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### Life Imitates Sierpinski

Posted: **June 17th, 2009, 2:38 pm**

by **Extrementhusiast**

I have found some examples of the Sierpinski gasket showing up in many forms of Life. In Conway's Game of Life and HighLife:

Game of Life and HighLife:

In Seeds (extend the diagonal):

Code: Select all

```
O.....
.O....
..O...
...O..
....O.
.....O
```

In Wolfram 22, Wolfram 60, Wolfram 90, Wolfram 102, B1/S12:

In HighLife (thanks, Nathaniel), down below.

More examples coming soon!

Updated as of 07:38 UTC.

### Re: Life Imitates Sierpinski

Posted: **June 22nd, 2009, 6:00 am**

by **Macbi**

I just realized that if you take a horizontal cross-section through some vertical lines in life, the row you're watching just obeys rule 22, hence the appearance of Sierpinski at the end of the lines.

### Re: Life Imitates Sierpinski

Posted: **June 22nd, 2009, 6:51 am**

by **Nathaniel**

Also, the

2x2 rule can imitate Rule 90 (and thus the Sierpinski triangle), though in a much more hidden way (it doesn't show up explicitly as a visual, but governs the period of certain oscillators). See

here and

here.

### Re: Life Imitates Sierpinski

Posted: **July 12th, 2009, 2:37 pm**

by **Nathaniel**

Here's a

HighLife replicator-based Sierpinski triangle breeder:

Code: Select all

```
#C replicator breeder
#C David I. Bell, 1994
x = 41, y = 37, rule = B36/S23
bb3o$bobbo$o3bo$obbo$3o6$33bo$32b3o$31b3obo$30boobb3o$29boo3boo$28b3o
bboo$29bob3o6bo$30b3o7bo$31bo8bo5$36b3o$35bo3bo$35bo3bo$35bo3bo$36b3o
5$34b3o$33bobbo$32bo3bo$32bobbo$32b3o!
```

### Re: Life Imitates Sierpinski

Posted: **July 12th, 2009, 10:42 pm**

by **dvgrn**

A couple of years back I received an email from Bart Wisialowski, who had put together an

interesting summary page of results for Game of Life Sierpinski patterns, including calculations of their fractal dimension.

[Meant to post this earlier, but the link didn't seem to be working the last time I tried it...]

### Re: Life Imitates Sierpinski

Posted: **October 16th, 2009, 2:29 pm**

by **Axaj**

A gigantic filled square does the job quite nicely. Also, I found these two in seeds quite interesting:

Code: Select all

```
x = 121, y = 2, rule = B2/S
o7bo7bo7bo7bo7bo7bo7bo7bo7bo7bo7bo7bo7bo7bo7bo$o7bo7bo7bo7bo7bo7bo7bo
7bo7bo7bo7bo7bo7bo7bo7bo!
```

Code: Select all

```
x = 128, y = 128, rule = B2/S
o$bo$2bo$3bo$4bo$5bo$6bo$7bo$8bo$9bo$10bo$11bo$12bo$13bo$14bo$15bo$16b
o$17bo$18bo$19bo$20bo$21bo$22bo$23bo$24bo$25bo$26bo$27bo$28bo$29bo$30b
o$31bo$32bo$33bo$34bo$35bo$36bo$37bo$38bo$39bo$40bo$41bo$42bo$43bo$44b
o$45bo$46bo$47bo$48bo$49bo$50bo$51bo$52bo$53bo$54bo$55bo$56bo$57bo$58b
o$59bo$60bo$61bo$62bo$63bo$64bo$65bo$66bo$67bo$68bo$69bo$70bo$71bo$72b
o$73bo$74bo$75bo$76bo$77bo$78bo$79bo$80bo$81bo$82bo$83bo$84bo$85bo$86b
o$87bo$88bo$89bo$90bo$91bo$92bo$93bo$94bo$95bo$96bo$97bo$98bo$99bo$
100bo$101bo$102bo$103bo$104bo$105bo$106bo$107bo$108bo$109bo$110bo$111b
o$112bo$113bo$114bo$115bo$116bo$117bo$118bo$119bo$120bo$121bo$122bo$
123bo$124bo$125bo$126bo$127bo!
```

### Re: Life Imitates Sierpinski

Posted: **October 13th, 2017, 2:28 am**

by **Koiti Kimura**

Does "chaos" that produces multi-D Sierpinski imitate multi-D Life in reverse/to the contrary? I'd like to know if all the multi-D Life-like-CAs are equivalent to Life in terms of their UTM-ness.

### Re: Life Imitates Sierpinski

Posted: **October 13th, 2017, 6:48 am**

by **Macbi**

I suppose you could divide 3D space up into vertical lines each crossing the horizontal plane at one point. Then if each of those lines is either all off or all on, the rule will imitate a 2D cellular automaton. In particular, if an alive line has two or three alive neighbours, then each of its cells has 8 or 11 alive neighbours. And if a dead line touches three alive lines, each of its cells has 9 alive neighbours. So B9/S8(11) would simulate life. If you just used very long lines rather than infinite ones then the ends would froth and collapse in, leaving an approximate record of the history of the life universe.

Something something d-brane something holographic principle.

EDIT: I think you need at least B7 or lower in order to have spaceships (Imagine if you were stuck on one side of the plane x+y+z=0 and wanted to get (0,0,0) to turn on). The rule B5 is sort of the equivalent of B3 in two dimensions: B4 and lower explode, so B5 is the first interesting birth rule. I've been playing around with

3DLife.py and it seems like 3D rules are very sensitive. Anything with enough Survival rules to not die out tends to explode.

### Re: Life Imitates Sierpinski

Posted: **October 13th, 2017, 8:20 pm**

by **Koiti Kimura**

I got it!: Life IS the "fractals"-producing "chaos". Since even LG, one of the "orderly" UTMs, is equivalent to (is the necessary and sufficient condition of) "chaotic-wake" results, all the CAs are too: they are all equivalent discrete/approximated-"fractal"-arithmetic-geometry-containing UTMs; they all share only-seemingly-"chaotic" UTM-ness! It's actually no wonder at all, is it?: all CAs have neat orderly rules, so they must give neat orderly results including neat orderly computation results; both of the rules and the results only seemed "chaotic" in the past. I don't mean by "chaos" the "statistically-self-similar"(which are in fact just "self-analogous")-wakes-leaving motions, but the motions leaving wakes of self-similarity in the rigid Euclidean sense. The cognition of the former "chaotic" motions needs an inductive experimental/statistical probability theory, and that of the latter a deductive theoretical/"logical" probability theory = the general catastrophe theory.

### Re: Life Imitates Sierpinski

Posted: **October 18th, 2017, 8:44 pm**

by **Koiti Kimura**

I take back my previous assertion. It is true that if Life = chaos, then all CAs = UTMs, but that is not the case. The reasons are twofold:

1. You lack any prospects for the proof of the antecedent: that Life is the necessary and sufficient condition of fractals-generations.

2. You fail to experiment and verify the consequent on the potentially infinite number of all CAs.

### Re: Life Imitates Sierpinski

Posted: **October 22nd, 2017, 1:38 am**

by **Koiti Kimura**

Here are two disproofs of my previous assertion that all CAs are equivalent to chaotic/fractals-generating CAs and to Life UTMs:

1. Even neat orderly rules can give some CAs a neat orderly trait of the lack of logical gates.

2. Chaotic CAs, non-chaotic CAs, Lifes, non-Life CAs make a cross classification: they fail to form the same set.