MarBlocks + emulation

[muzik]

This thread is to consolidate all information relevant to rules that support certain isotropic Margolus simulations exhibited by higher cellular automata. There's quite a few of these known, so I want to keep track of them as well as encourage the sharing of further types.

The most known class of such block CA are detailed in this blog post (sourced from this forum post) by Nathaniel: http://www.njohnston.ca/2009/05/rectang ... automaton/

and are explored further in the following paper: http://www.njohnston.ca/wp-content/uplo ... 01/2x2.pdf

A starter list of known media:

Margolus rule being simulated by all five possible media (empty doesn't count) coexisting in a single rule:

Code: Select all

x = 84, y = 12, rule = B2c3i4i5i/S2c3i4i5i
2o18b2o20bo19bo19bo$bo18b2o20bo40bo$2o18b2o20bo19bo19bo$bo18b2o20bo40b
o$2o18b2o20bo19bo19bo$bo18b2o20bo40bo$2o18b2o20bo19bo19bo$bo18b2o20bo
40bo$2o18b2o20bo19bo19bo$bo18b2o20bo40bo$2o18b2o20bo19bo19bo$bo18b2o
20bo40bo!

I'm surprised that the young members of the forum did not explore the rule introduced here:

http://www.conwaylife.com/forums/viewto ... 125#p32973

A beautiful period 4194302 oscillator:

Code: Select all

x = 48, y = 48, rule = B2ae3i/S
.A.A3.A.A.A5.A.A.A7.A.A3.A5.A.A.A$A3.A.A5.A3.A5.A5.A3.A.A.A3.A5.A$9.
A3.A.A3.A3.A3.A7.A.A.A3.A3.A$A3.A.A.A.A7.A.A3.A.A5.A3.A5.A.A$.A.A5.
A.A.A.A.A.A.A9.A.A5.A.A.A.A.A$6.A3.A7.A.A.A.A.A3.A.A.A.A.A3.A.A$.A.
A.A.A5.A.A3.A.A5.A.A.A.A9.A3.A$A5.A.A.A.A3.A3.A3.A5.A.A3.A.A7.A$3.A
3.A9.A5.A.A.A3.A3.A3.A3.A.A$A.A.A5.A.A5.A.A.A.A3.A5.A5.A.A$3.A.A.A.
A3.A9.A5.A.A.A11.A$A3.A9.A3.A.A19.A.A.A.A$.A5.A.A5.A.A3.A.A5.A.A.A5.
A5.A.A$2.A.A.A3.A13.A3.A5.A3.A3.A3.A$11.A3.A.A3.A3.A.A3.A3.A.A3.A.A
$2.A.A.A5.A.A3.A.A.A7.A.A7.A.A.A.A$.A5.A13.A.A.A.A.A.A.A.A.A.A.A.A3.
A$A3.A3.A3.A.A3.A3.A7.A.A7.A.A$3.A.A3.A.A3.A.A.A5.A.A3.A3.A.A3.A5.A
$A.A.A.A11.A.A.A.A3.A5.A3.A7.A$3.A.A.A.A.A3.A3.A9.A.A.A5.A.A3.A$A3.
A.A5.A.A.A5.A.A17.A.A$.A3.A3.A5.A.A.A.A3.A3.A.A.A5.A$2.A5.A.A.A3.A9.
A.A5.A3.A.A.A.A$3.A.A.A.A3.A5.A.A9.A3.A.A.A5.A$8.A5.A.A.A3.A3.A.A.A
.A5.A3.A3.A$3.A.A17.A.A5.A.A.A5.A.A3.A$2.A3.A.A5.A.A.A9.A3.A3.A.A.A
.A.A$.A7.A3.A5.A3.A.A.A.A11.A.A.A.A$A5.A3.A.A3.A3.A.A5.A.A.A3.A.A3.
A.A$5.A.A7.A.A7.A3.A3.A.A3.A3.A3.A$A3.A.A.A.A.A.A.A.A.A.A.A.A13.A5.
A$.A.A.A.A7.A.A7.A.A.A3.A.A5.A.A.A$4.A.A3.A.A3.A3.A.A3.A3.A.A3.A$.A
3.A3.A3.A5.A3.A13.A3.A.A.A$A.A5.A5.A.A.A5.A.A3.A.A5.A.A5.A$.A.A.A.A
19.A.A3.A9.A3.A$2.A11.A.A.A5.A9.A3.A.A.A.A$5.A.A5.A5.A3.A.A.A.A5.A.
A5.A.A.A$2.A.A3.A3.A3.A3.A.A.A5.A9.A3.A$.A7.A.A3.A.A5.A3.A3.A3.A.A.
A.A5.A$A3.A9.A.A.A.A5.A.A3.A.A5.A.A.A.A$3.A.A3.A.A.A.A.A3.A.A.A.A.A
7.A3.A$A.A.A.A.A5.A.A9.A.A.A.A.A.A.A5.A.A$3.A.A5.A3.A5.A.A3.A.A7.A.
A.A.A3.A$A3.A3.A.A.A7.A3.A3.A3.A.A3.A$.A5.A3.A.A.A3.A5.A5.A3.A5.A.A
3.A$2.A.A.A5.A3.A.A7.A.A.A5.A.A.A3.A.A!

Brian Prentice

Code: Select all

x = 64, y = 10, rule = B2e/S
63bo$42bobo17bo$25bobo13bo3bo15bo$12bobo9bo3bo11bo5bo13bo$3bobo5bo3bo
7bo5bo9bo7bo11bo$obo3bo3bo5bo5bo7bo7bo9bo9bo$7bobo7bo3bo9bo5bo11bo7bo$
18bobo11bo3bo13bo5bo$33bobo15bo3bo$52bobo!

Is there any more interesting rule than muzik's example with the five media?

For margolus media that can have the cool oscillators, I know of three other types.

Code: Select all

x = 91, y = 77, rule = B2i3ai4qt5i/S1e2ik3ij4w5i
4o17bo3bo14b12o9bo3bo3bo3bo3bo3bo$4o16b3ob3o13b12o8b3ob3ob3ob3ob3ob3o$
21bo3bo35bo3bo3bo3bo3bo3bo18$8o13bo3bo3bo3bo$8o12b3ob3ob3ob3o$21bo3bo
3bo3bo8$40b16o5bo3bo3bo3bo3bo3bo3bo3bo$40b16o4b3ob3ob3ob3ob3ob3ob3ob3o
$61bo3bo3bo3bo3bo3bo3bo3bo8$8o13bo3bo3bo3bo$8o12b3ob3ob3ob3o$8o13bo3bo
3bo3bo$8o15bo7bo$21bo3bo3bo3bo$20b3ob3ob3ob3o$21bo3bo3bo3bo24$40b16o5b
o3bo3bo3bo3bo3bo3bo3bo$40b16o4b3ob3ob3ob3ob3ob3ob3ob3o$40b16o5bo3bo3bo
3bo3bo3bo3bo3bo$40b16o7bo7bo7bo7bo$61bo3bo3bo3bo3bo3bo3bo3bo$60b3ob3ob
3ob3ob3ob3ob3ob3o$61bo3bo3bo3bo3bo3bo3bo3bo!

Code: Select all

x = 266, y = 10, rule = B2en3ijq4w6aen/S2cek3aceny4ejqr5jk6a
b2o4b2o22b2o4b2o4b2o4b2o20b2o4b2o4b2o4b2o20b2o4b2o4b2o4b2o4b2o4b2o18b
2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o16b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o$o2bo2b
o2bo20bo2bo2bo2bo2bo2bo2bo2bo18bo2bo2bo2bo2bo2bo2bo2bo18bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo16bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
14bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o2bo2bo2bo20bo2bo2bo
2bo2bo2bo2bo2bo18bo2bo2bo2bo2bo2bo2bo2bo18bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo2bo16bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo14bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$b2o4b2o22b2o4b2o4b2o4b2o20b2o4b2o
4b2o4b2o20b2o4b2o4b2o4b2o4b2o4b2o18b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o16b
2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o3$71b2o4b2o4b2o4b2o130b2o4b2o4b2o4b2o4b
2o4b2o4b2o4b2o$70bo2bo2bo2bo2bo2bo2bo2bo128bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo$70bo2bo2bo2bo2bo2bo2bo2bo128bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo$71b2o4b2o4b2o4b2o130b2o4b2o4b2o4b2o4b2o4b
2o4b2o4b2o!

Other media that can support M10:

Code: Select all

x = 55, y = 25, rule = DeficientSeeds
A.A19.A.A27.A.A6$A.A19.A.A27.A.A6$22.A.A27.A.A6$22.A.A27.A.A6$52.A.A!

Code: Select all

x = 25, y = 9, rule = B1c/S1c|B2a/S1c2e
24bo$21bo$18bo$15bo$12bo$9bo$6bo$3bo$o!

Here's another rule that exhibits that same kind of Margolus behaviour:

Code: Select all

 x = 48, y = 48, rule = R2,C0,M0,S5..5,B3..3,NN
bo$obo$bo$4bo$3bobo$4bobo$5bobo$6bo$9bo$8bobo$9bobo$10bobo$11bobo$12bo
bo$13bo$16bo$15bobo$16bobo$17bobo$18bobo$19bobo$20bobo$21bobo$22bo$25b
o$24bobo$25bobo$26bobo$27bobo$28bobo$29bobo$30bobo$31bobo$32bobo$33bo$
36bo$35bobo$36bobo$37bobo$38bobo$39bobo$40bobo$41bobo$42bobo$43bobo$
44bobo$45bobo$46bo!

The following is a non-totalistic rule that can simulate these block CA at twice the speed:

B3i4int5ey6k7e/S1e2k3ey4irt5i

Code: Select all

x = 319, y = 95, rule = B3i4int5ey6k7e_S1e2k3ey4irt5i
17b2o19b2o54b2o23b2o25b2o71b2o22b2o27b2o3b2o26b2o$bo17bo20bo21bo2bo27b
o24bo26bo2bo66bo2bo26bo2bo22bo7bo24bo5bo3bo$bo17bo20bo21bo2bo27bo24bo
26bo2bo66bo2bo26bo2bo22bo7bo24bo5bo3bo$17b2o19b2o23b2o29b2o23b2o98b2o
22b2o27b2o3b2o26b2o$bo14bo23bo24bo30bo21bo2bo26bo66bo5bo23bo2bo22bo2bo
4bo27bo2bo3bo$bo14bo23bo24bo30bo21bo2bo26bo66bo5bo23bo2bo22bo2bo4bo27b
o2bo3bo$17b2o19b2o54b2o23b2o98b2o22b2o27b2o3b2o26b2o14$2o14b4o17b4o20b
7o26b2o20b8o20b6o61b14o16b10o19b12o19b18o$16b4o40bo7bo25b2o20b8o$60bo
7bo25b2o$60bo7bo25b2o$61b7o26b2o$89b5o2b5o$88bo5b2o5bo$87bob5o2b5obo$
88bo5b2o5bo$89b5o2b5o$94b2o$94b2o$94b2o$94b2o$94b2o$37b4o$37b4o$37b4o$
34b3o4b3o$33bo3b4o3bo$32bob3o4b3obo97b8o59b16o$33bo3b4o3bo$34b3o4b3o$
37b4o$37b4o$37b4o5$92b6o$92b6o$90b2o6b2o$90b2o6b2o$90b2o6b2o$87b3o2b6o
2b3o$87b3o2b6o2b3o$87b3o2b6o2b3o$84b3o3b2o6b2o3b3o$84b3o3b2o6b2o3b3o$
84b3o3b2o6b2o3b3o$84b3o3b2o6b2o3b3o$80b4o3b3o2b6o2b3o3b4o$80b4o3b3o2b
6o2b3o3b4o$78b2o4b3o3b2o6b2o3b3o4b2o$77bo2b4o3b3o2b6o2b3o3b4o2bo$77bo
2b4o3b3o2b6o2b3o3b4o2bo$77bo2b4o3b3o2b6o2b3o3b4o2bo$74b3ob2o4b3o3b2o6b
2o3b3o4b2ob3o$73bo3bo2b4o3b3o2b6o2b3o3b4o2bo3bo$72bob3ob2o4b3o3b2o6b2o
3b3o4b2ob3obo$71bobo3bo2b4o3b3o2b6o2b3o3b4o2bo3bobo$70bobob3ob2o4b3o3b
2o6b2o3b3o4b2ob3obobo$71bobo3bo2b4o3b3o2b6o2b3o3b4o2bo3bobo$72bob3ob2o
4b3o3b2o6b2o3b3o4b2ob3obo$73bo3bo2b4o3b3o2b6o2b3o3b4o2bo3bo$74b3ob2o4b
3o3b2o6b2o3b3o4b2ob3o$77bo2b4o3b3o2b6o2b3o3b4o2bo$77bo2b4o3b3o2b6o2b3o
3b4o2bo$77bo2b4o3b3o2b6o2b3o3b4o2bo$78b2o4b3o3b2o6b2o3b3o4b2o$80b4o3b
3o2b6o2b3o3b4o$80b4o3b3o2b6o2b3o3b4o$84b3o3b2o6b2o3b3o$84b3o3b2o6b2o3b
3o$84b3o3b2o6b2o3b3o$84b3o3b2o6b2o3b3o$87b3o2b6o2b3o$87b3o2b6o2b3o$87b
3o2b6o2b3o$90b2o6b2o$90b2o6b2o$90b2o6b2o$92b6o$92b6o!

EDIT:

Code: Select all

x = 193, y = 84, rule = B3i4int5ey6k7e_S1e2k3ey4irt5i
b2o13bo2bo13b2o21b2o52b2o65b2o$3bo12bo2bo12bo22bo2bo50bo2bo60bo2bo2bo$
3bo13b2o13bo22bo2bo50bo2bo60bo2bo2bo$b2o16bo13b2o21b2o52b2o$o18bo12bo
2bo19bo2bo53bo60bo2bo2bo$o31bo2bo19bo2bo53bo60bo2bo2bo$b2o30b2o21b2o
52b2o65b2o6$b3o10b8o$o3bo9b8o$b3o10b8o9b7o$14b8o8bo7bo$31b7o4$50b15o
43b6o57b11o$49bo15bo41bo6bo55bo11bo$49bo15bo41bo6bo55bo11bo$49bo15bo
41bo6bo55bo11bo$49bo15bo38b3ob6ob3o52bo11bo$49bo15bo38b3ob6ob3o52bo11b
o$49bo15bo36b2o3bo6bo3b2o50bo11bo$49bo15bo36b2o3bo6bo3b2o50bo11bo$50b
15o37b2o3bo6bo3b2o43b7ob11ob7o$102b2o3bo6bo3b2o42bo7bo11bo7bo$102b2o3b
o6bo3b2o41bob7ob11ob7obo$97b5o2b3ob6ob3o2b5o35bobo7bo11bo7bobo$96bo5b
2o3bo6bo3b2o5bo35bob7ob11ob7obo$96bo5b2o3bo6bo3b2o5bo36bo7bo11bo7bo$
94b2ob5o2b3ob6ob3o2b5ob2o35b7ob11ob7o$93bo2bo5b2o3bo6bo3b2o5bo2bo41bo
11bo$93bo2bo5b2o3bo6bo3b2o5bo2bo41bo11bo$93bo2bo5b2o3bo6bo3b2o5bo2bo
41bo11bo$90b3ob2ob5o2b3ob6ob3o2b5ob2ob3o38bo11bo$89bo3bo2bo5b2o3bo6bo
3b2o5bo2bo3bo37bo11bo$88bob3ob2ob5o2b3ob6ob3o2b5ob2ob3obo36bo11bo$88bo
b3ob2ob5o2b3ob6ob3o2b5ob2ob3obo36bo11bo$86b2obo3bo2bo5b2o3bo6bo3b2o5bo
2bo3bob2o35b11o$85bo2bob3ob2ob5o2b3ob6ob3o2b5ob2ob3obo2bo$84bob2obo3bo
2bo5b2o3bo6bo3b2o5bo2bo3bob2obo$83bobo2bob3ob2ob5o2b3ob6ob3o2b5ob2ob3o
bo2bobo$82bobob2obo3bo2bo5b2o3bo6bo3b2o5bo2bo3bob2obobo$81bobobo2bob3o
b2ob5o2b3ob6ob3o2b5ob2ob3obo2bobobo$80bobobob2obo3bo2bo5b2o3bo6bo3b2o
5bo2bo3bob2obobobo$81bobobo2bob3ob2ob5o2b3ob6ob3o2b5ob2ob3obo2bobobo$
49b16o17bobob2obo3bo2bo5b2o3bo6bo3b2o5bo2bo3bob2obobo$49b16o18bobo2bob
3ob2ob5o2b3ob6ob3o2b5ob2ob3obo2bobo$49b16o19bob2obo3bo2bo5b2o3bo6bo3b
2o5bo2bo3bob2obo$49b16o20bo2bob3ob2ob5o2b3ob6ob3o2b5ob2ob3obo2bo$49b
16o21b2obo3bo2bo5b2o3bo6bo3b2o5bo2bo3bob2o38b4o$49b16o23bob3ob2ob5o2b
3ob6ob3o2b5ob2ob3obo40b4o$49b16o23bob3ob2ob5o2b3ob6ob3o2b5ob2ob3obo40b
4o$49b16o24bo3bo2bo5b2o3bo6bo3b2o5bo2bo3bo41b4o$90b3ob2ob5o2b3ob6ob3o
2b5ob2ob3o42b4o$93bo2bo5b2o3bo6bo3b2o5bo2bo45b4o$93bo2bo5b2o3bo6bo3b2o
5bo2bo45b4o$93bo2bo5b2o3bo6bo3b2o5bo2bo45b4o$94b2ob5o2b3ob6ob3o2b5ob2o
46b4o$96bo5b2o3bo6bo3b2o5bo48b4o$96bo5b2o3bo6bo3b2o5bo38b10o4b10o$97b
5o2b3ob6ob3o2b5o39b10o4b10o$102b2o3bo6bo3b2o42b2o10b4o10b2o$102b2o3bo
6bo3b2o42b2o10b4o10b2o$102b2o3bo6bo3b2o40b2o2b10o4b10o2b2o$102b2o3bo6b
o3b2o40b2o2b10o4b10o2b2o$102b2o3bo6bo3b2o42b2o10b4o10b2o$104b3ob6ob3o
44b2o10b4o10b2o$104b3ob6ob3o46b10o4b10o$107bo6bo49b10o4b10o$107bo6bo
59b4o$107bo6bo59b4o$108b6o60b4o$174b4o$174b4o$174b4o$174b4o$174b4o$
174b4o$174b4o!

[muzik]

There is also a 3-state equivalent to the usual 2D margolus rules, which follow the analogous 1D rule, break down according to a different sequence than the powers of two, and generally do not have lower mods than periods.

Here are the 3-state equivalents of the block CA found in 2x2:

Code: Select all

x = 268, y = 2, rule = marg3
2A2.4A2.6A2.8A2.10A2.12A2.14A2.16A2.18A2.20A2.22A2.24A2.26A2.28A2.30A
$2A2.4A2.6A2.8A2.10A2.12A2.14A2.16A2.18A2.20A2.22A2.24A2.26A2.28A2.
30A!

Currently working on the rule so that any arrangement of blocks can work. Sometimes it breaks down with more complex ones but we can see the gist of the effects here. I haven't been able to find any of these in the wild yet. Yet to see anyone try to generalize these up to any (prime) number - I assume it'd be easy since it's just the modulo of the sum of neighbouring cells.

Comparison to 2 state:

Code: Select all

x = 268, y = 2, rule = B3/S5
2o2b4o2b6o2b8o2b10o2b12o2b14o2b16o2b18o2b20o2b22o2b24o2b26o2b28o2b30o$
2o2b4o2b6o2b8o2b10o2b12o2b14o2b16o2b18o2b20o2b22o2b24o2b26o2b28o2b30o!

Some squares, p4n 4nx4n:

Code: Select all

 x = 40, y = 12, rule = marg3
28.12A$28.12A$10.8A10.12A$10.8A10.12A$4A6.8A10.12A$4A6.8A10.12A$4A6.
8A10.12A$4A6.8A10.12A$10.8A10.12A$10.8A10.12A$28.12A$28.12A!

A diagonal version of the 3-state version of the commonly encountered margolus rule which follows rule 6:

Code: Select all

x = 50, y = 61, rule = marg3diag
13.A.A$12.A3.A21.A.A$11.A5.A19.A3.A$10.A7.A17.A5.A$9.A9.A15.A7.A$8.A
11.A13.A9.A$7.A13.A11.A11.A$6.A15.A9.A13.A$5.A17.A7.A15.A$4.A19.A5.A
17.A$3.A21.A3.A19.A$2.A23.A.A$.A47.A$48.A$.A45.A$2.A43.A$3.A41.A$4.A
39.A$5.A37.A$6.A35.A$7.A33.A$8.A17.A.A$9.A15.A3.A9.A$10.A13.A5.A7.A$
11.A11.A13.A$12.A17.A5.A$13.A7.A7.A5.A$14.A5.A13.A$19.A7.A5.A$14.A3.A
13.A$13.A3.A$12.A17.A$11.A5.A11.A$10.A7.A9.A$9.A9.A7.A$8.A11.A5.A$7.A
13.A3.A$6.A15.A.A$5.A$4.A$3.A$2.A$.A$A$33.A$A31.A$.A29.A$2.A27.A$3.A
25.A$4.A23.A$5.A21.A$6.A19.A$7.A17.A$8.A15.A$9.A13.A$10.A11.A$11.A9.A
$12.A7.A$13.A5.A$14.A3.A$15.A.A!

This one follows 3 state rule 8229. Failed oscillators follow A048473 and split into multiple smaller oscillators.

Can someone make a ruletable for the 5-state or 7-state versions of this?

[wwei47]

Code: Select all

x = 118, y = 88, rule = B1e2i3er4e5i6cn/S2ac:T120,90
3bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo
5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo5bo5bo5b
o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo5bo5bo5bo5bo
5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5b
o5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo
5bo5bo5bo5bo5bo3$3bo5bo11bo5bo11bo5bo5bo5bo11bo5bo5bo11bo5bo5bo5bo5bo
3$o5bo5bo11bo5bo11bo5bo5bo17bo5bo5bo11bo5bo5bo5bo3$3bo5bo5bo5bo5bo5bo
11bo5bo5bo17bo5bo5bo11bo5bo5bo5bo3$o5bo5bo5bo5bo5bo5bo11bo5bo5bo17bo5b
o5bo11bo5bo5bo3$3bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo11bo5bo5bo5bo11bo5bo
5bo3$o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo11bo5bo3$3bo5bo
5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo
5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo5bo5bo5bo5bo5b
o11bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo5bo5bo5bo17bo5bo5bo
5bo5bo5bo5bo5bo5bo5bo3$3bo5bo11bo5bo5bo5bo5bo17bo5bo5bo5bo5bo5bo5bo5bo
5bo5bo3$o5bo5bo11bo5bo5bo5bo5bo17bo5bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo
11bo5bo5bo5bo5bo17bo5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo11bo5bo5bo5bo
5bo17bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo5bo11bo5bo5bo5bo5bo17bo5bo5bo5b
o5bo5bo5bo5bo3$o5bo5bo5bo5bo11bo5bo5bo5bo5bo17bo5bo5bo5bo5bo5bo5bo3$3b
o5bo5bo5bo5bo11bo5bo5bo5bo5bo11bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo
5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo5bo5bo5bo5bo5b
o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo
5bo5bo5bo5bo5bo5bo5bo5bo5bo3$3bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5b
o5bo5bo5bo5bo5bo5bo3$o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo
5bo5bo5bo3$3bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5b
o3$o5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo!

[muzik]

This post and five posts that follow it mention a lot regarding these media:

Something that I realized about these XOR oscillators: They're all actually 2x2 block oscillators, it's just that the 2x2 blocks have different contents. For the original 2x2 rule, the 2x2 blocks are completely full; for B2c/S and analogues, the blocks consist of one on cell and 3 off cells; and for rules such as that in the above post, the blocks look like this:

Code: Select all

oo
..

Obviously, they aren't all Margolus automata (at least range 1) — unlike 2x2 — but I feel like they still classify as 2x2 block oscillators, if not automata.

Here are two more families I've found based on this idea:

Code: Select all

x = 96, y = 2, rule = B2c/S2c
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobo$bobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobobobobobo!

Code: Select all

x = 90, y = 2, rule = B2c3i5i/S3i4i
90o$bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobo!

Unfortunately, patterns in rules supporting the latter family cannot escape their bounding box [EDIT: actually diamond].

EDIT 6-25-2020: Variants of the last kind of oscillator can actually function in rules where patterns can escape their bounding diamond, albeit following block automata with multiple types of off states:

Code: Select all

x = 90, y = 2, rule = B2ce3iy4a5i/S3i4i5y
90o$bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobo!

Something that I realized about these XOR oscillators: They're all actually 2x2 block oscillators, it's just that the 2x2 blocks have different contents. For the original 2x2 rule, the 2x2 blocks are completely full; for B2c/S and analogues, the blocks consist of one on cell and 3 off cells; and for rules such as that in the above post, the blocks look like this:

Code: Select all

oo
..

Yup, and clearly the dynamics of all these rules have to be exactly isomorphic in some sense, or the recognizable behavior couldn't happen.

I had this rule confused with Snowflakes for a while there, and posted the following in the wrong thread. Putting it here now with appropriate edits, so as not to waste it --

There seem to be oscillators out there with period (2^N-2)*2^k for any N>2 and k>=0. And then there are the exceptions-that-prove-the-rule ones at 174762 and 45 billion [one third of "standard" numbers, both -- (2^19-2)/3 and (2^37-2)/3 ]...

Code: Select all

#C period 45812984490 oscillator
x = 188, y = 1, rule = B2ci3ai4ci8/S02ae3eijkq4iz5a6i7e
188o!

... So can the pointy ends of these diamonds make gliders and suchlike, the way AbhpzTa did with the horiship guns in B2cek3i/S12cei?

Something that I realized about these XOR oscillators: They're all actually 2x2 block oscillators, it's just that the 2x2 blocks have different contents. For the original 2x2 rule, the 2x2 blocks are completely full; for B2c/S and analogues, the blocks consist of one on cell and 3 off cells; and for rules such as that in the above post, the blocks look like this:

I found this out a good few months ago. I tried looking for a rule where the last family worked though, with no success.

Here's more:

Code: Select all

x = 23, y = 6, rule = B2e/S
o3bo5bo6bo$bo3bo5bo6bo$6bo5bo6bo$7bo5bo6bo$14bo6bo$22bo!

The thread also mentioned this one:

Code: Select all

x = 65, y = 32, rule = B2-a3/S01c5i
o19bo19bo19bo$21bo19bo19bo$22bo19bo19bo$43bo19bo$64bo16$2o18b2o18b2o
18b2o$2o18b2o18b2o18b2o$2o18b2o18b2o18b2o$2o18b2o18b2o18b2o$20b2o18b2o
18b2o$20b2o18b2o18b2o$20b2o18b2o18b2o$20b2o18b2o18b2o$40b2o18b2o$40b2o
18b2o$60b2o$60b2o!

(referencing this post and this post)

A type R(?) replicator, which fits in at least six of the seven rules:

Code: Select all

x = 1, y = 1, rule = B1c2n3c4c/S
o!

Code: Select all

x = 2, y = 1, rule = B1c2a3q6i/S
2o!

Code: Select all

x = 2, y = 2, rule = B1c2n/S1c2n4c
bo$o!

Code: Select all

x = 2, y = 2, rule = B1c2a3aq4qw8/S2a3q6i
2o$bo!

Code: Select all

x = 2, y = 2, rule = B1c2a4w/S3a4q8
2o$2o!

Code: Select all

x = 1, y = 1, rule = B1e2i3e4e/S
o!

It also appears that the "off" block state can be replaced under some circumstances allowing for oscillators that are similar if not exactly the same:

Code: Select all

x = 15, y = 15, rule = B4c/S02en3ce4ci5e
15o$obobobobobobobo$2obobobobobob2o$obobobobobobobo$2obobobobobob2o$ob
obobobobobobo$2o9bob2o$obobobobobobobo$2obobobobobob2o$obobobobobobobo
$2obobobobobob2o$obobobobobobobo$2obobobobobob2o$obobobobobobobo$15o!

Code: Select all

x = 15, y = 15, rule = B4t5ey6ci/S2ek3cnr4cy5e
4b2o3b2o$b2obobobobob2o$bobobobobobobo$2bobobobobobo$2obobobobobob2o$o
bobobobobobobo$bob2obobobo2bo$2bobobobobobo$bobobobobobobo$obobobobobo
bobo$2obobobobobob2o$2bobobobobobo$bobobobobobobo$b2obobobobob2o$4b2o
3b2o!

Code: Select all

x = 17, y = 17, rule = B3a5i8/S3i4i6ci7e8
bobobobobobobobo$17o$bobobobobobobobo$17o$bobobobobobobobo$17o$bobobob
obobobobo$17o$b13obo$17o$bobobobobobobobo$17o$bobobobobobobobo$17o$bob
obobobobobobo$17o$bobobobobobobobo!

This one seems to work for any even number not divisible by 4:

Code: Select all

x = 36, y = 16, rule = B2e3a4w/S1c3a4q
2o5b2o11b2o$2o5b2o11b2o$2b2o5b2o11b2o$2b2o5b2o11b2o$11b2o11b2o$11b2o
11b2o$13b2o11b2o$13b2o11b2o$28b2o$28b2o$30b2o$30b2o$32b2o$32b2o$34b2o$
34b2o!

Also, bilateral rules can allow for this:

Code: Select all

x = 31, y = 31, rule = B3i6i/S2i5i
2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$
2o28bo$2o28bo9$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo$2o28bo
$2o28bo$2o28bo$2o28bo!

[muzik]

Found in the alternating CA thread:

Code: Select all

x = 67, y = 1, rule = B12345678/S|B/S3
77bo3$76bo3$45bo29bo3$44bo29bo3$21bo21bo29bo3$20bo21bo29bo3$5bo13bo21b
o29bo3$4bo13bo21bo29bo10$o3bobo3bobobo3bobobobo3bobobobobo3bobobobobob
o3bobobobobobobo3bobobobobobobobo10$o9bo17bo25bo3$11bo17bo25bo3$12bo
17bo25bo3$31bo25bo3$32bo25bo3$59bo3$60bo!

And one I randomly came up wiTh while thinking about this:

Code: Select all

x = 148, y = 1, rule = B1/S|B2c/S
o4bo3bo4bo3bo3bo4bo3bo3bo3bo4bo3bo3bo3bo3bo4bo3bo3bo3bo3bo3bo4bo3bo3bo
3bo3bo3bo3bo4bo3bo3bo3bo3bo3bo3bo3bo!

Obvious variant:

Code: Select all

x = 148, y = 1, rule = B1/S0|B2c/S
o4bo3bo4bo3bo3bo4bo3bo3bo3bo4bo3bo3bo3bo3bo4bo3bo3bo3bo3bo3bo4bo3bo3bo
3bo3bo3bo3bo4bo3bo3bo3bo3bo3bo3bo3bo!

And this preserves the oblique versions

Code: Select all

x = 67, y = 1, rule = B12345678/S0|B/S4a
77bo3$76bo3$45bo29bo3$44bo29bo3$21bo21bo29bo3$20bo21bo29bo3$5bo13bo21b
o29bo3$4bo13bo21bo29bo10$o3bobo3bobobo3bobobobo3bobobobobo3bobobobobob
o3bobobobobobobo3bobobobobobobobo10$o9bo17bo25bo3$11bo17bo25bo3$12bo
17bo25bo3$31bo25bo3$32bo25bo3$59bo3$60bo!

[muzik]

Arbitrary-period Margolus media, using Andrew's alternating rule script:

Code: Select all

x = 16, y = 10, rule = B123/S12345678|B123/S12345678|B123/S12345678|B123/S12345678|B123/S12345678|B123/S12345678|B2c/S
o13bo13bo13bo13bo13bo13bo13bo13bo!

[muzik]

The 5-state Margolus rule is now completed. Here are all three (2, 3, 5) prime modulo-based margolus rules so far:

Code: Select all

x = 96, y = 95, rule = B2e/S
14$14bobo57bo$13bo3bo21bobo33bo$12bo5bo19bo3bo33bo$11bo7bo17bo5bo33bo$
10bo9bo15bo7bo33bo$9bo11bo13bo9bo33bo$8bo13bo11bo11bo33bo$7bo15bo9bo
13bo33bo$6bo17bo7bo15bo33bo$5bo19bo5bo17bo33bo$4bo21bo3bo19bo33bo$3bo
23bobo55bo$2bo47bo35bo$49bo37bo$2bo45bo39bo$3bo43bo41bo$4bo41bo43bo$5b
o39bo45bo$6bo37bo47bo$7bo35bo49bo$8bo33bo51bo$9bo17bobo$10bo15bo3bo9bo
53bo$11bo13bo5bo7bo53bo$12bo11bo13bo53bo$13bo17bo5bo53bo$14bo7bo7bo5bo
53bo$15bo5bo13bo53bo$20bo7bo5bo53bo$15bo3bo13bo53bo$14bo3bo67bo$13bo
17bo53bo$12bo5bo11bo53bo$11bo7bo9bo53bo$10bo9bo7bo53bo$9bo11bo5bo53bo$
8bo13bo3bo53bo$7bo15bobo53bo$6bo71bo$5bo71bo$4bo71bo$3bo71bo$2bo$bo71b
o$34bobo35bo$bo31bo3bo33bo$2bo29bo5bo31bo$3bo27bo7bo29bo$4bo25bo9bo27b
o$5bo23bo11bo25bo$6bo21bo13bo23bo$7bo19bo15bo21bo$8bo17bo17bo19bo$9bo
15bo19bo17bo$10bo13bo21bo15bo$11bo11bo23bo13bo$12bo9bo25bo11bo$13bo7bo
27bo9bo$14bo5bo29bo7bo$15bo3bo31bo5bo$16bobo33bo3bo$53bobo!

Code: Select all

x = 94, y = 62, rule = marg3diag
13.A.A57.A$12.A3.A21.A.A33.A$11.A5.A19.A3.A33.A$10.A7.A17.A5.A33.A$9.
A9.A15.A7.A33.A$8.A11.A13.A9.A33.A$7.A13.A11.A11.A33.A$6.A15.A9.A13.A
33.A$5.A17.A7.A15.A33.A$4.A19.A5.A17.A33.A$3.A21.A3.A19.A33.A$2.A23.A
.A55.A$.A47.A35.A$48.A37.A$.A45.A39.A$2.A43.A41.A$3.A41.A43.A$4.A39.A
45.A$5.A37.A47.A$6.A35.A49.A$7.A33.A51.A$8.A17.A.A$9.A15.A3.A9.A53.A$
10.A13.A5.A7.A53.A$11.A11.A13.A53.A$12.A17.A5.A53.A$13.A7.A7.A5.A53.A
$14.A5.A13.A53.A$19.A7.A5.A53.A$14.A3.A13.A53.A$13.A3.A67.A$12.A17.A
53.A$11.A5.A11.A53.A$10.A7.A9.A53.A$9.A9.A7.A53.A$8.A11.A5.A53.A$7.A
13.A3.A53.A$6.A15.A.A53.A$5.A71.A$4.A71.A$3.A71.A$2.A71.A$.A$A71.A$
33.A.A35.A$A31.A3.A33.A$.A29.A5.A31.A$2.A27.A7.A29.A$3.A25.A9.A27.A$
4.A23.A11.A25.A$5.A21.A13.A23.A$6.A19.A15.A21.A$7.A17.A17.A19.A$8.A
15.A19.A17.A$9.A13.A21.A15.A$10.A11.A23.A13.A$11.A9.A25.A11.A$12.A7.A
27.A9.A$13.A5.A29.A7.A$14.A3.A31.A5.A$15.A.A33.A3.A$52.A.A!

Code: Select all

x = 94, y = 62, rule = marg5diag
13.A.A57.A$12.A3.A21.A.A33.A$11.A5.A19.A3.A33.A$10.A7.A17.A5.A33.A$9.
A9.A15.A7.A33.A$8.A11.A13.A9.A33.A$7.A13.A11.A11.A33.A$6.A15.A9.A13.A
33.A$5.A17.A7.A15.A33.A$4.A19.A5.A17.A33.A$3.A21.A3.A19.A33.A$2.A23.A
.A55.A$.A47.A35.A$48.A37.A$.A45.A39.A$2.A43.A41.A$3.A41.A43.A$4.A39.A
45.A$5.A37.A47.A$6.A35.A49.A$7.A33.A51.A$8.A17.A.A$9.A15.A3.A9.A53.A$
10.A13.A5.A7.A53.A$11.A11.A13.A53.A$12.A17.A5.A53.A$13.A7.A7.A5.A53.A
$14.A5.A13.A53.A$19.A7.A5.A53.A$14.A3.A13.A53.A$13.A3.A67.A$12.A17.A
53.A$11.A5.A11.A53.A$10.A7.A9.A53.A$9.A9.A7.A53.A$8.A11.A5.A53.A$7.A
13.A3.A53.A$6.A15.A.A53.A$5.A71.A$4.A71.A$3.A71.A$2.A71.A$.A$A71.A$
33.A.A35.A$A31.A3.A33.A$.A29.A5.A31.A$2.A27.A7.A29.A$3.A25.A9.A27.A$
4.A23.A11.A25.A$5.A21.A13.A23.A$6.A19.A15.A21.A$7.A17.A17.A19.A$8.A
15.A19.A17.A$9.A13.A21.A15.A$10.A11.A23.A13.A$11.A9.A25.A11.A$12.A7.A
27.A9.A$13.A5.A29.A7.A$14.A3.A31.A5.A$15.A.A33.A3.A$52.A.A!

Like how 2-state follows rule 6 and 3 state follows rule 8229, 5 states supports 205464118576052930. And where 2-state oscillators break down for the Mersenne numbers and 3 states for A048473, the 5 state oscillators appear to break down according to A081655 (thus implying h states break down at every 2*h^n - 1):

Code: Select all

x = 249, y = 249, rule = marg5diag
248.A$247.A$246.A$245.A$244.A$243.A$242.A$241.A$240.A$239.A$238.A$
237.A$236.A$235.A$234.A$233.A$232.A$231.A$230.A$229.A$228.A$227.A$
226.A$225.A$224.A$223.A$222.A$221.A$220.A$219.A$218.A$217.A$216.A$
215.A$214.A$213.A$212.A$211.A$210.A$209.A$208.A$207.A$206.A$205.A$
204.A$203.A$202.A$201.A$200.A$199.A$198.A$197.A$196.A$195.A$194.A$
193.A$192.A$191.A$190.A$189.A$188.A$187.A$186.A$185.A$184.A$183.A$
182.A$181.A$180.A$179.A$178.A$177.A$176.A$175.A$174.A$173.A$172.A$
171.A$170.A$169.A$168.A$167.A$166.A$165.A$164.A$163.A$162.A$161.A$
160.A$159.A$158.A$157.A$156.A$155.A$154.A$153.A$152.A$151.A$150.A$
149.A$148.A$147.A$146.A$145.A$144.A$143.A$142.A$141.A$140.A$139.A$
138.A$137.A$136.A$135.A$134.A$133.A$132.A$131.A$130.A$129.A$128.A$
127.A$126.A$125.A$124.A$123.A$122.A$121.A$120.A$119.A$118.A$117.A$
116.A$115.A$114.A$113.A$112.A$111.A$110.A$109.A$108.A$107.A$106.A$
105.A$104.A$103.A$102.A$101.A$100.A$99.A$98.A$97.A$96.A$95.A$94.A$93.
A$92.A$91.A$90.A$89.A$88.A$87.A$86.A$85.A$84.A$83.A$82.A$81.A$80.A$
79.A$78.A$77.A$76.A$75.A$74.A$73.A$72.A$71.A$70.A$69.A$68.A$67.A$66.A
$65.A$64.A$63.A$62.A$61.A$60.A$59.A$58.A$57.A$56.A$55.A$54.A$53.A$52.
A$51.A$50.A$49.A$48.A$47.A$46.A$45.A$44.A$43.A$42.A$41.A$40.A$39.A$
38.A$37.A$36.A$35.A$34.A$33.A$32.A$31.A$30.A$29.A$28.A$27.A$26.A$25.A
$24.A$23.A$22.A$21.A$20.A$19.A$18.A$17.A$16.A$15.A$14.A$13.A$12.A$11.
A$10.A$9.A$8.A$7.A$6.A$5.A$4.A$3.A$2.A$.A$A!

[muzik]

7-state Margolus is now also done, resulting in four fundamental rule types having been constructed:

Code: Select all

x = 147, y = 165, rule = B2e/S
77bobo$76bo3bo$75bo5bo$74bo7bo$73bo9bo$72bo11bo$71bo13bo$70bo15bo$69bo
17bo$68bo19bo$67bo21bo$66bo23bo$65bo25bo$64bo27bo$63bo29bo$62bo31bo$
61bo33bo$60bo35bo$59bo37bo$58bo39bo$57bo41bo$56bo43bo$55bo45bo$54bo$
103bo$52bo51bo$51bo53bo$50bo55bo$49bo57bo$48bo59bo$47bo61bo$46bo63bo$
45bo65bo$44bo67bo$43bo69bo$42bo71bo$41bo73bo$40bo75bo$39bo77bo$38bo79b
o$37bo81bo$36bo83bo$35bo85bo$34bo87bo$33bo89bo$32bo91bo$31bo$30bo35bob
o57bo$29bo35bo3bo21bobo33bo$28bo35bo5bo19bo3bo33bo$63bo7bo17bo5bo33bo$
26bo35bo9bo15bo7bo33bo$25bo35bo11bo13bo9bo33bo$24bo35bo13bo11bo11bo33b
o$23bo35bo15bo9bo13bo33bo$22bo35bo17bo7bo15bo33bo$21bo35bo19bo5bo17bo
33bo$20bo35bo21bo3bo19bo33bo$19bo35bo23bobo55bo$18bo35bo47bo35bo$17bo
83bo37bo$16bo37bo45bo39bo$15bo39bo43bo41bo$14bo41bo41bo43bo$13bo43bo
39bo45bo$12bo45bo37bo47bo$11bo47bo35bo49bo$10bo49bo33bo51bo$9bo51bo17b
obo$8bo53bo15bo3bo9bo53bo$7bo55bo13bo5bo7bo53bo$6bo57bo11bo13bo53bo$5b
o59bo17bo5bo53bo$4bo61bo7bo7bo5bo53bo$3bo63bo5bo13bo53bo$2bo69bo7bo5bo
53bo$bo65bo3bo13bo53bo$66bo3bo67bo$bo63bo17bo53bo$2bo61bo5bo11bo53bo$
3bo59bo7bo9bo53bo$4bo57bo9bo7bo53bo$5bo55bo11bo5bo53bo$6bo53bo13bo3bo
53bo$7bo51bo15bobo53bo$8bo49bo71bo$9bo47bo71bo$10bo45bo71bo$11bo43bo
71bo$12bo41bo$13bo39bo71bo$14bo71bobo35bo$15bo37bo31bo3bo33bo$16bo37bo
29bo5bo31bo$17bo37bo27bo7bo29bo$18bo37bo25bo9bo27bo$19bo37bo23bo11bo
25bo$20bo37bo21bo13bo23bo$21bo37bo19bo15bo21bo$22bo37bo17bo17bo19bo$
23bo37bo15bo19bo17bo$24bo37bo13bo21bo15bo$25bo37bo11bo23bo13bo$26bo37b
o9bo25bo11bo$27bo37bo7bo27bo9bo$66bo5bo29bo7bo$27bo39bo3bo31bo5bo$26bo
41bobo33bo3bo$25bo79bobo$24bo$23bo$22bo$21bo$20bo$19bo$18bo$17bo$16bo$
15bo$14bo$13bo$12bo$11bo$10bo$9bo$8bo$7bo$6bo$5bo$4bo$3bo$2bo$bo$o$59b
obo$o57bo3bo$bo55bo5bo$2bo53bo7bo$3bo51bo9bo$4bo49bo11bo$5bo47bo13bo$
6bo45bo15bo$7bo43bo17bo$8bo41bo19bo$9bo39bo21bo$10bo37bo23bo$11bo35bo
25bo$12bo33bo27bo$13bo31bo29bo$14bo29bo31bo$15bo27bo33bo$16bo25bo35bo$
17bo23bo37bo$18bo21bo39bo$19bo19bo41bo$20bo17bo43bo$21bo15bo45bo$22bo
13bo47bo$23bo11bo49bo$24bo9bo51bo$25bo7bo53bo$26bo5bo55bo$27bo3bo57bo$
28bobo59bo$91bo!

Code: Select all

x = 147, y = 165, rule = marg3diag
77bobo$76bo3bo$75bo5bo$74bo7bo$73bo9bo$72bo11bo$71bo13bo$70bo15bo$69bo
17bo$68bo19bo$67bo21bo$66bo23bo$65bo25bo$64bo27bo$63bo29bo$62bo31bo$
61bo33bo$60bo35bo$59bo37bo$58bo39bo$57bo41bo$56bo43bo$55bo45bo$54bo$
103bo$52bo51bo$51bo53bo$50bo55bo$49bo57bo$48bo59bo$47bo61bo$46bo63bo$
45bo65bo$44bo67bo$43bo69bo$42bo71bo$41bo73bo$40bo75bo$39bo77bo$38bo79b
o$37bo81bo$36bo83bo$35bo85bo$34bo87bo$33bo89bo$32bo91bo$31bo$30bo35bob
o57bo$29bo35bo3bo21bobo33bo$28bo35bo5bo19bo3bo33bo$63bo7bo17bo5bo33bo$
26bo35bo9bo15bo7bo33bo$25bo35bo11bo13bo9bo33bo$24bo35bo13bo11bo11bo33b
o$23bo35bo15bo9bo13bo33bo$22bo35bo17bo7bo15bo33bo$21bo35bo19bo5bo17bo
33bo$20bo35bo21bo3bo19bo33bo$19bo35bo23bobo55bo$18bo35bo47bo35bo$17bo
83bo37bo$16bo37bo45bo39bo$15bo39bo43bo41bo$14bo41bo41bo43bo$13bo43bo
39bo45bo$12bo45bo37bo47bo$11bo47bo35bo49bo$10bo49bo33bo51bo$9bo51bo17b
obo$8bo53bo15bo3bo9bo53bo$7bo55bo13bo5bo7bo53bo$6bo57bo11bo13bo53bo$5b
o59bo17bo5bo53bo$4bo61bo7bo7bo5bo53bo$3bo63bo5bo13bo53bo$2bo69bo7bo5bo
53bo$bo65bo3bo13bo53bo$66bo3bo67bo$bo63bo17bo53bo$2bo61bo5bo11bo53bo$
3bo59bo7bo9bo53bo$4bo57bo9bo7bo53bo$5bo55bo11bo5bo53bo$6bo53bo13bo3bo
53bo$7bo51bo15bobo53bo$8bo49bo71bo$9bo47bo71bo$10bo45bo71bo$11bo43bo
71bo$12bo41bo$13bo39bo71bo$14bo71bobo35bo$15bo37bo31bo3bo33bo$16bo37bo
29bo5bo31bo$17bo37bo27bo7bo29bo$18bo37bo25bo9bo27bo$19bo37bo23bo11bo
25bo$20bo37bo21bo13bo23bo$21bo37bo19bo15bo21bo$22bo37bo17bo17bo19bo$
23bo37bo15bo19bo17bo$24bo37bo13bo21bo15bo$25bo37bo11bo23bo13bo$26bo37b
o9bo25bo11bo$27bo37bo7bo27bo9bo$66bo5bo29bo7bo$27bo39bo3bo31bo5bo$26bo
41bobo33bo3bo$25bo79bobo$24bo$23bo$22bo$21bo$20bo$19bo$18bo$17bo$16bo$
15bo$14bo$13bo$12bo$11bo$10bo$9bo$8bo$7bo$6bo$5bo$4bo$3bo$2bo$bo$o$59b
obo$o57bo3bo$bo55bo5bo$2bo53bo7bo$3bo51bo9bo$4bo49bo11bo$5bo47bo13bo$
6bo45bo15bo$7bo43bo17bo$8bo41bo19bo$9bo39bo21bo$10bo37bo23bo$11bo35bo
25bo$12bo33bo27bo$13bo31bo29bo$14bo29bo31bo$15bo27bo33bo$16bo25bo35bo$
17bo23bo37bo$18bo21bo39bo$19bo19bo41bo$20bo17bo43bo$21bo15bo45bo$22bo
13bo47bo$23bo11bo49bo$24bo9bo51bo$25bo7bo53bo$26bo5bo55bo$27bo3bo57bo$
28bobo59bo$91bo!

Code: Select all

x = 147, y = 165, rule = marg5diag
77bobo$76bo3bo$75bo5bo$74bo7bo$73bo9bo$72bo11bo$71bo13bo$70bo15bo$69bo
17bo$68bo19bo$67bo21bo$66bo23bo$65bo25bo$64bo27bo$63bo29bo$62bo31bo$
61bo33bo$60bo35bo$59bo37bo$58bo39bo$57bo41bo$56bo43bo$55bo45bo$54bo$
103bo$52bo51bo$51bo53bo$50bo55bo$49bo57bo$48bo59bo$47bo61bo$46bo63bo$
45bo65bo$44bo67bo$43bo69bo$42bo71bo$41bo73bo$40bo75bo$39bo77bo$38bo79b
o$37bo81bo$36bo83bo$35bo85bo$34bo87bo$33bo89bo$32bo91bo$31bo$30bo35bob
o57bo$29bo35bo3bo21bobo33bo$28bo35bo5bo19bo3bo33bo$63bo7bo17bo5bo33bo$
26bo35bo9bo15bo7bo33bo$25bo35bo11bo13bo9bo33bo$24bo35bo13bo11bo11bo33b
o$23bo35bo15bo9bo13bo33bo$22bo35bo17bo7bo15bo33bo$21bo35bo19bo5bo17bo
33bo$20bo35bo21bo3bo19bo33bo$19bo35bo23bobo55bo$18bo35bo47bo35bo$17bo
83bo37bo$16bo37bo45bo39bo$15bo39bo43bo41bo$14bo41bo41bo43bo$13bo43bo
39bo45bo$12bo45bo37bo47bo$11bo47bo35bo49bo$10bo49bo33bo51bo$9bo51bo17b
obo$8bo53bo15bo3bo9bo53bo$7bo55bo13bo5bo7bo53bo$6bo57bo11bo13bo53bo$5b
o59bo17bo5bo53bo$4bo61bo7bo7bo5bo53bo$3bo63bo5bo13bo53bo$2bo69bo7bo5bo
53bo$bo65bo3bo13bo53bo$66bo3bo67bo$bo63bo17bo53bo$2bo61bo5bo11bo53bo$
3bo59bo7bo9bo53bo$4bo57bo9bo7bo53bo$5bo55bo11bo5bo53bo$6bo53bo13bo3bo
53bo$7bo51bo15bobo53bo$8bo49bo71bo$9bo47bo71bo$10bo45bo71bo$11bo43bo
71bo$12bo41bo$13bo39bo71bo$14bo71bobo35bo$15bo37bo31bo3bo33bo$16bo37bo
29bo5bo31bo$17bo37bo27bo7bo29bo$18bo37bo25bo9bo27bo$19bo37bo23bo11bo
25bo$20bo37bo21bo13bo23bo$21bo37bo19bo15bo21bo$22bo37bo17bo17bo19bo$
23bo37bo15bo19bo17bo$24bo37bo13bo21bo15bo$25bo37bo11bo23bo13bo$26bo37b
o9bo25bo11bo$27bo37bo7bo27bo9bo$66bo5bo29bo7bo$27bo39bo3bo31bo5bo$26bo
41bobo33bo3bo$25bo79bobo$24bo$23bo$22bo$21bo$20bo$19bo$18bo$17bo$16bo$
15bo$14bo$13bo$12bo$11bo$10bo$9bo$8bo$7bo$6bo$5bo$4bo$3bo$2bo$bo$o$59b
obo$o57bo3bo$bo55bo5bo$2bo53bo7bo$3bo51bo9bo$4bo49bo11bo$5bo47bo13bo$
6bo45bo15bo$7bo43bo17bo$8bo41bo19bo$9bo39bo21bo$10bo37bo23bo$11bo35bo
25bo$12bo33bo27bo$13bo31bo29bo$14bo29bo31bo$15bo27bo33bo$16bo25bo35bo$
17bo23bo37bo$18bo21bo39bo$19bo19bo41bo$20bo17bo43bo$21bo15bo45bo$22bo
13bo47bo$23bo11bo49bo$24bo9bo51bo$25bo7bo53bo$26bo5bo55bo$27bo3bo57bo$
28bobo59bo$91bo!

Code: Select all

x = 147, y = 165, rule = marg7diag
77bobo$76bo3bo$75bo5bo$74bo7bo$73bo9bo$72bo11bo$71bo13bo$70bo15bo$69bo
17bo$68bo19bo$67bo21bo$66bo23bo$65bo25bo$64bo27bo$63bo29bo$62bo31bo$
61bo33bo$60bo35bo$59bo37bo$58bo39bo$57bo41bo$56bo43bo$55bo45bo$54bo$
103bo$52bo51bo$51bo53bo$50bo55bo$49bo57bo$48bo59bo$47bo61bo$46bo63bo$
45bo65bo$44bo67bo$43bo69bo$42bo71bo$41bo73bo$40bo75bo$39bo77bo$38bo79b
o$37bo81bo$36bo83bo$35bo85bo$34bo87bo$33bo89bo$32bo91bo$31bo$30bo35bob
o57bo$29bo35bo3bo21bobo33bo$28bo35bo5bo19bo3bo33bo$63bo7bo17bo5bo33bo$
26bo35bo9bo15bo7bo33bo$25bo35bo11bo13bo9bo33bo$24bo35bo13bo11bo11bo33b
o$23bo35bo15bo9bo13bo33bo$22bo35bo17bo7bo15bo33bo$21bo35bo19bo5bo17bo
33bo$20bo35bo21bo3bo19bo33bo$19bo35bo23bobo55bo$18bo35bo47bo35bo$17bo
83bo37bo$16bo37bo45bo39bo$15bo39bo43bo41bo$14bo41bo41bo43bo$13bo43bo
39bo45bo$12bo45bo37bo47bo$11bo47bo35bo49bo$10bo49bo33bo51bo$9bo51bo17b
obo$8bo53bo15bo3bo9bo53bo$7bo55bo13bo5bo7bo53bo$6bo57bo11bo13bo53bo$5b
o59bo17bo5bo53bo$4bo61bo7bo7bo5bo53bo$3bo63bo5bo13bo53bo$2bo69bo7bo5bo
53bo$bo65bo3bo13bo53bo$66bo3bo67bo$bo63bo17bo53bo$2bo61bo5bo11bo53bo$
3bo59bo7bo9bo53bo$4bo57bo9bo7bo53bo$5bo55bo11bo5bo53bo$6bo53bo13bo3bo
53bo$7bo51bo15bobo53bo$8bo49bo71bo$9bo47bo71bo$10bo45bo71bo$11bo43bo
71bo$12bo41bo$13bo39bo71bo$14bo71bobo35bo$15bo37bo31bo3bo33bo$16bo37bo
29bo5bo31bo$17bo37bo27bo7bo29bo$18bo37bo25bo9bo27bo$19bo37bo23bo11bo
25bo$20bo37bo21bo13bo23bo$21bo37bo19bo15bo21bo$22bo37bo17bo17bo19bo$
23bo37bo15bo19bo17bo$24bo37bo13bo21bo15bo$25bo37bo11bo23bo13bo$26bo37b
o9bo25bo11bo$27bo37bo7bo27bo9bo$66bo5bo29bo7bo$27bo39bo3bo31bo5bo$26bo
41bobo33bo3bo$25bo79bobo$24bo$23bo$22bo$21bo$20bo$19bo$18bo$17bo$16bo$
15bo$14bo$13bo$12bo$11bo$10bo$9bo$8bo$7bo$6bo$5bo$4bo$3bo$2bo$bo$o$59b
obo$o57bo3bo$bo55bo5bo$2bo53bo7bo$3bo51bo9bo$4bo49bo11bo$5bo47bo13bo$
6bo45bo15bo$7bo43bo17bo$8bo41bo19bo$9bo39bo21bo$10bo37bo23bo$11bo35bo
25bo$12bo33bo27bo$13bo31bo29bo$14bo29bo31bo$15bo27bo33bo$16bo25bo35bo$
17bo23bo37bo$18bo21bo39bo$19bo19bo41bo$20bo17bo43bo$21bo15bo45bo$22bo
13bo47bo$23bo11bo49bo$24bo9bo51bo$25bo7bo53bo$26bo5bo55bo$27bo3bo57bo$
28bobo59bo$91bo!

This is based on rule 206968711034748555365673889122449783979099.

Indeed, it breaks down aborting to 2*7^n-1 (A198480):

Code: Select all

x = 113, y = 97, rule = marg7diag
112.A$111.A$110.A$109.A$108.A$107.A$106.A$105.A$104.A$103.A$102.A$
101.A$100.A$99.A$98.A$97.A$96.A$95.A$94.A$93.A$92.A$91.A$90.A$89.A$
88.A$87.A$86.A$85.A$84.A$83.A$82.A$81.A$80.A$79.A$78.A$77.A$76.A$75.A
$74.A$73.A$72.A$71.A$70.A$69.A$68.A$67.A$66.A$65.A$64.A$63.A$62.A$61.
A$60.A$59.A$58.A$57.A$56.A$55.A$54.A$53.A$52.A$51.A$50.A$49.A$48.A$
47.A$46.A$45.A$44.A$43.A$42.A$41.A$40.A$39.A$38.A$37.A$36.A$35.A$34.A
$33.A$32.A$31.A$30.A$29.A$14.A13.A$13.A13.A$12.A13.A$11.A13.A$10.A13.
A$9.A13.A$8.A13.A$7.A13.A$6.A13.A$5.A13.A$4.A13.A$3.A13.A$A.A13.A!

[wwei47]

How do you construct these rules? How do you decide what happens when 3 or more cells meet?

[muzik]

How do you construct these rules?

Notepad++ and a lot of trial and error.

How do you decide what happens when 3 or more cells meet?

Most of that is pretty much undefined since I'm currently only interested in simulating single diagonal line oscillators, and I only define those which are integral to the functioning of those.

[wwei47]

Most of that is pretty much undefined since I'm currently only interested in simulating single diagonal line oscillators, and I only define those which are integral to the functioning of those.

Okay then. How do you decide the important ones?

[muzik]

Most of that is pretty much undefined since I'm currently only interested in simulating single diagonal line oscillators, and I only define those which are integral to the functioning of those.

Okay then. How do you decide the important ones?

If the absence of the transition would result in diagonal line oscillators not working, then it has to be added (and follow the expected rules).

[wwei47]

If the absence of the transition would result in diagonal line oscillators not working, then it has to be added (and follow the expected rules).

Which state do you pick for it to transition to though?

[muzik]

If the absence of the transition would result in diagonal line oscillators not working, then it has to be added (and follow the expected rules).

Which state do you pick for it to transition to though?

By checking what the next generation should look like, as each generation of such an oscillator should look like a set of overlapping rectangles. For 2 states the rectangles XOR each other, for 3 and up the XOR is generalised as being modulo n.

For example (using the 2x2 blocks for clarity), for the 2 state case, generation 0 of a 2x12 rectangle is, via an arbitrary convention called the "identity property", a 2x12 rectangle:

Code: Select all

x = 2, y = 12, rule = B3/S5
2o$2o$2o$2o$2o$2o$2o$2o$2o$2o$2o$2o!

The second generation is a 4x10 rectangle:

Code: Select all

x = 4, y = 10, rule = B3/S5
4o$4o$4o$4o$4o$4o$4o$4o$4o$4o!

The third generation is the XOR of a 2x12 rectangle and a 6x8 rectangle (1+1=2, 2 mod 2=0):

Code: Select all

x = 6, y = 12, rule = B3/S5
2b2o$2b2o$2o2b2o$2o2b2o$2o2b2o$2o2b2o$2o2b2o$2o2b2o$2o2b2o$2o2b2o$2b2o
$2b2o!

For our 3 state case, we start out again with a 2x12 rectangle:

Code: Select all

x = 2, y = 12, rule = marg3
2o$2o$2o$2o$2o$2o$2o$2o$2o$2o$2o$2o!

This again evolves into a 4x10:

Code: Select all

x = 4, y = 10, rule = marg3
4A$4A$4A$4A$4A$4A$4A$4A$4A$4A!

In the next generation, since we use modulo 3 instead of modulo 2, 1+1 no longer equals 0, but equals 2. As a result, the cells in the middle no longer die, and instead transition to state 2:

Code: Select all

x = 6, y = 12, rule = marg3
2.2A$2.2A$2A2B2A$2A2B2A$2A2B2A$2A2B2A$2A2B2A$2A2B2A$2A2B2A$2A2B2A$2.
2A$2.2A!

[wwei47]

Not exactly a margolus media, but this shocked me:

Code: Select all

#C Emulates B02468/S02468|B02468/S0246
x = 12, y = 12, rule = B1357/S1357|B2468/S02468
10b2o$10b2o$8b2o$8b2o$6b2o$6b2o$4b2o$4b2o$2b2o$2b2o$2o$2o!

Code: Select all

x = 6, y = 6, rule = B246/S0246
o$bo$2bo$3bo$4bo$5bo!

The rule and the original imperfect form also have near-perfect emulation in 1x2 bars, which I noticed when JLS searching the original form for (4,1)c/4. Then Laundrypizza03 posted a 2c/2 spaceship in blocks, and that led me to this rule. Converting it to non-B0 for LifeViewer seems to have shined some light on what's going on-look at the first half of the rule!!!
EDIT: The exact rule emulated seems to be B24-e68/S0246.
Edit 2: I mean B24-e68/S024-c6

[LaundryPizza03]

B0 rule containing emulations of all 5 basic Margolus media for the same Margolus rule.

Code: Select all

x = 88, y = 46, rule = B01e2cei3ei4eit5i6c/S01e2cei3ei4eit5i6c
20bo5bo33bo5bo$20bo5bo33b2o4b2o$20bo5bo33bo5bo$20bo5bo33b2o4b2o$20bo
39bo$20bo39b2o$20bo39bo$20bo39b2o$24bobo37bobo$24bobo37b4o$24bobo37bob
o$24bobo37b4o9$o5bo13b2o4b2o12bo5bo13b2o4b2o12b2o4b2o$41bo5bo13bo5bo
12b2o4b2o$o5bo13b2o4b2o12bo5bo13b2o4b2o12b2o4b2o$41bo5bo13bo5bo12b2o4b
2o$o19b2o18bo19b2o18b2o$41bo19bo18b2o$o19b2o18bo19b2o18b2o$41bo19bo18b
2o$4bobo17b4o16bobo17b4o16b4o$45bobo17bobo16b4o$4bobo17b4o16bobo17b4o
16b4o$45bobo17bobo16b4o9$4bobo17b4o16bobo17b4o16b4o$45bobo16bobo17b4o$
2bobobo15b6o14bobobo15b6o14b6o$43bobobo14bobobo15b6o$o19b2o18bo19b2o
18b2o$41bo18bo19b2o!

[muzik]

Another obvious set originating from the alternating rulespace:

Code: Select all

x = 49, y = 45, rule = B1/S|B4w/S
o9bo9bo9bo9bo2$12bo9bo9bo9bo2$24bo9bo9bo2$36bo9bo2$48bo2$o19bo2$2bo19b
o2$4bo19bo2$6bo19bo2$8bo19bo2$10bo19bo2$32bo8$o2$2bo2$4bo2$6bo2$8bo2$
10bo2$12bo2$14bo!

Code: Select all

x = 49, y = 45, rule = B1/S0|B4w/S
o9bo9bo9bo9bo2$12bo9bo9bo9bo2$24bo9bo9bo2$36bo9bo2$48bo2$o19bo2$2bo19b
o2$4bo19bo2$6bo19bo2$8bo19bo2$10bo19bo2$32bo8$o2$2bo2$4bo2$6bo2$8bo2$
10bo2$12bo2$14bo!

Code: Select all

x = 49, y = 45, rule = B12n/S|B5a/S
o9bo9bo9bo9bo2$12bo9bo9bo9bo2$24bo9bo9bo2$36bo9bo2$48bo2$o19bo2$2bo19b
o2$4bo19bo2$6bo19bo2$8bo19bo2$10bo19bo2$32bo8$o2$2bo2$4bo2$6bo2$8bo2$
10bo2$12bo2$14bo!

Code: Select all

x = 49, y = 45, rule = B12n/S0|B5a/S
o9bo9bo9bo9bo2$12bo9bo9bo9bo2$24bo9bo9bo2$36bo9bo2$48bo2$o19bo2$2bo19b
o2$4bo19bo2$6bo19bo2$8bo19bo2$10bo19bo2$32bo8$o2$2bo2$4bo2$6bo2$8bo2$
10bo2$12bo2$14bo!

Something a bit higher period:

Code: Select all

x = 197, y = 1, rule = B12c/S0|B2c4w/S3e5i8
o9bo5bo13bo5bo5bo7bo5bo5bo5bo11bo5bo5bo5bo5bo15bo5bo5bo5bo5bo5bo9bo5b
o5bo5bo5bo5bo5bo!

[muzik]

I decided to take some time to try and create ruletables for the 2x2 block versions of the modulo-5 and modulo-7 MarBlocks rules. Both are now completed, so we now have all prime examples up to 7:

Code: Select all

x = 142, y = 20, rule = B3/S5
140b2o$140b2o$120b2o18b2o$120b2o18b2o$100b2o18b2o18b2o$100b2o18b2o18b
2o$80b2o18b2o18b2o18b2o$80b2o18b2o18b2o18b2o$60b2o18b2o18b2o18b2o18b2o
$60b2o18b2o18b2o18b2o18b2o$40b2o18b2o18b2o18b2o18b2o18b2o$40b2o18b2o
18b2o18b2o18b2o18b2o$30b2o8b2o18b2o18b2o18b2o18b2o18b2o$30b2o8b2o18b2o
18b2o18b2o18b2o18b2o$20b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$20b2o8b2o
8b2o18b2o18b2o18b2o18b2o18b2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b
2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o
18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o!

Code: Select all

x = 142, y = 20, rule = MarBlocks3-OT-2x2-minimal
140b2o$140b2o$120b2o18b2o$120b2o18b2o$100b2o18b2o18b2o$100b2o18b2o18b
2o$80b2o18b2o18b2o18b2o$80b2o18b2o18b2o18b2o$60b2o18b2o18b2o18b2o18b2o
$60b2o18b2o18b2o18b2o18b2o$40b2o18b2o18b2o18b2o18b2o18b2o$40b2o18b2o
18b2o18b2o18b2o18b2o$30b2o8b2o18b2o18b2o18b2o18b2o18b2o$30b2o8b2o18b2o
18b2o18b2o18b2o18b2o$20b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$20b2o8b2o
8b2o18b2o18b2o18b2o18b2o18b2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b
2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o
18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o!

Code: Select all

x = 142, y = 20, rule = MarBlocks5-OT-2x2-minimal
140b2o$140b2o$120b2o18b2o$120b2o18b2o$100b2o18b2o18b2o$100b2o18b2o18b
2o$80b2o18b2o18b2o18b2o$80b2o18b2o18b2o18b2o$60b2o18b2o18b2o18b2o18b2o
$60b2o18b2o18b2o18b2o18b2o$40b2o18b2o18b2o18b2o18b2o18b2o$40b2o18b2o
18b2o18b2o18b2o18b2o$30b2o8b2o18b2o18b2o18b2o18b2o18b2o$30b2o8b2o18b2o
18b2o18b2o18b2o18b2o$20b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$20b2o8b2o
8b2o18b2o18b2o18b2o18b2o18b2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b
2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o
18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o!

Code: Select all

x = 142, y = 20, rule = MarBlocks7-OT-2x2-minimal
140b2o$140b2o$120b2o18b2o$120b2o18b2o$100b2o18b2o18b2o$100b2o18b2o18b
2o$80b2o18b2o18b2o18b2o$80b2o18b2o18b2o18b2o$60b2o18b2o18b2o18b2o18b2o
$60b2o18b2o18b2o18b2o18b2o$40b2o18b2o18b2o18b2o18b2o18b2o$40b2o18b2o
18b2o18b2o18b2o18b2o$30b2o8b2o18b2o18b2o18b2o18b2o18b2o$30b2o8b2o18b2o
18b2o18b2o18b2o18b2o$20b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$20b2o8b2o
8b2o18b2o18b2o18b2o18b2o18b2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b
2o$10b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o
18b2o18b2o18b2o18b2o$2o8b2o8b2o8b2o8b2o18b2o18b2o18b2o18b2o18b2o!

Unlike 3-state rules, I don't see 5-state or 7-state outer-totalistic rules getting native support in any major CA programs any time soon (feel free to prove me wrong on that, though), so these will probably stay ruletables.

Also, I created the diagonal emulation of the modulo-4 MarBlocks rule as well, which I had earlier skipped due to it not being prime and therefore not being as clean as others would be:

Code: Select all

x = 64, y = 50, rule = marg4diag
63.A$52.A9.A$42.A8.A9.A$33.A7.A8.A9.A$25.A6.A7.A8.A9.A$18.A5.A6.A7.A
8.A9.A$12.A4.A5.A6.A7.A8.A9.A$7.A3.A4.A5.A6.A7.A8.A9.A$3.A2.A3.A4.A5.
A6.A7.A8.A9.A$A.A2.A3.A4.A5.A6.A7.A8.A9.A11$63.B$52.B9.B$42.B8.B9.B$
33.B7.B8.B9.B$25.B6.B7.B8.B9.B$18.B5.B6.B7.B8.B9.B$12.B4.B5.B6.B7.B8.
B9.B$7.B3.B4.B5.B6.B7.B8.B9.B$3.B2.B3.B4.B5.B6.B7.B8.B9.B$B.B2.B3.B4.
B5.B6.B7.B8.B9.B11$63.C$52.C9.C$42.C8.C9.C$33.C7.C8.C9.C$25.C6.C7.C8.
C9.C$18.C5.C6.C7.C8.C9.C$12.C4.C5.C6.C7.C8.C9.C$7.C3.C4.C5.C6.C7.C8.C
9.C$3.C2.C3.C4.C5.C6.C7.C8.C9.C$C.C2.C3.C4.C5.C6.C7.C8.C9.C!

Note how state 4 is basically 2-state, as 2n mod 4 is ultimately behaviourally the same as n mod 2. Any other rule which relies on a Pascal-triangle-mod-4 (the following 1D rule: https://www.wolframalpha.com/input/?i=t ... 4%2C+k%3D4) can be clearly seen to exhibit the same effects:

Code: Select all

x = 3, y = 3, rule = Fredkin_mod4_vonNeumann
$.A!
[[ AUTOFIT ]]

Code: Select all

x = 3, y = 3, rule = Fredkin_mod4_vonNeumann
$.B!
[[ AUTOFIT ]]

Code: Select all

x = 3, y = 3, rule = Fredkin_mod4_vonNeumann
$.C!
[[ AUTOFIT ]]

[GUYTU6J]

Why are the transitions in Rule:marg{x}diag inconsistent with theoretical results determined by XOR-ing neighbour state numbers? For example, the ruletables for the two cases with x=3 and x=5 respectively are copied below, with what I think should be otherwise noted after double slash:

Code: Select all

0,0,0,1,1,1//2
0,0,2,1,2,1//2
0,2,2,2,2,1//2
0,0,0,2,2,2//1
0,0,1,2,1,2//1
0,1,1,1,1,2//1
1,0,0,0,0,0
2,0,0,0,0,0

Code: Select all

0,0,0,1,1,1//2
0,0,2,1,4,1//2
0,0,3,1,3,1//2
0,2,3,4,3,1//2
0,2,2,4,4,1//2
0,3,3,3,3,1//2
0,0,0,2,2,2//4
0,0,4,2,3,2//4
0,0,1,2,1,2//4
0,4,1,3,1,2//4
0,4,4,3,3,2//4
0,1,1,1,1,2//4
0,0,0,3,3,3//1
0,0,1,3,2,3//1
0,0,4,3,4,3//1
0,1,4,2,4,3//1
0,1,1,2,2,3//1
0,4,4,4,4,3//1
0,0,0,4,4,4//3
0,0,3,4,1,4//3
0,0,2,4,2,4//3
0,3,2,1,2,4//3
0,3,3,1,1,4//3
0,2,2,2,2,4//3
1,0,0,0,0,0
2,0,0,0,0,0
3,0,0,0,0,0
4,0,0,0,0,0

EDIT: and also, why does the former exclude transitions like 0,1,1,1,2,2 and 0,1,2,2,2,1?

[muzik]

Why are the transitions in Rule:marg{x}diag inconsistent with theoretical results determined by XOR-ing neighbour state numbers?

EDIT: and also, why does the former exclude transitions like 0,1,1,1,2,2 and 0,1,2,2,2,1?

Those are the results I got for overlaying and moduloing rectangles as Nathaniel demonstrated for the 2-state case. Are there any behavioural differences for the "correct"(?) version?

[GUYTU6J]

Well, um... I think by "XOR-ing neighbour state numbers" nearly half a year ago, I meant "taking sum of neighbour state numbers and taking its modulo total state count". At that time I did not understand the "overlapping rectangle" method. In retrospect it is doubtful/questionable if my idea is actually "correct" with the notion of thinking modulo as non-binary "XOR".

...Are there any behavioural differences for the "correct"(?) version?

The states are swapped with parity thoroughly, so for 0, 0, 0, x, x with x running from 1 to (n-1) and total state count n being odd, the resulting state is no longer x itself from 1 to (n-1), but 2, 4, ..., (n-1), 1, 3, ..., (n-2). There are also additional transitions as I have said, but they seem to not appear in a Margolus oscillator for small cases. Besides these, I have no clear idea.
---
Self-promotion alert: why did I even look into this in the first place? The reason can be found in the new HashLife/Metacell folder of Golly pattern collection.

[muzik]

This 2014 BSFK rule may be another potential medium:

Code: Select all

x = 457, y = 90, rule = B2SF2K1
31$431.A$432.A$429.A$430.A$377.A49.A$325.A52.A49.A$326.A48.A49.A$323.
A52.A49.A$273.A50.A48.A49.A$274.A46.A52.A49.A$231.A39.A50.A48.A49.A$
232.A39.A46.A52.A49.A$189.A39.A39.A50.A48.A49.A$147.A42.A39.A39.A46.A
52.A49.A$148.A38.A39.A39.A50.A48.A49.A$115.A29.A42.A39.A39.A46.A52.A
49.A$116.A29.A38.A39.A39.A50.A48.A49.A$113.A29.A42.A39.A39.A46.A52.A
49.A$114.A29.A38.A39.A39.A50.A48.A49.A$111.A29.A42.A39.A39.A46.A52.A
49.A$81.A30.A29.A38.A39.A39.A50.A48.A49.A$82.A26.A29.A42.A39.A39.A46.
A52.A49.A$59.A19.A30.A29.A38.A39.A39.A50.A48.A49.A$60.A19.A26.A29.A
42.A39.A39.A46.A52.A49.A$37.A19.A19.A30.A29.A38.A39.A39.A50.A48.A49.A
$38.A19.A19.A26.A29.A42.A39.A39.A46.A52.A49.A$35.A19.A19.A30.A29.A38.
A39.A39.A50.A48.A49.A$36.A19.A19.A26.A29.A42.A39.A39.A46.A52.A49.A$
23.A9.A19.A19.A30.A29.A38.A39.A39.A50.A48.A49.A$24.A9.A19.A19.A26.A
29.A42.A39.A39.A46.A52.A49.A$21.A9.A19.A19.A30.A29.A38.A39.A39.A50.A
48.A49.A$22.A9.A19.A19.A99.A39.A39.A99.A49.A!

[GUYTU6J]

I was revisiting my old posts when I noticed this new (?) Margolus medium.

Wow

Code: Select all

x = 4, y = 4, rule = B2ae3a4ek/S
3bo$3bo$2bo$2o!

Compare the following three patterns:

Code: Select all

x = 593, y = 113, rule = B2e/S
537bo$536bobo$535bobobo$534bobobo$533bobobo$532bobobo$431bo99bobobo$
430bobo97bobobo$429bobobo95bobobo$428bobobo95bobobo$427bobobo95bobobo$
426bobobo95bobobo$335bo89bobobo95bobobo$334bobo87bobobo95bobobo$333bob
obo85bobobo95bobobo$332bobobo85bobobo95bobobo$331bobobo85bobobo95bobob
o$330bobobo85bobobo95bobobo$239bo89bobobo85bobobo95bobobo$238bobo87bob
obo85bobobo95bobobo$237bobobo85bobobo85bobobo95bobobo$236bobobo85bobob
o85bobobo95bobobo$235bobobo85bobobo85bobobo95bobobo$234bobobo85bobobo
85bobobo95bobobo$163bo69bobobo85bobobo85bobobo95bobobo$162bobo67bobobo
85bobobo85bobobo95bobobo$161bobobo65bobobo85bobobo85bobobo95bobobo$
160bobobo65bobobo85bobobo85bobobo95bobobo$159bobobo65bobobo85bobobo85b
obobo95bobobo$158bobobo65bobobo85bobobo85bobobo95bobobo$97bo59bobobo
65bobobo85bobobo85bobobo95bobobo$96bobo57bobobo65bobobo85bobobo85bobob
o95bobobo$95bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$94bobobo55b
obobo65bobobo85bobobo85bobobo95bobobo$93bobobo55bobobo65bobobo85bobobo
85bobobo95bobobo$92bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$41bo
49bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$40bobo47bobobo55bobob
o65bobobo85bobobo85bobobo95bobobo$39bobobo45bobobo55bobobo65bobobo85bo
bobo85bobobo95bobobo$38bobobo45bobobo55bobobo65bobobo85bobobo85bobobo
95bobobo$37bobobo45bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$36bo
bobo45bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$5bo29bobobo45bobo
bo55bobobo65bobobo85bobobo85bobobo95bobobo$4bobo27bobobo45bobobo55bobo
bo65bobobo85bobobo85bobobo95bobobo$3bobobo25bobobo45bobobo55bobobo65bo
bobo85bobobo85bobobo95bobobo$2bobobo25bobobo45bobobo55bobobo65bobobo
85bobobo85bobobo95bobobo$bobobo25bobobo45bobobo55bobobo65bobobo85bobob
o85bobobo95bobobo$obobo25bobobo45bobobo55bobobo65bobobo85bobobo85bobob
o95bobobo$bobo27bobo47bobo57bobo67bobo87bobo87bobo97bobo$2bo29bo49bo
59bo69bo89bo89bo99bo11$12bo39bo49bo69bo79bo89bo99bo99bo$11bobo37bobo
47bobo67bobo77bobo87bobo97bobo97bobo$10bobobo35bobobo45bobobo65bobobo
75bobobo85bobobo95bobobo95bobobo$11bobobo35bobobo45bobobo65bobobo75bob
obo85bobobo95bobobo95bobobo$12bobobo35bobobo45bobobo65bobobo75bobobo
85bobobo95bobobo95bobobo$13bobobo35bobobo45bobobo65bobobo75bobobo85bob
obo95bobobo95bobobo$14bobobo35bobobo45bobobo65bobobo75bobobo85bobobo
95bobobo95bobobo$15bobobo35bobobo45bobobo65bobobo75bobobo85bobobo95bob
obo95bobobo$16bobobo35bobobo45bobobo65bobobo75bobobo85bobobo95bobobo
95bobobo$17bobo37bobobo45bobobo65bobobo75bobobo85bobobo95bobobo95bobob
o$18bo39bobobo45bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$59bobob
o45bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$60bobobo45bobobo65bo
bobo75bobobo85bobobo95bobobo95bobobo$61bobobo45bobobo65bobobo75bobobo
85bobobo95bobobo95bobobo$62bobobo45bobobo65bobobo75bobobo85bobobo95bob
obo95bobobo$63bobo47bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$64b
o49bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$115bobobo65bobobo75b
obobo85bobobo95bobobo95bobobo$116bobobo65bobobo75bobobo85bobobo95bobob
o95bobobo$117bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$118bobobo
65bobobo75bobobo85bobobo95bobobo95bobobo$119bobo67bobobo75bobobo85bobo
bo95bobobo95bobobo$120bo69bobobo75bobobo85bobobo95bobobo95bobobo$191bo
bobo75bobobo85bobobo95bobobo95bobobo$192bobobo75bobobo85bobobo95bobobo
95bobobo$193bobobo75bobobo85bobobo95bobobo95bobobo$194bobobo75bobobo
85bobobo95bobobo95bobobo$195bobo77bobobo85bobobo95bobobo95bobobo$196bo
79bobobo85bobobo95bobobo95bobobo$277bobobo85bobobo95bobobo95bobobo$
278bobobo85bobobo95bobobo95bobobo$279bobobo85bobobo95bobobo95bobobo$
280bobobo85bobobo95bobobo95bobobo$281bobo87bobobo95bobobo95bobobo$282b
o89bobobo95bobobo95bobobo$373bobobo95bobobo95bobobo$374bobobo95bobobo
95bobobo$375bobobo95bobobo95bobobo$376bobobo95bobobo95bobobo$377bobo
97bobobo95bobobo$378bo99bobobo95bobobo$479bobobo95bobobo$480bobobo95bo
bobo$481bobobo95bobobo$482bobobo95bobobo$483bobo97bobobo$484bo99bobobo
$585bobobo$586bobobo$587bobobo$588bobobo$589bobo$590bo!

Code: Select all

x = 593, y = 113, rule = B2e4e/S
537bo$536bobo$535bobobo$534bobobo$533bobobo$532bobobo$431bo99bobobo$
430bobo97bobobo$429bobobo95bobobo$428bobobo95bobobo$427bobobo95bobobo$
426bobobo95bobobo$335bo89bobobo95bobobo$334bobo87bobobo95bobobo$333bob
obo85bobobo95bobobo$332bobobo85bobobo95bobobo$331bobobo85bobobo95bobob
o$330bobobo85bobobo95bobobo$239bo89bobobo85bobobo95bobobo$238bobo87bob
obo85bobobo95bobobo$237bobobo85bobobo85bobobo95bobobo$236bobobo85bobob
o85bobobo95bobobo$235bobobo85bobobo85bobobo95bobobo$234bobobo85bobobo
85bobobo95bobobo$163bo69bobobo85bobobo85bobobo95bobobo$162bobo67bobobo
85bobobo85bobobo95bobobo$161bobobo65bobobo85bobobo85bobobo95bobobo$
160bobobo65bobobo85bobobo85bobobo95bobobo$159bobobo65bobobo85bobobo85b
obobo95bobobo$158bobobo65bobobo85bobobo85bobobo95bobobo$97bo59bobobo
65bobobo85bobobo85bobobo95bobobo$96bobo57bobobo65bobobo85bobobo85bobob
o95bobobo$95bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$94bobobo55b
obobo65bobobo85bobobo85bobobo95bobobo$93bobobo55bobobo65bobobo85bobobo
85bobobo95bobobo$92bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$41bo
49bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$40bobo47bobobo55bobob
o65bobobo85bobobo85bobobo95bobobo$39bobobo45bobobo55bobobo65bobobo85bo
bobo85bobobo95bobobo$38bobobo45bobobo55bobobo65bobobo85bobobo85bobobo
95bobobo$37bobobo45bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$36bo
bobo45bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$5bo29bobobo45bobo
bo55bobobo65bobobo85bobobo85bobobo95bobobo$4bobo27bobobo45bobobo55bobo
bo65bobobo85bobobo85bobobo95bobobo$3bobobo25bobobo45bobobo55bobobo65bo
bobo85bobobo85bobobo95bobobo$2bobobo25bobobo45bobobo55bobobo65bobobo
85bobobo85bobobo95bobobo$bobobo25bobobo45bobobo55bobobo65bobobo85bobob
o85bobobo95bobobo$obobo25bobobo45bobobo55bobobo65bobobo85bobobo85bobob
o95bobobo$bobo27bobo47bobo57bobo67bobo87bobo87bobo97bobo$2bo29bo49bo
59bo69bo89bo89bo99bo11$12bo39bo49bo69bo79bo89bo99bo99bo$11bobo37bobo
47bobo67bobo77bobo87bobo97bobo97bobo$10bobobo35bobobo45bobobo65bobobo
75bobobo85bobobo95bobobo95bobobo$11bobobo35bobobo45bobobo65bobobo75bob
obo85bobobo95bobobo95bobobo$12bobobo35bobobo45bobobo65bobobo75bobobo
85bobobo95bobobo95bobobo$13bobobo35bobobo45bobobo65bobobo75bobobo85bob
obo95bobobo95bobobo$14bobobo35bobobo45bobobo65bobobo75bobobo85bobobo
95bobobo95bobobo$15bobobo35bobobo45bobobo65bobobo75bobobo85bobobo95bob
obo95bobobo$16bobobo35bobobo45bobobo65bobobo75bobobo85bobobo95bobobo
95bobobo$17bobo37bobobo45bobobo65bobobo75bobobo85bobobo95bobobo95bobob
o$18bo39bobobo45bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$59bobob
o45bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$60bobobo45bobobo65bo
bobo75bobobo85bobobo95bobobo95bobobo$61bobobo45bobobo65bobobo75bobobo
85bobobo95bobobo95bobobo$62bobobo45bobobo65bobobo75bobobo85bobobo95bob
obo95bobobo$63bobo47bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$64b
o49bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$115bobobo65bobobo75b
obobo85bobobo95bobobo95bobobo$116bobobo65bobobo75bobobo85bobobo95bobob
o95bobobo$117bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$118bobobo
65bobobo75bobobo85bobobo95bobobo95bobobo$119bobo67bobobo75bobobo85bobo
bo95bobobo95bobobo$120bo69bobobo75bobobo85bobobo95bobobo95bobobo$191bo
bobo75bobobo85bobobo95bobobo95bobobo$192bobobo75bobobo85bobobo95bobobo
95bobobo$193bobobo75bobobo85bobobo95bobobo95bobobo$194bobobo75bobobo
85bobobo95bobobo95bobobo$195bobo77bobobo85bobobo95bobobo95bobobo$196bo
79bobobo85bobobo95bobobo95bobobo$277bobobo85bobobo95bobobo95bobobo$
278bobobo85bobobo95bobobo95bobobo$279bobobo85bobobo95bobobo95bobobo$
280bobobo85bobobo95bobobo95bobobo$281bobo87bobobo95bobobo95bobobo$282b
o89bobobo95bobobo95bobobo$373bobobo95bobobo95bobobo$374bobobo95bobobo
95bobobo$375bobobo95bobobo95bobobo$376bobobo95bobobo95bobobo$377bobo
97bobobo95bobobo$378bo99bobobo95bobobo$479bobobo95bobobo$480bobobo95bo
bobo$481bobobo95bobobo$482bobobo95bobobo$483bobo97bobobo$484bo99bobobo
$585bobobo$586bobobo$587bobobo$588bobobo$589bobo$590bo!

Code: Select all

x = 593, y = 113, rule = B2e4e/S
491bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobobo$490bo3bobo3bobo3bobo3bobo
3bobo3bobo3bobo3bobo$535bobobo$490bo3bobo3bobo3bobo3bobo3bobo3bobo3bob
obobobo$491bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobobobo$532bobobo$391bo
bo3bobo3bobo3bobo3bobo3bobo3bobobo59bobo3bobo3bobo3bobo3bobo3bobo3bobo
bobobo3bo$390bo3bobo3bobo3bobo3bobo3bobo3bobo3bobo57bo3bobo3bobo3bobo
3bobo3bobo3bobo3bobobobobo$429bobobo95bobobo$390bo3bobo3bobo3bobo3bobo
3bobo3bobobobobo57bo3bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo$391bobo
3bobo3bobo3bobo3bobo3bobo3bobobobo57bobo3bobo3bobo3bobo3bobo3bobo3bobo
bobobo3bo$426bobobo95bobobo$301bobo3bobo3bobo3bobo3bobo3bobobo55bobo3b
obo3bobo3bobo3bobo3bobobobobo3bo57bobo3bobo3bobo3bobo3bobo3bobobobobo
3bobo3bo$300bo3bobo3bobo3bobo3bobo3bobo3bobo53bo3bobo3bobo3bobo3bobo3b
obo3bobobobobo57bo3bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo$333bobobo
85bobobo95bobobo$300bo3bobo3bobo3bobo3bobo3bobobobobo53bo3bobo3bobo3bo
bo3bobo3bobobobobo3bobo57bo3bobo3bobo3bobo3bobo3bobobobobo3bobo3bobo$
301bobo3bobo3bobo3bobo3bobo3bobobobo53bobo3bobo3bobo3bobo3bobo3bobobob
obo3bo57bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo3bo$330bobobo85bobobo
95bobobo$211bobo3bobo3bobo3bobo3bobobo61bobo3bobo3bobo3bobo3bobobobobo
3bo53bobo3bobo3bobo3bobo3bobobobobo3bobo3bo57bobo3bobo3bobo3bobo3bobob
obobo3bobo3bobo3bo$210bo3bobo3bobo3bobo3bobo3bobo59bo3bobo3bobo3bobo3b
obo3bobobobobo53bo3bobo3bobo3bobo3bobo3bobobobobo3bobo57bo3bobo3bobo3b
obo3bobo3bobobobobo3bobo3bobo$237bobobo85bobobo85bobobo95bobobo$210bo
3bobo3bobo3bobo3bobobobobo59bo3bobo3bobo3bobo3bobobobobo3bobo53bo3bobo
3bobo3bobo3bobobobobo3bobo3bobo57bo3bobo3bobo3bobo3bobobobobo3bobo3bob
o3bobo$211bobo3bobo3bobo3bobo3bobobobo59bobo3bobo3bobo3bobo3bobobobobo
3bo53bobo3bobo3bobo3bobo3bobobobobo3bobo3bo57bobo3bobo3bobo3bobo3bobob
obobo3bobo3bobo3bo$234bobobo85bobobo85bobobo95bobobo$141bobo3bobo3bobo
3bobobo47bobo3bobo3bobo3bobobobobo3bo59bobo3bobo3bobo3bobobobobo3bobo
3bo53bobo3bobo3bobo3bobobobobo3bobo3bobo3bo57bobo3bobo3bobo3bobobobobo
3bobo3bobo3bobo3bo$140bo3bobo3bobo3bobo3bobo45bo3bobo3bobo3bobo3bobobo
bobo59bo3bobo3bobo3bobo3bobobobobo3bobo53bo3bobo3bobo3bobo3bobobobobo
3bobo3bobo57bo3bobo3bobo3bobo3bobobobobo3bobo3bobo3bobo$161bobobo65bob
obo85bobobo85bobobo95bobobo$140bo3bobo3bobo3bobobobobo45bo3bobo3bobo3b
obobobobo3bobo59bo3bobo3bobo3bobobobobo3bobo3bobo53bo3bobo3bobo3bobobo
bobo3bobo3bobo3bobo57bo3bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo$141bo
bo3bobo3bobo3bobobobo45bobo3bobo3bobo3bobobobobo3bo59bobo3bobo3bobo3bo
bobobobo3bobo3bo53bobo3bobo3bobo3bobobobobo3bobo3bobo3bo57bobo3bobo3bo
bo3bobobobobo3bobo3bobo3bobo3bo$158bobobo65bobobo85bobobo85bobobo95bob
obo$81bobo3bobo3bobobo43bobo3bobo3bobobobobo3bo45bobo3bobo3bobobobobo
3bobo3bo59bobo3bobo3bobobobobo3bobo3bobo3bo53bobo3bobo3bobobobobo3bobo
3bobo3bobo3bo57bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bo$80bo3bobo3b
obo3bobo41bo3bobo3bobo3bobobobobo45bo3bobo3bobo3bobobobobo3bobo59bo3bo
bo3bobo3bobobobobo3bobo3bobo53bo3bobo3bobo3bobobobobo3bobo3bobo3bobo
57bo3bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo$95bobobo55bobobo65bobobo
85bobobo85bobobo95bobobo$80bo3bobo3bobobobobo41bo3bobo3bobobobobo3bobo
45bo3bobo3bobobobobo3bobo3bobo59bo3bobo3bobobobobo3bobo3bobo3bobo53bo
3bobo3bobobobobo3bobo3bobo3bobo3bobo57bo3bobo3bobobobobo3bobo3bobo3bob
o3bobo3bobo$81bobo3bobo3bobobobo41bobo3bobo3bobobobobo3bo45bobo3bobo3b
obobobobo3bobo3bo59bobo3bobo3bobobobobo3bobo3bobo3bo53bobo3bobo3bobobo
bobo3bobo3bobo3bobo3bo57bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bo$
92bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$31bobo3bobobo39bobo3b
obobobobo3bo41bobo3bobobobobo3bobo3bo45bobo3bobobobobo3bobo3bobo3bo59b
obo3bobobobobo3bobo3bobo3bobo3bo53bobo3bobobobobo3bobo3bobo3bobo3bobo
3bo57bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo3bo$30bo3bobo3bobo37bo3bo
bo3bobobobobo41bo3bobo3bobobobobo3bobo45bo3bobo3bobobobobo3bobo3bobo
59bo3bobo3bobobobobo3bobo3bobo3bobo53bo3bobo3bobobobobo3bobo3bobo3bobo
3bobo57bo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo$39bobobo45bobobo55b
obobo65bobobo85bobobo85bobobo95bobobo$30bo3bobobobobo37bo3bobobobobo3b
obo41bo3bobobobobo3bobo3bobo45bo3bobobobobo3bobo3bobo3bobo59bo3bobobob
obo3bobo3bobo3bobo3bobo53bo3bobobobobo3bobo3bobo3bobo3bobo3bobo57bo3bo
bobobobo3bobo3bobo3bobo3bobo3bobo3bobo$31bobo3bobobobo37bobo3bobobobob
o3bo41bobo3bobobobobo3bobo3bo45bobo3bobobobobo3bobo3bobo3bo59bobo3bobo
bobobo3bobo3bobo3bobo3bo53bobo3bobobobobo3bobo3bobo3bobo3bobo3bo57bobo
3bobobobobo3bobo3bobo3bobo3bobo3bobo3bo$36bobobo45bobobo55bobobo65bobo
bo85bobobo85bobobo95bobobo$bobobo25bobobobobo3bo37bobobobobo3bobo3bo
41bobobobobo3bobo3bobo3bo45bobobobobo3bobo3bobo3bobo3bo59bobobobobo3bo
bo3bobo3bobo3bobo3bo53bobobobobo3bobo3bobo3bobo3bobo3bobo3bo57bobobobo
bo3bobo3bobo3bobo3bobo3bobo3bobo3bo$o3bobo23bo3bobobobobo37bo3bobobobo
bo3bobo41bo3bobobobobo3bobo3bobo45bo3bobobobobo3bobo3bobo3bobo59bo3bob
obobobo3bobo3bobo3bobo3bobo53bo3bobobobobo3bobo3bobo3bobo3bobo3bobo57b
o3bobobobobo3bobo3bobo3bobo3bobo3bobo3bobo$3bobobo25bobobo45bobobo55bo
bobo65bobobo85bobobo85bobobo95bobobo$obobobo23bobobobo3bobo37bobobobo
3bobo3bobo41bobobobo3bobo3bobo3bobo45bobobobo3bobo3bobo3bobo3bobo59bob
obobo3bobo3bobo3bobo3bobo3bobo53bobobobo3bobo3bobo3bobo3bobo3bobo3bobo
57bobobobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo$bobobobo23bobobobobo3bo
37bobobobobo3bobo3bo41bobobobobo3bobo3bobo3bo45bobobobobo3bobo3bobo3bo
bo3bo59bobobobobo3bobo3bobo3bobo3bobo3bo53bobobobobo3bobo3bobo3bobo3bo
bo3bobo3bo57bobobobobo3bobo3bobo3bobo3bobo3bobo3bobo3bo$obobo25bobobo
45bobobo55bobobo65bobobo85bobobo85bobobo95bobobo$bobo3bo23bobo3bobo3bo
37bobo3bobo3bobo3bo41bobo3bobo3bobo3bobo3bo45bobo3bobo3bobo3bobo3bobo
3bo59bobo3bobo3bobo3bobo3bobo3bobo3bo53bobo3bobo3bobo3bobo3bobo3bobo3b
obo3bo57bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo3bo$2bobobo25bobobo3bob
o39bobobo3bobo3bobo43bobobo3bobo3bobo3bobo47bobobo3bobo3bobo3bobo3bobo
61bobobo3bobo3bobo3bobo3bobo3bobo55bobobo3bobo3bobo3bobo3bobo3bobo3bob
o59bobobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo11$12bobobo3bo31bobobo3bob
o3bo35bobobo3bobo3bobo3bo49bobobo3bobo3bobo3bobo3bo53bobobo3bobo3bobo
3bobo3bobo3bo57bobobo3bobo3bobo3bobo3bobo3bobo3bo61bobobo3bobo3bobo3bo
bo3bobo3bobo3bobo3bo55bobobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo3bo$11b
obo3bobo31bobo3bobo3bobo35bobo3bobo3bobo3bobo49bobo3bobo3bobo3bobo3bob
o53bobo3bobo3bobo3bobo3bobo3bobo57bobo3bobo3bobo3bobo3bobo3bobo3bobo
61bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo55bobo3bobo3bobo3bobo3bobo3bo
bo3bobo3bobo3bobo$10bobobo3bo31bobobo3bo5bo35bobobo3bo5bo5bo49bobobo3b
o5bo5bo5bo53bobobo3bo5bo5bo5bo5bo57bobobo3bo5bo5bo5bo5bo5bo61bobobo3bo
5bo5bo5bo5bo5bo5bo55bobobo3bo5bo5bo5bo5bo5bo5bo5bo$11bobobobobo31bobob
obobo3bobo35bobobobobo3bobo3bobo49bobobobobo3bobo3bobo3bobo53bobobobob
o3bobo3bobo3bobo3bobo57bobobobobo3bobo3bobo3bobo3bobo3bobo61bobobobobo
3bobo3bobo3bobo3bobo3bobo3bobo55bobobobobo3bobo3bobo3bobo3bobo3bobo3bo
bo3bobo$10bobobobo3bo29bobobobo3bobo3bo33bobobobo3bobo3bobo3bo47bobobo
bo3bobo3bobo3bobo3bo51bobobobo3bobo3bobo3bobo3bobo3bo55bobobobo3bobo3b
obo3bobo3bobo3bobo3bo59bobobobo3bobo3bobo3bobo3bobo3bobo3bobo3bo53bobo
bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo3bo$13bobobo35bobobo45bobobo65b
obobo75bobobo85bobobo95bobobo95bobobo$10bo3bobobobo29bo3bobobobobo3bo
33bo3bobobobobo3bobo3bo47bo3bobobobobo3bobo3bobo3bo51bo3bobobobobo3bob
o3bobo3bobo3bo55bo3bobobobobo3bobo3bobo3bobo3bobo3bo59bo3bobobobobo3bo
bo3bobo3bobo3bobo3bobo3bo53bo3bobobobobo3bobo3bobo3bobo3bobo3bobo3bobo
3bo$11bobobobobo31bobobobobo3bobo35bobobobobo3bobo3bobo49bobobobobo3bo
bo3bobo3bobo53bobobobobo3bobo3bobo3bobo3bobo57bobobobobo3bobo3bobo3bob
o3bobo3bobo61bobobobobo3bobo3bobo3bobo3bobo3bobo3bobo55bobobobobo3bobo
3bobo3bobo3bobo3bobo3bobo3bobo$12bo3bobobo31bo3bobobo3bo37bo3bobobo3bo
5bo51bo3bobobo3bo5bo5bo55bo3bobobo3bo5bo5bo5bo59bo3bobobo3bo5bo5bo5bo
5bo63bo3bobobo3bo5bo5bo5bo5bo5bo57bo3bobobo3bo5bo5bo5bo5bo5bo5bo$11bob
o3bobo31bobo3bobobobobo35bobo3bobobobobo3bobo49bobo3bobobobobo3bobo3bo
bo53bobo3bobobobobo3bobo3bobo3bobo57bobo3bobobobobo3bobo3bobo3bobo3bob
o61bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo55bobo3bobobobobo3bobo3bobo
3bobo3bobo3bobo3bobo$10bo3bobobo31bo3bobobobobo3bo33bo3bobobobobo3bobo
3bo47bo3bobobobobo3bobo3bobo3bo51bo3bobobobobo3bobo3bobo3bobo3bo55bo3b
obobobobo3bobo3bobo3bobo3bobo3bo59bo3bobobobobo3bobo3bobo3bobo3bobo3bo
bo3bo53bo3bobobobobo3bobo3bobo3bobo3bobo3bobo3bobo3bo$59bobobo45bobobo
65bobobo75bobobo85bobobo95bobobo95bobobo$50bo3bobo3bobobobo33bo3bobo3b
obobobobo3bo47bo3bobo3bobobobobo3bobo3bo51bo3bobo3bobobobobo3bobo3bobo
3bo55bo3bobo3bobobobobo3bobo3bobo3bobo3bo59bo3bobo3bobobobobo3bobo3bob
o3bobo3bobo3bo53bo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo3bo$51bobo
3bobobobobo35bobo3bobobobobo3bobo49bobo3bobobobobo3bobo3bobo53bobo3bob
obobobo3bobo3bobo3bobo57bobo3bobobobobo3bobo3bobo3bobo3bobo61bobo3bobo
bobobo3bobo3bobo3bobo3bobo3bobo55bobo3bobobobobo3bobo3bobo3bobo3bobo3b
obo3bobo$52bo5bo3bobobo35bo5bo3bobobo3bo51bo5bo3bobobo3bo5bo55bo5bo3bo
bobo3bo5bo5bo59bo5bo3bobobo3bo5bo5bo5bo63bo5bo3bobobo3bo5bo5bo5bo5bo
57bo5bo3bobobo3bo5bo5bo5bo5bo5bo$51bobo3bobo3bobo35bobo3bobo3bobobobob
o49bobo3bobo3bobobobobo3bobo53bobo3bobo3bobobobobo3bobo3bobo57bobo3bob
o3bobobobobo3bobo3bobo3bobo61bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo
55bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo$50bo3bobo3bobobo35bo3b
obo3bobobobobo3bo47bo3bobo3bobobobobo3bobo3bo51bo3bobo3bobobobobo3bobo
3bobo3bo55bo3bobo3bobobobobo3bobo3bobo3bobo3bo59bo3bobo3bobobobobo3bob
o3bobo3bobo3bobo3bo53bo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo3bo$
115bobobo65bobobo75bobobo85bobobo95bobobo95bobobo$100bo3bobo3bobo3bobo
bobo47bo3bobo3bobo3bobobobobo3bo51bo3bobo3bobo3bobobobobo3bobo3bo55bo
3bobo3bobo3bobobobobo3bobo3bobo3bo59bo3bobo3bobo3bobobobobo3bobo3bobo
3bobo3bo53bo3bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bo$101bobo3bobo
3bobobobobo49bobo3bobo3bobobobobo3bobo53bobo3bobo3bobobobobo3bobo3bobo
57bobo3bobo3bobobobobo3bobo3bobo3bobo61bobo3bobo3bobobobobo3bobo3bobo
3bobo3bobo55bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo3bobo$102bo5bo5bo
3bobobo49bo5bo5bo3bobobo3bo55bo5bo5bo3bobobo3bo5bo59bo5bo5bo3bobobo3bo
5bo5bo63bo5bo5bo3bobobo3bo5bo5bo5bo57bo5bo5bo3bobobo3bo5bo5bo5bo5bo$
101bobo3bobo3bobo3bobo49bobo3bobo3bobo3bobobobobo53bobo3bobo3bobo3bobo
bobobo3bobo57bobo3bobo3bobo3bobobobobo3bobo3bobo61bobo3bobo3bobo3bobob
obobo3bobo3bobo3bobo55bobo3bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo$
100bo3bobo3bobo3bobobo49bo3bobo3bobo3bobobobobo3bo51bo3bobo3bobo3bobob
obobo3bobo3bo55bo3bobo3bobo3bobobobobo3bobo3bobo3bo59bo3bobo3bobo3bobo
bobobo3bobo3bobo3bobo3bo53bo3bobo3bobo3bobobobobo3bobo3bobo3bobo3bobo
3bo$191bobobo75bobobo85bobobo95bobobo95bobobo$170bo3bobo3bobo3bobo3bob
obobo51bo3bobo3bobo3bobo3bobobobobo3bo55bo3bobo3bobo3bobo3bobobobobo3b
obo3bo59bo3bobo3bobo3bobo3bobobobobo3bobo3bobo3bo53bo3bobo3bobo3bobo3b
obobobobo3bobo3bobo3bobo3bo$171bobo3bobo3bobo3bobobobobo53bobo3bobo3bo
bo3bobobobobo3bobo57bobo3bobo3bobo3bobobobobo3bobo3bobo61bobo3bobo3bob
o3bobobobobo3bobo3bobo3bobo55bobo3bobo3bobo3bobobobobo3bobo3bobo3bobo
3bobo$172bo5bo5bo5bo3bobobo53bo5bo5bo5bo3bobobo3bo59bo5bo5bo5bo3bobobo
3bo5bo63bo5bo5bo5bo3bobobo3bo5bo5bo57bo5bo5bo5bo3bobobo3bo5bo5bo5bo$
171bobo3bobo3bobo3bobo3bobo53bobo3bobo3bobo3bobo3bobobobobo57bobo3bobo
3bobo3bobo3bobobobobo3bobo61bobo3bobo3bobo3bobo3bobobobobo3bobo3bobo
55bobo3bobo3bobo3bobo3bobobobobo3bobo3bobo3bobo$170bo3bobo3bobo3bobo3b
obobo53bo3bobo3bobo3bobo3bobobobobo3bo55bo3bobo3bobo3bobo3bobobobobo3b
obo3bo59bo3bobo3bobo3bobo3bobobobobo3bobo3bobo3bo53bo3bobo3bobo3bobo3b
obobobobo3bobo3bobo3bobo3bo$277bobobo85bobobo95bobobo95bobobo$250bo3bo
bo3bobo3bobo3bobo3bobobobo55bo3bobo3bobo3bobo3bobo3bobobobobo3bo59bo3b
obo3bobo3bobo3bobo3bobobobobo3bobo3bo53bo3bobo3bobo3bobo3bobo3bobobobo
bo3bobo3bobo3bo$251bobo3bobo3bobo3bobo3bobobobobo57bobo3bobo3bobo3bobo
3bobobobobo3bobo61bobo3bobo3bobo3bobo3bobobobobo3bobo3bobo55bobo3bobo
3bobo3bobo3bobobobobo3bobo3bobo3bobo$252bo5bo5bo5bo5bo3bobobo57bo5bo5b
o5bo5bo3bobobo3bo63bo5bo5bo5bo5bo3bobobo3bo5bo57bo5bo5bo5bo5bo3bobobo
3bo5bo5bo$251bobo3bobo3bobo3bobo3bobo3bobo57bobo3bobo3bobo3bobo3bobo3b
obobobobo61bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo55bobo3bobo3bobo3bo
bo3bobo3bobobobobo3bobo3bobo$250bo3bobo3bobo3bobo3bobo3bobobo57bo3bobo
3bobo3bobo3bobo3bobobobobo3bo59bo3bobo3bobo3bobo3bobo3bobobobobo3bobo
3bo53bo3bobo3bobo3bobo3bobo3bobobobobo3bobo3bobo3bo$373bobobo95bobobo
95bobobo$340bo3bobo3bobo3bobo3bobo3bobo3bobobobo59bo3bobo3bobo3bobo3bo
bo3bobo3bobobobobo3bo53bo3bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo3bo$
341bobo3bobo3bobo3bobo3bobo3bobobobobo61bobo3bobo3bobo3bobo3bobo3bobob
obobo3bobo55bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo3bobo$342bo5bo5bo
5bo5bo5bo3bobobo61bo5bo5bo5bo5bo5bo3bobobo3bo57bo5bo5bo5bo5bo5bo3bobob
o3bo5bo$341bobo3bobo3bobo3bobo3bobo3bobo3bobo61bobo3bobo3bobo3bobo3bob
o3bobo3bobobobobo55bobo3bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo$340bo
3bobo3bobo3bobo3bobo3bobo3bobobo61bo3bobo3bobo3bobo3bobo3bobo3bobobobo
bo3bo53bo3bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo3bo$479bobobo95bobob
o$440bo3bobo3bobo3bobo3bobo3bobo3bobo3bobobobo53bo3bobo3bobo3bobo3bobo
3bobo3bobo3bobobobobo3bo$441bobo3bobo3bobo3bobo3bobo3bobo3bobobobobo
55bobo3bobo3bobo3bobo3bobo3bobo3bobobobobo3bobo$442bo5bo5bo5bo5bo5bo5b
o3bobobo55bo5bo5bo5bo5bo5bo5bo3bobobo3bo$441bobo3bobo3bobo3bobo3bobo3b
obo3bobo3bobo55bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobobobobo$440bo3bob
o3bobo3bobo3bobo3bobo3bobo3bobobo55bo3bobo3bobo3bobo3bobo3bobo3bobo3bo
bobobobo3bo$585bobobo$540bo3bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobobob
o$541bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobobobobo$542bo5bo5bo5bo5bo5b
o5bo5bo3bobobo$541bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo3bobo$540bo3b
obo3bobo3bobo3bobo3bobo3bobo3bobo3bobobo!

Introducing B4e in the 2-state case gives the oscillators a different appearance, and also allows for a variant running in a checkerboard-y duoplet agar. How to generalize this to higher state counts?

[LaundryPizza03]

Another natural Margolus emulator, in a JustFriends relative:

Code: Select all

x = 140, y = 20, rule = B2-a3ae4nr5e/S123en4nr
bo3bo5bo7bo9bo11bo13bo15bo17bo19bo27bo$o3bo5bo7bo9bo11bo13bo15bo17bo
19bo27bo$7bo5bo7bo9bo11bo13bo15bo17bo19bo23bo3bo$6bo5bo7bo9bo11bo13bo
15bo17bo19bo23bo3bo$15bo7bo9bo11bo13bo15bo17bo19bo23bo$14bo7bo9bo11bo
13bo15bo17bo19bo23bo$25bo9bo11bo13bo15bo17bo19bo15bo3bo$24bo9bo11bo13b
o15bo17bo19bo15bo3bo$37bo11bo13bo15bo17bo19bo$36bo11bo13bo15bo17bo19bo
$51bo13bo15bo17bo19bo$50bo13bo15bo17bo19bo$67bo15bo17bo19bo$66bo15bo
17bo19bo$85bo17bo19bo$84bo17bo19bo$105bo19bo$104bo19bo$127bo$126bo!

[muzik]

Two interweaved diagonal oscillators produce a period-2092034 oscillator:

Code: Select all

x = 29, y = 32, rule = B2e3ajk4akqw5ajk6e/S2e3ajk4akqw5ajk6e
5$2bo$3bo$4bo16bo$5bo14bo$6bo12bo$7bo10bo$8bo8bo$9bo6bo$10bo4bo$11bo2b
o$12b2o$12b2o$11bo2bo$10bo4bo$9bo6bo$8bo8bo$7bo10bo$6bo12bo$5bo14bo$4b
o16bo$22bo$23bo!