[math-fun] Life on the Edge

20 Feb 2007

      I have discovered an interesting cellular automaton.  This automaton is 
defined on the edges of a square grid: each edge has six neighbors, 
three at each end.

Like Conway's Life, each edge can be in one of two states: on or off; 
and its state in the next generation depends only on the current state 
of the edge itself and how many of its neighbors are on.  The rule for 
this automaton is that an edge that was off becomes on when exactly two 
of its neighbors are on; and an edge that was on remains on only if 
exactly one of its neighbors is on.  Or, more simply put, an edge is on 
in the next generation iff exactly two of the edges in its seven edge 
neighborhood - including the edge itself - are on.

Small patterns with this automaton are a bit more fragile than in Life. 
  For example, a row of two edges (you'll need a fixed-width font to 
properly view these pictures):

-.-

goes in the next step to

_|_
.|.

which then dies out.

The alternator is a simple period 2 pattern, alternating between

|.|

and

._
._

It is difficult to construct stable patterns, as they must be composed 
of pairs of lines, with their growth blocked off.  I believe the 
smallest is

...|.....
_|.|.|_..
.._....._
...|.|.|.
.....|...

However, the really interesting part is what happens when we start with 
3 sides of a square:

..._
..|_

This goes in the next generation to:

._|_.
._._|
..|..

which looks like an airplane (which is what I call this pattern).  The 
next generation it seems to fall apart:

....|_
|...._
....|.

but the next generation is

..._|_.
..._._|
....|..

and it flies!

Can anyone find other interesting patterns?  It would be especially 
interesting if someone can find an "airplane gun", which generates 
airplanes at regular intervals.  Any other patterns that move, and any 
other periodic patterns - especially with a period greater than two - 
would also be of interest.

Franklin T. Adams-Watters

P.S. Does anyone know of a forum where cellular automata are being 
actively discussed?  I see that Yahoo has a group, but it has been 
inactive for several years.

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franktaw＠netscape.net

Schroeppel, Richard

Edwin Clark

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