## B0 hyper-relativistic speeds

For discussion of other cellular automata.

### 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:
x = 3, y = 6, rule = B02357/S23456b2o$o$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: x = 32, y = 31, rule = B023456-a7/S01c234562$27b2o$26bo$26bo$24b2o$23bo$23bo$21b2o$20bo$20bo$18b2o$17bo$17bo$15b2o$14bo$14bo$12b2o$11bo$11bo$9b2o$8bo$8bo$6b2o$5bo$5bo! 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) AforAmpere Posts: 1041 Joined: July 1st, 2016, 3:58 pm ### Re: B0 hyper-relativistic speeds AforAmpere wrote:EDIT, a 5c/8 wave, not stablizable: x = 32, y = 31, rule = B023456-a7/S01c234562$27b2o$26bo$26bo$24b2o$23bo$23bo$21b2o$20bo$20bo$18b2o$17bo$17bo$15b2o$14bo$14bo$12b2o$11bo$11bo$9b2o$8bo$8bo$6b2o$5bo$5bo! It's more of a self-replicating ripple. It's not a wave because it has C2 symmetry. x = 4, y = 3, rule = B3-q4z5y/S234k5j2b2o$b2o$2o! LaundryPizza03 at Wikipedia LaundryPizza03 Posts: 457 Joined: December 15th, 2017, 12:05 am Location: Unidentified location "https://en.wikipedia.org/wiki/Texas" ### Re: B0 hyper-relativistic speeds Here's a very similar, but slightly smaller, example which also travels at 3c/4 diagonal x = 4, y = 3, rule = B02ace3r5kq/S2aek3n5a6akb2o$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. x = 9, y = 9, rule = B023ir4i57/S234562o$2o$2o$3b2o$3b2o$3b2o$6b2o$3b3o2bo$3b4obo! Asymmetric version of the 3c/4 in rules with B4k added: x = 6, y = 3, rule = B023ir4ik57/S234563b2o$3o2bo$4obo! Edit 2: Some bilaterally symmetric 3c/4 diagonal ships: x = 74, y = 74, rule = B023457/S02345610b2o5b2o$10b2o5b2o$6b3ob2o5b2o$6b3o7bobo$8bo7bo$8b3o6b2o$2b2o4b3o$2b2o$2b5o$5b2o6bo11b2o5b2o$3o2b2o18b2o5b2o$3o8b3o3b3o12b2o$11bo3b5o11bobo$9bobo2b4o13bo$13b5o14b2o$12b3o2b8obo$3b2o7b3obob9o$3o2bo5b5ob4o2bob2o$4obo5b2o2b4ob4ob3o$11b2o2b3ob5ob2o5b2o$15b7ob7o2b2o$15b2ob11ob4o$15b2ob2obob9obo$15b10ob6o$15b2o3b9obob2o$9b2o5b7obo3bobo$9b2o4b10o$18bob5o$20b6o$20bob2o$21b5o$12b2o7b3o$9b3o2bo4b3o2bo23b2o$9b4obo4b4obo20b3o2bo$37b4ob7obo$37b6o$41b3o$34b2o5b3o$34b2o6bo$34b2o5b3o11b2o5b2o$34b2o19b2o5b2o$35b3obo2b2o3b3o12b2o$34b6obob7o11bobo$34bob2obob7o13bo$34bo7b2o2b2o14b2o$33b2o7b2o3b8obo$33b2o7b3obob9o$33b2o6b5ob4o2bob2o$32bobo6b2o2b4ob4ob3o$32bo8b2o2b3ob5ob2o5b2o$33b2o10b7ob7o2b2o$45b2ob11ob4o$45b2ob2obob9obo$45b10ob6o$45b2o3b9obob2o$39b2o5b7obo3bobo$39b2o4b10o$48bob5o$50b6o$50bob2o$51b5o$42b2o7b3o18b2o$39b3o2bo4b3o2bo17b2o$39b4obo4b4obo17b2o$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)
> ./gfind /b023457s023456/d4n3ul240

A 3c/4 diagonal wickstretcher:

4c/6 diagonal:
x = 6, y = 6, rule = B0123jkqr4-ckyz5inqr6-k7c/S01e2-an3-ain4-enrt5-aek6-ci7cbo3bo$ob2o$bob3o$b3o$2bobo$obo! (5,2)c/6: x = 6, y = 6, rule = B012-an3-ijqr4-akw5aejy6ack/S012cen3-eknq4ceiryz5ceiny6cn72bo2bo$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) AforAmpere Posts: 1041 Joined: July 1st, 2016, 3:58 pm ### Re: B0 hyper-relativistic speeds (5,4)c/6, new record high speed: x = 5, y = 3, rule = B02-en3-einq4cejy5aknqy6eik7e/S01c2ack3aein4aeijknr5cjknq6ab3o$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): x = 5, y = 4, rule = B012-ae3-aj4ceikqry5er6ikn7/S01c2c3cjnr4ekwz5acjkn7o3bo$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

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### 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:
x = 3, y = 3, rule = B02aik3anr4aw5ae6ai7e/S01e3a5aq6c7eo$2bo$bo!

(3,2)c/4:
x = 2, y = 4, rule = B02-en3-ciqy4ackny5aijq6a7c/S12ac3ajk4c6n7ebo$o2$bo!

(3,1)c/4:
x = 3, y = 3, rule = B01c2aci3acinr4acrt5ijn6c7e/S1c23aein4irz5ci6ace2bo2$2o! 5c/6: x = 2, y = 3, rule = B01c2ikn3aceir4aitz5ackr6ikn/S02ik3ek4ceinqz5iry6-kbo$o$bo! 4c/6 diagonal: x = 3, y = 4, rule = B01e2cin3ajkny4ajty5nqr6en8/S1c2ack3eknqr4aj5ar6ao2$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) AforAmpere Posts: 1041 Joined: July 1st, 2016, 3:58 pm ### 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): x = 4, y = 8, rule = B012-ai3-ajqr4ikqrty5kry6eik/S012acn3ik4-jnqrw5k6en7o2bo4$3bo2$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

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### 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.

77topaz

Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: B0 hyper-relativistic speeds

5c/8 d:
x = 5, y = 5, rule = B012n3-ein4eijkt5cjq6cen7e8/S02-ae3-acn4jkqtwy5acjry6ckn7bo2bo$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)
AforAmpere

Posts: 1041
Joined: July 1st, 2016, 3:58 pm

### 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.

vyznev

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