## Smaller sawtooth

For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.
chris_c
Posts: 938
Joined: June 28th, 2014, 7:15 am

### Smaller sawtooth

Here is a sawtooth with minimum repeating population 252. It uses exactly the same concept as Sawtooth260 except that a blocker and a snake are used to kill the glider pairs. If the population is 252 at generation N then it is also 252 at generation 4 * N + 1536 (exactly the same as in Sawtooth260).

Code: Select all

``````x = 165, y = 111, rule = B3/S23
46bo\$46bo\$48bo5bo\$47bo6bobo\$46bo3bo2bo\$47bo2bobob2o\$52bob2o\$66b2o\$66b
2o7\$74b2o\$74b2o\$55b2o3bo\$53b3o4b2o\$48b3obo8b2o\$52bobo2b5o\$53bo3b3o3\$
40bo\$40bo7bo\$40bo5b3o\$39b2o3bobo\$45bo2bo\$44bo2b2o10\$27bo\$24b4o4\$27bobo
\$28bo\$2o2bo21b2o\$3bobo20bo2bo\$2bo16bo5b2ob2o\$19bo\$4b2o13bo2\$5b2o12b2o\$
4bo13bo2bo\$2b2ob2o11bobo\$5b2o10b2o\$3bo13bo\$20b2o\$20b2o\$20b2o\$17b2obo\$
18b3o\$19bo112b2o\$125b2o5b2o\$125b2o2\$7b2o\$7b2o118b2o27b2o\$96bo30b2o18b
2obo5b2o\$91b3ob2o2b2o20b2o24bob2o\$91b4o4b2o20b2o\$95b2o65b2o\$162b2o\$
158b2o\$15b2o79b2obo58b2o\$15b2o79bob2o3\$163b2o\$163b2o7\$128b2o19b3o\$127b
2o21bo\$129bo20b3o5\$116b2o39b2o\$116b2o40bo\$157bo\$157b2o2\$121b2o\$121b2o
27b2o\$117b2o\$117b2o\$158b2o\$130b2obo24b2o\$123b2o5bob2o18b2o\$123b2o27b2o
3\$154b2o\$147b2o5b2o\$147b2o!
``````

Freywa
Posts: 620
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Location: Singapore
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### Re: Smaller sawtooth

Though it may be a new record, it is rather trivial.
Princess of Science, Parcly Taxel

codeholic
Moderator
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Location: Hamburg, Germany

### Re: Smaller sawtooth

@Freywa Your comment is rude, and chris_c's sawtooth is not trivial, though it's based on prior art, because there is a non-trivial idea behind it and at least one reaction that was found specifically for this pattern.

@chris_c Congrats!
Ivan Fomichev

simsim314
Posts: 1733
Joined: February 10th, 2014, 1:27 pm

### Re: Smaller sawtooth

chris_c wrote:Here is a sawtooth with minimum repeating population 252
Nice! and congrats.

I was lately thinking if my 2 switch engine puffer could somehow allow to build a smaller sawtooth:

Code: Select all

``````x = 61, y = 42, rule = B3/S23
28bo\$27bobo\$26b2ob2o\$3b3o20b2ob2o\$2bo3bo18b3o\$bo4bo18b3o3bo12b2o\$o3bo
20b3o4bo11b2o\$o2bob3o18b2o5bo\$o7bo17b3o\$2bo3bobo18b2o\$2bo3bob2o18b2o\$
4b3ob2o33b2o\$20b2o\$20b2o6bo14b3o6b2o\$27b3o13bobo6b2o\$26bo3bo\$4b2o24b2o
13bo\$4b2o23b2o13b2o\$5bo38b2o2\$5bo22b2o13b2o\$28b2o13b3o\$41b3o\$3bo5b3o
21b3o5bo\$3bo7bo\$3bo5bobo10b2o16b3o\$9b2o10bo2bo15bo\$20bo3bo15b3o\$22b2o
35b2o\$59b2o\$16bo\$14b2o\$8b2o2b2o5bo\$8b2o4b2o\$18bo\$15b4o\$14bobo\$13bo2b2o
bo\$14bo4bo\$15bo\$16bo2bo\$16b2o!
``````
Freywa wrote:it is rather trivial
So why didn't you find it?

Scorbie
Posts: 1443
Joined: December 7th, 2013, 1:05 am

### Re: Smaller sawtooth

chris_c wrote:Here is a sawtooth with minimum repeating population 252.
Congrats for the new record! I think the pattern "may" have room for newer recordbreakers, as what simsim314 has shown with his minimum population quadratic growth pattern. And if newer records are made, I'm pretty sure it would be made by you.
Best wishes to you, Scorbie

dvgrn
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### Re: Smaller sawtooth

Scorbie wrote:I think the pattern "may" have room for newer recordbreakers, as what simsim314 has shown with his minimum population quadratic growth pattern.
Like this?

Code: Select all

``````#C Sawtooth with minimum population 246
#C Found with a technique that Calcyman calls "corollary sniping"
#C ( https://cp4space.wordpress.com/2013/05/15/simultaneous-proofs/ )
x = 165, y = 111, rule = B3/S23
46bo\$46bo\$48bo5bo\$47bo6bobo\$46bo3bo2bo\$47bo2bobob2o\$52bob2o\$66b2o\$66b
2o7\$74b2o\$74b2o\$55b2o3bo\$53b3o4b2o\$48b3obo8b2o\$52bobo2b5o\$53bo3b3o3\$
40bo\$40bo7bo\$40bo5b3o\$39b2o3bobo\$45bo2bo\$44bo2b2o10\$27bo\$24b4o4\$27bobo
\$28bo\$2o2bo21b2o\$3bobo20bo2bo\$2bo16bo5b2ob2o\$19bo\$4b2o13bo2\$5b2o12b2o\$
4bo13bo2bo\$2b2ob2o11bobo\$5b2o10b2o\$3bo13bo\$20b2o\$20b2o\$20b2o\$17b2obo\$
18b3o\$19bo112b2o\$125b2o5b2o\$125b2o2\$7b2o\$7b2o118b2o27b2o\$127b2o18b2obo
5b2o\$96b2o23b2o24bob2o\$92b2o4b4o19b2o\$92b2o2b2ob3o60b2o\$96bo65b2o\$158b
2o\$15b2o141b2o\$15b2o3\$163b2o\$163b2o7\$128b2o19b3o\$127b2o21bo\$129bo20b3o
5\$116b2o39b2o\$116b2o40bo\$157bo\$157b2o2\$121b2o\$121b2o27b2o\$117b2o\$117b
2o\$158b2o\$130b2obo24b2o\$123b2o5bob2o18b2o\$123b2o27b2o3\$154b2o\$147b2o5b
2o\$147b2o!``````

simsim314
Posts: 1733
Joined: February 10th, 2014, 1:27 pm

### Re: Smaller sawtooth

Congrats! Nice improvement. I guess optimization space in this setup is getting smaller and smaller.

dvgrn
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### Re: Smaller sawtooth

simsim314 wrote:Congrats! Nice improvement. I guess optimization space in this setup is getting smaller and smaller.
Yes, if this cheap trick hadn't worked, it was going to be necessary to work pretty hard -- either

A) find a still life or oscillator with less than 16 cells that could do the blocker's job or some similar trick, and/or

B) dig up a new two-, three- or four-engine Cordership with a lower minimum population, that can also reflect a glider 180 degrees onto a non-interfering nearby lane.

The first seems fairly unlikely; the second... well, you never know. The search that Paul Tooke used to find the three-engine Cordership wasn't exhaustive. And t's certainly not impossible that the leftover junk from some two-switch-engine puffer or other might be a cleanly burnable c/12 fuse. We wouldn't find something like that with a simple search for Corderships, only with some very determined searching along the lines of the fuse-finder script.

Have you tried adding one more switch engine to your two-engine block puffer, to see if you get anything interesting? Looks like the minimum population without the blocks is 105, so there are only a few cells to spare before it would be heavier than the 134-cell Cordership in the current sawtooth. But again, you never know.

simsim314
Posts: 1733
Joined: February 10th, 2014, 1:27 pm

### Re: Smaller sawtooth

dvgrn wrote:Have you tried adding one more switch engine to your two-engine block puffer, to see if you get anything interesting?
My search was also random based. I took 2 switch engines on different stages and placed them together to see whether anything with c/12 come up. With 2 engines it worked nicely, and came up with few pretty clean puffers. But with three it gave no results with all three engines surviving. It was strange, probably one need some more sophisticated tricks to make sure all the engines survive.

As for just adding a switch engine - I didn't try, but from my experience you can't just "add" switch engine, it's more about configuration of all three engines together, many times the initial smoke is also relevant. Anyway this should be simple test, and might work, I just never thought of it.

As for the property of reflecting the glider 180 degree on a non-colliding lane, it's also pretty rare.

---

I also tried to search with bellman a way to replace the oscillator with no success.

---

Taking it into account, there are other ways to utilize a sawtooth, like parabolic sawtooth, and puffer based sawtooth. So once we minimize the current approach, we might find something else from some other direction.

Kazyan
Posts: 925
Joined: February 6th, 2014, 11:02 pm

### Re: Smaller sawtooth

simsim314 wrote:Taking it into account, there are other ways to utilize a sawtooth, like parabolic sawtooth, and puffer based sawtooth. So once we minimize the current approach, we might find something else from some other direction.
One approach would be to WLS up a c/5 diagonal spaceship that can reflect a sideways glider so that the two are going in opposite directions. Then, you can set up a glider loop between a Gosper gun, 58P5H1V1 (reflecting forwards gliders to sideways ones), the 'magic' ship we don't have yet, and a buckaroo. It might be smaller.
Tanner Jacobi

chris_c
Posts: 938
Joined: June 28th, 2014, 7:15 am

### Re: Smaller sawtooth

Kazyan wrote:One approach would be to WLS up a c/5 diagonal spaceship that can reflect a sideways glider so that the two are going in opposite directions. Then, you can set up a glider loop between a Gosper gun, 58P5H1V1 (reflecting forwards gliders to sideways ones), the 'magic' ship we don't have yet, and a buckaroo. It might be smaller.
If you are reflecting gliders off the back of a c/5 spaceship then the returned glider stream is 9 times thinner than outgoing glider stream, so to make a sawtooth you would need to find a reaction where one incoming glider destroys 9 outgoing gliders. Sounds difficult, but the fact that a Gosper gun needs only 36 cells could be an advantage.

Other possibilities are reflecting off the back of a c/8 spaceship and each incoming glider destorying 3 outgoing gliders or reflecting off the back of a c/6 spaceship and each incoming glider destroying 5 outgoing gliders.

[ My formula for these calculations is (1/4 - v) / (1/4 + v) ]

Kazyan
Posts: 925
Joined: February 6th, 2014, 11:02 pm

### Re: Smaller sawtooth

Code: Select all

``````#C Sawtooth with minimum repeating population of 213
# Tanner Jacobi, April 13, 2015
x = 55, y = 76, rule = B3/S23
49b2o\$49b2o2\$23b2o5b2o\$23b2o5b2o5\$43bo5bo\$42b3o3b3o\$41b2obo3bob2o3\$23b
o7bo12bo3bo\$22bo2bo3bo2bo11bo3bo\$26bobo\$26bobo\$26bobo\$22bo2bo3bo2bo\$
23b3o3b3o7\$43bo\$43bobo\$43b2o\$30b2o\$30b2o2\$10b2o\$8b2ob2o8bobo\$8bo2bo10b
o\$8bo2bo\$9b2o33b2obo3bob2o\$32bo11bo2bo3bo2bo\$9b2o20bo13b3o3b3o\$8bo2bo
19b3o\$2o6bo2bo\$2o6b2ob2o\$10b2o9bobo\$21b2o\$22bo2\$45b2o\$45b2o2\$20bo\$20bo
bo\$20b2o\$9b4o21b2o\$7b2o4b2o18bobo15b2o\$7b2o5bo18bo17b2o\$9b2obobo18b3o\$
14bo\$10bo3bo\$10bo4bo\$12b3o3bo14b3o\$12b2o4bo5b2o7bo17b2o\$18b2o4b2o7bobo
15b2o\$20bo13b2o\$20b3o3\$23bo\$22bob5o\$21b2o5bo\$21b2o3bo2bo\$29bo\$23b2obo
2bo\$26bo2bo\$27b2o\$27b2o!
``````
This thing probably has room for optimization, to boot. The population graph has sawtooth shape, but only hits 213 at the bottom of every fifth tooth due to mismatching periods of the c/5 spaceship and the p46 shuttles. So, the next recurrence is at generation 112,232,640...
Tanner Jacobi

chris_c
Posts: 938
Joined: June 28th, 2014, 7:15 am

### Re: Smaller sawtooth

Kazyan wrote:Sawtooth with minimum repeating population of 213
Nice! I think this one definitely qualifies as "non-trivial". Congratulations!

Scorbie
Posts: 1443
Joined: December 7th, 2013, 1:05 am

### Re: Smaller sawtooth

Way to go!! Great to see another one of your delightful discoveries.
Best wishes to you, Scorbie

calcyman
Posts: 2118
Joined: June 1st, 2009, 4:32 pm

### Re: Smaller sawtooth

Trivial reduction to 201:

Code: Select all

``````#C Sawtooth with minimum population 201.
#C By Adam P. Goucher, inspired by a 213-population sawtooth by Tanner Jacobi.
x = 55, y = 81, rule = B3/S23
49b2o\$49b2o2\$23b2o5b2o\$23b2o5b2o5\$43bo5bo\$42b3o3b3o\$41b2obo3bob2o3\$23b
o7bo12bo3bo\$22bo2bo3bo2bo11bo3bo\$26bobo\$26bobo\$26bobo\$22bo2bo3bo2bo\$
23b3o3b3o7\$43bo\$43bobo\$43b2o\$30b2o\$30b2o2\$10b2o\$8b2ob2o8bobo\$8bo2bo10b
o\$8bo2bo\$9b2o33b2obo3bob2o\$32bo11bo2bo3bo2bo\$9b2o20bo13b3o3b3o\$8bo2bo
19b3o\$2o6bo2bo\$2o6b2ob2o\$10b2o9bobo\$21b2o\$22bo2\$45b2o\$45b2o2\$20bo\$20bo
bo\$20b2o4\$23b2o\$23b2o\$4b4o15b2o\$2b2o4b2o13bo\$2b2o5bo12bobo\$4b2obobo12b
ob2o\$9bo\$5bo3bo\$5bo4bo12b2o\$7b3o3bo9b2o\$7b2o4bo\$13b2o\$15bo\$15b3o3\$18bo
\$17bob5o\$16b2o5bo\$16b2o3bo2bo\$24bo\$18b2obo2bo\$21bo2bo\$22b2o\$22b2o!``````
What do you do with ill crystallographers? Take them to the mono-clinic!

simsim314
Posts: 1733
Joined: February 10th, 2014, 1:27 pm

### Re: Smaller sawtooth

Congrats on the new record!

dvgrn
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### Re: Smaller sawtooth

Kazyan wrote:This thing probably has room for optimization, to boot. The population graph has sawtooth shape, but only hits 213 at the bottom of every fifth tooth due to mismatching periods of the c/5 spaceship and the p46 shuttles. So, the next recurrence is at generation 112,232,640...
Could somebody double-check the recurrence? Looks to me like the population hits bottom every fourth cycle, at T=0, T=11232640, T=547659593220480, T=2672404111505817299520, and so on, with an expansion factor asymptotic to 47^4 = 4879681. For the second expansion it's exactly 47^4+1, and for the third it seems to be 47^4+(1/47^4).

-- Did I get all that right? The LifeWiki article wants to know...

Calcyman's new version has the same expansion factor, but the spaceship starts just a little farther from the shotgun, so it takes over twice as long to finish each cycle: T=0, T=234224640, T=1142941759764480, etc.

Kazyan
Posts: 925
Joined: February 6th, 2014, 11:02 pm

### Re: Smaller sawtooth

dvgrn wrote:Could somebody double-check the recurrence? Looks to me like the population hits bottom every fourth cycle, at T=0, T=11232640, T=547659593220480, T=2672404111505817299520, and so on, with an expansion factor asymptotic to 47^4 = 4879681. For the second expansion it's exactly 47^4+1, and for the third it seems to be 47^4+(1/47^4).

-- Did I get all that right? The LifeWiki article wants to know...

Calcyman's new version has the same expansion factor, but the spaceship starts just a little farther from the shotgun, so it takes over twice as long to finish each cycle: T=0, T=234224640, T=1142941759764480, etc.
You know, considering I built the 213 version, it's kind of surprising I don't know what I'm talking about. It indeed recurs every 4th cycle...I has assumed that the spaceship would be in a different phase every cycle. And, yes, you seem to be correct about the expansion factor instead of what I wrote in the article. Feel free to correct it.
Tanner Jacobi

Freywa
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### Re: Smaller sawtooth

42 is not a multiple of 8 (see the LifeWiki page). What a silly mistake, though I surmise that the blocker-catalysed block deletion works at all even phases.
Princess of Science, Parcly Taxel

calcyman
Posts: 2118
Joined: June 1st, 2009, 4:32 pm

### Re: Smaller sawtooth

...though I surmise that the blocker-catalysed block deletion works at all even phases.
No, it only works in that one phase and relies on the p46 glider stream not being a multiple of 8. (When I tried using a p184 glider stream, the previous pull reaction interfered with the blocker.) It's actually incredibly serendipitous that it works at all.
What do you do with ill crystallographers? Take them to the mono-clinic!

dvgrn
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### Re: Smaller sawtooth

calcyman wrote:
...though I surmise that the blocker-catalysed block deletion works at all even phases.
No, it only works in that one phase and relies on the p46 glider stream not being a multiple of 8. (When I tried using a p184 glider stream, the previous pull reaction interfered with the blocker.) It's actually incredibly serendipitous that it works at all.
I noticed that the blocker could be moved four cells to the NE, and that the spaceship could be moved in by the same amount to cut down on the initial cycle time -- but if you do that, the block deletion hasn't quite happened by minimum-population time for the gun, so you don't get a population-201 sawtooth. There may well be a bounding-box reduction in there somewhere, though.

I don't have time to figure out all the math this week, but I hope someone does it -- there's some fairly subtle and interesting stuff going on there. The blocker-mediated restart mechanism is different from the p46 one, so the replacement seems not really as "trivial" as advertised: the blocker makes an extra block to absorb a glider, where the p46 makes a bigger spark for the equivalent glider to hit.

It seems just barely possible that a very small still life or lower-period oscillator (blinker? toad? clock?) might manage a trick similar to the blocker, allowing a sub-200-cell record. I did have a quick look for obvious stable catalysts, and there don't seem to be any such.

Kazyan
Posts: 925
Joined: February 6th, 2014, 11:02 pm

### Re: Smaller sawtooth

Wow, I messed up every step of the way when writing that article. I'm just going to be quiet until I figure out another thing, for something else, then.
Tanner Jacobi

dvgrn
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### Re: Smaller sawtooth

I took out the part of the LifeWiki article that didn't make sense, but haven't replaced it with anything else -- because I'm not clear yet exactly what does make sense!

The blocker and ship can move two cells closer, anyway, if the blocker's phase is changed by 4. Oddly enough, this change also allows the population to return to 201 on every cycle instead of every fourth cycle:

Code: Select all

``````#C slightly smaller pop-201 sawtooth
#C Population = 201 at T=0, 1840, 88320, 4152880, 195187200, 9173800240,
#C  431168613120, etc.
#C  -- asymptotically approaching (from above) an expansion factor of 47
x = 55, y = 79, rule = B3/S23
49b2o\$49b2o2\$23b2o5b2o\$23b2o5b2o5\$43bo5bo\$42b3o3b3o\$41b2obo3bob2o3\$23b
o7bo12bo3bo\$22bo2bo3bo2bo11bo3bo\$26bobo\$26bobo\$26bobo\$22bo2bo3bo2bo\$
23b3o3b3o7\$43bo\$43bobo\$43b2o\$30b2o\$30b2o2\$10b2o\$8b2ob2o8bobo\$8bo2bo10b
o\$8bo2bo\$9b2o33b2obo3bob2o\$32bo11bo2bo3bo2bo\$9b2o20bo13b3o3b3o\$8bo2bo
19b3o\$2o6bo2bo\$2o6b2ob2o\$10b2o9bobo\$21b2o\$22bo2\$45b2o\$45b2o2\$20bo\$20bo
bo\$20b2o2\$25b2o\$25b2o\$6b4o15b2o\$4b2o4b2o13bo\$4b2o5bo12bobo\$6b2obobo12b
ob2o\$11bo\$7bo3bo\$7bo4bo12b2o\$9b3o3bo9b2o\$9b2o4bo\$15b2o\$17bo\$17b3o3\$20b
o\$19bob5o\$18b2o5bo\$18b2o3bo2bo\$26bo\$20b2obo2bo\$23bo2bo\$24b2o\$24b2o!
[[ AUTOSTART ZOOM 3 STEP 4 ]]``````

towerator
Posts: 328
Joined: September 2nd, 2013, 3:03 pm

### Re: Smaller sawtooth

Purely out of boredom, I trivially optimized the pattern to 163 cells:

Code: Select all

``````x = 55, y = 79, rule = B3/S23
49bo\$49b2o2\$24bo5bo\$23b2o5b2o5\$43bo5bo\$42b3o3b3o\$41b2obo3bob2o5\$26bobo
\$26bobo\$26bobo\$22bo2bo3bo2bo\$23b3o3b3o10\$31bo\$30b2o2\$10b2o\$8b2ob2o\$8bo
2bo\$8bo2bo\$9b2o33b2obo3bob2o\$44bo2bo3bo2bo\$9b2o34b3o3b3o\$8bo2bo\$2o6bo
2bo\$bo6b2ob2o\$10b2o4\$45b2o\$45bo6\$25b2o\$24bobo\$6b4o13b3o\$4b2o4b2o11b2o\$
4b2o5bo14b2o\$6b2obobo13b3o\$11bo\$7bo3bo\$7bo4bo12b2o\$9b3o3bo9bo\$9b2o4bo\$
15b2o\$17bo\$17b3o3\$20bo\$19bob5o\$18b2o5bo\$18b2o3bo2bo\$26bo\$20b2obo2bo\$
23bo2bo\$24b2o\$24b2o!
``````
This is game of life, this is game of life!
Loafin' ships eaten with a knife!

biggiemac
Posts: 504
Joined: September 17th, 2014, 12:21 am
Location: California, USA

### Re: Smaller sawtooth

towerator wrote:Purely out of boredom, I trivially optimized the pattern to 163 cells:

Code: Select all

``````x = 55, y = 79, rule = B3/S23
49bo\$49b2o2\$24bo5bo\$23b2o5b2o5\$43bo5bo\$42b3o3b3o\$41b2obo3bob2o5\$26bobo
\$26bobo\$26bobo\$22bo2bo3bo2bo\$23b3o3b3o10\$31bo\$30b2o2\$10b2o\$8b2ob2o\$8bo
2bo\$8bo2bo\$9b2o33b2obo3bob2o\$44bo2bo3bo2bo\$9b2o34b3o3b3o\$8bo2bo\$2o6bo
2bo\$bo6b2ob2o\$10b2o4\$45b2o\$45bo6\$25b2o\$24bobo\$6b4o13b3o\$4b2o4b2o11b2o\$
4b2o5bo14b2o\$6b2obobo13b3o\$11bo\$7bo3bo\$7bo4bo12b2o\$9b3o3bo9bo\$9b2o4bo\$
15b2o\$17bo\$17b3o3\$20bo\$19bob5o\$18b2o5bo\$18b2o3bo2bo\$26bo\$20b2obo2bo\$
23bo2bo\$24b2o\$24b2o!
``````
So you've made a 163-cell predecessor to the sawtooth, but still only with a minimum repeating population of 201. Although something goes wrong with your pattern after two sawtooth iterations, it explodes..

Edit: Putting them side by side I see that your pattern deleted the gliders, making the phase different, and ultimately leading to instability.
Physics: sophistication from simplicity.