works in many of the isotropic rules the well known 3c/4 works in, but none of the semi-totalistic ones. Code: Select all x = 9, y = 9, rule = B023ir4i57/S23456 2o$2o$2o$3b2o$3b2o$3b2o$6b2o$3b3o2bo$3b4obo! Asymmetric version of the 3c/4 in rules with B4k added: Code: Select all x = 6, y = 3, rule = B023ir4ik57/S23456 3b2o$3o2bo$4obo! Edit 2: Some bilaterally symmetric 3c/4 diagonal ships: Code: Select all x = 74, y = 74, rule = B023457/S023456 10b2o5b2o$10b2o5b2o$6b3ob2o5b2o$6b3o7bobo$8bo7bo$8b3o6b2o$2b2o4b3o$2b
2o$2b5o$5b2o6bo11b2o5b2o$3o2b2o18b2o5b2o$3o8b3o3b3o12b2o$11bo3b5o11bob o$9bobo2b4o13bo$13b5o14b2o$12b3o2b8obo$3b2o7b3obob9o$3o2bo5b5ob4o2bob
2o$4obo5b2o2b4ob4ob3o$11b2o2b3ob5ob2o5b2o$15b7ob7o2b2o$15b2ob11ob4o$15b2ob2obob9obo$15b10ob6o$15b2o3b9obob2o$9b2o5b7obo3bobo$9b2o4b10o$18b
ob5o$20b6o$20bob2o$21b5o$12b2o7b3o$9b3o2bo4b3o2bo23b2o$9b4obo4b4obo20b
3o2bo$37b4ob7obo$37b6o$41b3o$34b2o5b3o$34b2o6bo$34b2o5b3o11b2o5b2o$34b 2o19b2o5b2o$35b3obo2b2o3b3o12b2o$34b6obob7o11bobo$34bob2obob7o13bo$34b o7b2o2b2o14b2o$33b2o7b2o3b8obo$33b2o7b3obob9o$33b2o6b5ob4o2bob2o$32bob o6b2o2b4ob4ob3o$32bo8b2o2b3ob5ob2o5b2o$33b2o10b7ob7o2b2o$45b2ob11ob4o$45b2ob2obob9obo$45b10ob6o$45b2o3b9obob2o$39b2o5b7obo3bobo$39b2o4b10o$
48bob5o$50b6o$50bob2o$51b5o$42b2o7b3o18b2o$39b3o2bo4b3o2bo17b2o$39b4ob
o4b4obo17b2o$71bobo$65b2o4bo$65b2o5b2o5$64b2o$61b3o2bo$61b4obo!
There seems to be a bug in gfind because it doesn't find any of these ships when searching with bilateral symmetry (gfind v4.9 unmodified)

Code: Select all

> ./gfind /b023457s023456/d4n3ul240
A 3c/4 diagonal wickstretcher:

Code: Select all

x = 54, y = 54, rule = B023457/S023456
11b3o$11b3o$9b5o$9b8o$11b2ob3o$11b6o$12b8o$14b2ob3o$8b2o4b6o$2b2o4b2o 5b8o$2b2o13b2ob3o$6o11b6o$7o11b8o$4ob2o13b2ob3o$3b6o5b2o4b6o$3b7o4b2o 5b8o$3b4ob2o13b2ob3o$6b6o11b6o$6b7o11b8o$6b4ob2o13b2ob3o10b2o$9b6o5b2o
4b6o10b2o$9b7o4b2o5b5o10b2o$9b4ob2o13b4o4b2o2bobo$12b6o11b3o5b3obo$12b
7o11b2o5b2o3b2o$12b4ob2o18b3o$15b6o5b2o8b2o$15b7o4b2o7bo$15b4ob2o12b3o
$18b6o5b2o2b3o9b2o5b2o$18b7o4b2o3bo10b2o5b2o$18b7o7b3o2b3o12b2o$22bo8b
9o11bobo$29bob3ob3o13bo$28b5ob4o14b2o$27b3o2b4ob8obo$26bobo3b3obob9o$22b5o4b5ob4o2bob2o$22b4o5b2o2b4ob4ob3o$23bobo5b2o2b3ob5ob2o5b2o$35b7ob
7o2b2o$22b2o11b2ob11ob4o$19b3o2bo10b2ob2obob9obo$19b4obo10b10ob6o$35b
2o3b9obob2o$29b2o5b7obo3bobo$29b2o4b10o$38bob5o$40b6o$40bob2o$41b5o$32b2o7b3o$29b3o2bo4b3o2bo$29b4obo4b4obo! I tried to find a stabilisation for the p8 wick with JLS but was unsuccessful. p12 3c/4 diagonal frontend. Looks promising but I was unable to complete it. gfind not working for B0 searches with bilateral symmetry is annoying. Code: Select all x = 41, y = 41, rule = B023457/S023456 12b3o3b2o$10bob3o3b2o$9b2ob4ob3o$9b2o4b3o$9b3o3b3o5bo$10b2o3bo$10b2o2b 3obobo$7b5o4bobobo$7b5o4b5o2b3o$2b3o2b5obo2b6ob3o$b10ob2o2b2ob3o$4b6o
2b9o4b2o$3o7b4o2b5o6bo$3o6b4ob3ob4o3b3o3b2o$3o3bo4bob5o2bobo2b3o2b3o$
2b5o4bob3ob6o2b7o$3b2ob9o2b4o4b3obobo$2b3o3b5ob7o4b3ob3o6b2o$3o3b4ob3o b3ob2o9b3o2b3o2bo$3o5b6ob13o3b2o2b4obo$6b23o2b3o$9b2o2bobo3b6ob3o$14b 2o3b3o2bobob4o$4bo3b2o9b3ob4o2b3o$8b2o9b9ob3o$8b2obob5ob2o2b5ob3ob2o$11bob5ob10ob5o$12b6ob3o2b11o$15bo4b3o3b9o$15b3o4b4ob4obo$14b2ob2o3b8o 3b2o$13b8ob7o$13b2o3b3o5b4o$20bo4b4obo$25b4obo$18b2o$18b2o$18b2o$17bob o$17bo$18b2o! Edit 3: Added "diagonal" to text in several places for clarity The latest version of the 5S Project contains over 226,000 spaceships. There is also a GitHub mirror of the collection. Tabulated pages up to period 160 (out of date) are available on the LifeWiki. AforAmpere Posts: 1077 Joined: July 1st, 2016, 3:58 pm ### Re: B0 hyper-relativistic speeds (3,2)c/4: Code: Select all x = 2, y = 3, rule = B02aei3-enq4nrz5ijnqr6aek7c/S1c2-an3-ajqy4iknqrty5ijk6ak bo$2o$2o! (4,3)c/6: Code: Select all x = 6, y = 6, rule = B01c2-kn34-atyz5-knq6cek7e/S012-cn3-ejny4-ejk5iy6e7 2ob2o$o3bo$2o3bo$2bo$2bobo$2bo2bo!

4c/6 diagonal:

Code: Select all

x = 6, y = 6, rule = B0123jkqr4-ckyz5inqr6-k7c/S01e2-an3-ain4-enrt5-aek6-ci7c
bo3bo$ob2o$bob3o$b3o$2bobo$obo!  (5,2)c/6: Code: Select all x = 6, y = 6, rule = B012-an3-ijqr4-akw5aejy6ack/S012cen3-eknq4ceiryz5ceiny6cn7 2bo2bo$o$5bo2$o3b2o$b3o!  I and wildmyron manage the 5S project, which collects all known spaceship speeds in Isotropic Non-totalistic rules. Things to work on: - Find a (7,1)c/8 ship in a Non-totalistic rule - Finish a rule with ships with period >= f_e_0(n) (in progress) A for awesome Posts: 1949 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: ### Re: B0 hyper-relativistic speeds (5,4)c/6, new record high speed: Code: Select all x = 5, y = 3, rule = B02-en3-einq4cejy5aknqy6eik7e/S01c2ack3aein4aeijknr5cjknq6a b3o$2obo$5o!  (5,4)c/6 = √(41)/6c = 1.067187372...c > 1.060660171...c = √(18)/4c = (3,3)c/4. EDIT: Smaller (in gen 5): Code: Select all x = 5, y = 4, rule = B012-ae3-aj4ceikqry5er6ikn7/S01c2c3cjnr4ekwz5acjkn7 o3bo$4bo$3b2o$o2bo!

Last edited by A for awesome on February 2nd, 2018, 10:37 pm, edited 2 times in total.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

AforAmpere
Posts: 1077
Joined: July 1st, 2016, 3:58 pm

### Re: B0 hyper-relativistic speeds

A for awesome wrote:new record high speed
What do you mean? I found a bunch of 3-cell ships:

3c/4 diagonal:

Code: Select all

x = 3, y = 3, rule = B02aik3anr4aw5ae6ai7e/S01e3a5aq6c7e
o$2bo$bo!
(3,2)c/4:

Code: Select all

x = 2, y = 4, rule = B02-en3-ciqy4ackny5aijq6a7c/S12ac3ajk4c6n7e
bo$o2$bo!
(3,1)c/4:

Code: Select all

x = 3, y = 3, rule = B01c2aci3acinr4acrt5ijn6c7e/S1c23aein4irz5ci6ace
2bo2$2o! 5c/6: Code: Select all x = 2, y = 3, rule = B01c2ikn3aceir4aitz5ackr6ikn/S02ik3ek4ceinqz5iry6-k bo$o$bo! 4c/6 diagonal: Code: Select all x = 3, y = 4, rule = B01e2cin3ajkny4ajty5nqr6en8/S1c2ack3eknqr4aj5ar6a o2$bo$2bo! I and wildmyron manage the 5S project, which collects all known spaceship speeds in Isotropic Non-totalistic rules. Things to work on: - Find a (7,1)c/8 ship in a Non-totalistic rule - Finish a rule with ships with period >= f_e_0(n) (in progress) A for awesome Posts: 1949 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: ### Re: B0 hyper-relativistic speeds AforAmpere wrote: A for awesome wrote:new record high speed What do you mean? See my edit. I was referring to the Euclidean norm of the speed. EDIT: Even faster, (7,5)c/8 (4 cells in gen. 1; Euclidean speed of 1.075290658...c): Code: Select all x = 4, y = 8, rule = B012-ai3-ajqr4ikqrty5kry6eik/S012acn3ik4-jnqrw5k6en7 o2bo4$3bo2$bobo$o2bo!

The ultimate limit, of course, is (2,1)c/2, which corresponds to a Euclidean speed of 1.118033988...c. That speed may or may not actually be attainable, and it may also be attainable only in limit, with speeds infinitely (for all practical purposes) approaching, but not reaching, (2,1)c/2.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

77topaz
Posts: 1467
Joined: January 12th, 2018, 9:19 pm

### Re: B0 hyper-relativistic speeds

I don't think the Euclidean norm is an entirely accurate/representative way of measuring spaceship speeds, though, because it doesn't entirely fit with the Moore neighbourhood. This is because the Euclidean distance between two diagonally touching cells is √2 that between two orthogonally touching cells, but in the Moore neighbourhood they are treated as equally distant. This discrepancy causes the Euclidean norm to produce more obviously discrepant results at high speeds, such as suggesting a (7,5)c/8 ship using the Moore neighbourhood is moving faster than c, when this is by the definition of the Moore neighbourhood and c impossible.

AforAmpere
Posts: 1077
Joined: July 1st, 2016, 3:58 pm

### Re: B0 hyper-relativistic speeds

5c/8 d:

Code: Select all

x = 5, y = 5, rule = B012n3-ein4eijkt5cjq6cen7e8/S02-ae3-acn4jkqtwy5acjry6ckn7
bo2bo$o2bo$3bo$b3o$o3bo!

I and wildmyron manage the 5S project, which collects all known spaceship speeds in Isotropic Non-totalistic rules.

Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule
- Finish a rule with ships with period >= f_e_0(n) (in progress)

vyznev
Posts: 27
Joined: April 23rd, 2016, 4:08 am

### Re: B0 hyper-relativistic speeds

I just posted a bunch of small ships in B0 rules in another thread.

The linked post includes ships with slopes (X,Y)c/2P for all 0 ≤ YX < 2P ≤ 8, with the exception of (5,5)c/6 and (7,3..7)c/8, as well as orthogonal (2P,0)c/2P photons for all 0 < 2P ≤ 8.

As I also noted elsewhere, at least (2,1)c/2 appears to be impossible. Off the top of my head I'd conjecture that isotropic 8-neighbor CA rules with B0 can probably support ships with all sublight speeds and slopes, as well as orthogonal photons, but that diagonal and oblique photons are impossible.

I don't (yet) have a proof for the impossibility of oblique lightspeed ships in B0 rules, but I do suspect that, as in the linked LLS-assisted impossibility proof for (2,1)c/2, it might be possible to prove this by considering the outermost live cells in the first and last rows/columns of the ship (in the direction where the ship moves at lightspeed), and showing that these corner cells cannot both advance and retreat obliquely at lightspeed.

EDIT: The conjecture I made above is wrong; as Majestas32 pointed out in the other thread, 3c/4 is the fastest diagonal speed possible, since any rule where the leading corner can advance diagonally on both even and odd generations will explode. The same argument, of course, rules out oblique (X,Y)c/2P ships with X + Y > 3P, as the best you can possibly do is have the leading corner advance by a knight's move every two generations.