L.S.

The Life Rule: B37/S23 apparently gives rise to easily generated chaotic
universes.

Two simple patterns that do this are:

ooo
o..
ooo
o..

and

o.....
..oo..
.o..o.
..oo..
o....o

These universes call for a completely different set of theorems than the
  "constructionist" ones (i.e., those filled with "engines" that
propagate a certain regular "pattern").

Questions to be asked about the B37/S23 Universe fillers and their
progenitors are:

1. Is the growth initiated by the above patterns boundless (in both
    space and time) ?
2. How do we determine its growth (increase in cells live w.r.t to
    "enclosed" cells) ?
3. What is it's asymptotical speed-of-growth ?
4. What is it's asymptotical density (live cells / total cells enclosed)
5. Are these properties dependent on the progenitor patterns,or are they
    determined by the rules ?
6. What is the internal "rate of combustion", i.e. the rate of
    cells changed from life-to-death and the other way around divided
    by the total number) ?
7. Does the rate of combustion have an asymptotic limit ?
8. Radiation.  As far as I have been able to determine, these universes
    only radiate simple gliders.  Can this be proven to be true ?
9. What is the radiation density limit (i.e. what is the number of
    gliders released per generation, and is there an asymptotic limit
    to them) ?

I hope this post generates interest in this (and other) chaotic
universes in Life.

Kind regards,

--
Toon Moene - e-mail: t...@moene.indiv.nluug.nl - phone: +31 346 214290
Saturnushof 14, 3738 XG  Maartensdijk, The Netherlands
At home: http://moene.indiv.nluug.nl/~toon/
Progress of GNU Fortran: http://gcc.gnu.org/ml/gcc/2008-01/msg00009.html

The "pop-plot" script in Golly [ http://golly.sourceforge.net ] may be
useful in getting some rough estimates for these.  Running 30,000
ticks on the seven-cell pattern

 ..oo.
 oo.oo
 ...o.

gave a rate of increase averaging a bit over three cells per tick over
that time.  End population was 101,764 cells, but with some
significant spikes and drops along the way.  The expansion starts out
slowly -- e.g., more like 1 cell/tick over the first 10,000 ticks --
and creeps up from there.

If you don't mind a little symmetry, the six-cell

 oo..
 .oo.
 ..oo

is also explosive, with a growth rate closer to 4 cells per tick over
30,000 ticks -- end population 112,896.  (I borrowed these two
starting patterns from early investigations of B37/S23 "methuselahs"
by several Life enthusiasts in March 1994.)

The above seems to be about the limit of my patience for collecting
arbitrary population statistics, but clearly the starting pattern can
make a difference.  In these two cases, growth rates look like they're
getting more predictable with time, but exceptions to any rule are
very likely.  Spacefiller patterns might possibly even be within reach
of current search programs; these would have much more predictable
(and higher) growth rates from the start.

Unlikely to be provable, I would think.  The usual non-glider suspects
from Conway's Life (lightweight, middleweight, and heavyweight
spaceships) all fail here due to central 7-neighbor OFF cells.  But
otherwise B37/S23 has fairly good prospects for glider constructions,
and can probably be shown to be universal -- somehow.  It has glider
guns and reflectors, at least:

#C P60 pre-pulsar-based glider gun, thinned out to p480 so that it
#C could be used with (hypothetical) p32 spinner-based technology.
#C See also http://entropymine.com/jason/life/alt/ .
#C Jason Summers, 11 October 2001
x = 88, y = 96, rule = B37/S23
29bo5bo$29boo3boo$29boo3boo$22boo6bo3bo$23bo$23bobo$24boo18bo$42bobbo$
42bobbo$31bobo3bobo3bo$30bobbo3bobbo$10boo3boo14bobo3bobo3bo$9bo7bo24b
obbo$12bobo27bobbo$10boo3boo27bo$$41boo$40bobbo$3o25bo10bobobo$b3o22bo
bo9b3obo$27boo9b3o$$9b3o3b3o36bo$b3o4bo3bobo3bo33b3o$3o5bo3bobo3bo3boo
27bo$8bo3bobo3bo3bobo26boo$9b3o3b3o4bo$3boo54bobo$bbobbo52bo$bbobobo
52bobbo$3bobbo52bobobo$7bo52bobbo$4bobo54boo$43bo4b3o3b3o$13boo26bobo
3bo3bobo3bo$14bo27boo3bo3bobo3bo5b3o$11b3o33bo3bobo3bo4b3o$11bo36b3o3b
3o$$25b3o9boo$23bob3o9bobo22b3o$22bobobo10bo25b3o$22bobbo$23boo$$21bo
27boo3boo$20bobbo27bobo$20bobbo24bo7bo$22bo3bobo3bobo14boo3boo$25bobbo
3bobbo$22bo3bobo3bobo$20bobbo$20bobbo16bo$21bo19bo$39b3o$53b3o$31bo3bo
16bobbo$30boo3boo18boo$30boo3boo14bob3o$30bo5bo14bo4bo$52bo$53b3o$39b
oo$35boo4b4o$35boobboob3o$39bo5$87bo$85b3o$84bo$84boo18$75bo$71boo$71b
ooboo$71boobo$73bo!

In any case, you never know if one of the following spaceships *might*
(with breathtakingly low probability) emerge from random soup:

#C Random collection of orthogonal Conway's Life spaceships that
#C also work in B37/S23 -- plus a larger, rule-specific spaceship
#C by Dean Hickerson from October 1998, a distant relative of
#C David Eppstein's "Puddle Jumper". For other B3/S23 spaceships,
#C see http://fano.ics.uci.edu/ca/rules/b37s23/ .
x = 141, y = 114, rule = B37/S23
59b3o15b3o$58bo3bo13bo3bo$57b2o4bo11bo4b2o$56bobob2ob2o3b3o3b2ob2obobo
$55b2obo4bob2ob3ob2obo4bob2o$54bo4bo3bo4bobo4bo3bo4bo$66bo5bo$54b2o7b
2o9b2o7b2o3$68b3o$67bo3bo$67b2ob2o$68bobo$65b3obobo2b3o$70bo3b3o$65b2o
7bo2bo$74bo$75bo$74bobo4$71bo$64b2ob3ob3o$64b2o4bo2b2ob2o$63bo2bob2o3b
o2b2o$75bo2bo4$69b2o$65b2obo2bob2o$65b2o6b2o$65bobo4bobo$67b2o2b2o$66b
2ob2ob2o$68bo2bo$66bo6bo$66bo6bo2$4b3o7b3o49b8o50b3o7b3o$4bo2bo5bo2bo
48b2o6b2o49bo2bo5bo2bo$3bo3bo5bo3bo6b3o7b3o67b3o7b3o6bo3bo5bo3bo$4bo2b
o5bo2bo7bo2bo5bo2bo67bo2bo5bo2bo7bo2bo5bo2bo$4bobo7bobo6bo3bo5bo3bo6b
3o7b3o27b3o7b3o6bo3bo5bo3bo6bobo7bobo$24bo2bo5bo2bo7bo2bo5bo2bo27bo2bo
5bo2bo7bo2bo5bo2bo$24bobo7bobo6bo3bo5bo3bo6b3o7b3o6bo3bo5bo3bo6bobo7bo
bo$44bo2bo5bo2bo7bo2bo5bo2bo7bo2bo5bo2bo$44bobo7bobo6bo3bo5bo3bo6bobo
7bobo$64bo2bo5bo2bo$4b3o7b3o2b3o42bobo7bobo42b3o2b3o7b3o$6bo7bo111bo7b
o$4bobo7bobo2bobo2b3o7b3o2b3o57b3o2b3o7b3o2bobo2bobo7bobo$20bo5bo7bo
71bo7bo5bo$24bobo7bobo2bobo2b3o7b3o2b3o17b3o2b3o7b3o2bobo2bobo7bobo$
21bo18bo5bo7bo31bo7bo5bo18bo$21bo22bobo7bobo2bobo2b3o7b3o2bobo2bobo7bo
bo22bo$21bo19bo18bo5bo7bo5bo18bo19bo$5b2o34bo22bobo7bobo22bo34b2o$4bo
2bo33bo19bo17bo19bo33bo2bo$4bo2bo53bo17bo53bo2bo$5b2o54bo17bo54b2o$9bo
121bo$8b2o121b2o$8bobo18bo81bo18bobo$28b2o81b2o$28bobo18bo41bo18bobo$
48b2o41b2o$15bo32bobo13bo12bo12bobo32bo$2b3o9b2o47b3o10b3o46b2o9b3o$
14bobo18bo27b3o10b3o26bo18bobo$34b2o69b2o$8bo25bobo67bobo25bo$8bo57bo
3b2o3bo56bo$bo3bo15bo42bob4o2b4obo41bo15bo3bo$3o17b2o42bob4o2b4obo41b
2o17b3o$7bo12bobo95bobo12bo$5b2obo59b6o58bob2o$3bo2bobo58b2o4b2o57bobo
2bo$bo2bo62b2o4b2o61bo2bo$2b3o131b3o4$67bo6bo$66bobo4bobo$66bobo4bobo$
67bo6bo2$66b3o4b3o$66b4o2b4o$69bo2bo$66b3ob2ob3o$66b2o6b2o$65b2o8b2o2$
64bo12bo$62bo2bo10bo2bo$62bo2bo10bo2bo2$65b3o6b3o$65bo10bo$67bo6bo$65b
3o6b3o2$68bob2obo$69bo2bo$65bob2ob2ob2obo$65b3o6b3o2$65bobo6bobo$65bob
o6bobo$63b3ob3o2b3ob3o!

Besides the standard glider, many other c/4 diagonal spaceships in B3/
S23 also work in B37/S23.

A glider-emitting breeder pattern can probably also be constructed in
this rule, which would out-radiate any chaotic pattern by a
considerable margin.

Keep the cheer,

Dave Greene

Just happened to run into a semi-counterexample -- not an orthogonal
spaceship, but at least a natural c/2 orthogonal puffer seed:

 ooo.o
 o...o
.o...o

This eight-bit pattern turns into a B-heptomino pair, plus some fading
junk, in 23 ticks.  It showed up 25K+ ticks into a random pattern I
was running in Golly:

#C 3obo$o2bo$o2bo! puffer appears in 25438 ticks from this seed
x = 123, y = 77, rule = B37/S23
37b2o$36bo2bo$37b2o2$31b3o$18bo$18bo$18bo5$49b2o$49b2o$42bo$42bo$42bo
22b2o$38b2o24bo2bo$38b2o25bobo$66bob2o$42bo24bo2bo$42bo25bobo$42bo26bo
3$77b2o$76bo2bo22b2o$77b2o22bo2bo$101bo2bo$102b2o2$27b3o31b2o$60bo2bo$
61b2o$55b2o$55b2o3$80b2o$80b2o2$60bo$58bo2bob2o$58bo4b2o$27b3o27bo3b2o
$58b5o9bo$62b2o6bo3bo$62b2o6bo4bo$70bobob2o5b3o$81b3o$59bob2o18bo2bo$
59b3o11b2o7b3o$60bo12b2o5b2o13b2o$79b5o11b2o$78b3o13bobo$79bo15bo$69b
2o8bo14bobo$bo7bo29b2o15bo7bo4b2o10bo10b2ob2o$obo5bobo28b2o14bobo5bobo
12bobo11b3ob2ob2o$bo7bo46bo7bo14b2o2b2obo4bobo3b2obo5bo6bo7bo$80b2o3b
2o3bo2bo6bo4bobo4bobo5bobo$81bo3bo8b2obo8bo6bo7bo$39b2o42b2o4bo3b3obo$
39b2o51bo$90b2o$89bo$82b2o$81bo2bo$82b2o8$27b3o20b3o!

Possibly a convoy of these puffers could be arranged to clean up their
combined junk, but this rule is explosive enough that it might not be
particularly easy.

Keep the cheer,

DaveG

Sorry, that picture might not copy well to a CA editor -- the Flying
Spaghetti Monster seems to have added an extra period in an awkward
spot.  I meant:

 ooo.o
 o...o
 o...o

Keep the cheer,

DaveG

Argh!  _Three_ extra periods -- drat that rascally FSM.  I even
remember testing the second pattern I sent, so I'm not sure *what*
happened exactly.

Third time's the charm.  The 8-bit seed I was trying to send was

 ooo.o
 o..o
 o..o

In B37/S23, it goes symmetrical after 10 ticks, and produces a
familiar pair of B-heptominoes after 23 ticks (the next generation
after this can be found in the rotor of a standard B3/S23 p46
oscillator):

 o.oo.oo.o
 ooo...ooo
 .o.....o..

Might be interesting to see if the debris from this puffer can be
suppressed somehow -- assuming somebody hasn't already done that, but
I don't see any sign of previous attempts.  Maybe escorting it with
some number of Glider 10239s (http://fano.ics.uci.edu/ca/rules/b37s23/
g2.html) would work... there are some vaguely promising-looking sparks
toward the back, but B37/S23 seems to be just enough more prolific
that what in Conway's Life would be "suppressing sparks" are now
"forest-fire-starting sparks" instead.

Keep the cheer,

Dave