Sierpiński triangle in Life

2008-06-18
2013-06-05
  • Just curious if anyone else has seen this.  I can generate a fractal triangle by making a really long line, and I hadn't heard it mentioned elsewhere.  It needs to be a minimum of ~4000 long, longer the better.  10,000 is clear, 200,000 long is just cool to watch.  I hadn't seen any comment on it on the web, and was curious if anyone else knew of any fractal patterns like that.

     
    • Dave Greene
      Dave Greene
      2008-06-19

      I'm going to be lazy here and paste in a quote from some email correspondence from last year.  This was a response I wrote to an email that came in to my LifeNews email address, I think... looks like the link still works, and it's a good one:

      ---beginquote---
      On 25 June 2007, Bart Wisialowski wrote:
      > I'm an amateur Game of Life hobbyist and I've discovered an interesting
      > fractal-production phenomenon in GoL.  I have no idea whether this
      > phenomenon has already been examined by more knowledgeable enthusiasts
      > and automaton researchers.
      >
      > I've posted my "discovery," analyzing it to the best of my ability, at
      > https://webfiles.uci.edu/bwisialo/www/gameoflife2.html
      >
      > I'm interested to know whether this phenomenon has already been
      > discovered and examined by others.

      Well, let's see... very similar patterns have certainly been seen in
      many other cellular automata, both one- and two-dimensional.  Here's
      an unusual sample that I particularly like, from a rule neighboring
      Conway's Life:

      #C Sierpinski-triangle breeder pattern in HighLife
      x = 41, y = 37, rule = B36/S23
      2b3o$bo2bo$o3bo$o2bo$3o6$33bo$32b3o$31b3obo$30b2o2b3o$29b2o3b2o$28b3o
      2b2o$29bob3o6bo$30b3o7bo$31bo8bo5$36b3o$35bo3bo$35bo3bo$35bo3bo$36b3o
      5$34b3o$33bo2bo$32bo3bo$32bo2bo$32b3o!

      But in Conway's Life, I think you may be the first to nail down the
      best lengths of line to approximate the mirrored-Sierpinski-triangle
      fractal behavior.  It looks like you choose 16370 instead of the
      nearby power of two, 16384, because the two triangles happen to meet
      a little more cleanly with a length-16370 line?  I saw some
      irregularities in the escaping patterns of gliders at other lengths.
      Besides that, it wasn't quite clear why simple powers of two wouldn't
      be the lengths of choice...?

      Bart Wisialowski wrote:
      > The patterns in the images I used developed from horizontal lines, not
      > vertical lines... I should have clarified that on the webpage.  The only
      > difference is a 90 degree rotation of the final pattern.
      >
      > Yes, with lengths that are powers of 2 minus 14, the squares forming in the
      > center meet cleanly -- they don't bump into or interact with one another in
      > any way.  2^x - 14 yields the numbers closest to 2^x where the squares don't
      > interact.
      >
      > With lengths that are powers of 2, the two largest squares forming in the
      > middle bump into one another, as do the squares forming within those
      > squares, creating altered patterns that reduce internal self-similarity to
      > some extent.  I'd have to use images to pinpoint what I'm referring to...
      > too difficult to describe.  In the bigger picture, if one used lengths that
      > were powers of 2, the overall analysis wouldn't be very different.  One
      > would still be able to formulate functions for the mean final population,
      > etc., and the fractal dimension (D) would end up being the same.
      ---endquote---

      Let me know if any of this fails to make sense -- I think it pretty well covers what little I know of this topic.  Keep the cheer,

      Dave Greene

       
    • I kinda figured I wasn't the first to discover it. :)  I did see some pretty interesting final shapes as I extended the line:

      41 - First pulsar
      56 - First time gliders escape
      96 - First toads
      102 - H oscillator
      135 - Double pulsars, was up to 4 but 2 get eaten

      Giant plus sign (count total cells):
      157 - total still life
      535 - 4 pulsars

      I also did some searches through X's, both even and odd (one cell in center or two cells in center) and V, although my favorite is a filled triangle with a total cellcount of 1849.  I call it a Pulsar Butterfly, since it ends with two pulsars in the middle of a butterfly.