Variations on Conway Life.

[Harold V. McIntosh]

The past two weeks have seen a discussion of variants on Life,
including an evaluation that "there is only one good rule set,"
referring to semitotalistic rule 2;3,4 on a hexagonal lattice.

The rule has its merits, although it is not the Golay rule nor one of
the Preston rules. Those are fully symmetric, but not semitotalistic;
the main difference lies, among other things, in whether a pair of
diametrically opposed live cells should give birth to a new cell.

Rules wherein no cell survives to the second generation but are born
from two or three parents typically show a lot of activity; one of the
most famous rules of this type is the "phantom fishtank;" also called
"Silverman's rule." (More likely fishtank was the name of the program,
whose repertoire contained that particular rule.) Such rules are
fascinating to watch, but nearly hopeless for cataloging their myriad
life forms.

Giving the cells a pretext to survive for a while improves the
situation; at least it provides us with the rules under discussion. It
is impossible to resist remarking that all their mean field curves
exhibit diagonal tangencies, especially hex st2;3,4 which sticks very
closely to the diagonal.

Lest one think that divinations based on the mean field curve are
entirely capricious, consider the fate of the 3,4 totalistic Moore Life
rule: there are two very stable fixed points at zero and half density.
As long as densities are fairly low, they remain low with many nice
artifacts, a glider, and what not. But there are monsters lurking
about, which remain fairly stable, growing only slowly --- for a while.
Once they get big enough, though, they fill up everything in sight and
keep on expanding. And of course large regions at the higher densities
never reduce themselves spontaneously.

There must be more rules with two basins of attraction, with varying
kinds of coexistence; it would be interesting to see one in which the
two regions, if they existed, could maintain their identities for an
arbitrarily long time.

One way to explore cellular automata is to generate random
configurations and then watch them evolve; another is to seed the field
with known small artifacts, again to sit back and watch. Within the
limits of computer power, it is also possible to map out all the ways
that small configurations can join together to produce a particular
result. Tracing paths through the map will reveal larger configurations
with the same property.

As an example, consider a strip of length 6 in hex st2;3,4 with respect
to transversal motion. Suppose further that the strip begins and ends
with zero cells, and that the strips are stacked vertically (actually,
running off at 120 degrees in a true hex lattice). Transversal motion
refers to speed-of-light fuses surrounded by quiescent cells, burning
along the direction of the stack. The fuse could be infinite in both
directions, giving an infinitely long light-speed glider.

The following graph completely describes all such fuses or gliders.

000 000 - ...... ...... ...... 000 010 - ...... ...... ...1..
010 020 - ...... ...1.. ...... 020 010 - ...1.. ...... ...1..
000 040 - ...... ...... ..1... 040 080 - ...... ..1... ......
040 0D0 - ...... ..1... ..11.. 080 040 - ..1... ...... ..1...
000 100 - ...... ...... .1.... 000 110 - ...... ...... .1.1..
010 160 - ...... ...1.. .11... 020 140 - ...1.. ...... .11...
0D0 1E0 - ..1... ..11.. .11... 100 200 - ...... .1.... ......
100 210 - ...... .1.... ...1.. 110 260 - ...... .1.1.. ..1...
140 280 - ...... .11... ...... 160 280 - ...1.. .11... ......
1E0 280 - ..11.. .11... ...... 110 320 - ...... .1.1.. .1....
100 340 - ...... .1.... .11... 140 380 - ...... .11... .1....
260 080 - .1.1.. ..1... ...... 2A0 010 - .111.. ...... ...1..
280 040 - .11... ...... ..1... 200 100 - .1.... ...... .1....
200 110 - .1.... ...... .1.1.. 210 160 - .1.... ...1.. .11...
000 400 - ...... ...... 1..... 000 410 - ...... ...... 1..1..
010 420 - ...... ...1.. 1..... 020 410 - ...1.. ...... 1..1..
000 440 - ...... ...... 1.1... 0D0 4E0 - ..1... ..11.. 1.1...
040 580 - ...... ..1... 11.... 080 500 - ..1... ...... 11....
080 510 - ..1... ...... 11.1.. 380 600 - .11... .1.... 1.....
380 610 - .11... .1.... 1..1.. 320 740 - .1.1.. .1.... 111...
340 780 - .1.... .11... 11.... 400 800 - ...... 1..... ......
400 810 - ...... 1..... ...1.. 410 820 - ...... 1..1.. ......
420 810 - ...1.. 1..... ...1.. 400 840 - ...... 1..... ..1...
440 980 - ...... 1.1... .1.... 440 9D0 - ...... 1.1... .111..
4E0 980 - ..11.. 1.1... .1.... 500 A00 - ...... 11.... ......
500 A10 - ...... 11.... ...1.. 510 A60 - ...... 11.1.. ..1...
580 A00 - ..1... 11.... ...... 580 A10 - ..1... 11.... ...1..
600 800 - .1.... 1..... ...... 600 810 - .1.... 1..... ...1..
610 820 - .1.... 1..1.. ...... 600 840 - .1.... 1..... ..1...
740 A80 - .1.... 111... ...... 780 A00 - .11... 11.... ......
780 A10 - .11... 11.... ...1.. 980 200 - 1.1... .1.... ......
980 210 - 1.1... .1.... ...1.. 9D0 2A0 - 1.1... .111.. ......
A60 080 - 11.1.. ..1... ...... A80 040 - 111... ...... ..1...
A00 100 - 11.... ...... .1.... A00 110 - 11.... ...... .1.1..
A10 160 - 11.... ...1.. .11... 800 400 - 1..... ...... 1.....
800 410 - 1..... ...... 1..1.. 810 420 - 1..... ...1.. 1.....
820 410 - 1..1.. ...... 1..1.. 800 440 - 1..... ...... 1.1...
840 580 - 1..... ..1... 11....

The stacking rules are that any group of three lines are to be placed
on top of one another, in that order. The triplet labelled by x,y
can be placed on top of the three labelled y,z as though they were
dominoes. The last line of the top group must match and overlay the
first line of the bottom group. For example, the group 000 000
generates a null fuse, which indeed meets the requirement.

The pair 010 020 and 020 010 generate a barber pole --- a line in which
live cells alternate quiescent cells. In each generation, all the live
cells die, but create a similar set of replacements with opposite
parity. (This string would be missing in the Golay automaton.) If the
string begins 000 000 followed by 000 010 followed by the cycle, there
is a fuse; it cannot end similarly because the direction of motion
would be wrong. Nevertheless an interesting artifact results.

Note that no group ends with 000 000, consistent with the nonexistence
of finite light-speed space ships or gliders.

Giving the fullblown treatment to more than a single generation is
computationally prohibitive, but there are alternative or partial
treatments available; apparently there are persons actively using them
here and there. It would be interesting to hear about the search
schemes which are in use.
---
Harold V. McIntosh |Depto. de Aplicacion de Microcomputadoras
mcin...@unamvm1.dgsca.unam.mx |Inst. de Ciencias, UAP, Apdo. Postal 461
MCINTOSH@UNAMVM1 (BITNET) |72000 Puebla, Pue., Mexico

[Torben AEgidius Mogensen]

mcin...@uapnx1.dgsca.unam.mx (Harold V. McIntosh) writes:

>Lest one think that divinations based on the mean field curve are
>entirely capricious, consider the fate of the 3,4 totalistic Moore Life
>rule: there are two very stable fixed points at zero and half density.
>As long as densities are fairly low, they remain low with many nice
>artifacts, a glider, and what not. But there are monsters lurking
>about, which remain fairly stable, growing only slowly --- for a while.
>Once they get big enough, though, they fill up everything in sight and
>keep on expanding. And of course large regions at the higher densities
>never reduce themselves spontaneously.

>There must be more rules with two basins of attraction, with varying
>kinds of coexistence; it would be interesting to see one in which the
>two regions, if they existed, could maintain their identities for an
>arbitrarily long time.

I once made some rules for "symmetric" life, based on a square grid
with 8 neighbours exactly like the original Life. The symmetry comes
from the fact that a pattern of filled cells in an empty field behaves
exactly as the same pattern of holes in a filled field. I succeded in
finding rules that didn't cause patterns to "explode". In fact, it was
difficult to come up with small patterns that would grow to fill very
large areas before dissolving into small stable or periodic bits.

As far as I recall the rules are {3,5,7,8} for birth and {2,4,6,7,8}
for survival. This gives, among other things, the following periodics:

XXX (2) XXX (4)
XX

Torben Mogensen (tor...@diku.dk)