Periods by Rotor Minpop

For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.
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83bismuth38
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Periods by Rotor Minpop

Post by 83bismuth38 » September 23rd, 2021, 9:10 am

yesterday, I brought up an interesting question about potentially finding a p19 through brute force. see, because 19<2^5, the smallest rotor a p19 could possibly have would be 5 cells. it was quickly decided that it was probably more than 5 cells, but it started a discussion about the smallest rotors for known periods. here's the list of smallest known up to 13:

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x = 157, y = 15, rule = B3/S23
87b2o$88bo$75bo11bo$36bo37bobo9bob3o23b2ob2o11bobo$35bobo37bo10bobo2bo
23bobobo10b2obob2o$22b2o11b2o25bo20b2obo2b2obo22bo4bo12bobobo$21bobo9b
2o18b2o6bobo9b5o6bobo3bobo8b2ob2o6b2obo5bo8b2obo3bo2bo7b2ob2o$2ob2o6b
2o8bo10bo3b3o13bobo5bobobo5bobo5bobo3bob3obob2o6bobob2o6b2obob2o3bo3b
2obobob2ob2obobo5bobobobo$o3bo2b2obobo5b2ob2o8bo2b2o3bobo10bo7bobobo5b
2obo3bob2o4bo2b3o2bo4b3obo12bobobo3bo2b2obobobo3bo2bo6bo5bo$b3o3b2obo
7b2obobo3b2o3b2o4bob2o2b2o5b2o4b2obo3bob2o5bo3bo8b2o3bo5bo4b4o9bo2b2o
2b2o5bo3bobobo6b2obobobobob2o$10bo10bobo4bo5bobobo5bo2b3o7bobo4bob2o5b
o3bo10b3o5bo2b2obo2bo10bo12bo5b2o7bobo2bobob2obo$10bo2bo7bo2b4o6bo3bo
7b2obob2o7b4o10b3o7b3o10b2obo3bo3b2o7b7o6b4o13b2o3bo$b3o7b2obo5b2o13b
3o11b2o2bo30bo2b2o9bo3bobobo2bo13bo10b2obo10bobobo$o3bo9bo9b2o11bobo5b
ob2o3b2o8b2o10b2obo9bobo8bobobobob2o11b2o7b5obob2o9bo2b2o$2ob2o9b2o8b
2o12b2o5b2obo13b2o10bob2o10bo10b2ob2o15b2o7bo2bobo13b2o!
that 'smallest power of 2' lower bound was nice, but it's obviously not accurate in the slightest. p2, p3, p4, p5, and p6 each have 1 rotor cell extra for the period (2 for 2, 3 for 3, etc.), but p7 and onwards have some oddities where the size of the rotor is less than the period, such as p7 with 5 cells or p13 with 10. personally, I was wondering what the smallest rotor for every period is, as everything up to p4 have been proven to be minimal (by hotdogpi, which I will allow to explain because I really don't understand it fully). 5:4, 6:5, and 8:7 seem possible if not likely.

there was also a suggested 'smallest stator' counterpart suggested. it just contains phoenix 1, the strictly volitle p3, mazing, etc. until 7, which is this, as far as I can tell:

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x = 21, y = 21, rule = B3/S23
4b5o3b5o$4bo2bobobobo2bo$4bo2bobobobo2bo$4bo3bo3bo3bo$4o3bo5bo3b4o$o6b
o5bo6bo$o6bo5bo6bo$3ob3obo3bob3ob3o$o2bo3bo5bo3bo2bo$b2o15b2o2$b2o15b
2o$o2bo3bo5bo3bo2bo$3ob3obo3bob3ob3o$o6bo5bo6bo$o6bo5bo6bo$4o3bo5bo3b
4o$4bo3bo3bo3bo$4bo2bobobobo2bo$4bo2bobobobo2bo$4b5o3b5o!
looking to support hive five (n+18) and charity's oboo reaction (2n+18)

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x = 28, y = 13, rule = B3/S23
19bo$3bo15bo4b2o$2bobo14bo4bobo$2bobo20b2o$3bo11b3o2$25b3o$b2o22b3o$o
2bo$b2o12b2o$10b2o2bobo$bo8b2o2b2o$obo7b2o!

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wwei47
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Joined: February 18th, 2021, 11:18 am

Re: Periods by Rotor Minpop

Post by wwei47 » September 23rd, 2021, 9:22 am

83bismuth38 wrote:
September 23rd, 2021, 9:10 am
there was also a suggested 'smallest stator' counterpart suggested. it just contains phoenix 1, the strictly volitle p3, mazing, etc. until 7, which is this, as far as I can tell:

Code: Select all

x = 21, y = 21, rule = B3/S23
4b5o3b5o$4bo2bobobobo2bo$4bo2bobobobo2bo$4bo3bo3bo3bo$4o3bo5bo3b4o$o6b
o5bo6bo$o6bo5bo6bo$3ob3obo3bob3ob3o$o2bo3bo5bo3bo2bo$b2o15b2o2$b2o15b
2o$o2bo3bo5bo3bo2bo$3ob3obo3bob3ob3o$o6bo5bo6bo$o6bo5bo6bo$4o3bo5bo3b
4o$4bo3bo3bo3bo$4bo2bobobobo2bo$4bo2bobobobo2bo$4b5o3b5o!
I can't shrink the stator, but I can increase the volatility.

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x = 31, y = 31, rule = Life
10bo9bo$10bo9bo$10bo9bo3$9b5o3b5o$9bo2bobobobo2bo$9bo2bobobobo2bo$9bo
3bo3bo3bo$5b4o3bo5bo3b4o$3o2bo6bo5bo6bo2b3o$5bo6bo5bo6bo$5b3ob3obo3bo
b3ob3o$5bo2bo3bo5bo3bo2bo$6b2o15b2o2$6b2o15b2o$5bo2bo3bo5bo3bo2bo$5b3o
b3obo3bob3ob3o$5bo6bo5bo6bo$3o2bo6bo5bo6bo2b3o$5b4o3bo5bo3b4o$9bo3bo3b
o3bo$9bo2bobobobo2bo$9bo2bobobobo2bo$9b5o3b5o3$10bo9bo$10bo9bo$10bo9b
o!
Help me find high-period c/2 technology!
My guide: https://bit.ly/3uJtzu9
My c/2 tech collection: https://bit.ly/3qUJg0u
Overview of periods: https://bit.ly/3LwE0I5
Most wanted periods: 76,116

hotdogPi
Posts: 1587
Joined: August 12th, 2020, 8:22 pm

Re: Periods by Rotor Minpop

Post by hotdogPi » September 23rd, 2021, 10:56 am

Why p4 with 3 rotor cells is impossible:

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x = 3, y = 4, rule = LifeHistory
2.A$.A$ADC$3B!
[[ THUMBNAIL ]]
Someone else proved that they can't all touch each other, so they must form a straight or crooked line. Each cell must have a number of stator cells next to it, and the only way it works for p4 is 3 on the edges and 1 in the middle (shown by someone else, but not that hard to verify).

In the diagram above, green is rotor, and white is one of the stator cells.
The red cell must be off, as it is adjacent to 0, 1, or 2 on rotor cells at different times; this cannot allow it to be on and stay on, so it must be permanently off. If 0, 1, or 2 blue cells are stator cells, the red cell will turn on with 2, 1, or 0 adjacent rotor cells on. If all three blue cells are stator cells, the middle of the three blue cells will die from having 4 neighbors.
User:HotdogPi/My discoveries

Periods discovered: 5-16,⑱,⑳G,㉑G,㉒㉔㉕,㉗-㉛,㉜SG,㉞㉟㊱㊳㊵㊷㊹㊺㊽㊿,54G,55G,56,57G,60,62-66,68,70,73,74S,75,76S,80,84,88,90,96
100,02S,06,08,10,12,14G,16,17G,20,26G,28,38,47,48,54,56,72,74,80,92,96S
217,486,576

S: SKOP
G: gun

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