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2012 August 25

Life Variants
Penrose tilings and Ready

My apologies for the lack of recent postings. There are rumours that the LifeNews server may be turning off in the immediate future, so I was tentative about uploading something new. Nevertheless, Andrew Trevorrow convinced me that this deserves to be published on, and I understand this to currently be the most direct way of doing so.

The developers of Golly have recently turned their attentions to creating a new piece of software capable of supporting reaction-diffusion systems and cellular automata on arbitrary meshes. This has been discussed on Complex Projective 4-Space and The Aperiodical, amongst other places.

There has also been some work on cellular automata on Penrose tilings. Nick Owens and Susan Stepney investigated B3/S23 a while ago, writing a chapter about the topic in Adamatzky's Game of Life Cellular Automata. This summer, a couple of related, independent and almost simultaneous discoveries were made. One of these was a weakly universal cellular automaton on a Penrose tiling; the other was a glider.

The glider attracted a lot more attention, not least because a $100 prize was offered for its discovery. Again, this was mentioned on Complex Projective 4-Space, New Scientist and The Aperiodical. Even more recently, it was investigated on the kite-and-dart Penrose tiling, where it follows fractal paths and symmetrical loops. An image is displayed below:

The lengths of loops (half the period of the resulting oscillators, since the glider is c/2) are tabulated in the sequence A215878 of the Online Encyclopedia of Integer Sequences.

2006 March 07

Life Variants
Colored Life

forward-shooting switchengine interacts with a dirty Schickoidal
backrake to produce complex long-term behavior
Bill Gosper, 26 February 2006
Bill Gosper has been doing some fascinating original research on patterns with 'simple' initial states but complex long-term behavior, using the new cross-platform hashlife-capable Game of Life simulator, Golly.

slight variation on previous pattern
emits an unprecedented glider burst at 5e10.
Bill Gosper, 26 February 2006

At 2.3e12 (Golly screenshot here) this pattern seems
to have regularized: the last twelve bursts from the left formed
a geometric progression of trapezoids. But the top two corners
of the next "trapezoid" are missing, as were several previous.
To see that the trapezoid progression is exactly doubling,
click with the up/down scaling cursor on the limit point.
Bill Gosper, 27 Feb 2006
The following are Gosper's detailed notes on the "Gotts-puzzle" pattern shown at right:

...[O]bserve that the bursts forming the bottom-left corners are acute Vees (inset) flying backwards! In fact, the inset shows two backwards mirror image Vees flying apart. This is seriously weird by the following argument.

Experiments on this scale nearly always take on a characteristic expanding star shape, with typically eight or fewer "limit" points, each receding with the velocity vector of a hypothetical glider or spaceship launched from the Origin by a Big Bang. (Any finite initial configuration (Origin) is infinitesimal compared with with these billions-wide star diagrams.) The diagrams are often crisscrossed by numerous linear features at many different angles, e.g., the patterns on Nick Gotts' weblog or the pattern shown below.

Unlimited novelty:  nick-gotts-g at 2,000,000 ticks
Golly screen shot of 'nick-gotts-g' around generation 2,000,000

'nick-gotts-g': 2006-02-27-hashlife-puzzle2.rle
nick-gotts-n type rake interaction with apparently unlimited novelty
The interaction of the downward puffer's trail with the first
bounce of its backrake produces two very sparse LWSS waves.
Bill Gosper, 27 Feb 2006
The above screenshot shows the complex long-term behavior of the 'nick-gotts-g' pattern shown at right, which consists of two interacting rakes. Here is a larger screenshot of the same pattern at 4.9e8 ticks.

The components of each line have equal velocity vectors, easily determined by which of the star-points radiates the extended line. The Origin is a ninth point which radiates all lines of immobile objects. (The production of particles with a nonstandard velocity would create an additional star-point.) Lines joining two star-points are guns or puffers. Thus the little tick designated by the red arrow in gottspuzz.png can only be a wave of a million or so spaceships (MWSS in fact) pointing toward the N spaceship limit, which unfortunately is graphically inapparent because no spaceships are actually escaping. (They can't be heading S because there was a single pixel NW-SE jog in that tick that my screenshot was too slow to catch.)

Thus when several rays emanate from the vertex of an internal wave crossing, it is easy to see the chain of causality, i.e. which lines are inputs and which are collision products, because components along a ray are older toward its star-point.

In other words, Vees fly frontwards.

And one of the two arms (apparently the sparser) of these backwards Vees heads at an "inexplicable" angle, i.e., not through a star-point!