Polybishops

For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.
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Tropylium
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Polybishops

Post by Tropylium » September 23rd, 2021, 6:22 pm

By analogy with the synonym "polyking" for polyplets, let a polybishop be a polyplet where all squares are connected diagonally. Equivalently, these are polyominoes rotated 45° so that they now occupy only every other square in a lattice. Nothing new combinatorically, but in CA…

We can note that it's easily proved that in CGoL there are arbitrarily large polybishops that either cleanly die off (fuse segments) or are still lives (barges).

Here are the polybishops up to 7 cells:

Code: Select all

x = 402, y = 301, rule = B3/S23
310bo$310b2o$269bo$268b2o19$111bo157bo$110b2o38bo117b2o$150b2o6$150bo
9bo$150b2o8b2o13$137bo$136bobo$135bo$134bo$135bo$136bo$119bo109bo$118b
2o108b2o4$284bobo$281bo3bo$282bobo$283bo14$160b2o$160bo4$118b2o$119bo
24$300b2o$300bo17$350bo49bo$350b2o48b2o9$158bo49bo49bo49bo$158b2o48b2o
48b2o48b2o13$137bo188bo49bo$136bo44bo44b2o48b2o45bobo47bobo$126b2o3bob
obo40b2o4bo3bo40bo4bo44bo4bo3bo37bo49bo$127bo4bobo42bo5bobo47bobobo45b
obo37bo49bo$184bobo47bobo47bobo35bo49bo$235bo49bo37bo47bo4$110b2o2bo$
114bo2$113bo$112bo2$112bo225b2o48b2o$338bo49bo$112bo9$bo$2o4$126bo89bo
$125bo29bobo27bo29bo$124bobo27bobo27bobo25bobobo$123bobo27bobo27bo27bo
bo$122bo29bo31bo$185bo123bo29bo$186bo121b2o28b2o4$13bob2o29bo217bo$12b
o12bo5b2o12bobo16bobo16bobobo175bobo48bobo27bobo$24bobo4bo12bo18bobobo
16bobo175bo3bo46bobobo25bobobo$12bo10bo3bo15bobo16bo20bo179bobo46bobo
29bo$15bo10bo316bo$12bo3bo2$12bo3bo$13bobo$27bo$26bo19bo17bobo9bo$25bo
19bobo17bobo7bobo16bo$24bo17bobo19bo9bo18bobobo$23bo19bo19bo9bo18bo3bo
$22bo49bo5$137bo$26bo19bobo17bo30bo27bo10bo19bo39bo7bo17bo3bo19bo9bo8b
o$25bo8bobo8bobo15bobobo17bo10bo29bo8bo9bo9bo9bobo7bo7bo3bo7bobo7bo9bo
5bobobobo4bobo10bobo7bo8bo$24bobo6bobo8bo19bo17bobobo8bobo25bobo8bo11b
o9bo5bobobo7bobo5bobobo7bo9bobo7bobo7bo8bobobo4bobo9bobo6bobo$23bo3bo
4bo10bo19bo19bo10bo27bobobo6bo9bobobo5bobo5bobo7bobo3bo3bo3bo5bobo9bo
7bobo18bo3bo6bo9bo8bo3bo$33bo50bo8bo38bo9bobo7bobo17bo19bo9bobo7bobo
27bo9bo14bo$131bo119bo3$264bo$265bo$34bobo7bo18bo10bo10bo50bo9bo49bo
57bo9bobo$24bobo8bo9bo7bobo8bobo6bobo8bobo8bobo37bobo7bobo4bo11bobo6bo
13bo7bo9bo36bobobo8bobo5bobo$23bobo8bobo7bobo7bobo6bobo6bobo8bobobo4bo
bobo26bo10bo7bobobo6bo3bo5bobobo6bobo7bobobo5bobo7bobobo32bobobobo4bob
obo5bo$22bo3bo6bo9bo9bobo6bo12bo17bo28bobobo6bobo7bo10bobo5bo12bobo5bo
bo7bo9bo3bo45bo$42bo78bo3bobo8bo18bo7bo10bo7bo9bobo7bo$156bo16bo3$254b
o$127bo125bo90bo$25bo10bo29bo7bobo49bobo7bobo17bo11bo14bobo7bobo8bo11b
o9bo25bobobo19bo7bo58bobo$24bobo8bo8bobo8bo9bo7bobo9bo7bobobo27bobo7bo
bo7bo9bobobo3bobobobo14bo9bobo8bo9bo9bobo7bobo8bobo2bo3bo17bobo9bo58bo
$23bobo8bobo6bobo6bobobo7bo11bo7bobo5bobobo27bo9bo3bo5bobo7bo3bo5bobo
16bobo7bo3bo6bobo5bobobo7bobo7bobo8bobo26bo9bobo58bo$24bo8bobo6bobo8bo
bo7bobo9bo7bo3bo37bo9bo7bo11bo26bo3bo5bo10bo9bo3bo5bo9bo3bo4bobo28bobo
7bo60bobo$62bo19bo59bo59bo19bo13bo6bo28bo9bo$141bobo59bo77bo4$25bo159b
obo87bobo$24bobo19bo10bo18bo48bo10bo8bo18bo9bo9bobo9bo18bo10bo17bo9bo
11bo9bo$23bobo17bobo8bobo18bobo46bobo8bo8bo9bobo8bobo5bobo7bo11bobo16b
obo8bobo15bobobo5bobobo5bobo9bobo$22bo21bobo6bobo18bo48bobo8bo8bobo7bo
bo8bobo5bobobo5bo11bo18bobo8bo19bobo5bobobo5bobobo7bo$43bo8bo20bo50bob
o6bobo6bobobo5bobo8bo9bo7bo11bo18bobo8bobo17bo23bo5bo$74bo57bo3bo18bo
6bo29bo29bo$193bo3$175bobo$145bo28bobo$133bobo8bo9bobo9bo6bo21bobo$
125bo8bobo8bo9bobo5bobobo4bo21bobo141b2o$124bobo6bo8bobobo7bo9bo8bo19b
o145bo$123bo3bo4bobo8bo9bo9bobo26bobo$122bo31bo$121bo5$24bo99bo9bo9bo
19bobo9bo9bo$23bobo97bo9bo9bo10bo10bobo7bobo7bo$22bo3bo97bobo5bobobo5b
obobo6bobo8bo9bo9bobo6bobo6bobobo$123bobo9bo7bobo6bo3bo6bo9bo9bo3bo4bo
bobo6bobo$10b2ob2o107bo11bo16bo5bo4bo9bo13bo4bo5bo4bo3bo$171bo2$10bo
13bo24bo$10bo2bo9bo5bo5bobo2bobo5bobo$14bo7bo3bobobo3bobo4bobo3bo$21bo
5bo5bo6bo5bo$14bo5bo$10b2obo$21bo11bo12bo5bo$22bo3bobo3bo4bobo5bo5bobo
$21bobo3bo3bobo2bobobo3bobo3bo$20bo5bobo3bo10bo7bo4$163bobo18bobo17bob
o19bo36bo3bo14bo3bo20bo$10bo3bo108bobo17bo20bobo16bobo17bobo19bo17bobo
bo16bobo16bobo18bobo$10bo3bo107bobobo15bobobo16bobo18bobo15bobobo15bob
obo17bobo16bobo18bobo16bobo17bo3bo$23bo8bo9bo80bobo17bobobo14bo20bo39b
o19bo18bo20bo18bo19bobobo$22bo4bo3bo4bobo2bobo178bo19bo60bo19bobo$10b
2ob2o6bo4bobobobo2bobo4bo$20bo4bo2$14bo$14bo11$154bo91bo$128bo8bo15bob
o19bo21bo16bobo26bobo41bo37bo$123bobobo4bobobo15bo3bo15bo3bo17bobo16bo
bo28bobo37bobo35bobobo$122bobobo6bobobo17bo17bobobo15bobo16bo30bo39bob
obo35bo3bo$154bo19bo17bo20bo28bo39bo39bo$191bo22bo!
Nothing particularly special in terms of their final evolutions though. These seem to be a bit less fecund than polyominoes. There is just one polybishop at ≤5 cells that takes more than 6 generations to settle (a T-tetromino parent), just two at 6 cells than take more than 20 (a C-heptomino parent and a LoM precedessor). At 7 cells one evolves into glider + R in 6 generations, while another is just slightly more long-lived, taking 1280 gens to stabilize.

It might be of some interest that a lot of polybishops initially evolve into polyominoes: 3/5 = 60% of tetrabishops, 8/12 = 67% pentabishops, 18/35 = 51% hexabishops, 55/108 = 51% heptabishops. Since every polybishop is a rotation of a corresponding polyomino, we could ask how this iterates? One fixed point is the Z-pentabishop which evolves into a Z-pentomino; there might be others.
(This should be somewhat easier to investigate than my old question about polyominoes that evolve into polyominoes.)

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Moosey
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Re: Polybishops

Post by Moosey » September 27th, 2021, 9:28 pm

Terminology comment: "Polybishop" would to me suggest that a bishop has a path where its moves place it on squares in the polybishop such that it can move between any two squares in it (i.e. a connected graph is produced if you follow the algorithm where when any two cells are along the same diagonal, you put an edge between them) which is rather stronger than polybishops as defined here (although still not quite as strong as "collections of cells which are the same color on a checkerboard pattern").

Personally, I'd suggest the term "polyferz" as an alternate term at least (although this pays the price of making sense to many)

Anyway, conjecture, which should be pretty easy to prove or disprove: if (but not only if) every cell in a polyferz has a VN neighbor which will be born by B3e, and no cell has the S4c neighborhood (i.e. nobody's going to die) then it will evolve into a polyomino

(here's the fairly trivial counterexample to what I almost conjectured, which didn't have the S4c condition: )

Code: Select all

x = 3, y = 3, rule = B3/S23
obo$bo$obo!
here's the trivial example which doesn't meet this criteria but evolves into a polyomino:

Code: Select all

x = 3, y = 3, rule = B3/S23
o$bo$2bo!
Semi-relatedly, I was going to conjecture that for any polyomino there is a polyferz which evolves into it, but I don't think any polyferz evolves into the blinker? There's a trivial polybishop (under the wider definition) which does:

Code: Select all

x = 4, y = 3, rule = B3/S23
bobo$o$bo!
I think there should be a proof via some form of case analysis, but I'm not sure:

Code: Select all

x = 26, y = 10, rule = LifeHistory
.A.A7.A$ADCDA5.ADCDA$13.A5$A3.A5.A$.CDC7.CDC8.CDC$2.A9.A.A6.A.A.A!
My CA rules can be found here

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MathAndCode
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Re: Polybishops

Post by MathAndCode » September 27th, 2021, 11:56 pm

Moosey wrote:
September 27th, 2021, 9:28 pm
Anyway, conjecture, which should be pretty easy to prove or disprove: if (but not only if) every cell in a polyferz has a VN neighbor which will be born by B3e, and no cell has the S4c neighborhood (i.e. nobody's going to die) then it will evolve into a polyomino
Here is a counterexample.

Code: Select all

x = 6, y = 6, rule = B3/S23
2bo$3bo$obobo$bobobo$2bo$3bo!


Edit: Here is a smaller counterexample.

Code: Select all

x = 5, y = 6, rule = B3/S23
4bo$3bo$obobo$bobo$2bo$3bo!


Another edit: Here is a smaller counterexample.

Code: Select all

x = 3, y = 5, rule = B3/S23
2bo$bo$obo$bo$o!

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Tropylium
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Re: Polybishops

Post by Tropylium » September 28th, 2021, 8:06 am

Moosey wrote:
September 27th, 2021, 9:28 pm
"Polybishop" would to me suggest that a bishop has a path where its moves place it on squares in the polybishop such that it can move between any two squares in it (i.e. a connected graph is produced if you follow the algorithm where when any two cells are along the same diagonal, you put an edge between them) which is rather stronger than polybishops as defined here (although still not quite as strong as "collections of cells which are the same color on a checkerboard pattern").

Personally, I'd suggest the term "polyferz" as an alternate term at least (although this pays the price of making sense to many)
"Bishop" seems sufficient to me, given how the usual standard is also to describe polyominos as "rookwise connected" and not "wazirwise connected". Both rooks and bishops can besides only move thru squares they could also stop at / cannot jump over occupied squares; usually also not "non-board squares" whenever operating on a nonconvex board (though this is, of course, not defined within the usual rules of chess).
Moosey wrote:
September 27th, 2021, 9:28 pm
Semi-relatedly, I was going to conjecture that for any polyomino there is a polyferz which evolves into it, but I don't think any polyferz evolves into the blinker?
It doesn't seem to me that any width 1 length ≥3 line polyomino, or any polyomino with this kind of a segment, has a polybishop ancestor. Almost any polyomino that does, seems to be coverable with L-triminoes, though some cases like this can have diagonally meeting domino ends:

Code: Select all

x = 5, y = 5, rule = B3/S23
o3bo$bobo$obo$bobo$obobo!

hotdogPi
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Re: Polybishops

Post by hotdogPi » September 28th, 2021, 8:10 am

I don't consider knights to jump over pieces, either — they move between them. If you take a straight line between two squares a knight's move away, it doesn't encounter any other square centers on the way. There's enough space to fit a knight between adjacent pieces while the knight is (temporarily) not bound to the grid.
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