dvgrn wrote:See the Life Lexicon definition for starters.
Thanks — I have to say, though, I didn't find the "diagram that illustrates the idea" at all enlightening.
dvgrn wrote:See the Life Lexicon definition for starters.
Rich Holmes wrote:dvgrn wrote:See the Life Lexicon definition for starters.
Thanks — I have to say, though, I didn't find the "diagram that illustrates the idea" at all enlightening.
x = 47, y = 206, rule = LifeHistory
37.2D$37.D.D$37.D60$32.A$18.A12.3A$17.3A4.3A3.2A.A$10.3A3.2A.A3.A2.A
3.3A4.2C$9.A2.A3.3A7.A4.2A4.C.C$12.A4.2A3.A3.A10.C$8.A3.A9.A3.A$.3A4.
A3.A13.A$A2.A8.A10.A.A$3.A5.A.A$3.A$A.A3$11.A$10.3A$9.2A.A$9.3A$10.2A
3$16.3A$15.A2.A5.A5.3A$18.A4.3A3.A2.A5.A$18.A4.A.2A5.A4.3A$5.3A7.A.A
6.3A5.A4.A.2A$4.A2.A16.3A2.A.A6.3A$7.A16.3A11.3A$7.A16.2A12.3A$4.A.A
31.2A$44.A$43.3A$43.A.2A$44.3A$44.2A7$42.A$31.3A7.3A$17.3A5.A5.A2.A6.
A.2A$11.A5.A2.A3.3A4.A10.3A$10.3A4.A5.2A.A4.A10.3A$9.2A.A4.A5.3A6.A.A
7.2A$9.3A6.A.A2.3A$2.A6.3A11.3A$.3A5.3A12.2A$2A.A6.2A$3A40.A$.2A39.3A
$42.A.2A$43.3A$43.2A$10.3A$10.A2.A$10.A$10.A$11.A.A2$17.A$16.3A12.A$
15.2A.A4.3A4.3A$15.3A4.A2.A3.2A.A4.3A$6.A9.2A7.A3.3A4.A2.A$5.3A13.A3.
A4.2A7.A$4.2A.A13.A3.A9.A3.A$4.3A18.A9.A3.A$5.2A15.A.A14.A$36.A.A$43.
3A$42.A2.A$45.A$45.A$42.A.A7$32.A8.3A$18.A12.3A6.A2.A$17.3A4.3A4.A.2A
8.A$10.3A4.A.2A3.A2.A4.3A4.A3.A$10.A2.A4.3A3.A7.2A9.A$10.A7.2A4.A3.A
11.A.A$10.A3.A9.A3.A$.3A6.A3.A9.A$.A2.A5.A14.A.A$.A9.A.A$.A40.3A$2.A.
A36.A2.A$44.A$44.A$11.A29.A.A$10.3A$10.A.2A$11.3A$11.2A3$16.3A$16.A2.
A4.A5.3A$16.A6.3A4.A2.A4.A$16.A5.2A.A4.A6.3A$5.3A9.A.A2.3A5.A5.2A.A$
5.A2.A13.3A6.A.A2.3A$5.A16.3A11.3A$5.A17.2A11.3A$6.A.A28.2A$44.A$43.
3A$42.2A.A$42.3A$43.2A7$42.A$31.3A7.3A$17.3A5.A4.A2.A6.2A.A$11.A4.A2.
A4.3A6.A6.3A$10.3A6.A4.A.2A5.A6.3A$10.A.2A5.A5.3A2.A.A8.2A$11.3A2.A.A
6.3A$2.A8.3A11.3A$.3A7.3A11.2A$.A.2A6.2A$2.3A38.A$2.2A38.3A$41.2A.A$
41.3A$42.2A$10.3A$9.A2.A$12.A$12.A$9.A.A!
[[ STOP 213 ]]
gmc_nxtman wrote:Would it be possible to search for a small glider gun of period < 20 by searching for oscillators which emit gliders as "sparks"? As in, a small billiard-ish oscillator-like thing emitting gliders, akin to oscillators emitting large sparks.
Rich Holmes wrote:Where is there a writeup of the symmetry codes (e.g. C1, D8_1, etc.) used in Catagolue?
blah wrote:For sawtooths, why can't you extend the idea of having a gun releasing fast ships interacting with slower ships (like copperheads) to those slower ships being arbitrarily slow manufactured ships, like gemini? As the target ships being slower brings the EF closer to 1, you could have a manufactured spaceship which can travel at any arbitrary speed desired, meaning an EF arbitrarily close to 1. Is there something wrong with that?
x = 338, y = 361, rule = B3/S23
270bo$270b2o$269bobo16$282bo$281bobo$281b2o5$262b2o$262b2o$266b2o$266b
o$264bobo$264b2o7b2o4b2o$273b2o4b2o8bo$279b2o7bobo$280b2o7b2o$243b2o
35b2o3b2o$243b2o34bo2bo2b2o7b2o$278bob2o11bobo3bo$267b2o3b2o4bob2o12bo
3bobo$266bobo3b4o3bo18b2o$266bo3bo2bo2bo$266b2o3bo4bo6b4o$271bo3bo7bo
3bo$270bo16bo$283b2obo$283bo$279bo2bo$279bo2bo$279bo2bo$279bo2bo$253b
2o25b2o$253b2o3$258b2o$258b2o3$266b2o$265bobo$265bo$264b2o2$282bo$281b
obo$281b2o3$274b2o19bo17b2o$274b2o7b2o11bo16b2o$283bo10b3o$281bobo$
281b2o$264b2o$264b2o20b2o24b2o$254b2o30b2o24b2o$255bo$255bobo$245bo10b
2o$245b3o20b3o$248bo18bo3bo$247b2o3b2o12bo5bob3o41b2o$252b2o12bo5bobo
2b2o39b2o$266bo5bo5bo$267bo3bo3bo2bo$268b3o5b2o$274bo$260b2o11b3o$251b
o7bobo11bo2bo48b2o$250bobo6bo15b2o48bobo$250bobo5b2o32b2o32bo$251bo19b
3o17bobo$271b3o18bo4$275b2o$256b2o17b2o39b2o$256b2o9b2o46bobo$268bo45b
obo$265b3o47bo$265bo2$266bo$265bobo47b2o$265bobo47bobo$263b3ob2o47bo$
262bo$263b3ob2o$265bob2o$313b2o22bo$275b2o16b2o18bobo19bobo$275b2o7b2o
7bobo18bo21b2o$261bo22bo9bo$260bobo19bobo$260bobo12b3o4b2o$261bo13bo$
276bo2$262b2o$262b2o37b2o$300bobo$301bo3$278bo$277bobo$268b2o7bobo$
268b2o8bo$279b3o$281bo8$281b2o$281b2o30$332b2o$332bobo$332bo188$3o$2bo
$bo!
muzik wrote:Could a gun similar to this one exist in normal life?Code: Select allx = 22, y = 21, rule = B38/S23
4$4b3o$4bo3bo$3b2o3bo$2b2o5bo$2b2o2b2o2bo$8bobo$9bo!
PHPBB12345 wrote:No.
NotLiving wrote:So... In normal Life an agar cannot have an average density (over space and time) above 1/2, if I read correctly.
Life Lexicon wrote::density The density of a pattern is the limit of the proportion of live cells in a (2n+1)x(2n+1) square centred on a particular cell as n tends to infinity, when this limit exists. (Note that it does not make any difference what cell is chosen as the centre cell. Also note that if the pattern is finite then the density is zero.) There are other definitions of density, but this one will do here.
In 1994 Noam Elkies proved that the maximum density of a stable pattern is 1/2, which had been the conjectured value. See the paper listed in the bibliography. Marcus Moore provided a simpler proof in 1995, and in fact proves that a still life with an m x n bounding box has at most (mn+m+n)/2 cells.
But what is the maximum average density of an oscillating pattern? The answer is conjectured to be 1/2 again, but this remains unproved. The best upper bound so far obtained is 8/13 (Hartmut Holzwart, September 1992).
The maximum possible density for a phase of an oscillating pattern is also unknown. An example with a density of 3/4 is known (see agar), but densities arbitrarily close to 1 may perhaps be possible.
NotLiving wrote:What about patterns that are aperiodic, be it in space, time, or both? Does this still hold?
NotLiving wrote:Also: is there an equivalent to Penrose tilings for Life?
NotLiving wrote:Do we know that bounded-periodic oscillatory patterns have an upper-bound 1/2 density limit on average, at least? Seems easier to prove than the aperiodic case. Although still way above my pay grade.
NotLiving wrote:Each tile edge is composed of the presence or absence of 2 tables or absence thereof - use a binary encoding for the colors. So each tile is 14 x 14 - 15x15 once you include the gap between adjacent tiles. Use this set of Wang tiles.
x = 47, y = 47, rule = LifeHistory
5.C2.C.C2.C5.C2.C.C2.C5.C2.C.C2.C$5.4C.4C5.4C.4C5.4C.4C2$5.4A.4A5.4A.
4A5.4A.4A$5.A2.A.A2.A5.A2.A.A2.A5.A2.A.A2.A$2C.2A9.2A.2A9.2A.2A9.2A.
2C$.C.A11.A.A11.A.A11.A.C$.C.A11.A.A11.A.A11.A.C$2C.2A9.2A.2A9.2A.2A
9.2A.2C2$2C.2A9.2A.2A9.2A.2A9.2A.2C$.C.A11.A.A11.A.A11.A.C$.C.A11.A.A
11.A.A11.A.C$2C.2A9.2A.2A9.2A.2A9.2A.2C$5.A2.A.A2.A5.A2.A.A2.A5.A2.A.
A2.A$5.4A.4A5.4A.4A5.4A.4A2$5.4A.4A3.2B4AB4A3B2.4A.4A$5.A2.A.A2.A3.2B
A2BABA2BA3B2.A2.A.A2.A$2C.2A9.2A.2A9B2AB2A9.2A.2C$.C.A11.A.A11BABA11.
A.C$.C.A11.A.A11BABA11.A.C$2C.2A9.2A.2A9B2AB2A9.2A.2C$17.14B$2C.2A9.
2A.2A9B2AB2A9.2A.2C$.C.A11.A.A11BABA11.A.C$.C.A11.A.A11BABA11.A.C$2C.
2A9.2A.2A9B2AB2A9.2A.2C$5.A2.A.A2.A3.2BA2BABA2BA3B2.A2.A.A2.A$5.4A.4A
3.2B4AB4A3B2.4A.4A$17.14B$5.4A.4A5.4A.4A5.4A.4A$5.A2.A.A2.A5.A2.A.A2.
A5.A2.A.A2.A$2C.2A9.2A.2A9.2A.2A9.2A.2C$.C.A11.A.A11.A.A11.A.C$.C.A
11.A.A11.A.A11.A.C$2C.2A9.2A.2A9.2A.2A9.2A.2C2$2C.2A9.2A.2A9.2A.2A9.
2A.2C$.C.A11.A.A11.A.A11.A.C$.C.A11.A.A11.A.A11.A.C$2C.2A9.2A.2A9.2A.
2A9.2A.2C$5.A2.A.A2.A5.A2.A.A2.A5.A2.A.A2.A$5.4A.4A5.4A.4A5.4A.4A2$5.
4C.4C5.4C.4C5.4C.4C$5.C2.C.C2.C5.C2.C.C2.C5.C2.C.C2.C!
x = 13, y = 13, rule = LifeHistory
2.4D.4D$2.D2.D.D2.D$2D9.2D$D11.D$D3.2E.2E3.D$2D2.2E.2E2.2D2$2D2.2E.2E
2.2A$D3.2E.2E3.A$D11.A$2D9.2A$2.D2.D.D2.D$2.4D.4D!
dvgrn wrote:NotLiving wrote:Do we know that bounded-periodic oscillatory patterns have an upper-bound 1/2 density limit on average, at least? Seems easier to prove than the aperiodic case. Although still way above my pay grade.
Hasn't been proven yet, as far as I can recall. I think everyone suspects that 1/2 is in fact the highest possible average density, but that doesn't mean it's easy to come up with a bulletproof proof.
dvgrn wrote:NotLiving wrote:Each tile edge is composed of the presence or absence of 2 tables or absence thereof - use a binary encoding for the colors. So each tile is 14 x 14 - 15x15 once you include the gap between adjacent tiles. Use this set of Wang tiles.
Ah, there's an 11-tile four-color aperiodic set, and they've proved that 10-tile sets and three-color sets aren't enough! My information was out of date, and for some reason that 2015 tile set didn't show up in my searches. Somebody should update the Wikipedia article.
dvgrn wrote:The tiles you're describing would be 14x14, though of course they could be made bigger
dvgrn wrote:<interesting 14x14 tiling cell>
#CXRLE Pos=-14,-14 Gen=11
x = 27, y = 27, rule = B3/S23:T28,28
4ob4o2b2ob2o2b4ob4o$o2bobo2bo3bobo3bo2bobo2bo$12bobo$11b2ob2o$2o23b2o$
o10b2ob2o10bo$o11bobo11bo$2o10bobo10b2o$11b2ob2o$2o23b2o$o25bo$o3bo2bo
bo2bobo2bobo2bo3bo$2o2b4ob4ob4ob4o2b2o2$2o2b4ob4ob4ob4o2b2o$o3bo2bobo
2bobo2bobo2bo3bo$o25bo$2o23b2o$11b2ob2o$2o10bobo10b2o$o11bobo11bo$o10b
2ob2o10bo$2o23b2o$11b2ob2o$12bobo$o2bobo2bo3bobo3bo2bobo2bo$4ob4o2b2ob
2o2b4ob4o!
dvgrn wrote:It might be a good idea to add some extra decoration, like four blocks, in the middle of the tile. That way even if there's a tile that is made up of mostly the no-tables color, and there's just one unpaired table, the tile will still reliably cause some interesting chaos. A table by itself with enough empty space around it just collapses into vacuum, so you'd end up with a stable pattern again immediately
dvgrn wrote:However, it might be possible to come up with tiles smaller than 14x14, by mixing tables with other lengths of induction coil (?) I suppose the ON cells would have to be either mirror-symmetric or would have to be stable even without an exact match between induction coils. Would have to be careful that none of the combinations with other "colors" is also stable -- a length-2 offset wouldn't be good, unless it was two copies of the same color.
NotLiving wrote:dvgrn wrote:The tiles you're describing would be 14x14, though of course they could be made bigger
Smaller, you mean?
NotLiving wrote:Ah, so there is some room for it to be made smaller. (Also: what are you using to create said images?)
NotLiving wrote:I was thinking of this, actually... which has the downside that every second tile has to be mirrored...
NotLiving wrote:I've tossed a PBSAT solver at it. I shall see what happens.
(Actually two encodings: one that enforces a 2x2 blank at each corner that's substantially faster, and one that does not but is substantially slower. The corner tiles as far as I can tell require 11^4 clauses to cover - "yay?". Any suggestions for how to deal with them in a sane manner?)
x = 15, y = 37, rule = LifeHistory
7B.7B$7B.7B$7BAC6B$7BAC6B$7B.7B$7B.7B$7B.7B4$7B.7B$7B.7B$7B.7B$7BAC6B
$7BAC6B$7B.7B$7B.7B4$7B.7B$7B.7B$7B.7B$6B2AC6B$7B.7B$7B.7B$7B.7B4$7B.
7B$7B.7B$7BA7B$7BA7B$7BA7B$7B.7B$7B.7B!
x = 108, y = 28, rule = LifeHistory
3B2A3B12.3BA4B12.3B2A3B12.2B2A4B12.3BA4B$8B12.3BA4B12.8B12.8B12.3BA4B
$8B12.8B12.7BA12.8B12.8B$7BA12.7BA12.7BA12.A5B2A12.A5B2A$7BA12.7BA12.
7BA12.8B12.8B$8B12.8B12.8B12.8B12.8B$8B12.8B12.8B12.8B12.8B$3B2A3B12.
3BA4B12.8B12.3B2A3B12.2B2A4B13$2B2A4B12.3B2A3B12.3BA4B12.3BA4B12.3B3A
2B12.3B3A2B$8B12.8B12.3BA4B12.3BA4B12.8B12.8B$A6BA12.A6BA12.7BA12.8B
12.7BA12.7BA$A6BA12.A6BA12.A6BA12.A6BA12.A6BA12.7BA$8B12.7BA12.A7B12.
A6BA12.A6BA12.7BA$8B12.8B12.8B12.8B12.8B12.8B$8B12.8B12.8B12.8B12.8B
12.8B$3B2A3B12.3BA4B12.3BA4B12.2B2A4B12.3BA4B12.8B!
x = 191, y = 41, rule = LifeHistory
35.2C$7B.7B18.2B2A4B45.2C18.C19.2C17.2C19.C$7B.7B5.2E11.8B42.3B2A3B
12.3BA4B12.3B2A3B12.2B2A4B12.3BA4B$7BAC6B4.E2.E9.AC6BAC41.8B12.3BA4B
12.8B12.8B12.3BA4B$7BAC6B4.E2.E9.AC6BAC40.A8B11.A8B11.A7BA12.8B12.8B$
7B.7B4.E2.E10.8B41.A7BAC10.A7BAC10.A7BA10.2AC5B2AC9.2AC5B2AC$7B.7B4.E
2.E10.8B41.A7BAC10.A7BAC10.A7BA12.8B12.8B$7B.7B5.2E11.8B42.8B12.8B12.
8B12.8B12.8B$33.2B2C4B42.8B12.8B12.8B12.8B12.8B$35.2A9.2C35.3B2C3B12.
3BC4B12.8B12.3B2C3B12.2B2C4B$43.3B2A3B35.2A18.A19.3A17.2A17.2A$7B.7B
5.E22.8B55.A$7B.7B5.E22.8B$7B.7B5.E21.AC6BAC$7BAC6B5.E21.AC6BAC$7BAC
6B5.E22.8B$7B.7B5.E22.8B$7B.7B5.E22.3B2C3B$46.2A2$36.C$7B.7B3.3E12.3B
A4B44.2C19.2C18.C19.C$7B.7B6.E11.3BA4B42.2B2A4B12.3B2A3B12.3BA4B12.3B
A4B12.3B3A2B12.3B3A2B$7B.7B6.E11.8B42.8B12.8B12.3BA4B12.3BA4B12.8B12.
8B$6B2AC6B5.E10.2AC5B2AC40.AC6BAC10.AC6BA12.7BAC11.8B12.7BA11.A7BA$7B
.7B4.E13.8B41.AC6BAC10.AC6BA11.AC6BAC10.AC6BAC10.AC6BA11.A7BA$7B.7B3.
E14.8B42.8B12.7BA11.AC7B11.AC6BAC10.AC6BA11.A7BA$7B.7B3.4E11.8B42.8B
12.8B12.8B12.8B12.8B12.8B$33.3BC4B42.8B12.8B12.8B12.8B12.8B12.8B$36.A
46.3B2C3B12.3BC4B12.3BC4B12.2B2C4B12.3BC4B12.8B$36.A49.2A18.A19.A18.
2A19.A19.3A$7B.7B3.3E85.A19.A39.A$7B.7B6.E21.3B3A2B$7BA7B6.E21.8B$7BA
7B3.3E21.A7BA$7BA7B6.E20.A7BA$7B.7B6.E20.A7BA$7B.7B3.3E22.8B$43.8B$
43.8B$46.3A!
x = 32, y = 32, rule = B3/S23:T32,32
3.2A6.2A5.2A.A5.2A3.$32.$.A7.A8.2A.A10.$A.A5.A.A6.A.A.3A4.2A2.$.A3.2A2.A3.2A.A2.A4.A2.A2.A.$A3.A2.2A3.A2.2A.A2.3A3.2A2.A$4.2A6.2A4.2A.A10.$18.A2.A5.2A3.$19.2A6.2A3.$4.2A6.2A18.$A3.A2.2A3.A2.2A6.2A6.A$A4.2A.A4.2A2.A5.A.A5.A$16.2A6.2A6.$18.2A.A4.2A.A2.$2.A7.A7.A.2A4.A.2A2.$2.3A5.3A19.$5.A7.A18.$2.4A6.A5.2A6.2A4.$.A10.4A.A.A2.2A.A.A2.2A$.2A2.3A7.A.A.A.A.A.A.A.A.A$2.A2.A2.A3.2A.A.A.2A2.A.A.2A2.A$A5.2A3.A.A.2A.A4.2A.A4.A$10.A7.A7.A5.$3.2A5.3A5.3A5.3A3.$3.2A8.A7.A7.A2.$10.4A6.2A6.2A2.$6.2A.A22.$.A3.A.A.2A2.3A4.4A4.2A2.$A.A.A2.A2.A2.A2.A2.A4.A3.A.A.$A.A.2A.2A5.2A3.5A7.A$2.A2.A20.A5.$2.A2.A5.2A5.A.2A4.3A3.!
8 8 9 1
3 3 4 4
6 10 5 5
7 6 2 0
+---+---+---+---+
| 2 | 2 | 3 | 2 |
|181|181|193|311|
| 0 | 0 | 2 | 2 |
+---+---+---+---+
| 0 | 0 | 2 | 2 |
|232|232|242|242|
| 1 | 1 | 0 | 0 |
+---+---+---+---+
| 1 | 1 | 0 | 0 |
|063|3a0|050|050|
| 2 | 1 | 1 | 1 |
+---+---+---+---+
| 2 | 1 | 1 | 1 |
|170|063|323|301|
| 2 | 2 | 3 | 1 |
+---+---+---+---+
dvgrn wrote:It seems better to stay away from this idea, then, since if there are rules about when to use mirror images (or rotations) then the tiles are no longer really Wang tiles.
dvgrn wrote:What size tiles are you asking the PBSAT solver about?
dvgrn wrote:The only criterion is that if each color-A half pattern is matched with a color-B other-half-pattern, for A!=B, then the combination must _not_ be stable.
dvgrn wrote:I think that's technically doable with 8x8 tiles, as long as we can count blinkers as "P2 stable" -- i.e., your test is to run the pattern for two ticks, not just one, and then if anything is different then you must have mismatched a tile
dvgrn wrote:If P1 stable tiles are a requirement, then the initial upper bound seems to be 9x9, to be able to fit independent copies of four stable "color" objects at each of the four edges, without them interfering with each other.
It's quite possible that there's some delicately balanced set of 8x8 or smaller P1 stable tiles, with a lot more ON cells: the center would have to be customized for each of the 11 tiles so that it's internally stable.
NotLiving wrote:With the restriction that the 16 corner cells are blank and everything is still life, I have an 8x8 set of tiles, and 7x7 or smaller is not allowed (assuming my code was correct...
This is really hard to visually see as 8x8 cells.
x = 64, y = 64, rule = LifeHistory
3D2C3D3.2A3.2D2CDC2D3.2A3.3D2C3D3.2A3.2D2CDC2D3.2A$8D8.8D8.8D8.8D$DC
6D.A6.2D2CDC2D8.DC6D.A6.2D2CDC2D$CDC5DA.A5.DCDCD3C4.2A2.CDC5DA.A5.DCD
CD3C4.2A$DC3D2CD.A3.2A.C2DC4DA2.A2.A.DC3D2CD.A3.2A.C2DC4DA2.A2.A$C3DC
2DCA3.A2.ACDC2D3C3.2A2.AC3DC2DCA3.A2.ACDC2D3C3.2A2.A$4D2C2D4.2A2.2D2C
DC2D8.4D2C2D4.2A2.2D2CDC2D$8D8.2DC2DC2D3.2A3.8D8.2DC2DC2D3.2A$8.8D3.
2A3.3D2C3D8.8D3.2A3.3D2C3D$4.2A2.4D2C2D8.8D4.2A2.4D2C2D8.8D$A3.A2.AC
3DC2DCA6.AC6DCA3.A2.AC3DC2DCA6.AC6DC$A4.2A.C4D2CD.A5.ADC5DCA4.2A.C4D
2CD.A5.ADC5DC$8.8D2A6.2C6D8.8D2A6.2C6D$8.8D2.2A.A2.2D2CDC2D8.8D2.2A.A
2.2D2CDC2D$2.A5.2DC5D2.A.2A2.2DCD2C2D2.A5.2DC5D2.A.2A2.2DCD2C2D$2.3A
3.2D3C3D8.8D2.3A3.2D3C3D8.8D$5DC2D5.A2.8D8.5DC2D5.A2.8D$2D4C2D4.A3.2D
2C4D2.2A4.2D4C2D4.A3.2D2C4D2.2A$DC6D4.4ADCDC2D2C.A.A2.2ADC6D4.4ADCDC
2D2C.A.A2.2A$D2C2D3C7.ADCDCDCDC.A.A.A.AD2C2D3C7.ADCDCDCDC.A.A.A.A$2DC
2DC2DA3.2A.ADCD2C2DC.A.2A2.A2DC2DC2DA3.2A.ADCD2C2DC.A.2A2.A$C5D2C3.A.
A.ACDC4DCA.A4.AC5D2C3.A.A.ACDC4DCA.A4.A$8D2.A5.2DC5D2.A5.8D2.A5.2DC5D
2.A$3D2C3D2.3A3.2D3C3D2.3A3.3D2C3D2.3A3.2D3C3D2.3A$3.2A3.5DC2D5.A2.5D
C2D3.2A3.5DC2D5.A2.5DC2D$8.2D4C2D4.2A2.4D2C2D8.2D4C2D4.2A2.4D2C2D$6.
2ADC6D8.8D6.2ADC6D8.8D$.A3.A.AD2C2D3C4.4A4D2C2D.A3.A.AD2C2D3C4.4A4D2C
2D$A.A.A2.A2DC2DC2DA2.A4.A3DCDCDA.A.A2.A2DC2DC2DA2.A4.A3DCDCD$A.A.2A.
2A5D2C3.5A7DCA.A.2A.2A5D2C3.5A7DC$2.A2.A2.8D8.2DE5D2.A2.A2.8D8.2DE5D$
2.A2.A2.3D2C3D2.A.2A2.2D3E3D2.A2.A2.3D2C3D2.A.2A2.2D3E3D$3D2C3D3.2A3.
2D2CDC2D3.2E3.3D2C3D3.2A3.2D2CDC2D3.2E$8D8.8D8.8D8.8D$DC6D.A6.2D2CDC
2D8.DC6D.A6.2D2CDC2D$CDC5DA.A5.DCDCD3C4.2A2.CDC5DA.A5.DCDCD3C4.2A$DC
3D2CD.A3.2A.C2DC4DA2.A2.A.DC3D2CD.A3.2A.C2DC4DA2.A2.A$C3DC2DCA3.A2.AC
DC2D3C3.2A2.AC3DC2DCA3.A2.ACDC2D3C3.2A2.A$4D2C2D4.2A2.2D2CDC2D8.4D2C
2D4.2A2.2D2CDC2D$8D8.2DC2DC2D3.2A3.8D8.2DC2DC2D3.2A$8.8D3.2A3.3D2C3D
8.8D3.2A3.3D2C3D$4.2A2.4D2C2D8.8D4.2A2.4D2C2D8.8D$A3.A2.AC3DC2DCA6.AC
6DCA3.A2.AC3DC2DCA6.AC6DC$A4.2A.C4D2CD.A5.ADC5DCA4.2A.C4D2CD.A5.ADC5D
C$8.8D2A6.2C6D8.8D2A6.2C6D$8.8D2.2A.A2.2D2CDC2D8.8D2.2A.A2.2D2CDC2D$
2.A5.2DC5D2.A.2A2.2DCD2C2D2.A5.2DC5D2.A.2A2.2DCD2C2D$2.3A3.2D3C3D8.8D
2.3A3.2D3C3D8.8D$5DC2D5.A2.8D8.5DC2D5.A2.8D$2D4C2D4.A3.2D2C4D2.2A4.2D
4C2D4.A3.2D2C4D2.2A$DC6D4.4ADCDC2D2C.A.A2.2ADC6D4.4ADCDC2D2C.A.A2.2A$
D2C2D3C7.ADCDCDCDC.A.A.A.AD2C2D3C7.ADCDCDCDC.A.A.A.A$2DC2DC2DA3.2A.AD
CD2C2DC.A.2A2.A2DC2DC2DA3.2A.ADCD2C2DC.A.2A2.A$C5D2C3.A.A.ACDC4DCA.A
4.AC5D2C3.A.A.ACDC4DCA.A4.A$8D2.A5.2DC5D2.A5.8D2.A5.2DC5D2.A$3D2C3D2.
3A3.2D3C3D2.3A3.3D2C3D2.3A3.2D3C3D2.3A$3.2A3.5DC2D5.A2.5DC2D3.2A3.5DC
2D5.A2.5DC2D$8.2D4C2D4.2A2.4D2C2D8.2D4C2D4.2A2.4D2C2D$6.2ADC6D8.8D6.
2ADC6D8.8D$.A3.A.AD2C2D3C4.4A4D2C2D.A3.A.AD2C2D3C4.4A4D2C2D$A.A.A2.A
2DC2DC2DA2.A4.A3DCDCDA.A.A2.A2DC2DC2DA2.A4.A3DCDCD$A.A.2A.2A5D2C3.5A
7DCA.A.2A.2A5D2C3.5A7DC$2.A2.A2.8D8.2DC5D2.A2.A2.8D8.2DC5D$2.A2.A2.3D
2C3D2.A.2A2.2D3C3D2.A2.A2.3D2C3D2.A.2A2.2D3C3D!
NotLiving wrote:dvgrn wrote:It's quite possible that there's some delicately balanced set of 8x8 or smaller P1 stable tiles, with a lot more ON cells: the center would have to be customized for each of the 11 tiles so that it's internally stable.
Hence why I'm punting it to a SAT solver. I am no good at this sort of thing myself.
Edit: just saw that edit. Now attempting to minimize the number of live cells - it'll serve as a sanity check at the very least (if it gets a worse solution than yours, I have a problem).
Also going to run a run maximizing the number of live cells - should get something closer to that weird still life and interesting cascades I am hoping for.
NotLiving wrote:Is there an upper bound to the size of the bounding-box of the minimum-sized-bounding-box-period-N-oscillator?
A for awesome wrote:NotLiving wrote:Is there an upper bound to the size of the bounding-box of the minimum-sized-bounding-box-period-N-oscillator?
There are 2^x states possible in a bounding box of size x, so an absolute limit is log_2(N), taking N = x. Considering symmetry, this becomes log_2(N)+1, due to the fact that a pattern cannot become all 8 of its orientations in a square box or all 4 in a rectangular box. Beyond this, I don't know of any other limit, but there may be proofs that I haven't heard of.
x = 129, y = 106, rule = LifeHistory
97.A11.2A$64.A4.B26.A.A9.B2AB$62.3A3.2B2A24.A.A9.3B$61.A5.2BABAB21.3A
.2A9.B.B$55.2B4.2A4.3BA2B.4B15.A4.B8.5B$54.9B4.12B2.4B9.3AB2A6.6B$54.
7B7.11B.5B11.A.2A4.8B$54.9B5.18B12.13B$54.10B.3B.17B14.13B$54.2B2A28B
13.15B$53.2BA2BA27B2.B10.15B$53.2BA2BA32B5.B.17B$54.2B2A10BA46B$53.
14BABA16B2A13B2A13B$54.13BABA16B2A13B2A14B$54.4B.3B.5BA43B3.B2A$55.2B
5.49B4.A2.A$61.34B2.2B2.B3.6B5.2A.A$60.4B.12B3.7B2.4B13.6B7.A$59.4B2.
12B4.6B17.9B6.2A$58.4B7.7B6.3B19.2A4.4B10.2A$57.4B9.7B6.B21.A5.4B9.A$
56.4B10.6B26.3A7.4B10.A$55.4B12.6B25.A2.3A5.4B5.5A$54.4B12.7B27.A2.A
5.4B4.A$53.4B14.7B25.2A2.A.AB.7B2.B3A$52.4B14.9B29.2AB.7B3.2B.A$51.4B
16.9B30.12B4A$42.2A6.4B5.2A9.6B.4B29.7B2A3BAB2.2A$43.A6.4B5.A3.A7.6B.
4B28.7B2A2B.B3A2.A$43.A.AB.7B.BA.A2.A.A5.6B3.4B27.10B3.B.A.2A$44.2AB.
7B.B2A3.A.A6.6B3.4B25.8B8.A$46.5BA5B4.2A.3A3.6B5.4B23.9B7.2A$46.4BABA
4B5.B4.A3.6B5.4B21.4B2.3B$46.4BABA6B.B2AB3A3.6B7.4B19.4B3.5B$48.3BA7B
.B2A.A6.6B7.4B10.A6.4B7.2A$49.12B9.6B9.4B7.3A5.4B8.A$49.13B9.6B9.4B5.
A7.4B10.3A$49.13B2A2.2A2.6B11.4B4.2A5.4B13.A$50.12BA.A2.A3.6B11.9B4.
4B$48.10B.2B.B.A.A3.6B13.6B5.4B$48.2A3.6B4.2A.2A3.6B12.8B2.4B$49.A2.
6B6.A5.6B11.15B$46.3A4.6B3.A.A6.6B10.14B$46.A5.6B4.2A6.6B11.13B$53.6B
12.6B8.2AB.10B$52.6B12.6B8.A.AB3.B2A3B$53.6B12.6B7.A6.B2A3B$36.2A14.
6B12.6B7.2A6.4B$35.A.A15.6B12.6B15.3B$29.2A4.A16.6B12.6B17.2B.BA$27.A
2.A2.2A.4A13.6B12.6B15.B2ABA.A$27.2A.A.A.A.A2.A12.6B12.6B15.BABABA.A$
30.A.ABABAB15.6B12.6B12.A2.A.A.A.A.2A$30.A.AB2AB15.6B12.6B13.4A.2A2.A
2.A$31.AB.2B17.6B12.6B16.A4.2A$34.3B15.6B12.6B15.A.A$34.4B6.2A7.6B12.
6B14.2A$32.3B2AB6.A7.6B12.6B$32.3B2AB3.BA.A8.6B12.6B$30.10B.B2A8.6B
12.6B$29.13B11.6B6.2A4.6B5.A$28.14B10.6B6.A.A3.6B4.3A$27.15B11.6B5.A
6.6B2.A$26.4B2.8B12.6B3.2A.2A4.6B3.2A$25.4B5.6B13.6B3.A.A.B.2B.10B$
24.4B4.9B11.6B3.A2.A.A12B$9.A13.4B5.2A4.4B11.6B2.2A2.2A13B$9.3A10.4B
7.A5.4B9.6B9.13B$12.A8.4B5.3A7.4B9.6B9.12B$11.2A7.4B6.A10.4B7.6B6.A.
2AB.7BA3B$11.5B3.4B19.4B7.6B3.3AB2AB.6BABA4B$13.3B2.4B21.4B5.6B3.A4.B
5.4BABA4B$3.2A7.9B23.4B5.6B3.3A.2A4.5BA5B$3.A8.8B25.4B3.6B6.A.A3.2AB.
7B.B2A$2A.A.B3.10B27.4B3.6B5.A.A2.A.AB.7B.BA.A$A2.3AB.2B2A7B28.4B.6B
7.A3.A5.4B6.A$.2A2.BA3B2A7B29.4B.6B9.2A5.4B6.2A$3.4A12B30.9B16.4B$3.A
.2B3.7B.B2A29.9B14.4B$4.3AB2.7B.BA.A2.2A25.7B14.4B$7.A4.4B5.A2.A27.7B
12.4B$2.5A5.4B5.3A2.A25.6B12.4B$2.A10.4B7.3A26.6B10.4B$4.A9.4B5.A21.B
6.7B9.4B$3.2A10.4B4.2A19.3B6.7B7.4B$8.2A6.9B17.6B4.12B2.4B$9.A7.6B13.
4B2.7B3.12B.4B$9.A.2A5.6B3.B2.2B2.31BC2B$10.A2.A4.45B2C2B5.2B$11.2AB
3.43BA3B2C.3B.4B$12.14B2A13B2A16BABA13B$13.13B2A13B2A16BABA14B$14.46B
A10B2A2B$14.17B.B5.32BA2BA2B$15.15B10.B2.27BA2BA2B$15.15B13.28B2A2B$
16.13B14.17B.3B.10B$18.13B12.18B5.9B$17.8B4.2A.A11.5B.11B7.7B$17.6B6.
2AB3A9.4B2.12B4.9B$17.5B8.B4.A15.4B.2BA3B4.2A4.2B$17.B.B9.2A.3A21.BAB
A2B5.A$18.3B9.A.A24.2A2B3.3A$17.B2AB9.A.A26.B4.A$18.2A11.A!
A for awesome wrote:Another interesting question related to this might be what the highest finite period attainable by a pattern with a predecessor fitting in a certain-size bounding box.
NotLiving wrote:I'm looking for an upper bound [i]on the lower bound[i].
In other words, a minimum size such that "assuming there is a pN oscillator, there is guaranteed to be one that fits in an NxM box, where N and M are defined as follows...".
The concrete example works, but I'm specifically looking at the remaining no-known-pattern oscillator periods here.
x = 23, y = 18, rule = B3/S23
11bo3$5bo2bo5bo2bo$3b2o2bob5obo2b2o$5bo11bo$3bob2o9b2obo$bobo3b3o3b3o
3bobo$3obo13bob3o$10bobo$4b3o3bobo3b3o$4bo2bo3bo3bo2bo$3b4o2bobobo2b4o
$3b2o2b3obob3o2b2o$2bo3bo3bobo3bo3bo$3bo2bobobobobobo2bo$4bobob2o3b2ob
obo$5bo11bo!
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