Life Imitates Sierpinski

[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:

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OOOOOOOOOOOOOOOOOOO and so on

Game of Life and HighLife:

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OOOOO OOOOO OOOOO and so on

In Seeds (extend the diagonal):

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O.....
.O....
..O...
...O..
....O.
.....O


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

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O

In HighLife (thanks, Nathaniel), down below.

More examples coming soon!
Updated as of 07:38 UTC.

[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.

[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.

[Nathaniel]

Here's a HighLife replicator-based Sierpinski triangle breeder:

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#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!

[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...]

[Axaj]

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

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


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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!