Multicolor Life variants

For discussion of other cellular automata.
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fluffykitty
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Multicolor Life variants

Post by fluffykitty » December 23rd, 2020, 6:37 pm

People have made quite a few multicolor life variants, including Immigration and QuadLife. With the restrictions that live cells cannot change color and colors must be treated completely symmetrically, those and ordinary B3/S23 are the only possible variants. However, other variants should be possible if smaller symmetry groups are allowed. Which permutation groups can be realized with multicolor Life variants, and what interesting patterns exist in these?

MathAndCode
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Re: Multicolor Life variants

Post by MathAndCode » December 23rd, 2020, 7:03 pm

fluffykitty wrote:
December 23rd, 2020, 6:37 pm
With the restrictions that live cells cannot change color and colors must be treated completely symmetrically, those and ordinary B3/S23 are the only possible variants.
I can think of at least one other such variant: There are two colors, and each newly-born cell takes the color that an odd number of live neighbors have. One interesting question concerning this rule is whether patterns in this rule would tend to eventually all become one color,
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bubblegum
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Re: Multicolor Life variants

Post by bubblegum » December 23rd, 2020, 9:58 pm

MathAndCode wrote:
December 23rd, 2020, 7:03 pm
I can think of at least one other such variant: There are two colors, and each newly-born cell takes the color that an odd number of live neighbors have. One interesting question concerning this rule is whether patterns in this rule would tend to eventually all become one color,
I think (don't quote me on this) that the last possible variants for two-colour Life are "Even" and "Minority".
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MathAndCode
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Re: Multicolor Life variants

Post by MathAndCode » December 24th, 2020, 3:44 pm

Here are the four distinct gliders in Minority besides color inversions.

Code: Select all

x = 13, y = 13, rule = Minority
.A9.A$2A8.2A$A.A7.A.B8$.A9.A$AB8.AB$A.A7.A.B!
The two gliders in the top row each have period four, and the two gliders in the bottom row each have period 28. All four gliders are glide-reflective, and none are their own color inversions.
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Hunting
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Re: Multicolor Life variants

Post by Hunting » December 25th, 2020, 5:04 am

FYI, I'm the creator of Minority, because I had this idea a few days ago. It was originally created as a more aggressive variant of Immigration (to prefer attacks), but the ash turned out to be stripy.

Two LWSSes I know of so far:

Code: Select all

x = 5, y = 11, rule = Minority
A2BA$A3.B$B$.A2.B4$2ABA$A3.A$A$.A2.A!
They have impressive periods.

I also created Infect to get rid of the "living cells must not change color" restriction.

MathAndCode
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Re: Multicolor Life variants

Post by MathAndCode » December 25th, 2020, 8:59 pm

Hunting wrote:
December 25th, 2020, 5:04 am
Two LWSSes I know of so far:

Code: Select all

x = 5, y = 11, rule = Minority
A2BA$A3.B$B$.A2.B4$2ABA$A3.A$A$.A2.A!
In Minority, any two colorations of the same repeating pattern in ConwayLife can have their colorations XORed in all phases.

Code: Select all

x = 12, y = 10, rule = Minority
A2BA$A3.B$B$.A2.B2.AB2A$7.A3.B$7.B$2ABA4.A2.B$A3.A$A$.A2.A!
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Moosey
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Re: Multicolor Life variants

Post by Moosey » December 28th, 2020, 10:48 pm

You can also have variants in which, say, cells of a different color are born from three of the same color
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bubblegum
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Re: Multicolor Life variants

Post by bubblegum » December 28th, 2020, 10:49 pm

Moosey wrote:
December 28th, 2020, 10:48 pm
You can also have variants in which, say, cells of a different color are born from three of the same color
Those being what I originally meant when I said Even and Minority, though I can't complain about the interpretation.
Each day is a hidden opportunity, a frozen waterfall that's waiting to be realised, and one that I'll probably be ignoring
sonata wrote:
July 2nd, 2020, 8:33 pm
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Hunting
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Re: Multicolor Life variants

Post by Hunting » December 29th, 2020, 8:18 am

MathAndCode wrote:
December 25th, 2020, 8:59 pm
Hunting wrote:
December 25th, 2020, 5:04 am
Two LWSSes I know of so far:

Code: Select all

x = 5, y = 11, rule = Minority
A2BA$A3.B$B$.A2.B4$2ABA$A3.A$A$.A2.A!
In Minority, any two colorations of the same repeating pattern in ConwayLife can have their colorations XORed in all phases.

Code: Select all

x = 12, y = 10, rule = Minority
A2BA$A3.B$B$.A2.B2.AB2A$7.A3.B$7.B$2ABA4.A2.B$A3.A$A$.A2.A!

Code: Select all

x = 14, y = 14, rule = Minority
2A$A$.A.A2$3.A.A2$5.A.B2$7.A.A2$9.A.A2$11.A.A$12.2A!

MathAndCode
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Re: Multicolor Life variants

Post by MathAndCode » December 29th, 2020, 2:23 pm

Hunting wrote:
December 29th, 2020, 8:18 am

Code: Select all

x = 14, y = 14, rule = Minority
2A$A$.A.A2$3.A.A2$5.A.B2$7.A.A2$9.A.A2$11.A.A$12.2A!

Code: Select all

x = 25, y = 29, rule = Minority
2A9.2A$A10.A$.A.A8.A.B2$3.A.A8.A.A2$5.A.B8.A.B2$7.A.A8.A.A2$9.A.A8.A.
B2$11.A.A8.A.A$12.2A9.2A2$5.2A$5.A$6.A.B2$8.A.A2$10.A.A2$12.A.A2$14.A
.B2$16.A.A$17.2A!
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creeperman7002
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Re: Multicolor Life variants

Post by creeperman7002 » December 29th, 2020, 2:27 pm

MathAndCode wrote:
December 29th, 2020, 2:23 pm
Hunting wrote:
December 29th, 2020, 8:18 am

Code: Select all

x = 14, y = 14, rule = Minority
2A$A$.A.A2$3.A.A2$5.A.B2$7.A.A2$9.A.A2$11.A.A$12.2A!

Code: Select all

x = 25, y = 29, rule = Minority
2A9.2A$A10.A$.A.A8.A.B2$3.A.A8.A.A2$5.A.B8.A.B2$7.A.A8.A.A2$9.A.A8.A.
B2$11.A.A8.A.A$12.2A9.2A2$5.2A$5.A$6.A.B2$8.A.A2$10.A.A2$12.A.A2$14.A
.B2$16.A.A$17.2A!
It appears that is an emulation of Rule 150.
B2n3-jn/S1c23-y is an interesting rule. It has a replicator, a fake glider, an OMOS and SMOS, a wide variety of oscillators, and some signals. Also this rule is omniperiodic.
viewtopic.php?f=11&t=4856

smjg
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Re: Multicolor Life variants

Post by smjg » July 25th, 2021, 7:40 pm

It just so happens I've just been studying these. Specifically, I've been looking at the possibilities for colourified CAs subject to these constraints:
  • The underlying CA (colours flattened) is Life-like in the strictest sense - 2D square grid, two states, outer-totalistic, Moore neighbourhood.
  • Surviving cells don't change colour.
  • Colour-totalistic - colour of a born cell depends only on the bag of neighbour colours, not how they are arranged.
  • Colour-symmetric - if the colours are permuted the CA behaves the same.
With Conway's Life as the underlying CA, I have concluded that there are six possibilities:
  • Two live state colours: majority (Immigration), odd, even, minority
  • Four live state colours: remaining colour for 3 different colours in neighbourhood (necessary for colour-symmetry); majority (QuadLife) or odd for 1 or 2 different colours in neighbourhood.
For other base rules, the possibilities are different. Specifically, they depend on the B part of the rule string. Under B2, the only possibility is a three-colour analogue of QuadLife. Any base rule containing B6, such as HighLife, can't be colourified within these constraints at all, since:
  • There would need to be a unique third state for 3+3 to give birth to.
  • Thus there are at least three states, thus must be a unique fourth state for 2+2+2 to give birth to.
  • However, this fourth state means the aforementioned third state can't be unique.
I haven't yet studied possibilities for multiple dead cell colours. I can see that if dead cell colours are separate from live cell colours then this would work only in universes containing S8. But having the same set of colours for dead and live cells would open up possibilities.

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Entity Valkyrie 2
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Re: Multicolor Life variants

Post by Entity Valkyrie 2 » July 29th, 2021, 9:42 pm

By the way, my rule EV2QuadColor is a marked version of QuadLife.
Bx222 IS MY WORST ENEMY.

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My recent rules:
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smjg
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Re: Multicolor Life variants

Post by smjg » July 30th, 2021, 5:09 am

Entity Valkyrie 2 wrote:
July 29th, 2021, 9:42 pm
By the way, my rule EV2QuadColor is a marked version of QuadLife.
Is this something you've published somewhere? I can't seem to find it.

BTW, I recently made a QuadLife rule file for Golly (as yet unpublished), intended for study purposes, with Unknown and Influence states. Unknown means a live cell whose colour is unknown. Influence is the same with the added rule that, when a cell is born, if Unknown and Influence are both in the neighbourhood that gave birth to it then the born cell is Influence; as such, it's good for seeing what future cells might possibly be influenced by the colour of a specific cell.

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