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## Smallest Oscillators Supporting Specific Periods

For discussion of other cellular automata.

### Re: Smallest Oscillators Supporting Specific Periods

Macbi wrote:4 cell, period 4N+3 (N>1):
`x = 6, y = 4, rule = B2ei3ir4r/S012ce3j5bo\$2bo\$o\$2bo!`
This completes the proof that all periods can be achieved in at most 4 cells.
2718281828 wrote:4 cells for 4N+1 (N>2):
`x = 9, y = 22, rule = B2ei3i4r/S012cek3j4t5bo\$2bo\$o\$2bo3\$6bo\$2bo\$o\$2bo3\$7bo\$2bo\$o\$2bo3\$8bo\$2bo\$o\$2bo!`

2718281828's oscillators are p(4n+3), not p(4n+1), unfortunately.
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### Re: Smallest Oscillators Supporting Specific Periods

:-o

How's this?
`x = 15, y = 4, rule = B2ein3aciq4jkny5any6ck/S012ce3ejnr4jknrwy5-aiky6ci7e14bo\$2bo\$o2bo\$2bo!`

Macbi

Posts: 668
Joined: March 29th, 2009, 4:58 am

### Re: Smallest Oscillators Supporting Specific Periods

Macbi wrote::-o

How's this?
`x = 15, y = 4, rule = B2ein3aciq4jkny5any6ck/S012ce3ejnr4jknrwy5-aiky6ci7e14bo\$2bo\$o2bo\$2bo!`

Perfectly valid for p13+4k, so now we have all periods > 9 in at most four cells.
Of course, the specific lower bound doesn't matter much, considering the wide range of periods that now have 2-cell solutions.
LifeWiki: Like Wikipedia but with more spaceships. [citation needed]

Posts: 1875
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### Re: Smallest Oscillators Supporting Specific Periods

Current status:
`Period 0, 1 cellPeriods 2 - 104, 2 cellsEven periods 106 - 204, 2 cellsPeriods 109, 208, 212, 2 cellsPeriods 117, 119, 127, 139, 165, 189, 214, 216, 224, 232, 234, 240, 248, 263, 274, 286, 304, 314, 320, 344, 358, 368, 376, 392, 3 cellsAll other periods, 4 cells`

Macbi

Posts: 668
Joined: March 29th, 2009, 4:58 am

### Re: Smallest Oscillators Supporting Specific Periods

Some more results - the 5x9 search seems to be providing many more high period results than 9x5

p105, 2 cells
`x = 3, y = 1, rule = B2-ck3akqy4cijknry5iknqy6ac8/S01e2ik3-cijn4cjnqrty5-jkq6aci7eobo!`

p107, 2 cells
`x = 3, y = 1, rule = B2-ck3ak4ijnr5-cjqy6cei8/S01e2ik3-acik4-erwz5-ajkq6-kn7eobo!`

p111, 2 cells
`x = 3, y = 1, rule = B2aei3akqy4ijknry5ijnqy6ac8/S01e2eik3-aijn4-eikz5-ajk6-kn7eobo!`

p206, 2 cells
`x = 3, y = 1, rule = B2aei3aky4inqr5-ceq6ci8/S01e2ikn3-cikq4cijnqty5-acjk6cik7eobo!`

p210, 2 cells
`x = 3, y = 1, rule = B2-ck3akqy4iknqr5ijkny6ci/S01e2eik3aeknr4-eqrw5-aikq6-n7eobo!`

Edit:
p216, 2 cells
`x = 3, y = 1, rule = B2aei3-eikr4-acqwz5-aqry6-kn7e/S01e2-ac3enqry4ijknrtz5cer6-an8obo!`

Updated status:
`Period 0, 1 cellPeriods 2 - 112, 2 cellsEven periods 114 - 212, 2 cellsPeriod 216, 2 cellsPeriods 117, 119, 127, 139, 165, 189, 214, 224, 232, 234, 240, 248, 263, 274, 286, 304, 314, 320, 344, 358, 368, 376, 392, 3 cellsAll other periods, 4 cells`
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

I just noticed that I accidentally wrote "period 0" instead of "period 1" in that status. I guess since a period 0 pattern is any one which never repeats the smallest period 0 oscillator actually does have only one cell:
`x = 1, y = 1, rule = B1c/So!`
and technically the smallest period 1 oscillator has 0 cells:
`x = 0, y = 0, rule = B/S!`

So we have
`Period 0, 1 cellPeriod 1, 0 cellsPeriods 2 - 112, 2 cellsEven periods 114 - 212, 2 cellsPeriod 216, 2 cellsPeriods 117, 119, 127, 139, 165, 189, 214, 224, 232, 234, 240, 248, 263, 274, 286, 304, 314, 320, 344, 358, 368, 376, 392, 3 cellsAll other periods, 4 cells`

Macbi

Posts: 668
Joined: March 29th, 2009, 4:58 am

### Re: Smallest Oscillators Supporting Specific Periods

p214, 2 cells
`x = 3, y = 1, rule = B2aei3ajq4-acjkq5-akqr6ci/S01e2eik3-ain4ajkntw5cejny6-en7e8obo!`

The 5x9 search is now complete, p214 being the only new period found after p216. The 9x5 search looks like it will take much longer to complete. It's currently reporting 79.4314% (in the forward direction) and the output file is now larger than 6 GB. I added some filtering of low period oscillators to the code, but I don't want to restart the search so I'm stuck with letting that file grow very big. I've started a 9x5 search in the reverse direction, but considering the current rate of progress I'm not expecting to be able to completely cover that search space.

Edit: Included p214 which I had overlooked somehow.

The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

I tried adapting pop2osc for a 3-cell oscillator search using o.o.o as the starting pattern. It seems to work reasonably well, but search progress is much slower. I was able to complete 5x5 (no new record smallest oscillators found) but anything bigger seems like it's going to run for much longer and I didn't get any promising results. [Edit: I just realised I removed the B2a requirement but forgot about the exploding pattern test, so this result is unreliable.] Instead, I dusted off my Golly Python script for random rule search and ran it to find a few more 3-cell oscillators at high periods. Most of the results have o.o.o as the starting pattern but I also included a few other patterns in the search and there are some results with them as well. See below for results. I included B2a in the allowed transitions, but the highest period oscillator which used it was p98.

Patterns included in search:
`ooo     oo.o    o.o.o   o.o     oo      o.o                        .o      ..o     .                                        o`

The first one is the only one not represented in the results below. I'm considering adapting pop2osc.cpp to run this kind of random rule search with the o.o.o pattern - I suspect it would run a good deal faster than my Python script.

Updated summary:
`Periods 2 - 112, 2 cellsEven periods 114 - 216, 2 cellsOdd periods 113-119, 123, 127, 131, 137, 139, 163, 165, 189, 263, 279, 3 cellsEven periods 218, 220, 224, 228, 230-234, 240-248, 254, 258, 260, 266, 274, 280, 284-288, 294, 302, 304, 314, 320, 336, 344, 356, 358, 368, 376, 392, 396, 412, 3 cellsAll other periods, 4 cells`

Results from 3-cell random rule search:
Odd periods:
p113, 3 cells
`x = 5, y = 1, rule = B2-ae3aikny4ckqy5-ain6ek/S1e2i3-ceir4eij5j6iobobo!`

p115, 3 cells
`x = 3, y = 2, rule = B2-an3r4-cikw5-cinq8/S12k3aekr4act5aceny6k7cobo\$bo!`

p123, 3 cells
`x = 5, y = 1, rule = B2-ai3r4ijnqwyz5jnr6i/S012aik3acq4inqryz5jn6eiobobo!`

p131, 3 cells
`x = 5, y = 1, rule = B2cik3acj4ceiyz5-cijr/S02-cn3eqy4aitw5eky6ai8obobo!`

p137, 3 cells
`x = 5, y = 1, rule = B2cn3aeijk4jnwz5ajny6cik7e8/S01e2i3acknq4-ijkwz5c6k7eobobo!`

p163, 3 cells
`x = 5, y = 1, rule = B2-a3acky4cjnwyz5cnq6ci7/S1e3ejkqy4aez5an6cei7eobobo!`

p279, 3 cells
`x = 5, y = 1, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8obobo!`

Even periods:
p218, 3 cells
`x = 5, y = 1, rule = B2cen3eiy4centz5jqy6-ek/S01c2en3cjq4-nryz5einry6a7e8obobo!`

p220, 3 cells
`x = 5, y = 1, rule = B2cik3-ikn4-jrtyz5aiy6-ai7e8/S02in3anq4a5cei6ik8obobo!`

p228, 3 cells
`x = 3, y = 2, rule = B2-a3ajy4aejkwy5qr6n7/S01c2ce3eik4e5jn6kobo\$bo!`

p230, 3 cells
`x = 5, y = 1, rule = B2-a3acer4cinr5acery6aik7e/S1e3acry4ejr5-aeq6ein7e8obobo!`

p242, 3 cells
`x = 5, y = 1, rule = B2c3aijry4eiktyz5ejkr6ck8/S02ckn3inr4acent5-aeir6aik8obobo!`

p244, 3 cells
`x = 3, y = 2, rule = B2-ak3aq4ajw5ary6aik8/S13enqy4-irtw5aijr6ci7eobo\$bo!`

p246, 3 cells
`x = 3, y = 2, rule = B2-a3jr4acjrz5aejq6ai7/S12in3qy4acenry5knqr6ci7c8obo\$bo!`

p254, 3 cells
`x = 5, y = 1, rule = B2-an3ky4aejnrw5qry6cen7c/S1e2e3kny4aceknrt5i6i8obobo!`

p258, 3 cells
`x = 5, y = 1, rule = B2-ak3-anr4eijkqw5nq6ai7/S12n3kry4jqrtwyz5-ikq6ci7eobobo!`

p260, 3 cells
`x = 5, y = 1, rule = B2-a3aky4ijnqrtz5ikr6-ai7e8/S1e2-ai3acek4eiyz5-cjq6-knobobo!`

p266, 3 cells
`x = 5, y = 1, rule = B2cek3aknq4ciz5ajky6cn8/S01c2ekn3ckr4iy5iy6ain7cobobo!`

p280, 3 cells
`x = 5, y = 1, rule = B2-ae3acekq4jnqtyz5jn6ac7/S12ek3kry4iqy5ij6-kn7e8obobo!`

p284, 3 cells
`x = 3, y = 2, rule = B2-an3aq4c5iy6i/S01c2n3nqy4cejkwy5aeik6aik8obo\$bo!`

p288, 3 cells
`x = 5, y = 1, rule = B2-an3eikq4cn5eikqy6ik8/S2-n3ry4iy5aejkn6aen8obobo!`

p294, 3 cells
`x = 5, y = 1, rule = B2-an3acqr4eqtz5-jknr6ik7/S01c2ac3aei4cnqrtz5kry6ik8obobo!`

p302, 3 cells
`x = 5, y = 1, rule = B2-a3iry5-acey6e7/S1e2ein3-ijkr4aceiknz5-eknq6ei7cobobo!`

p336, 3 cells
`x = 5, y = 1, rule = B2-an3cikr4ekrtw5cinry6kn8/S1e2ein3ac4eijnrwy5ijk6ek7cobobo!`

p356, 3 cells
`x = 4, y = 1, rule = B2ce3-ackn4kntz5j6ae/S12i3ij4ijnqz5j6ei7c2obo!`

p396, 3 cells
`x = 3, y = 3, rule = B2eik3-eknq4ainqwyz5i6-in7c/S02k3acjqy4ajobo2\$o!`

p412, 3 cells
`x = 3, y = 2, rule = B2-ai3jk4eiqt5aciq6aek8/S12cen3-einr4ny5ary6aek82o\$2bo!`
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

wildmyron wrote:p228, 3 cells
`x = 3, y = 2, rule = B2-a3ajy4aejkwy5qr6n7/S01c2ce3eik4e5jn6kobo\$bo!`

A bit off-topic, but this rule somehow also manages to have a three-cell spaceship, of the somewhat unusual speed of 4c/28 diagonal to boot:
`x = 4, y = 4, rule = B2-a3ajy4aejkwy5qr6n7/S01c2ce3eik4e5jn6ko\$bo2\$3bo!`

wildmyron wrote:p279, 3 cells
`x = 5, y = 1, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8obobo!`

And this rule has a similarly sparky p49 that has a minimum of five cells:
`x = 5, y = 5, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a82bo2\$obobo2\$2bo!`

And a p71 with a minimum of six:
`x = 11, y = 11, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8o4\$5bo\$4bobo\$5bo4\$10bo!`

wildmyron wrote:p284, 3 cells
`x = 3, y = 2, rule = B2-an3aq4c5iy6i/S01c2n3nqy4cejkwy5aeik6aik8obo\$bo!`

This rule, meanwhile, has an impressive class F (?) replicator:
`x = 188, y = 186, rule = 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77topaz

Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Smallest Oscillators Supporting Specific Periods

Neat! There are so many of these things I often don't look at the rules themselves so it's nice to see some of the other interesting things that exist in them.

77topaz wrote:
wildmyron wrote:p279, 3 cells
`x = 5, y = 1, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8obobo!`

And this rule has a similarly sparky p49 that has a minimum of five cells:
`x = 5, y = 5, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a82bo2\$obobo2\$2bo!`

And a p71 with a minimum of six:
`x = 11, y = 11, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8o4\$5bo\$4bobo\$5bo4\$10bo!`

Hah, and you even apgsearched it. I'm surprised that this little c/3 didn't show up, though there were perhaps not enough soups for that.
`x = 7, y = 7, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a84bo\$2o3bo\$2b2o\$2bo3bo\$2b2o\$2o3bo\$4bo!`

There's also a 2c/5.
`x = 16, y = 15, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8o\$3o10bo\$bo2b2o7b2o\$o4bo8b2o\$2bo6b2o2b2o\$5bo3b3obo\$bo3b4ob2o\$b3o2bo3bo\$bo3b4ob2o\$5bo3b3obo\$2bo6b2o2b2o\$o4bo8b2o\$bo2b2o7b2o\$3o10bo\$o!`

c/2 seems viable, but I haven't been able to find one.

OK, that was a little diversion, apologies for extending it.
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

Nice! The rules with three-cell oscillators do seem to be generally more interesting than the previously posted two-cell ones - because all of those two-cell rules had B2a, and so were explosive.

77topaz

Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Smallest Oscillators Supporting Specific Periods

77topaz wrote:And this rule has a similarly sparky p49 that has a minimum of five cells:
`x = 5, y = 5, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a82bo2\$obobo2\$2bo!`

It has a phase with only four cells...
`x = 3, y = 3, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8bo\$obo\$bo!`

EDIT: Wow.
`x = 23, y = 6, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8bo\$obo\$bo\$21bo\$20bobo\$21bo!`
This post was brought to you by the Element of Magic.

Plz correct my grammar mistakes. I'm still studying English.

Working on:

Nothing.

Favorite gun ever:
`#C Favorite Gun. Found by me.x = 4, y = 6, rule = B2e3i4at/S1c23cijn4ao2bo\$4o3\$4o\$o2bo!`
Hunting

Posts: 1058
Joined: September 11th, 2017, 2:54 am
Location: Ponyville, Equestria

### Re: Smallest Oscillators Supporting Specific Periods

Hunting wrote:EDIT: Wow.
`x = 23, y = 6, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8bo\$obo\$bo\$21bo\$20bobo\$21bo!`

That's an interesting reaction. Unfortunately, because of the large maximum size of the oscillators, it appears it can't be turned into a wick as the different sections interfere with each other:
`x = 23, y = 12, rule = B2-a3ar4eiqz5-aiy6eik7e8/S02-c3eijn4tw5ein6a8:T0,126\$bo\$obo\$bo\$21bo\$20bobo\$21bo!`

77topaz

Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Smallest Oscillators Supporting Specific Periods

Another update after running my random rule search with maxGen set to 1000:

Summary:
`Periods 2 - 112, 2 cellsEven periods 114 - 216, 2 cellsOdd periods 113-119, 123, 127, 131, 137, 139, 163, 165, 189, 191, 263, 279, 3 cellsEven periods 218, 220, 224, 228, 230-234, 240-250, 254, 258, 260, 266, 274, 278, 280, 284-288, 292, 294, 302, 304, 310, 314, 316, 320, 336, 340-344, 348, 356, 358, 368, 376, 392, 396, 398, 412, 420, 434, 674, 788, 998, 3 cellsAll other periods, 4 cells`

New oscillators:

Odd periods:
p191, 3 cells
`x = 5, y = 1, rule = B2-an3cnqry4etwz5ciny6ckn7c/S1e2ek3cen4aknqz5cy6cek7obobo!`

Even periods:
p250, 3 cells
`x = 5, y = 1, rule = B2-ae3-jknr4ry5qy6cn7c/S02in3-cn4cejq5-cej6cinobobo!`

p278, 3 cells
`x = 5, y = 1, rule = B2-an3acy4cijkw5y/S02aik3jqry4aeikr5jy6eobobo!`

p292, 3 cells
`x = 3, y = 3, rule = B2-ae3aeikn4-jwy5cekr6k7/S02n3acikr4-anryz5ckr6e7e8obo2\$o!`

p310, 3 cells
`x = 3, y = 2, rule = B2-a3aeqry4acjkt5ein6ei7e/S1c2a3ejny4ceijrty5jy6ci7cobo\$bo!`

p316, 3 cells
`x = 4, y = 1, rule = B2-ak3acek4aetyz5cekr6k/S1e2ak3jn4ckn6ai2obo!`

p340, 3 cells
`x = 5, y = 1, rule = B2cik3aceiq4-cnrty5-ajqy6ac7/S1e3q4cijz5ir6cnobobo!`

p342, 3 cells
`x = 3, y = 2, rule = B2-a3i4eknw5-jn6in7/S02e3cei5aeqr6-anobo\$bo!`

p348, 3 cells
`x = 5, y = 1, rule = B2cin3-eknr4cijky5iq6acn7e/S02ai3e4irw5jkny6akobobo!`

p398, 3 cells
`x = 3, y = 3, rule = B2cen3eijkq4acejtw5ej6cin/S01c3kny4aikqryz5iq6ci7obo2\$bo!`

p420, 3 cells
`x = 5, y = 1, rule = B2-ae3acnry4acity5ajny6ce7c/S12ac3acik4en5cj8obobo!`

p434, 3 cells
`x = 3, y = 2, rule = B2-ai3cqry4-acerz5ckny6aek7c8/S12n3ejny4ceijtyz5ciny6cobo\$bo!`

p674, 3 cells
`x = 5, y = 1, rule = B2-ae3aikq4aeinr5jnr6ci/S13iy4ary5inobobo!`

p788, 3 cells
`x = 5, y = 1, rule = B2ci3-jkqy4acejtyz5cek6/S1e2i3aciky4er5ijy6-knobobo!`

p998, 3 cells
`x = 5, y = 1, rule = B2-ai3ikry4ceiryz5-cknr6ce8/S01c2i3cik4cejnwz5ejkny6an7e8obobo!`
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

A final update after a run with the maxGen set to 5000 - one oscillator with period > 1000 showed up. I think this will be my final update to this thread for some time. The 2-cell 9x5 search is still running, but both the forward and reverse searches seem to be bogged down in parts of the search space which are unproductive and I'm going to discontinue them. See the end of this post for progress.

Updated Smallest Oscillators summary:
`Periods 2 - 112, 2 cellsEven periods 114 - 216, 2 cellsOdd periods 113-119, 123, 127-143, 157, 163-167, 173, 189, 191, 193, 211, 225, 231, 263, 271, 279, 297, 313, 453, 843, 3 cellsEven periods 218-236, 240-250, 254, 258, 260-266, 270-278, 280, 284-288, 292, 294, 298, 302, 306, 310-316, 320, 328, 334-348, 356, 358, 364, 368, 376, 384, 390, 392, 396, 398, 412, 420, 422, 432, 434, 442, 444, 448, 460, 484, 502, 512, 550, 554, 626, 674, 698, 706, 788, 806, 832, 998, 1010, 3 cellsAll other periods, 4 cells`

Apologies for the very long post, but I'm not sure how else to share all the new 3-cell oscillators:

Odd periods:
p129, 3 cells
`x = 5, y = 1, rule = B2-an3-ijnr4ceintz5acknr6ikn78/S03ajy4aiknw5aiqr6aei7eobobo!`

p133, 3 cells
`x = 5, y = 1, rule = B2cin3acnr4-ckrwy5aijy6-ci7/S02a3jky4aijknt5cjkq7cobobo!`

p135, 3 cells
`x = 5, y = 1, rule = B2ckn3aciky4eijnrwz5-nqr6ae8/S02a3jnq4jnq5cny6cnobobo!`

p141, 3 cells
`x = 5, y = 1, rule = B2cik3aenr4eity5c6k7c8/S12ci3ceq4ceknz5ajqy6ekobobo!`

p143, 3 cells
`x = 5, y = 1, rule = B2cek3jkqr4cijqtz5cijnr67e8/S01c2ik3-jq4ckrt5-nqry6an78obobo!`

p157, 3 cells
`x = 5, y = 1, rule = B2-ae3-ejnq4enqr5cr6ce7c8/S02in3acikr4cijkqrz5ainy6aei78obobo!`

p167, 3 cells
`x = 5, y = 1, rule = B2-ai3cei4cqrtz5-cejq6-ei7c/S01c2e3q4kyz5inrobobo!`

p173, 3 cells
`x = 5, y = 1, rule = B2cik3aeijq4-jqwyz5-ijny6i7c/S1e3j4jtz5ry6iobobo!`

p193, 3 cells
`x = 5, y = 1, rule = B2cik3-eijy4ijrz5eikry6c7/S02ain3ajkqr4cijqry5acjnr6aik7eobobo!`

p211, 3 cells
`x = 5, y = 1, rule = B2-a3eikr4eikwyz5ikr6-ai/S02ei3air4ajkq5cenqr6ik8obobo!`

p225, 3 cells
`x = 5, y = 1, rule = B2-an3aejr4einqrw5-eknq6ae7/S02-ce4i5eir6ciobobo!`

p231, 3 cells
`x = 5, y = 1, rule = B2-ak3eik4cijq5aijq6acn8/S1e2akn3kn4cny5q8obobo!`

p271, 3 cells
`x = 5, y = 1, rule = B2-ae3aciqr4-aejy5nqr6ck7c8/S02-ac3eiry4aejqz5aeq6-n7obobo!`

p297, 3 cells
`x = 5, y = 1, rule = B2cin3aijq4jknqtw5akry6-kn7/S01e2ai3ijr4nqtwz5ejqr6ekn7obobo!`

p313, 3 cells
`x = 5, y = 1, rule = B2-an3a4acertwz5cqy6ei7c8/S1e2k3ceqy4ertwy5-cejq6c78obobo!`

p453, 3 cells
`x = 5, y = 1, rule = B2-ae3aeiqr4cinqwyz5ij6-ci/S02kn3acy4ar5ikqy6einobobo!`

p843, 3 cells
`x = 5, y = 1, rule = B2-an3eijr4eiqt5cey6ek/S1e2cen3jkqr4eiq5in6n7eobobo!`

Even periods:
p222, 3 cells
`x = 3, y = 3, rule = B2-a3aek4cenrtyz5-cenq6ak7/S02-ce4jkq5cikn6in7cobo2\$o!`

p226, 3 cells
`x = 5, y = 1, rule = B2-ak3cjq4ny5iq6ci7e/S1e2ik3anqr4eijz5robobo!`

p236, 3 cells
`x = 5, y = 1, rule = B2ein3ijkn4ckqrtw5-aejq6-en78/S01e2ei3aknq4-kqrw5ajnqr6aen7cobobo!`

p262, 3 cells
`x = 5, y = 1, rule = B2-a3-air4twz5aeknq6-k8/S12i3iq4ey5ackqr6cnobobo!`

p264, 3 cells
`x = 5, y = 1, rule = B2cin3aiqy4cry5cijry7e/S01e2k3acjkn4ijnqryz5cnry7obobo!`

p270, 3 cells
`x = 3, y = 2, rule = B2-an3ikn4cei5ejnr6aei/S1c2cei3aejnr4ey5cr6a7obo\$bo!`

p272, 3 cells
`x = 5, y = 1, rule = B2cen3-ajr4eijry5acikq6ci7c/S02k3cir4cj5cjkn6aenobobo!`

p276, 3 cells
`x = 5, y = 1, rule = B2cek3aekq4ekqw5aeiqr6eik7c/S02cin3-eikq4jqrtz5jn6ei78obobo!`

p298, 3 cells
`x = 5, y = 1, rule = B2-a3ceijr4cjnqyz5acnqr6cin7c/S03a4eknty5ceiqr6ak8obobo!`

p306, 3 cells
`x = 5, y = 1, rule = B2cin3aiq4-jqr5aeikr6kn7e8/S1e2ekn3ijkq4-acnwz5eiqy6ack8obobo!`

p312, 3 cells
`x = 3, y = 3, rule = B2-ak3akn4ciwy5aceny6k7e/S12acn3-aijn4aiknz5jknqy6ciobo2\$o!`

p328, 3 cells
`x = 5, y = 1, rule = B2-a3cnqry4-cijtw5ckqry6cek8/S1e2in3ci4ijq5cy8obobo!`

p334, 3 cells
`x = 5, y = 1, rule = B2-a3cn4ejqtwyz5ciny6acn7e/S1e2n3aeikq4kqtyz5acjy6ace7eobobo!`

p338, 3 cells
`x = 5, y = 1, rule = B2-ai3-acij4jkrtz5-cj6in7c8/S01c2aen3-aijq4ceijtw5ikny6-ci7cobobo!`

p346, 3 cells
`x = 5, y = 1, rule = B2cei3-ajkq4eqtz5-acek6eik7c8/S1e2ikn3anq4cenqty5ajry6ai8obobo!`

p364, 3 cells
`x = 3, y = 2, rule = B2-a3y4-ciknt5eqry6cn/S01c2akn3-akr4-aknq5-einr6-ikobo\$bo!`

p384, 3 cells
`x = 3, y = 2, rule = B2-ak3aq4etz5cejy6a7c/S12aen3cj4ajqtyz5-ciky6ci7cobo\$bo!`

p390, 3 cells
`x = 3, y = 3, rule = B2cei3ajry4rz5acikn6ae/S01c2-c3iqy4eiz5-eknr6-i78obo2\$o!`

p422, 3 cells
`x = 5, y = 1, rule = B2-ae3aeik4-aijrw5cikry6ein78/S1e2ein3ackny4cew5enq7e8obobo!`

p432, 3 cells
`x = 5, y = 1, rule = B2-ae3aenqr4-jknwz5ckn6akn/S02i3ajk4ijnq5ciny6eikobobo!`

p442, 3 cells
`x = 5, y = 1, rule = B2cen3er4acwz5-jqr/S01c2kn3ajnry4-ajnqt5kq6cin7obobo!`

p444, 3 cells
`x = 5, y = 1, rule = B2-an3acej4cekqrz5-q6ack/S1e2ci3jkny4aeqr5ackr6-ei7eobobo!`

p448, 3 cells
`x = 3, y = 2, rule = B2-a3iq4jnryz5ceiky6-en7c8/S1c2e3aenr4qtw5en6k7c8obo\$bo!`

p460, 3 cells
`x = 3, y = 3, rule = B2cik3aekn4ntw5cjqry6ae7c/S12ek3iqy4cijrw5aj6acn7c8obo2\$bo!`

p484, 3 cells
`x = 3, y = 2, rule = B2-ak3ijk4cijkq5eijy6aen7c8/S01c3aijy4cqr5ijqr6ck8obo\$bo!`

p502, 3 cells
`x = 5, y = 1, rule = B2eik3ik4-a5-ceky6a8/S01c2ack3-acny4cty5ceknq6kobobo!`

p512, 3 cells
`x = 3, y = 2, rule = B2-ac3-jnry4eityz5aknqy6n7c8/S2-k3inqr4cqrz5-ikny6in7cobo\$bo!`

p550, 3 cells
`x = 3, y = 2, rule = B2cei3ajny4eijnqwy5cr6ac7e8/S1c2-kn3ek4jnw5-ejry6cek8obo\$bo!`

p554, 3 cells
`x = 5, y = 1, rule = B2-ak3ceiry4acekrwz5cer6ik/S12i3cej4-aqtw5-jkqr6cn7c8obobo!`

p626, 3 cells
`x = 5, y = 1, rule = B2-ai3ae4-aeijt5-acen6-ek78/S02ek3-aeqy4kq5anr6eobobo!`

p698, 3 cells
`x = 3, y = 2, rule = B2-ac3aeiq4krtwz5r6i7e8/S1c2ae3eijy4airt5k6-k7eobo\$bo!`

p706, 3 cells
`x = 5, y = 1, rule = B2-a3aqr4-aejqt5-anqr6aik7/S1e2i3aeq4ejny5e6aiobobo!`

p806, 3 cells
`x = 5, y = 1, rule = B2cen3-aik4ceqrz5-inr6i7c/S02c3ejk4-ajy5ackqy6ai7e8obobo!`

p832, 3 cells
`x = 5, y = 1, rule = B2cei3iqry4kntwyz5cijk6acn7e/S12ein3-aijn4ejkrwy5ijnry6acn7e8obobo!`

p1010, 3 cells
`x = 5, y = 1, rule = B2-ai3eqry4knqtz5ackn6cik7e/S01c2c3-ckn4cnrt5ajry6ae7e8obobo!`

==========
Edit: Update on 9x5 2-cell oscillator search progress:

Forward search (no output filtering):
`Number of iterations = 614335 millionNumber of oscillators = 117248244Progress = 79.4404% (6/8, 2/8, 3/4, 5/16, 28/32, 1/16, 111/128, 26/32, 0/16, 3/4, 1/2, 29/32, 7/16, 1/2, 0/4, 1/4, 1/2, 0/4, 5/8, 0/2, 28/32, 0/2)Number of iterations = 614340 millionNumber of oscillators = 117248244Progress = 79.4404% (6/8, 2/8, 3/4, 5/16, 28/32, 1/16, 111/128, 26/32, 2/16, 4/8, 1/4, 3/8, 4/8, 0/1, 0/1, 1/4, 5/8, 1/2, 1/2, 3/4, 2/4, 2/8, 0/1, 1/4, 0/4, 0/1, 0/1, 1/4, 3/4, 1/4, 0/1, 0/2, 1/2, 0/2)`

Reverse search (results filtered to only include odd period >= 81 and even period >= 160 :
`Number of iterations = 919540 millionNumber of oscillators = 3Progress = 14.2759% (1/8, 1/8, 2/16, 11/64, 3/4, 1/4, 2/4, 4/8, 5/16, 0/8, 8/16, 2/8, 1/16, 1/2, 2/4, 1/4, 0/2, 8/16, 2/4, 1/2, 2/4, 0/8, 6/16, 1/4, 1/2, 0/2, 0/2, 1/2, 3/4, 0/4, 0/4)Number of iterations = 919560 millionNumber of oscillators = 3Progress = 14.2759% (1/8, 1/8, 2/16, 11/64, 3/4, 1/4, 2/4, 4/8, 5/16, 0/8, 8/16, 2/8, 3/16, 1/2, 1/2, 0/2, 0/4, 21/32, 1/2, 0/2, 0/8, 1/8, 0/1, 0/1, 0/4, 0/2, 1/2, 0/1, 0/1, 0/2, 1/2, 0/1, 0/1, 0/4, 0/2, 0/1, 1/2, 0/1, 0/1, 0/1, 0/1, 0/1, 0/1, 0/1, 1/4, 3/4, 0/2, 1/2, 2/4, 0/2)`
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

Wow! The first period over 1000! Congratulations!

The p1010 has the same minrule and maxrule, so it completely specifies the rule. It's quite a nice rule too, there's a nice 4 cell spaceship and some small oscillators:
`x = 34, y = 80, rule = B2-ai3eqry4knqtz5ackn6cik7e/S01c2c3-ckn4cnrt5ajry6ae7e8o7bo6bo\$2bo3bo6bo\$b3o4b2o5b2o\$2b2o4b3o4b3o\$8b2o6bo\$18bo5\$2bo\$b3o2\$b3o\$2bo5\$6bo\$bo3bo\$4bo\$3bo\$2bo\$bo3bo\$o5\$2bo3bo5bo7bo2bo\$bo3b4o2bob2o6b2o8bobo\$o4b4o3bobo5b4o6b3o\$7bo5bobo3b2o2b2o4bob2o\$13b2obo11b3o\$15bo13b2o\$28bo6\$2bobo\$b2o\$ob4o\$b2obo5\$o\$2bo2\$2bo\$o25\$obobo!`

Macbi

Posts: 668
Joined: March 29th, 2009, 4:58 am

### Re: Smallest Oscillators Supporting Specific Periods

wildmyron wrote:p460, 3 cells
`x = 3, y = 3, rule = B2cik3aekn4ntw5cjqry6ae7c/S12ek3iqy4cijrw5aj6acn7c8obo2\$bo!`

This rule has a natural 4c/30 orthogonal which pushes an octomino:
`x = 7, y = 3, rule = B2cik3aekn4ntw5cjqry6ae7c/S12ek3iqy4cijrw5aj6acn7c8bob4o\$5obo\$bob4o!`

wildmyron wrote:p843, 3 cells
`x = 5, y = 1, rule = B2-an3eijr4eiqt5cey6ek/S1e2cen3jkqr4eiq5in6n7eobobo!`

And this one has a natural c/7 orthogonal:
`x = 4, y = 5, rule = B2-an3eijr4eiqt5cey6ek/S1e2cen3jkqr4eiq5in6n7e3bo\$3bo\$3o\$3bo\$3bo!`

And a nice p9:
`x = 9, y = 3, rule = B2-an3eijr4eiqt5cey6ek/S1e2cen3jkqr4eiq5in6n7ebo5bo\$obo3bobo\$bo5bo!`

77topaz

Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Smallest Oscillators Supporting Specific Periods

Period 20200:
`x = 66, y = 8, rule = B345/S54bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo\$2bobo3bobo3bobo3bobo3bobobo5bobo3bobo3bobo3bobobo3bo\$2bobo3bobo3bobo3bobo3bobo7bobo3bobo3bobo3bobo7bo\$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo\$bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo\$bo3bobo3bobo3bobobobo3bobo3bobo3bobo3bobo3bobo7bobo\$3bobobobobobobobobobobo3bobo3bobo3bobo3bobo3bobo3bobobobo\$3bobo3bobobobo3bobobobobobobobobobobobobobobobobobobobo3bo!`

EDIT: 188 cells, so not that impressive.
Life is hard. Deal with it.
My favorite oscillator of all time:
`x = 7, y = 5, rule = B3/S2-i3-y4i4b3o\$6bo\$o3b3o\$2o\$bo!`

Hdjensofjfnen

Posts: 1093
Joined: March 15th, 2016, 6:41 pm
Location: r cis θ

### Re: Smallest Oscillators Supporting Specific Periods

Hdjensofjfnen wrote:EDIT: 188 cells, so not that impressive.

Precisely, given that that's quite a lot bigger than 4.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3301
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Smallest Oscillators Supporting Specific Periods

Hdjensofjfnen wrote:Period 20200:
`longlife`

EDIT: 188 cells, so not that impressive.

Isn’t that a trivial oscillator too? Or does some cell oscillate at that period
My rules:
They can be found here

Also, the tree game
Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

Moosey

Posts: 1546
Joined: January 27th, 2019, 5:54 pm
Location: A house, or perhaps the OCA board.

### Re: Smallest Oscillators Supporting Specific Periods

wildmyron wrote:A final update after a run with the maxGen set to 5000 - one oscillator with period > 1000 showed up. I think this will be my final update to this thread for some time. The 2-cell 9x5 search is still running, but both the forward and reverse searches seem to be bogged down in parts of the search space which are unproductive and I'm going to discontinue them. See the end of this post for progress.

Well, I couldn't bring myself to kill that search off completely, so I left the reverse search running. 4 days later and it has progressed from 14.2759% to 14.2806%, but this morning it finally reached an interesting part of the search space and the number of oscillators found has jumped from 3 to > 18000.

Here are some new periods for 2-cell oscillators:
p218, 2 cells
`x = 3, y = 1, rule = B2-cn3ejnq4ejntw5-jky6cek8/S01e2ikn3-in4-tyz5-ajnq6-knobo!`

p220, 2 cells
`x = 3, y = 1, rule = B2-ck3acek4ijktwy5-akq6ace/S01e2in3-ci4aejwy5-jqr6cei7eobo!`

p222, 2 cells
`x = 3, y = 1, rule = B2-cn3-aci4ejkny5cijn6c7e8/S01e2ein3-aikn4-tyz5-nq6cikobo!`

p226, 2 cells
`x = 3, y = 1, rule = B2aei3aceqy4cjknw5-ak6-cn7e/S01e2ikn3acejq4aeijnqy5-y6-an7eobo!`

p234, 2 cells
`x = 3, y = 1, rule = B2-ck3aceqy4ijktwy5-qr6cei/S01e2i3-cijy4-inqtw5ekny6-kn7eobo!`

I'm quite surprised by the jump from p226 to p234. Those two oscillators are the only 2 found with period greater than 222 and nearly all the other searches I've done have had only small gaps between the highest periods found as well as a large number of results at slightly lower periods before the new record high period was found. The 3-cell random rule search behaves quite differently, which I suspect is because it doesn't have the same bounding box restriction.

Updated Smallest Oscillators summary:
`Periods 2 - 112, 2 cellsEven periods 114 - 222, 226, 234, 2 cellsOdd periods 113-119, 123, 127-143, 157, 163-167, 173, 189, 191, 193, 211, 225, 231, 263, 271, 279, 297, 313, 453, 843, 3 cellsEven periods 224, 228-232, 236, 240-250, 254, 258, 260-266, 270-278, 280, 284-288, 292, 294, 298, 302, 306, 310-316, 320, 328, 334-348, 356, 358, 364, 368, 376, 384, 390, 392, 396, 398, 412, 420, 422, 432, 434, 442, 444, 448, 460, 484, 502, 512, 550, 554, 626, 674, 698, 706, 788, 806, 832, 998, 1010, 3 cellsAll other periods, 4 cells`
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

wildmyron wrote:Periods 2 - 112, 2 cells
Even periods 114 - 222, 226, 234, 2 cells
Odd periods 113-119, 123, 127-143, 157, 163-167, 173, 189, 191, 193, 211, 225, 231, 263, 271, 279, 297, 313, 453, 843, 3 cells
Even periods 224, 228-232, 236, 240-250, 254, 258, 260-266, 270-278, 280, 284-288, 292, 294, 298, 302, 306, 310-316, 320, 328, 334-348, 356, 358, 364, 368, 376, 384, 390, 392, 396, 398, 412, 420, 422, 432, 434, 442, 444, 448, 460, 484, 502, 512, 550, 554, 626, 674, 698, 706, 788, 806, 832, 998, 1010, 3 cells
All other periods, 4 cells

Hmm... so I propose a conjecture. Can someone prove or disprove it?
Every period of oscillator can be achieved in three cells or fewer.
Life is hard. Deal with it.
My favorite oscillator of all time:
`x = 7, y = 5, rule = B3/S2-i3-y4i4b3o\$6bo\$o3b3o\$2o\$bo!`

Hdjensofjfnen

Posts: 1093
Joined: March 15th, 2016, 6:41 pm
Location: r cis θ

### Re: Smallest Oscillators Supporting Specific Periods

Hdjensofjfnen wrote:Hmm... so I propose a conjecture. Can someone prove or disprove it?
Every period of oscillator can be achieved in three cells or fewer.

I'm fairly certain this conjecture is true, certain enough that I'm sure it couldn't be disproved. Proving that it is true requires showing that there are rules which contain 3 cell oscillators with adjustable period. I think the easiest proof would be to actually construct/discover said adjustable oscillators. The earlier posts by Macbi speculating on this topic, and A for Awesome showing a failed version of this idea are enough to convince me that it is possible to find such oscillators. It is worth noting that the quad wickstretcher and its interaction with the dot work in 2^61 rules. And there are many other possibilities for growing patterns plus interaction with a single dot which broaden the rulespace which could be used. Here are a few (non-adjustable) oscillators which show a dot hassling the quad wick-stretcher. They come from a brief random rule search I ran to explore this possibility. Neither of them actually has a 3-cell phase but that's mainly because of the way I set up the search.

`#C p37 oscillatorx = 8, y = 3, rule = B2cei3aq4-cjknq5q6i/S01c2ain3jkry4ejt5ceqbo\$obo4bo\$bo!`

`#C p56 oscillatorx = 10, y = 1, rule = B2cei3acr4-jknqy5ceiny7e/S01c2ai3cejqy4eint2obo5bo!`
The latest version of the 5S Project contains over 57,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1120
Joined: August 9th, 2013, 12:45 am

### Re: Smallest Oscillators Supporting Specific Periods

There's on obvious way to get 2-cell for every period, but we're only counting 2-state I'm assuming, so this wouldn't count:
`x = 3, y = 3, rule = 01c/2n/72.A2\$A![[ THEME Blues ]]`
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3301
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Smallest Oscillators Supporting Specific Periods

Here are some new periods
` B2-a3acjkq4cijqrz5aik6-n7c/S01c2ekn3-aeik4ijqtwy5acejy6ai7c8, obobo!, Period 181B2cei3-aeq4eiq5aejnr6ain7c8/S02cn3acery4aejknty5-cjky6-ek7e, obobo!, Period 365B2-an3-anqy4acnrtwy5enqy6-kn7/S02n3ekqy4cekn5-jnqr6c7c, obobo!, Period 229`

Edit:
p282
`x = 5, y = 1, rule = B2-a3ei4aikrtwy5acq6-ce8/S02ikn3aeiqy4ceijqy5-jky6-ai8obobo!`

p330
`x = 5, y = 1, rule = B2ci3-einq4ceiqtw5-ciqy6ikn7/S01c2ekn3iqry4-ejkqr5-cjny6-e7obobo!`

p238
`x = 5, y = 1, rule = B2-an3cjry4ajknry5ijkry6-cn7c8/S02ikn3-aekn4ciknr5ciqy6-k7cobobo!`

p498
`x = 5, y = 1, rule = B2cin3ajk4-cnrtz5ejkny6-k8/S02-n3ei4ejqrw5aekqr6-an7c8obobo!`

p185
`x = 5, y = 1, rule = B2ci3-nqry4aerz5ikr6aik7c8/S02cin3cij4cery5-akqr6ain7e8obobo!`

p380
`x = 5, y = 1, rule = B2-a3-aejn4cnwy5-einy6-n8/S01e2e3acery4ejkqrty5ceiy6aei7e8obobo!`

p505
`x = 5, y = 1, rule = B2cik3aeikq4aenr5-acn6cen7e8/S02ci3eijkr4acinwy5cij6a7eobobo!`

p145
`x = 5, y = 1, rule = B2ci3acqr4eijqr5-acjk6c7e8/S02-ck3-ckqr4cikntz5cknqr6-n7eobobo!`

p572
`x = 5, y = 1, rule = B2cin3aceir4aeijnwz5-ir6-ek78/S02ei3acek4aceqrtz5aer6ce8obobo!`

p161
`x = 5, y = 1, rule = B2ci3acky4cinry5-einr6a/S02eik3aijry4crz5-nry6-cn7cobobo!`

p149
`x = 5, y = 1, rule = B2-an3-aeiq4ceknryz5knqry6c8/S02ikn3-eikq4-aintw5ajnq6eik7c8obobo!`

p756
`x = 5, y = 1, rule = B2-ae3aekry4aeit5-cek6k/S01c2ein3-cnqy4ajnrwy5eky6aeiobobo!`

p322
`x = 5, y = 1, rule = B2-a3-aiy4ckt5aeqr6cei7c8/S01c2ac3kq4-crtz5-cqry6ae7eobobo!`

p199
`x = 5, y = 1, rule = B2cin3aeijq4aceqwz5ckr6-kn78/S02ik3-aeky4jtw5cjkn6cek7c8obobo!`

p145
`x = 5, y = 1, rule = B2cin3-cjnq4-acny5acijr6ckn8/S02aek3ckqy4ejknwz5enqr6aikobobo!`

p227
`x = 5, y = 1, rule = B2cin3aikq4aciqtw5jny6-en8/S01e2kn3eiq4aceijz5jnqry6-i78obobo!`

p988
`x = 5, y = 1, rule = B2cin3aikry4-anyz5jknqr6cn7e8/S02akn3aejr4einqw5aejnq6cek78obobo!`

p716
`x = 5, y = 1, rule = B2cik3ajqr4ceinq5jkr6-ce7e/S02eik3-akn4aekny5-ekny6-ac8obobo!`

p826
`x = 5, y = 1, rule = B2cik3aeknr4enqrty5-ikqr6-c7c8/S02ae3aceny4aceiqtz5-aeiq6ckobobo!`

`B2cik3aeikr4ijw5-nqr6ckn8/S02ain3-jkny4cjnqy5-aiqy6ck7c, obobo! Period 606B2cik3-ikny4aijknrz5-aejn8/S02-ae3cejky4ijqtw5knr6ae8, obobo! Period 678`