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Triangular neighbourhood rules

Posted: January 22nd, 2020, 9:12 pm
by muzik
2D CA are most popularly emulated on a square grid {4,4}, and occasionally on the hexagonal grid {6,3}. There exist one more proper tiling {3,6}, the triangular tiling, which can also be used to simulate cellular automata. LifeViewer has had support for range-1 triangular CA for almost a year now, but this feature hasn't seen nearly enough use, so it's probably a good idea to spark discussion of these rules so the implementation doesn't go to waste.

Obligatory parity/replication rules:

Code: Select all

 x = 1, y = 1, rule = B13579Y/S13579YL
o!

Code: Select all

 x = 1, y = 1, rule = B13579Y/S02468XZL
o!
For the Triangular Edges neighbourhood, it can be noted that these mimic the behaviour of the hexagonal parity rules on even generations:

Code: Select all

 x = 1, y = 1, rule = B13/S02LE
o!

Code: Select all

 x = 1, y = 1, rule = B135/S135H
o!

Code: Select all

 x = 1, y = 1, rule = B13/S13LE
o!

Code: Select all

 x = 1, y = 1, rule = B135/S0246H
o!
FredkinModN.py allows for generalisation of parity rules like with other neighbourhoods (even if attempting to generate higher order moore ones does murdery things to the computer):

Code: Select all

 x = 1, y = 1, rule = Fredkin mod3 triangularVonNeumann emulated
o!

Code: Select all

 x = 1, y = 1, rule = Fredkin mod5 triangularVonNeumann emulated
o!

Code: Select all

 x = 1, y = 1, rule = Fredkin mod7 triangularVonNeumann emulated
o!

Code: Select all

 x = 1, y = 1, rule = Fredkin mod11 triangularVonNeumann emulated
o!

Code: Select all

 x = 1, y = 1, rule = Fredkin mod13 triangularVonNeumann emulated
o!
Spaceships can exist in triangular rules like with most other rulespaces. For example, here are the spaceships from the trilife package on the golly rule repository:

Code: Select all

x = 4, y = 2, rule = B4/S456L
4o$4o!

Code: Select all

x = 7, y = 4, rule = B45/S23L
5bo$o3b3o$o3b3o$5bo!

Code: Select all

x = 8, y = 3, rule = B45/S34L
5bobo$3o2b3o$3o3b2o!
as well as the following ship provided in the triangular lifeviewer demo:

Code: Select all

x = 8, y = 6, rule = B456/S34L
bo3bo$bo4bo$b2o4bo$2bo5bo$2bo5bo$3b2ob2o!
The demo also shows a triangular generations rule with interesting dynamics including a common unloopable RRO:

Code: Select all

 x = 8, y = 6, rule = G10/B34/S34L
bo3bo$bo4bo$b2o4bo$2bo5bo$2bo5bo$3b2ob2o!
[[ THEME Blues ]]
I haven't tried creating an isotropic non-totalistic notation for the triangular grid, and it'd probably be significantly bigger than the current square moore non totalistic notation, and maybe bigger than range 2 vN and also the range 1 3 state notation I'm currently busy with. I might do so in the future though.

Please discuss further!

Re: Triangular neighbourhood rules

Posted: January 22nd, 2020, 9:52 pm
by Hdjensofjfnen
I apologize - is the neighborhood with the spaceships range-2 triangular?
EDIT: I see:

Code: Select all

x = 5, y = 3, rule = B/S0123456789L
bobo$2ob2o$b3o!
EDIT: Well, here is a p9 mod3:

Code: Select all

x = 5, y = 4, rule = B4/S345L
3bo$o$b4o$3bo!
and a p8 mod4:

Code: Select all

x = 9, y = 4, rule = B45/S23L
2b3ob2o$bo6bo$7bo$3bobo!

Re: Triangular neighbourhood rules

Posted: January 22nd, 2020, 11:31 pm
by bprentice
muzik wrote:
January 22nd, 2020, 9:12 pm
Please discuss further!
Look at these archives:

http://bprentice.webenet.net/Triangular%20Cell/

and search for "Triangular" in this board.

Brian Prentice

Re: Triangular neighbourhood rules

Posted: January 23rd, 2020, 5:28 am
by muzik
The p9 bundled with trilife:

Code: Select all

 x = 6, y = 4, rule = B4/S456L
3bo$b5o$b5o$3bo!
There's also a more complex p4 in the zip.

Re: Triangular neighbourhood rules

Posted: January 29th, 2020, 7:18 pm
by muzik
A question: can mod-3 2D Moore replicators exist on the triangular grid? I gave my computer a good hundred hours to generate such a rule, and it got back to me with this:

Code: Select all

x = 1, y = 1, rule = Fredkin_mod3_triangularMoore_emulated
o!
Either the script is at fault here, or there's something special about the triangular grid that stops these replication habits from existing, as there's definitely preserved symmetry and large amounts of death at 3^n generations.

Re: Triangular neighbourhood rules

Posted: January 29th, 2020, 8:06 pm
by muzik
p6 mod 1:

Code: Select all

 x = 4, y = 2, rule = B45/S34L
4o$o2bo!

Re: Triangular neighbourhood rules

Posted: January 29th, 2020, 11:19 pm
by bprentice
muzik wrote:
January 29th, 2020, 7:18 pm
A question: can mod-3 2D Moore replicators exist on the triangular grid?
What does "mod-3 2D Moore replicators" mean?

Triangular rules defined by these dialogs:

1.png
1.png (117.03 KiB) Viewed 9233 times
2.png
2.png (49.43 KiB) Viewed 9233 times

and generated by this step code:

Code: Select all

  public int step(int c, int r)
  {
    int cell = triangularCell.getCell(c, r);
    int count;
    if (!((c & 1) == 0) ^ ((r & 1) == 0))
    {
      count =
      stateWeights[triangularCell.getCell(c - 2, r + 1)] * neighborWeights[0][4] +
      stateWeights[triangularCell.getCell(c + 2, r + 1)] * neighborWeights[0][0] +
      stateWeights[triangularCell.getCell(c    , r - 1)] * neighborWeights[2][2] +
      stateWeights[triangularCell.getCell(c    , r + 1)] * neighborWeights[0][2] +
      stateWeights[triangularCell.getCell(c - 1, r - 1)] * neighborWeights[2][3] +
      stateWeights[triangularCell.getCell(c - 1, r    )] * neighborWeights[1][3] +
      stateWeights[triangularCell.getCell(c - 1, r + 1)] * neighborWeights[0][3] +
      stateWeights[triangularCell.getCell(c + 1, r    )] * neighborWeights[1][1] +
      stateWeights[triangularCell.getCell(c + 1, r + 1)] * neighborWeights[0][1] +
      stateWeights[triangularCell.getCell(c + 2, r    )] * neighborWeights[1][0] +
      stateWeights[triangularCell.getCell(c - 2, r    )] * neighborWeights[1][4] +
      stateWeights[triangularCell.getCell(c + 1, r - 1)] * neighborWeights[2][1] +
      stateWeights[cell] * neighborWeights[1][2];
    }
    else
    {
      count =
      stateWeights[triangularCell.getCell(c - 2, r - 1)] * neighborWeights[0][0] +
      stateWeights[triangularCell.getCell(c + 2, r - 1)] * neighborWeights[0][4] +
      stateWeights[triangularCell.getCell(c    , r - 1)] * neighborWeights[0][2] +
      stateWeights[triangularCell.getCell(c    , r + 1)] * neighborWeights[2][2] +
      stateWeights[triangularCell.getCell(c - 1, r - 1)] * neighborWeights[0][1] +
      stateWeights[triangularCell.getCell(c - 1, r    )] * neighborWeights[1][1] +
      stateWeights[triangularCell.getCell(c - 1, r + 1)] * neighborWeights[2][1] +
      stateWeights[triangularCell.getCell(c + 1, r    )] * neighborWeights[1][3] +
      stateWeights[triangularCell.getCell(c + 1, r + 1)] * neighborWeights[2][3] +
      stateWeights[triangularCell.getCell(c + 2, r    )] * neighborWeights[1][4] +
      stateWeights[triangularCell.getCell(c - 2, r    )] * neighborWeights[1][0] +
      stateWeights[triangularCell.getCell(c + 1, r - 1)] * neighborWeights[0][3] +
      stateWeights[cell] * neighborWeights[1][2];
    }
    if ((count >= 0) && (count < noCounts))
    {
      useTable[cell][count] = 1;
      return ruleTable[cell][count];
    }
    return 0;
  }
support linear replicators. Many of the guns in this archive:

http://bprentice.webenet.net/Triangular ... 20Guns.zip

use such replicators.

Brian Prentice

Re: Triangular neighbourhood rules

Posted: January 30th, 2020, 5:07 am
by muzik
bprentice wrote:
January 29th, 2020, 11:19 pm
muzik wrote:
January 29th, 2020, 7:18 pm
A question: can mod-3 2D Moore replicators exist on the triangular grid?
What does "mod-3 2D Moore replicators" mean?
Obviously a pattern that replicates in two dimensions every 3^n generations according to the Triangular Moore neighbourhood.

Re: Triangular neighbourhood rules

Posted: January 30th, 2020, 5:23 am
by muzik
One of the other mentioned spaceships, in an explosive rule:

Code: Select all

 x = 7, y = 4, rule = B456/S12L
2bo2b2o$o4b2o$5b2o$b2o2bo!
Are any more spaceships known across the supported rulespaces?

Re: Triangular neighbourhood rules

Posted: January 30th, 2020, 4:45 pm
by bprentice
The supported rulespaces include the following rule families:

1.png
1.png (40.36 KiB) Viewed 9179 times

Many example patterns in each of these rule families are included in the following archive:

http://bprentice.webenet.net/Triangular ... 20Cell.zip

A rule from the "Rule Table" family is shown in a previous post above.

muzik wrote:
January 30th, 2020, 5:07 am
Are any more spaceships known across the supported rulespaces?
Of course, there are many other known spaceships in the supported rulespaces! Here are just two that are included in the following archive:

http://bprentice.webenet.net/Triangular ... 20Guns.zip

2.png
2.png (817.57 KiB) Viewed 9179 times

3.png
3.png (721.99 KiB) Viewed 9179 times

Brian Prentice

Re: Triangular neighbourhood rules

Posted: January 31st, 2020, 10:15 pm
by bprentice
Ship:

Code: Select all

x = 158, y = 58, rule = B1XYZ/S1L
53.5A$53.A.A2.A$51.2A5.A$51.3A$52.2A.2A$50.2A$50.A.A3.2A$55.3A$47.2A
5.3A$47.A.A3.2A$52.3A$44.2A5.3A$44.A.A3.2A$49.3A$41.2A5.3A$41.A.A3.
2A$46.3A$38.2A5.3A$38.A.A3.2A$43.3A$35.2A5.3A$35.A.A3.2A$40.3A$32.2A
5.3A$32.A.A3.2A$37.3A$29.2A5.3A$29.A.A3.2A$34.3A$26.2A5.3A38.2A$26.
A.A3.2A40.A.A3.2A$31.3A45.3A$23.2A5.3A38.2A5.3A$23.A.A3.2A40.A.A3.2A
$28.3A45.3A$20.2A5.3A38.2A5.3A$20.A.A3.2A40.A.A3.2A$25.3A45.3A$17.2A
5.3A38.2A5.3A$17.A.A3.2A40.A.A3.2A$22.3A45.3A$14.2A5.3A38.2A5.3A$14.
A.A3.2A40.A.A3.2A$19.3A45.3A$11.2A5.3A38.2A5.3A$11.A.A3.2A40.A.A3.2A
$16.3A45.3A$8.2A5.3A38.2A5.3A36.5A$8.A.A3.2A40.A.A3.2A37.A2.A2.A33.
5A7.5A$13.3A45.3A36.2A3.3A33.A.A.A7.A.A.A$5.2A5.3A38.2A5.3A37.A5.A$
5.A.A3.2A40.A.A3.2A40.A.2A.A.5A30.A11.A$10.3A45.3A44.2A.A.A2.A28.3A
9.3A$2.5A2.3A45.3A46.2A5.A$2.A.A2.A98.3A$2A5.A99.2A.2A$3A$.2A.2A!
Brian Prentice

Re: Triangular neighbourhood rules

Posted: February 1st, 2020, 2:43 pm
by Layz Boi
bprentice wrote:
January 31st, 2020, 10:15 pm
Ship:

Code: Select all

x = 158, y = 58, rule = B1XYZ/S1L
53.5A$53.A.A2.A$51.2A5.A$51.3A$52.2A.2A$50.2A$50.A.A3.2A$55.3A$47.2A
5.3A$47.A.A3.2A$52.3A$44.2A5.3A$44.A.A3.2A$49.3A$41.2A5.3A$41.A.A3.
2A$46.3A$38.2A5.3A$38.A.A3.2A$43.3A$35.2A5.3A$35.A.A3.2A$40.3A$32.2A
5.3A$32.A.A3.2A$37.3A$29.2A5.3A$29.A.A3.2A$34.3A$26.2A5.3A38.2A$26.
A.A3.2A40.A.A3.2A$31.3A45.3A$23.2A5.3A38.2A5.3A$23.A.A3.2A40.A.A3.2A
$28.3A45.3A$20.2A5.3A38.2A5.3A$20.A.A3.2A40.A.A3.2A$25.3A45.3A$17.2A
5.3A38.2A5.3A$17.A.A3.2A40.A.A3.2A$22.3A45.3A$14.2A5.3A38.2A5.3A$14.
A.A3.2A40.A.A3.2A$19.3A45.3A$11.2A5.3A38.2A5.3A$11.A.A3.2A40.A.A3.2A
$16.3A45.3A$8.2A5.3A38.2A5.3A36.5A$8.A.A3.2A40.A.A3.2A37.A2.A2.A33.
5A7.5A$13.3A45.3A36.2A3.3A33.A.A.A7.A.A.A$5.2A5.3A38.2A5.3A37.A5.A$
5.A.A3.2A40.A.A3.2A40.A.2A.A.5A30.A11.A$10.3A45.3A44.2A.A.A2.A28.3A
9.3A$2.5A2.3A45.3A46.2A5.A$2.A.A2.A98.3A$2A5.A99.2A.2A$3A$.2A.2A!
!
Brian Prentice
Some useless oscillators:

Code: Select all

x = 101, y = 85, rule = B1xyz/S1L
22$45b5o$45b2ob2o$46b3o2b5o$51b2ob2o$49b6o$44bo14bo18b5o3bo4bo$29b5o44b
2ob3o$23bo5b2ob2o13bo31b3obo$30b3o$87bo$27b5o45bo$27b2ob2o52bo$20bo5b
5o$27b2o2$33bo14bo5bo$28bo14bo$62bo$49b3obo$48b7o20bo6bo$40bo4b4obob5o
4bo$47b4obob2o$21bo5b5o20b3o$27b2ob2o12bo37b2o$25b6o$26b3o$28bo49bo6b
o$22bo7b3o25bo$29b2ob2o$28b6o16b5o11bo$50b2ob2o3b3o$51b3o3b6o$28bo30b
2ob2o$59b5o$41b3o$40b2ob2o$40b5o$41bo2$37bo8bo17b5o4bo$64b2ob2o$59bo5b
3o$63bo20bo2$22bo4b5o33bo$27b2ob2o15bo16b2o14b5o$28b3o49b2ob2obo$27b2o
52b3o6bo$28b6o9b3o7bo$23bo5b2ob2o8b2ob2o18bo$30b3o9b6o33bo$47bo2$41bo
8bo!

Re: Triangular neighbourhood rules

Posted: February 1st, 2020, 7:15 pm
by muzik
Here are the alternating cases of the aforementioned Moore parity rules:

Code: Select all

 x = 1, y = 1, rule = B13579Y/S13579YL|B13579Y/S02468XZL
o!

Code: Select all

 x = 1, y = 1, rule = B13579Y/S02468XZL|B13579Y/S13579YL
o!
and vN:

Code: Select all

 x = 1, y = 1, rule = B13/S02LE|B13/S13LE
o!

Code: Select all

 x = 1, y = 1, rule = B13/S13LE|B13/S02LE
o!
Triangular Vertices does not seem to support replicators.

I really want to see this neighbourhood supported:

Code: Select all

 x = 7, y = 4, rule = B/S0123456789XYZL
7o$b5o$2b3o$3bo!
[[ GRID ]]

Re: Triangular neighbourhood rules

Posted: February 1st, 2020, 9:08 pm
by bprentice
Three more ships:

Code: Select all

x = 63, y = 36, rule = B19YL
33.5A$32.A4.A$32.2A$33.3A2$32.2A3.2A$32.2A3.2A$33.2A.2A4$38.2A3.2A$
38.2A3.2A$39.2A.2A4$44.2A3.2A$44.2A3.2A$45.2A.2A4$50.2A3.2A$50.2A3.
2A$51.2A.2A4$56.2A3.2A$7.5A7.5A7.5A7.5A7.A6.A$7.A4.A6.A4.A6.A4.A6.A
4.A6.A$.5A5.2A10.2A10.2A10.2A$A4.A3.3A9.3A9.3A9.3A9.3A$2A$.3A!

Code: Select all

x = 37, y = 40, rule = B19YL
3.5A$2.A4.A$2.2A$3.3A2$2A3.2A$2A3.2A$.2A.2A4$6.2A3.2A$6.2A3.2A$7.2A
.2A4$12.2A3.2A$12.2A3.2A$13.2A.2A4$18.2A3.2A$18.2A3.2A$19.2A.2A4$24.
2A3.2A$24.2A3.2A$25.2A.2A4$26.3A.2A3.2A$25.2A.A7.A$24.A$24.2A.2A$25.
3A3.3A!

Code: Select all

x = 77, y = 60, rule = B19YL
69.5A$68.A3.2A$67.2A3.A$67.3A5.2A$33.5A7.5A7.5A7.A6.A$33.A4.A6.A4.A
6.A4.A6.A$37.2A10.2A10.2A$35.3A9.3A9.3A9.3A$29.2A3.2A$29.2A3.2A$30.
2A.2A4$35.2A3.2A$35.2A3.2A$36.2A.2A4$41.2A3.2A$41.2A3.2A$42.2A.2A4$
47.2A3.2A$47.2A3.2A$48.2A.2A4$53.2A3.2A$4.5A7.5A7.5A7.5A7.A6.A$4.A4.
A6.A4.A6.A4.A6.A4.A6.A$8.2A10.2A10.2A10.2A$6.3A9.3A9.3A9.3A9.3A$2A3.
2A$2A3.2A$.2A.2A4$6.2A3.2A$6.2A3.2A$7.2A.2A4$12.2A3.2A$12.2A3.2A$13.
2A.2A4$14.3A.2A3.2A$13.2A.A7.A$12.A$12.2A.2A$13.3A3.3A!
Brian Prentice

Re: Triangular neighbourhood rules

Posted: February 1st, 2020, 10:04 pm
by bprentice
muzik wrote:
February 1st, 2020, 7:15 pm
Triangular Vertices does not seem to support replicators.
This four state rule:

1.png
1.png (115.65 KiB) Viewed 9044 times

supports replicators! See ../Patterns/Rule Table/Rule 129 in:

http://bprentice.webenet.net/Triangular ... 20Cell.zip

Brian Prentice

Re: Triangular neighbourhood rules

Posted: February 3rd, 2020, 12:23 am
by bprentice
It is interesting to watch these rules evolve:

Code: Select all

x = 16, y = 4, rule = G6/B2367/S8LV:T1000,1000
.E.D.C.B.A$.E.DECDBCAB.A$4.E.DECDBCAB.A$7.E.D.C.B.A!

Code: Select all

x = 7, y = 4, rule = G5/B238/S4LV:T1000,1000
.2D$2.2C$3.2B$4.2A!

Code: Select all

x = 8, y = 2, rule = G5/B25/S369LV:T1000,1000
.D.C.B.A$.D.C.B.A!
Play some random fills on a torus.

Brian Prentice

Re: Triangular neighbourhood rules

Posted: July 17th, 2020, 3:49 pm
by muzik
Tri6inner and Tri6outer are now supported!

Code: Select all

x = 1, y = 1, rule = B135/S135LI
o!

Code: Select all

x = 1, y = 1, rule = B135/S0246LI
o!

Code: Select all

x = 1, y = 1, rule = B135/S135LO
o!

Code: Select all

x = 1, y = 1, rule = B135/S0246LO
o!

Re: Triangular neighbourhood rules

Posted: July 18th, 2020, 11:09 am
by praosylen
muzik wrote:
July 17th, 2020, 3:49 pm
Tri6inner and Tri6outer are now supported!
[...]

Code: Select all

x = 1, y = 1, rule = B135/S135LO
o!

Code: Select all

x = 1, y = 1, rule = B135/S0246LO
o!
It's worth mentioning that Tri6outer is isomorphic to two interleaved copies of a hexagonal neighborhood, at least for outer-totalistic rules:

Code: Select all

x = 37, y = 8, rule = B2/S34LO
32bobobo$2bo24bo7bo$bo28bobo$8bobo3bo$bo3bobo16bobo3bo$14bo6bo$24bo$14b
o!

Code: Select all

x = 7, y = 8, rule = B2/S34H
2bo$4bo$2bo3bo$2bo$3bo$o2bo$bo$3bo!

Re: Triangular neighbourhood rules

Posted: July 19th, 2020, 8:26 am
by muzik
Here's something funny:

Code: Select all

x = 1, y = 1, rule = B135/S135LI
o!
[[ STOP 30 ZOOM 4 ]]

Code: Select all

x = 1, y = 1, rule = B135/S0246LI
o!
[[ STOP 30 ZOOM 4 ]]

Code: Select all

x = 9, y = 5, rule = R2,C2,S1,3,5,7,9,11,13,15,17,B1,3,5,7,9,11,13,15,17,NH
o!
[[ STOP 15 ZOOM 4 ]]

Code: Select all

x = 9, y = 5, rule = R2,C2,S0,2,4,6,8,10,12,14,16,18,B1,3,5,7,9,11,13,15,17,NH
o!
[[ STOP 15 ZOOM 4 ]]

Re: Triangular neighbourhood rules

Posted: August 1st, 2020, 11:02 am
by muzik
A bug from a triangular HROT rule:

Code: Select all

x = 31, y = 17, rule = R5,C2,M1,S51..87,B51..68,NL
4$15b3obo$12b9o$10b14o$9b15o$9b8o3b5o$9b7o7b2o$10b6o7b2o$10b7o5b2o$12b
6ob4o$15b5o!
[[ AUTOFIT ]]

Re: Triangular neighbourhood rules

Posted: April 19th, 2021, 9:25 pm
by muzik
Interesting triangular patterns from other threads:
muzik wrote:
August 1st, 2020, 10:53 am
Triangular Bugs:

Code: Select all

 x = 823, y = 833, rule = R5,C0,M1,S51..87,B51..67,NL
bo3b2ob2obo2b5ob2o2bo4b3obo3bo4b2o5b4obo2b4o3bo2bo2bob2o2b3o10b3obob3o
bo2b2o2bob5obob7o3bobobo3bob2obob2o3bo3bob6ob2o2b3o2b2o2bo3bobo2b2o3b
2o3bobobo3b4obob3obobob2o5bo3b3o2b2obob2o2bob2o3bob2o4bo2b3o2b4obo2bo$
2obobo3bob5obobo2b3obo3b4obo2bobo4bob8o5b4o2bo2b2o4bo3b2o3bob3ob3ob2o
3b4o3bob2o2b5ob2obo2b2o3bobo6b2o2bo2bo4bo3bobob3o2bo2bob2o4bo2bob3ob2o
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4bob3o5b2o5b2o2bob2ob3ob2o3b7o4bo3b2o3b2ob3obo4b2ob3obobobob4obobobobo
bo2b3obo4bo2bobo2b2obo5bobo$obob7o7b2o2b2o2bo3b2o4b3o2bob2obob2ob5o4b
5o3b4obo2bo2b2ob2o3bob2o3bo4bo2bobo2bo2bo7bo2b2o2b2obo2bobob2obo4b4ob
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3b2ob3ob2ob2obobobobob2ob4o2bobob2o3b2o3bobo3b3o5b2o2bob2obobob3ob2ob
2ob2obo2bobo3bobobo$2bo2b4obob2o3bo2bobob4o2bobo2b2obo2b3obob2o2b2o4b
3o4b3obob2ob2o2b4obo5bo5bobobobobo2b6obobo7b2o5bo4bo2b4o2b2o8b4o3bob2o
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4b5o2b4obobobobobob2o2bo5b2o2b2o2bo3b3ob3o3b2ob4ob6ob5o2bo5bobobo2b3o
2b2o$3ob7o3b2o8b2ob2obo2b3obobo6b2ob4ob2o4bobob3ob3ob2o2b2o2bobobo3b3o
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5o4b2ob2obo3b2o3bo3bo3bob2o3bo2b2ob2o4bo2b2o3b3obobobo3bo2bo4bo4b3obob
2obobob2obob2obobob2obo3b2ob6o3b4o3b3o2bo4b3o2bob5ob4obob3o8b2o$3bo2b
3o3bo2bo3b2o2bob3obob2o3b2obo4b2o5b3o4b4o2bo2bobo3b2ob2o3b2o4bobo2b4ob
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5bob2o2b8o2b3o3b2o2b2obobo2b2o2b2obo2b6ob2ob2ob2ob2obo3bob3obo2bo4bobo
bo2bo$bo2b3o2b2o4bo2bo5bo2bobo2bo4b3obo2b2o4bobo2b3o3bob2ob3obo2bob4ob
2ob6o2bobob2ob2o2b4o2b2obobob2ob2obob2obo2bo5bo2bo2b2obob3obo2b2o4bo2b
o4bobo3bo3bo3b2ob2o3b2ob6o6bobo3bo2b2o4bob2ob2o4bob3ob2ob3o2bo3b2obobo
5bobo2bobo$obobo3bobo2bob5ob2o2bobo4b2o2b3o5bo2b6o2bobo3b4o4bobobobo3b
3obo3bo2bob6o2b2o2bob3o4bo2bo2b3o2b2obob3o9b2ob3o2bo3b5o5bob2ob3ob2o2b
obobo5b3o2bo2b2o6b4obo2b2o4b3obo2bob2o3bobob3ob5o3bob6o3b2ob2o2b4o4bo$
bob3o2b2obo2bobobo5b2o2b12obobo3b5ob6o3bo2b2o4bo2b2o2b2o4b3o2bobo2bo2b
obo3b2o3b4o2bob2o3bo2bo2b2o4b5o2bob2o2b2o4bo4bo6b2ob2ob4obo2bo2bobo2bo
2b2o3b2o2b3obo4b3o6b3o2bob6obo4bo2b2o2bo2b3obob4obobobob4ob3ob2o$2ob4o
b2o2b3o4bo3b2ob4ob4o2bo5bo2bo3bo5bo4b2ob2ob2o5bob2ob2o3bobo4b2o2b2o2bo
bob3o2bo5bobo6bob2o8bobobobob2o3bo8bo2b2o3b5o3bobo3b4obo3bo3b3o4bob2ob
ob7o2bobob3o2b3ob8o4bob7o3bo2bob2obo2bo3b3obo$4o2bo2b2o2bob3ob2ob2o2bo
2bob2o2b3o2b2o2b3obobob2obob2ob3o2bob3obob2ob3o4bo2b2o2bo2b3o3bo2b2o3b
o2b6ob2o2b2ob3o4bo2b2o3bo3b4o2bo4b3o2bobo3b6ob2obo5b3o3b2ob4o2bobo3b2o
b2o4b4o3b2o2bo2b2obo2b6o2bobo3b2ob2ob2o2b2obob5o2b3obo$2o2b4o3b2ob2ob
3obo5bobobo2b3ob5ob5ob3obobobobobo4bo2b9o7b6o4b3o2b3obob4obo5b2obob4ob
o2b2ob2o4b2o3bo4b2obob3o4b2obob4obo5b3o2b3o4b2o3b3o2bob2o2bobobobo5b2o
2bob2o2b2o2b4ob2o5b4o2b2o3b2ob3o2bo4b5o$3b2obo2b9o7bo2b8o2b2ob2obobobo
bo2b2o2b3obo2b2o3bo4bobobob5obob3o5bo2b2o4b2o2b2o2b2obobobob3o3bo6bobo
b2o3b2obobo2bob3ob2o2b7ob2obob3o5b2o2b2o2bobobo3bo3b3o2bo3b3o4b2o2b5o
3b3o2bobob7obo3bo2bob5o2b9o$bo3b3o2bob2obobo3bobo3b2ob2obo2b2obo6b7o2b
4obo2bo6bo2b2o2b2obob2obo4bo4bob2o2bobobobobo5bo3b6obo2bobo2b3ob2obo2b
5ob2obo2b2o2b2o2b2obo3b2obobo2bo4bo2bo2bob3obo4bobob3ob6o3bob2o2bo2b9o
b3ob2o3b3o2b2obobob4o3bobo3bo$b2o3b3o2bo2bobobo4b2o4bobob3o2b6o3b4o7bo
2bo2b5o3b5o2b6ob3obo2bobo2bob3o4b4ob5o2bo4b3o2b2o3b3o2b5obo3b2o4b2obo
2b2ob2obo2b6o3bob2obobo2b2obobo2bob2ob4o5b5o2b2o2b4ob2o9bo2bo6bo2b3obo
bob3o3bo3bo2bo$3ob2ob2obob2o4bob2o3b2ob2obo2b2ob3obob2obob2o3bo5b3obo
2bo4b2ob2ob2ob2ob3ob2ob3obo3bo5b2o2bobo5bo2b2o2bob2obo2bo2bo2b2ob2o2b
3ob3obo4bo2bob2ob4obob3ob3obob3o3b2ob2o2bo3bo2b2obo5b3o4bo2bo4bob4o2bo
2b4o3bobo3bob3o2bo2bo4b6obo$2bobo9b2o5bo2b2obo2bo4b3obo3bo3bob3o3bobob
2o2b3o2b6o7bob2ob3ob7o3bo4b2o2b4ob3o2bobobo2b3o2b2o2b3ob2ob3o3b3ob3o7b
o4b4ob2o3bob3ob2o2bob3obobo3bobob2obo4b2o5bob2o2b2o2b2o2b3obo2b6obo2bo
3bo6b7o2b4o$ob2o4b3ob3o3b5o3b4o2b2ob2o3bo2bo3b2o2b2ob2obo2b3obob2obob
5obo6b3o4b6obo5bob3o3b3o4b2ob3o4bobob11o2bobo2b3obobob2o4b4ob5o2b2obo
5b2o2b2o3bob5obo3bo2bob3obob2o3b4o3bob2obobob2o2b2o3bo5bobo2bo2b5obo2b
2o$2bob2o2b2obo8b2ob2ob2o3bo2bobo2bo3bob2o2b3obo3b3obobo3b3o2b3obob2o
4b2ob2o4bob2o3bobo2b2o2bo3b2o2b2ob4o2b2ob2o2bob2ob2o2b2o5bo2bo2bo6bo5b
obobo2b8obob4o6bob2o2b4o2b2o3b3o2bob2ob2o3b3o3bo2b2o2bo6b3o4b2obob6o2b
2obo$3bo4bo2b3obobobobo2bobo2b4o2bobo3bo2bo2b5o2bobob2obo4bo2bob2o2bob
o3bob2o5bo2bob3ob2ob2o5b2ob7ob2o2b2obo2b5ob3o2bo2b3obob2o2b3o6bo3b2ob
3o2b2o3b3obobo2bobo2b4ob2o2bob2o3bobob3obo2bo3bobo2bob3ob2o6b2o2b2o2b
2obo5b3o$bob2ob4o7bo2b3o3bo3b2o2bobob5o3bo3bo2b2o2bobo3bo2bobob3obobo
4b3obobo4b2o3bo7bo2b3obobob4ob2o2b4ob2o2b2obob2o2bo2b3o3b2o2bob2o3b2ob
2o3b2ob2o2bob3obob2ob2ob2o2b5obobo3bo7b2o5bob2obo2b5o3b4o2b6o2bob2o3bo
bo3bob2obo$5ob3ob2ob3ob8o6bob5ob2o3b2ob2o4bo3b4obo3b3obo2b3o2b3obob2o
2b2o3bobo2b6o2bob2ob2o2bo2b3o2b2ob2o2b4o2b3o2b3o2bobob2o2bobob2o2bo2bo
2b2o2b2ob2obo6b2ob5ob5ob3ob3ob2ob3o2bo3bo4bo4b2o2bob2obobobo2bo2b2o4bo
2b7o2bob4o$obo3b4ob2ob3o2b4obo2b4obobo2b3o2b2o3b2obo3bob2ob3o4b4o3b2o
6b3o3b2ob2o2b3ob4o3b2o4b4obo2bob2o2bo2bobo7bobob3obo3bob2o2b3o2bob2o3b
o2b2obobob2o9bobobo2bo3bo2b3ob3o2bob2obo4bobob4ob2obo3bob3o2bob4o2b2o
4b2o2bobo2bo$3b2o2b2obob4ob3o2bob3ob2o2b3o2b2ob2obob5obo3b2obobo3b2o6b
ob3ob3o3bo4bo4b2ob2ob3obo2b3obo2bob4ob3o2b2o7bo4b3o5bobobobo3bo2bob4o
2b2obo3bob6ob2o4b3o4bob3obobo2bob2o2bobobobob2o2b3o2b2o2bob2o3b3o2b3ob
o2bo2b2o2b2ob2o3b2o$ob2o2bo2b3ob2ob5o2b5ob3o3bo2bob11ob7obobobob2o2b5o
2bobobo3bo2b2o2b3o2b3o3bob3ob4obo2b2o4b2o3b2o4bo2b2o2b3o3b4obo2bo3b2o
4bo2bo7b7ob4o2b4obobo2b2o4b4o3bobo9bo2bobobo3bo3bob7obo5bob3o4bo2b2obo
!
Saka wrote:
October 18th, 2020, 4:06 am
2c/15 triangular neighborhood solid ship

Code: Select all

x = 35, y = 20, rule = R10,C0,S230-392,B200-398,NL
11b13o$8b19o$7b23o$4b27o$3b29o$2b31o$b16ob16o$bob11o7b14o$b11o11b12o$
12o11b12o$12o11b12o$b11o11b12o$bob11o7b14o$b16ob16o$2b31o$3b29o$4b27o$
7b23o$8b19o$11b13o!

Re: Triangular neighbourhood rules

Posted: April 29th, 2021, 7:56 am
by muzik
muzik wrote:
February 1st, 2020, 7:15 pm
I really want to see this neighbourhood supported:

Code: Select all

 x = 7, y = 4, rule = B/S0123456789XYZL
7o$b5o$2b3o$3bo!
[[ GRID ]]
Created this manually as a custom neighbourhood. Oddly, it does not seem to support replicators:

Code: Select all

x = 5, y = 5, rule = R3,C0,S1,3,5,7,9,11,13,15,B1,3,5,7,9,11,13,15,N@0021F3D74000L
o!

Code: Select all

x = 5, y = 5, rule = R3,C0,S0,2,4,6,8,10,12,14,B1,3,5,7,9,11,13,15,N@0021F3D74000L
o!
The expected configuration for Generation 2 of each would be as follows:

Code: Select all

x = 13, y = 7, rule = B/S0123456789xyzL
obobobobobobo$bobobobobobo$2bobo3bobo$3bobobobo$4bobobo$5bobo$6bo!

Code: Select all

x = 13, y = 7, rule = B/S0123456789xyzL
obobobobobobo$bobobobobobo$2bobobobobo$3bobobobo$4bobobo$5bobo$6bo!
Alternating also does not seem to work:

Code: Select all

x = 5, y = 5, rule = R3,C0,S0,2,4,6,8,10,12,14,B1,3,5,7,9,11,13,15,N@0021F3D74000L|R3,C0,S1,3,5,7,9,11,13,15,B1,3,5,7,9,11,13,15,N@0021F3D74000L
o!

Code: Select all

x = 5, y = 5, rule = R3,C0,S1,3,5,7,9,11,13,15,B1,3,5,7,9,11,13,15,N@0021F3D74000L|R3,C0,S0,2,4,6,8,10,12,14,B1,3,5,7,9,11,13,15,N@0021F3D74000L
o!
Perhaps triangular neighbourhoods do not as readily support Fredkin's parity rules like other neighbourhoods do, perhaps also explaining the 3-state case from earlier... there are still clear cases of mass cell death at powers of 2 though (and of 3 for the 3-state case) implying that there's definitely something relating to it going on.

Re: Triangular neighbourhood rules

Posted: December 16th, 2021, 2:10 pm
by muzik
bprentice wrote:
January 29th, 2020, 11:19 pm
muzik wrote:
January 29th, 2020, 7:18 pm
A question: can mod-3 2D Moore replicators exist on the triangular grid?
What does "mod-3 2D Moore replicators" mean?
The following is a mod-2 2D Moore replicator:

Code: Select all

x = 1, y = 1, rule = B13579Y/S13579YL
o!
As can be seen, it:
* replicates infinitely, which, for this rule, is true of any pattern
* operates on modulo 2:
** the rule is 2-state
** the population returns to its minimum value every 2^n generations
** whether a cell is born or survives depends on the sum of its neighbors mod 2: if 0, no, if 1, yes
* is on a 2D grid
* operate on each cell's triangular Moore neighbourhood

For comparison, here are mod-2 and mod-3 cases on the square grid, with the same principles applied:

Code: Select all

x = 1, y = 1, rule = B1357/S1357
o!

Code: Select all

x = 1, y = 1, rule = BA020508101316212432354043516270_BB01040712152023263134425053_TA01040712152023263134425053_TB020508101316212432354043516270_SA020508101316212432354043516270_SB01040712152023263134425053
A!
The modulo 3 case can clearly be seen to differ by switching out the modulo 2 attributes with that of modulo 3:
* the rule is 3 state
* the population returns to its minimum every 3^n generations
* a cell's future state depends on the modulo of its neighbours, with dead being +0, state 1 +1 and state 2 +2:
** if the sum is 0 mod 3, it dies or is not born
** if the sum is 1 mod 3, it becomes state 1
** if the sum is 2 mod 3, it becomes state 2

My line of thinking was to apply this modulo-3 transformation to the known modulo-2 triangular replicator shown at the top of this post. Golly has a FredkinModN script that allows for the generation of these parity/replication rules on certain neighbourhoods, and using the script to try and generate a triangular-Moore rule with this modulo-3 behaviour produced something more akin to an uninteresting explosion, rather than pattern replication every 3^n generations.

Code: Select all

x = 1, y = 1, rule = Fredkin_mod3_triangularMoore_emulated
o!
However, significant cell death can still be noted at generations 3^n and significant multiples thereof, so there's definitely an essence of the modulo there, just no replication, hence why I'm skeptical. Triangular grids and neighbourhoods as a whole tend to not lend themselves to replication as much as hexagonal and square grids - the post above this one with a perfectly triangular neighbourhood also seems to explode mostly boringly rather than exhibit replication, even though significant cell death can also be seen periodically in it. This may be due to the surplus diagonal neighbours a triangular cell has which is absent from square and hexagonal neighbourhoods, which may also explain why hyperbolic grids also seem to not harbour replication rules either. I can't be sure of that though - if anyone has more insight into it, an explanation would be appreciated.

----

Since the script I mentioned for generating the rule is rather old, it handles triangular rules using a system in which each square cell is divided into 2 triangles, which differs from the currently used way which has a 1:1 square-triangle correspondence. As such, the geometry shown here differs a lot from how it would actually look with proper triangular-grid symmetry. For comparison's sake, here's the modulo-2 rule generated with that same script - note how it's ultimately simulating the same thing as the first pattern provided in this post:

Code: Select all

x = 1, y = 1, rule = Fredkin_mod2_triangularMoore_emulated
o!

Re: Triangular neighbourhood rules

Posted: December 16th, 2021, 11:06 pm
by muzik
Just to test things out, I tried making range-2 replicators for triangular Moore and they appear to work, proving that it isn't just a range-1 fluke:

Code: Select all

x = 1, y = 1, rule = R2,C0,S1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,B1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,NL
o!
[[ ZOOM 4 ]]

Code: Select all

x = 1, y = 1, rule = R2,C0,S0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,B1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,NL
o!
[[ ZOOM 4 ]]
Seems just fine for Range 3 as well:

Code: Select all

x = 1, y = 1, rule = R3,C0,S1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,B1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,NL
o!
[[ ZOOM 4 ]]

Code: Select all

x = 1, y = 1, rule = R3,C0,S0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,B1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,NL
o!
[[ ZOOM 4 ]]
So what is it that makes 3-state non-replicatory? Is the script bugged, or is there some intricacy someone far more qualified on the subject than I am can highlight that allows this to work for 2 states but not 3 even though switching to 3 on other grids doesn't impart such a restriction?

My current guess is since a triangular grid's cells invert every even number, and 2 (the modulo of this replicator) is also an even number, replication works out, but all higher primes, being odd, would have to place cells of reversed parity every power of n generations, which somehow forbids clean replication. But I cannot be certain on that.

EDIT: Making a comparatively more symmetric hexagon in the modulo 3 rule doesn't change much:

Code: Select all

x = 2, y = 2, rule = Fredkin_mod3_triangularMoore_emulated
DA$CD!

Re: Triangular neighbourhood rules

Posted: January 12th, 2022, 2:57 am
by muzik
Visually resembles a failed quadratic replicator:

Code: Select all

x = 9, y = 5, rule = B19/S2LV
2o!
[[ ZOOM 16 ]]
This rule has a few natural oscillators:

Code: Select all

x = 85, y = 39, rule = B139/S2LV
$3b2ob2ob2ob2obob5o8b2ob3obo3bob5o2bobo5b2o4bo2b2o2b7obobo2bo$b2o2bob
3o5b4ob4ob4ob3o5b3o3b2ob2ob2ob2ob3obob3obo4b2o2b2o2bob2obo$2b3obob7ob
4ob2o3b2o3bobo2b3o3b3o3b2ob3obob2ob4obo3bo3b2obobo3bobo$3b7obo2b2ob5o
3b3ob3obo3bo4b2ob4ob3obob3ob5ob3ob2o2b5o3bobo$b3obo5b5obo2b8o6bob6obo
bo3b3o2bobob2o2bobob2o4b3ob3ob2ob2o$bob2o2b2obob3o2b4o3bob3ob2o3b4ob4o
2bo6b2o3bob2o3b2o2b2o5b2obob2o$b2ob2obo2b2obo8bob2o3b2ob8obo3b3ob2obo
bo3b5ob2obobo3b2ob2o3bobo$b6ob2obo2bo3b4o2bobob2obo2bob2ob3ob2ob4ob3o
2b2ob2obobo5b3o3b3o$bo4bo2b2o3b2o3bob2obob2o2bobob2o3b2o2b6o7bo3bobob
2ob2ob2o3bob3o2bobo$b2ob2o3b2o2b3obo2bobo2b4o2bobo2b2obobobo2bo2b2o2b
o3b2ob3o3b3obo3b4o3b2o$bobobo2bobo3bo4bo2bo6b2o3bob2o4b6o4bob5o2bo6bo
b4o5bo3b2o$bob7obob2ob2obob3obo2bobob2ob2obob2obobo2b2o5b2obo3b3obobo
2bo2bob2o2b3obo$bo6b6ob2ob2o2b3o3bobo5bob5obobo2b3obobo2bobob2o2bo2bo
b2obob2o4b2o$3b8o4bo3bobob2o3bob3ob9o3b2o2b3ob2ob6ob4o2bo4bobobo2bo$b
o2bob2o2b7o3bo2b6obo2bo2b2o4bob2o2bo2bo3b2ob2o2b2o3bo2b2obob6obo$3b3o
bobob2o2bo3bob4o6b3o4b2ob2obo3bo3bo3bo2bo2b2o3bobob3obob2o2bob2o$bob3o
bo3bo2b2o2b6o6b4ob3obobo2b2ob5o2b7o2bob3o2b3o2bobob6o$4b3o2b3o3bo2b3o
2bo2bo2bo3bobobo3b2ob2o3b5o3bobo2bo2bo2b2ob2o2b5o2bobo$b2o2b8ob2o4bob
3ob3obo5b2o4b2obo2bob2obob2obo2b2o3bob2obo2bob4o2bo$3bobo2b3ob2o2bobo
bo4bo2b3o2b5ob2obo2b3o2b2o2bobo3b2o4bob3o3bo2bobo3bo$b2ob2obo2bobobo3b
o3bobo3bob2obo2bo2b2o3b3o4bob2o5bo8bob2obobob3o2bo$bo3b5obo3b2o4bob3o
b2o3bo2bo3b3ob2o2b2o5bo7bobobob4ob3o3b6o$2bo2bob3o3b2o2bo3bob3ob3o2b3o
b2obo2b2o3b4ob3obo2bob2ob2obo2bo2bo2b5o2b2o$2bo2b2obo2b6o4b5obo7bobob
o4b2o6b2o2bo2bobobo2b3o3bob4ob4obo$3b2o2bo2b3ob2o2b2obo2bobo2b2o3b2o2b
2ob3ob3o3b3o3b2o3bobo3b5o4b2ob3o$b2o6b2obobob7o4b2ob2ob2ob2ob2obo5b3o
b3ob3obo2b4o2b2obo3bo2bo2b3o$b2ob2o2b3o2b2o4b2o3b4obo2bo3b2o3bo3b3ob4o
3bo3b2o3bob6o6b2ob3o$b2o2b5o2bobob3o3b2obobo2b2o2bob3ob2obob4obob3o2b
obob2o4b3obo2bob2ob3ob2o$b4o4b3o4bo3bob2o2bo3bob3o3b5ob2o4bo6bobobobo
7bob2obobo2bo$bo2b2o2b2obob4o2b7ob3obo5b3ob5obo3bobo3b2o5b3o2bobob2ob
4o$2bobo4bobo2b5ob2o2bob2ob2ob2obob3o3bobobob2o2bo2b5o3bob2ob4ob4o3bo
b2o$b6o3bob3obo2bo2b7obo2bobobo4bobo2b2o2b2o2b2ob4o4b2o3bo2bob2ob2o2b
o$5b2ob3o2b3ob3o2b4ob3o2b2o6b2o2bo2bob2obo3b3o2b4o3bobobo2bo2b3o3bo$b
3o2b2obo2bob2ob6o2b2o3b2o4bob2obo2b4o2b3o4b4o2bo2bobobob2obobob2o2bo$
2bo2b2o5bobob2obo4bobob3obob2o2b3o4bob2obo2b2obobo2bo4bobo2bo2b5obo2b
o$3b5ob2obobo2b3obobo3bo5bobo3b4obo2bobo2b2ob2ob3o3b2obobo2b4o4b2o2bo
$4b2obobob3obob2o2b2o3bob2o2bob2o2b2o10bobob2obo2bobobo3bo2bobobob2o4b
2o$2bobobob3o2b2o2bo2b2obo2b2ob3obobobo2bobo4b2ob7obobo3b2o3b2o3bo5b3o
bo!