## Smaller sawtooth

### Re: Smaller sawtooth

Not Too sure, but I think the sawtooth could be even smaller by using the new p120 gun.

Best wishes to you, Scorbie

### Re: Smaller sawtooth

Here's an appropriate 6hd gun at p120 with 8 fewer cells (119) at minimum population than the twin bees version. If someone can line up the blocker/ship the same as before it could break 200.
Edit: upon further investigation, the blocker won't work if the block is pulled at p8n. The second to last block pull needs to happen with the blocker out of phase or it'll make a mess. So we need a different deletion mechanism. Anything promising at p120?

There's also a 99-cell 10hd 3fd-block-pull mechanism using the double barrel p120 gun, but it isn't compatible with the ship.
Re-edit: Ah, a pd, of course. Nice job chris, and of course excellent find simsim!

Code: Select all

```
x = 73, y = 71, rule = B3/S23
64b2o$64b2o3$61b2o$61b2o8b2o$5bob2o62bo$5b2obo55b2o6bo$64b2o5b2o3$54bo
$53b3o$2o5b2o10bo36bo$2o5b2o11b2o32b2o$15bobobo35bo$4b2o9bobo$4b2o4$
27b2o$27b2o2$24b2o5b2o20b2o$24b2o5b2o20b2o2$56b2o$56b2o3$23bo29b2o$21b
obo29b2o$22b2o4$30bo$20bo9bobo$20b3o7b2o$23bo$22b2o4$25bo$23b2ob2o2$
22bo5bo2$22b2obob2o9$24b2o$24b2o$53bo$51bobo$52b2o5$53bo$54b2o$53b2o!
```

There's also a 99-cell 10hd 3fd-block-pull mechanism using the double barrel p120 gun, but it isn't compatible with the ship.

Code: Select all

```
x = 54, y = 59, rule = B3/S23
$23b2o$23bo$11bobo7bobo$11bo2bo6b2o$2b2o10b2o$2b2o8bo3b2o$7b2o5b2o$6bo
4bo2bo$11bobo6$2bo$2b3o$5bo$4b2o$10bo4b2o5b2o10bo$11bo3b2o5b2o11b2o$9b
3o18bobobo$19b2o9bobo$19b2o4$42b2o$4b2o3b2o31b2o$6b3o$5bo3bo29b2o5b2o$
6bobo30b2o5b2o$7bo4$6b2o$6b2o30bo$36bobo$37b2o10$40bo$41bo$39b3o6$52b
2o$52b2o!
```

Last edited by biggiemac on April 28th, 2015, 7:29 pm, edited 2 times in total.

Physics: sophistication from simplicity.

### Re: Smaller sawtooth

So far I've got a Sawtooth183 that reaches minimum population at generation 108 * (121^n - 1). Kudos to simsim for finding the quite shocking reduction of the p120 gun.biggiemac wrote:Here's an appropriate 6hd gun at p120 with 8 fewer cells at minimum population than the twin bees version. If someone can line up the blocker/ship the same as before it could break 200.

Code: Select all

```
x = 86, y = 98, rule = B3/S23
5bob2o$5b2obo3$77b2obo$77bob2o$10bo$2o3bo4b2o$2o2b2o5b2o$5b2o4bo$6bo3b
o$77b2o5b2o$77b2o5b2o2$80b2o$27b2o51b2o$27b2o2$24b2o5b2o$24b2o5b2o24b
2o$17bo39b2o12bo$18b2o51bobo$17b2o34b2o5b2o9b3o$53b2o5b2o11bo21$48bo$
46b2o$47b2o$54bo$52b3o$51bo$51b2o$46bo$45bo$45b3o7$46b2o3b2o$48b3o$47b
o3bo$48bobo$49bo4$49b2o$49b2o3$22b2o12bo4bo$34b2ob4ob2o$25bo10bo4bo$
14b4o$12b2o4b2o$12b2o5bo$14b2obobo$19bo$15bo3bo$15bo4bo$17b3o3bo$17b2o
4bo$23b2o$25bo$25b3o3$28bo$27bob5o$26b2o5bo$26b2o3bo2bo$34bo$28b2obo2b
o$31bo2bo$32b2o$32b2o!
```

Code: Select all

```
x = 78, y = 87, rule = B3/S23
5bob2o$5b2obo3$69b2obo$69bob2o$6bo$2o3bo3b3o$2o2bo6bo$5bo5bo$6b2ob2o$
69b2o5b2o$69b2o5b2o2$72b2o$27b2o43b2o$27b2o2$24b2o5b2o$24b2o5b2o16b2o$
18bo30b2o9bobo$16bobo41bobobo$17b2o26b2o5b2o11b2o$45b2o5b2o10bo20$50bo
$39bo8b3o$39bobo5bo$39b2o6b2o3$45bo$44bobo$39bo3bo3bo$37b2o5b3o$38b2o
2b2o3b2o11$17b4o24b2o$15b2o4b2o22b2o$15b2o5bo$17b2obobo$22bo9bo4bo$18b
o3bo7b2ob4ob2o$18bo4bo8bo4bo$20b3o3bo$20b2o4bo$26b2o$28bo$28b3o3$31bo$
30bob5o$29b2o5bo$29b2o3bo2bo$37bo$31b2obo2bo$34bo2bo$35b2o$35b2o!
```

### Re: Smaller sawtooth

There's actually quite a relation between the period of the gun and the expansion factor. Sawtooth 213/201 had p46 guns and an expansion factor of 46 + 1; sawtooth 183/181 uses p120 guns and has an expansion factor of 120 + 1. Both of them use the (-1, -1) two-glider block retraction.chris_c wrote:So far I've got a Sawtooth183 that reaches minimum population at generation 108 * (121^n - 1). Kudos to simsim for finding the quite shocking reduction of the p120 gun.

EDIT: Now Sawtooth181 with minimum population at 58 * (121^n - 1).

Princess of Science, Parcly Taxel

### Re: Smaller sawtooth

Strangely, I had made adjustings on the unix to get back to stability... whatever. Please ignore this post.biggiemac wrote: 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.

I thought the record was "Smallest pattern to exhibit sawtooth mechanism".

This is game of life, this is game of life!

Loafin' ships eaten with a knife!

Loafin' ships eaten with a knife!

### Re: Smaller sawtooth

Better later than never...congrats to Chris for finding Sawtooth 181! We should update Wiki.

I was checking Wiki being sure I'll find Chris's sawtooth and couldn't find it...

I was checking Wiki being sure I'll find Chris's sawtooth and couldn't find it...

### Re: Smaller sawtooth

Shameless corollary snipe to re-establish myself at the head of the Sawtooth Hall of Fame:

Code: Select all

```
#C Sawtooth 179
#C Adam P. Goucher, 2015-10-25
x = 72, y = 83, rule = B3/S23
63b2obo$63bob2o6$63b2o5b2o$63b2o5b2o$7b2o$7b2o57b2o$66b2o2$10b2o$2o8b
2o$bo41b2o$o6b2o34b2o9bobo$2o5b2o45bobobo$39b2o5b2o11b2o$39b2o5b2o10bo
$18bo$17b3o$16bo$17b2o$17bo9$18b2o$18b2o2$15b2o$15b2o2$44bo$18b2o13bo
8b3o$18b2o13bobo5bo$33b2o6b2o3$39bo$38bobo$33bo3bo3bo$31b2o5b3o$32b2o
2b2o3b2o11$11b4o24b2o$9b2o4b2o22b2o$9b2o5bo$11b2obobo$16bo9bo4bo$12bo
3bo7b2ob4ob2o$12bo4bo8bo4bo$14b3o3bo$14b2o4bo$20b2o$22bo$22b3o3$25bo$
24bob5o$23b2o5bo$23b2o3bo2bo$31bo$25b2obo2bo$28bo2bo$29b2o$29b2o!
```

What do you do with ill crystallographers? Take them to the

**!***mono*-clinic### Re: Smaller sawtooth

Congrats! We should update wiki once again.calcyman wrote:Shameless corollary snipe to re-establish myself at the head of the Sawtooth Hall of Fame

### Re: Smaller sawtooth

Well, corollary sniping is fun!calcyman wrote:Shameless corollary snipe to re-establish myself at the head of the Sawtooth Hall of Fame:Code: Select all

`rle`

Code: Select all

```
#C Sawtooth 177
#C thunk, 2015-10-27
#C Pop=177 at T=15, 6975...
x = 68, y = 76, rule = B3/S23
59b2o$59b2o3$56b2o$56b2o8b2o$7b2o57bo$7b2o50b2o6bo$59b2o5b2o2$10b2o54b
2o$2o8b2o55bo$bo61bo$o6b2o53bo3bo$2o5b2o55b2o2$2o$o$4bo43b3o$bo3bo41bo
2bo$2b2o43bo3bo$46b2obobo$46b2ob2o$47b3o$17b3o$17bo2bo$16bo3bo29b2o$
16bobob2o27bo2bo$17b2ob2o25b5o$18b3o25b2ob3o$46b3o$47bobo$16b2o22bo7b
2o$15bo2bo10bo8b3o$16b5o8bobo5bo$16b3ob2o7b2o6b2o$19b3o$18bobo$18b2o
15bo$34bobo$29bo3bo3bo$27b2o5b3o$28b2o2b2o3b2o11$7b4o24b2o$5b2o4b2o22b
2o$5b2o5bo$7b2obobo$12bo9bo4bo$8bo3bo7b2ob4ob2o$8bo4bo8bo4bo$10b3o3bo$
10b2o4bo$16b2o$18bo$18b3o3$21bo$20bob5o$19b2o5bo$19b2o3bo2bo$27bo$21b
2obo2bo$24bo2bo$25b2o$25b2o!
```

EDIT: Fixed second minimum population time.

"What's purple and commutes?

The Evanston Express."

The Evanston Express."

### Re: Smaller sawtooth

If I got it right, all the components are in its minimal phase, so there won't be any more improvements with

*trivial*(i.e. without change of mechanism) collorary sniping, right??Best wishes to you, Scorbie

### Re: Smaller sawtooth

Very good. It seemed impossible to me to get the phases matched up like that. Well done! Note that I would suggest the following as the canonical version of your pattern. Population is 177 at 48 * (121^n-1).thunk wrote: Well, corollary sniping is fun!Code: Select all

`#C Sawtooth 177 #C thunk, 2015-10-27`

Code: Select all

```
x = 68, y = 77, rule = B3/S23
59b2o$59b2o3$56b2o$56b2o8b2o$7b2o57bo$7b2o50b2o6bo$59b2o5b2o2$10b2o$2o
8b2o$bo46b3o$o6b2o40bo$2o5b2o40b3o4$17b3o$18bo$16b3o4$48b2o$48b2o2$51b
2o$51b2o2$18b2o$18b2o28b2o$40bo7b2o$15b2o21b3o$15b2o20bo$37b2o2$18b2o
5bobo$18b2o5b2o$26bo5$24bo$24bobo5b2o3b2o$24b2o8b3o$33bo3bo$34bobo$35b
o4$35b2o$6b4o25b2o$4b2o4b2o$4b2o5bo$6b2obobo10bo4bo$11bo8b2ob4ob2o$7bo
3bo10bo4bo$7bo4bo$9b3o3bo$9b2o4bo$15b2o$17bo$17b3o3$20bo$19bob5o$18b2o
5bo$18b2o3bo2bo$26bo$20b2obo2bo$23bo2bo$24b2o$24b2o!
```

### Re: Smaller sawtooth

My record didn't last very long. Many congratulations to you!thunk wrote:Well, corollary sniping is fun!

I concur; the canonical version should have the minimum expansion factor (121 in this case) and attain its minimum population in generation 0. Subject to those constraints, it's fashionable to minimise the bounding box.chris_c wrote:Well done! Note that I would suggest the following as the canonical version of your pattern. Population is 177 at 48 * (121^n-1).

I imagine this record will be extremely difficult to improve further. We either need a smaller shotgun, or a smaller spaceship, or a completely different tractor-beam mechanism.

What do you do with ill crystallographers? Take them to the

**!***mono*-clinic### Re: Smaller sawtooth

Just to make the list complete, a smaller 90-degree reflector is also a theoretical possibility. Any decade now, Catagolue might turn up a new [p10|p12|p15|p18|p20|p24] oscillator with a glider-turning spark, that happens to have less than 23 ON cells in the key phase.calcyman wrote:I imagine this record will be extremely difficult to improve further. We either need a smaller shotgun, or a smaller spaceship, or a completely different tractor-beam mechanism.

Or CatForce might turn up a five-block constellation that recovers after turning a glider, or maybe it will be a mixed constellation including a blinker or two.

All of these are impressively low-probability, I must admit.

### Re: Smaller sawtooth

Well, thunk, I implore to know what your real name and (optionally) location is for the sake of posterity. Conway's Life is still very much academic in nature.thunk wrote:...

Princess of Science, Parcly Taxel

### Re: Smaller sawtooth

Thanks. I would agree that that should be the canonical version, I just wasn't sure what the usual conventions for posting sawtooths were.chris_c wrote:Very good. It seemed impossible to me to get the phases matched up like that. Well done! Note that I would suggest the following as the canonical version of your pattern. Population is 177 at 48 * (121^n-1).

I think so too. The only B60 phases with smaller population are 39 and 40 ticks after the initial Herschel position, and none of those work in the sawtooth--at 40 ticks, there end up being two sets of gliders in the pattern when the block is finally deleted (10 cell penalty), and at 100 ticks, the FNGs have not yet been deleted by the boat-bits.Scorbie wrote:If I got it right, all the components are in its minimal phase, so there won't be any more improvements with trivial (i.e. without change of mechanism) collorary sniping, right??

Just so you know, demanding to out people in public is rather rude. Many people have reasons for not wanting to divulge their birth name, and I'm not the first (Guam comes to mind) to prefer pseudonymity.Freywa wrote: Well, thunk, I implore to know what your real name and (optionally) location is for the sake of posterity. Conway's Life is still very much academic in nature.

Anyway, a nym didn't prevent this guy from being published.

"What's purple and commutes?

The Evanston Express."

The Evanston Express."

### Re: Smaller sawtooth

Yes, I'm pretty certain it is optimal based on present technology. As such, I've created a wiki page for your discovery: http://conwaylife.com/wiki/Sawtooth_177thunk wrote:I think so too. The only B60 phases with smaller population are 39 and 40 ticks after the initial Herschel position, and none of those work in the sawtooth--at 40 ticks, there end up being two sets of gliders in the pattern when the block is finally deleted (10 cell penalty), and at 100 ticks, the FNGs have not yet been deleted by the boat-bits.Scorbie wrote:If I got it right, all the components are in its minimal phase, so there won't be any more improvements with trivial (i.e. without change of mechanism) collorary sniping, right??

I tend to adhere to Gaetane's advice on this matter, but it's entirely a matter of personal preference: http://security.stackexchange.com/quest ... 6631#46631thunk wrote:Just so you know, demanding to out people in public is rather rude. Many people have reasons for not wanting to divulge their birth name, and I'm not the first (Guam comes to mind) to prefer pseudonymity.

Or indeed Satoshi Nakamoto.thunk wrote:Anyway, a nym didn't prevent this guy from being published.

What do you do with ill crystallographers? Take them to the

**!***mono*-clinic### Re: Smaller sawtooth

Using the other side of the spaceship allows for some bounding box reduction.

On the right is the standard Sawtooth 177, at size 74*60 (63 with pd sparks). Its minimum population is now achieved in gens 28*(121^n-1).

On the left is the compactified variant, at size 57*63 (60*63 with pd sparks). Its minimum population is 189(+10 at odd n), at gens 23*(121^n-1). The odd multiplier means that the minimum population only occurs every other cycle.

EDIT: Forgot to count the pentadecathlon sparks. Though the twin bees sparks weren't counted in Sawtooth 213 or 201.

Code: Select all

```
x = 158, y = 74, rule = B3/S23
54b2obo$54bob2o2$12b2o$12b2o$46bo$46b2o$15b2o28bobo6b2o5b2o$5b2o8b2o
37b2o5b2o$6bo$5bo6b2o22bo20b2o$5b2o5b2o23b2o18b2o$36bobo2$149b2o$22b3o
10b2o112b2o$23bo11b3o$21b3o11b2ob2o$30b2o3b2ob2o106b2o$30b2o4bob2o106b
2o8b2o$37bo59b2o57bo$97b2o50b2o6bo$149b2o5b2o2$100b2o$90b2o8b2o$91bo
46b3o$23b2o65bo6b2o40bo$23b2o65b2o5b2o40b3o$45bo$20b2o21b3o$20b2o20bo$
2b4o36b2o63b3o$2o4b2o100bo$2o5bo15b2o5bobo73b3o$2b2obobo15b2o5b2o$7bo
23bo$3bo3bo$3bo4bo129b2o$5b3o3bo126b2o$5b2o4bo$11b2o16bo111b2o$13bo15b
obo5b2o3b2o97b2o$13b3o13b2o8b3o$38bo3bo65b2o$39bobo66b2o28b2o$16bo23bo
89bo7b2o$15bob5o83b2o21b3o$14b2o5bo83b2o20bo$14b2o3bo2bo104b2o$22bo17b
2o44b4o$16b2obo2bo17b2o42b2o4b2o16b2o5bobo$19bo2bo61b2o5bo16b2o5b2o$
20b2o64b2obobo24bo$20b2o5bo4bo58bo$25b2ob4ob2o52bo3bo$27bo4bo54bo4bo$
89b3o3bo$89b2o4bo18bo$95b2o17bobo5b2o3b2o$97bo16b2o8b3o$97b3o23bo3bo$
124bobo$125bo$100bo$99bob5o$98b2o5bo$98b2o3bo2bo18b2o$106bo18b2o$100b
2obo2bo$103bo2bo$104b2o6bo4bo$104b2o4b2ob4ob2o$112bo4bo!
```

On the left is the compactified variant, at size 57*63 (60*63 with pd sparks). Its minimum population is 189(+10 at odd n), at gens 23*(121^n-1). The odd multiplier means that the minimum population only occurs every other cycle.

EDIT: Forgot to count the pentadecathlon sparks. Though the twin bees sparks weren't counted in Sawtooth 213 or 201.

"What's purple and commutes?

The Evanston Express."

The Evanston Express."

### Re: Smaller sawtooth

Yup, the bounding box of Sawtooth 201 would be larger (79x59) if sparks were included in the same way that they are traditionally included in glider gun bounding boxes.thunk wrote:EDIT: Forgot to count the pentadecathlon sparks. Though the twin bees sparks weren't counted in Sawtooth 213 or 201.

Luckily, sawteeth are not glider guns -- their bounding boxes are always increasing in any case. And clearly there's something of an established tradition to measure the bounding box without any transitory sparks, unless they're actually present in the T=0 generation. So I've added the new Sawtooth 189 record as 63x57 in the LifeWiki and left all the old bounding-box measurements unchanged.

There's a potentially difficult case that comes up here, where a sawtooth with a smaller bounding box can be constructed if the minimum-population "canonical phase" occurs at T>0. Luckily I think that's not an issue for Sawtooth 189, since the spaceship is already as close as it can safely get to the pentadecathlon.

### Re: Smaller sawtooth

Ah, okay.dvgrn wrote:Luckily, sawteeth are not glider guns -- their bounding boxes are always increasing in any case. And clearly there's something of an established tradition to measure the bounding box without any transitory sparks, unless they're actually present in the T=0 generation. So I've added the new Sawtooth 189 record as 67x57 in the LifeWiki and left all the old bounding-box measurements unchanged.

Anyway, I forgot to mention: Since we don't need the buckaroo to be synchronized with the pd, the pd can be moved one cell further in, resulting in a bounding box of 62*56.

Code: Select all

```
x = 62, y = 56, rule = B3/S23
53b2obo$53bob2o2$11b2o$11b2o3$14b2o28b2o7b2o5b2o$4b2o8b2o37b2o5b2o$5bo
37bobo$4bo6b2o28bo14b2o$4b2o5b2o24b2obob2o12b2o$38bob2o$39bo2$33b2o$8b
2o23b2o$8bobo$8bo20b2o5b2o$29b2o5b2o4$21b3o$21bo2bo$21bob2o4$20b2o12bo
9bo$20bo2bo10bobo5b3o$2b4o15b3o10b2o5bo$2o4b2o33b2o$2o5bo$2b2obobo14b
2o$7bo14b2o$3bo3bo26bo4bo$3bo4bo23b2o5bo$5b3o3bo21b2o3bobo$5b2o4bo25b
2ob2o$11b2o23bo5bo$13bo25bo$13b3o20b2o3b2o3$16bo21bo$15bob5o16bo$14b2o
5bo15bo$14b2o3bo2bo$22bo$16b2obo2bo16b2o$19bo2bo16b2o$20b2o$20b2o5bo4b
o$25b2ob4ob2o$27bo4bo!
```

"What's purple and commutes?

The Evanston Express."

The Evanston Express."

### Re: Smaller sawtooth

That doesn't seem particularly unfortunate. If you're optimizing the bounding box, the minimum population doesn't matter, and vice versa -- unless you can hold one metric constant and reduce the other, anyway.thunk wrote:Since we don't need the buckaroo to be synchronized with the pd, the pd can be moved one cell further in, resulting in a bounding box of 62*56...

Unfortunately, the minimum population is increased still further to 195, achieved in gens 38*(121^n-1) (every cycle now).

I've done the necessary quick fixes to the LifeWiki, and am now trying to pretend that the almost-smallest almost-lowest-population Sawtooth 189 was never mentioned. Someone can create a page for Sawtooth 195 if they want, but I'm pretty happy just calling it a variant of Sawtooth 177 and providing an external link to the pattern, here in this thread.

There isn't a separate section in Sawtooth 177 showing Sawtooth 195 (yet -- that would be another option). So for now I just added SneakyWikiLinks: "[[Sawtooth 177|Sawtooth 195]]".

### Re: Smaller sawtooth

Where do you stand on the following variant of Sawtooth 177? It attains its minimum repeating population of 177 in generation 0, so it's canonical, and is 74-by-58 instead of 74-by-60. But it feels incredibly disingenuous in a way that is impossible to capture mathematically:dvgrn wrote:Luckily, sawteeth are not glider guns -- their bounding boxes are always increasing in any case. And clearly there's something of an established tradition to measure the bounding box without any transitory sparks, unless they're actually present in the T=0 generation.

Code: Select all

```
#C Canonical Sawtooth 177 with reduced bounding box:
x = 74, y = 58, rule = B3/S23
64b4o2$66b2o$62b2o$62b2o8b2o$13b2o57bo$13b2o50b2o6bo$65b2o5b2o2$16b2o$
6b2o8b2o$7bo46b3o$6bo6b2o40bo$6b2o5b2o40b3o4$23b3o$24bo$22b3o4$54b2o$
54b2o2$57b2o$57b2o2$24b2o$24b2o28b2o$46bo7b2o$21b2o21b3o$21b2o20bo$43b
2o$2b4o$2o4b2o16b2o5bobo$2o5bo16b2o5b2o$2b2obobo24bo$7bo$3bo3bo$3bo4bo
$5b3o3bo$5b2o4bo18bo$11b2o17bobo5b2o3b2o$13bo16b2o8b3o$13b3o23bo3bo$
40bobo$41bo$16bo$15bob5o$14b2o5bo$14b2o3bo2bo18b2o$22bo18b2o$16b2obo2b
o$19bo2bo$20b2o$20b2o4b10o!
```

What do you do with ill crystallographers? Take them to the

**!***mono*-clinic### Re: Smaller sawtooth

Bah, humbug. I mean... I was definitely thinking about tricks like this this morning, but figured I wouldn't speak up and give anyone any horrible ideas.calcyman wrote:Where do you stand on the following variant of Sawtooth 177? It attains its minimum repeating population of 177 in generation 0, so it's canonical, and is 74-by-58 instead of 74-by-60. But it feels incredibly disingenuous in a way that is impossible to capture mathematically...

I think every sawtooth up to now has returned to its exact initial configuration every time it has returned to its minimum population. This unfortunately isn't required by the definition of a sawtooth... but I think we could safely say that "canonical" implies "the phase of the sawtooth that you can compare to every future minimum phase, and see that it's the same except for the position of the spaceship."calcyman wrote:If anyone can see any reason why this shouldn't be considered the canonical Sawtooth 177, speak now...

Also the T=1 population doesn't match the pattern that all future sawtooth cycles will follow -- there are a couple of extra cells still present in the pre-pentadecathlon.

... If the T=1 population-graph argument doesn't hold, then another possible reason that this isn't really the canonical Sawtooth 177, is that someone might well be able to make an even more evil and disingenuous version that cuts off another few rows or columns. There are at least a dozen cells available that could be harmlessly subtracted from their starting location and added anywhere they're needed, to make pre-blocks or a pre-snake.

I'm not

*sure*that this can actually be done, and I'm trying hard not to think about it. It's a valid optimization problem, I suppose, not so different from shrinking a spacefiller down into its smallest possible bounding box, or curling up the spiral-growth recipe gliders so they fit into the initial diamond shape... but I think it's worth defending the definition of "canonical" so that it continues to mean the 74x60 Sawtooth 177.

### Re: Smaller sawtooth

Unfortunately the population graph can be made to match exactly, with hardly any trouble at all:dvgrn wrote:Also the T=1 population doesn't match the pattern that all future sawtooth cycles will follow -- there are a couple of extra cells still present in the pre-pentadecathlon.

Code: Select all

```
x = 74, y = 58, rule = B3/S23
64b4o2$66b2o$62b2o$62b2o8b2o$13b2o57bo$13b2o50b2o6bo$65b2o5b2o2$16b2o$
6b2o8b2o$7bo46b3o$6bo6b2o40bo$6b2o5b2o40b3o4$23b3o$24bo$22b3o4$54b2o$
54b2o2$57b2o$57bo$59bo$24b2o$24b2o28b2o$46bo7bo$21b2o21b3o9bo$21b2o20b
o$43b2o$2b4o$2o4b2o16b2o5bobo$2o5bo16b2o5b2o$2b2obobo24bo$7bo$3bo3bo$
3bo4bo$5b3o3bo$5b2o4bo18bo$11b2o17bobo5b2o3b2o$13bo16b2o8b3o$13b3o23bo
3bo$40bobo$41bo$16bo$15bob5o$14b2o5bo$14b2o3bo2bo18b2o$22bo18b2o$16b2o
bo2bo$19bo2bo$20b2o$20b2o4b10o!
```

### Re: Smaller sawtooth

Suppose PatternA and PatternB are identical after N ticks. Let S be the set of cells that become periodic in both patterns. We can say that PatternA <= PatternB if each cell in S becomes periodic in PatternA not later than the time that it becomes periodic in PatternB.dvgrn wrote: So I'm afraid the definition of "canonical" is going to have to carry all the weight of defending sawtooth initial bounding boxes against optimizations from the Dark Side. Anyone else have a better defense?

That should cut out the tricks mentioned so far. Maybe there are other problems with the definition but I'm trying not to look too hard for them.

### Re: Smaller sawtooth

calcyman wrote:Yes, I'm pretty certain it is optimal based on present technology.thunk wrote:I think so too...Scorbie wrote:If I got it right, all the components are in its minimal phase, so there won't be any more improvements with trivial (i.e. without change of mechanism) collorary sniping, right??

Um... for Sawtooth 177, however, it's perfectly possible to move the spaceship one diagonal step closer, cutting the bounding box by one column. No Dark Side trickery, really, just Yet Another Corollary-Snipe:dvgrn wrote:Luckily I think that's not an issue for Sawtooth 189, since the spaceship is already as close as it can safely get to the pentadecathlon.

Code: Select all

```
#C population 177 at T=0, 336720, 4930254240, ... 23*(121^2N-1)...
x = 73, y = 60, rule = B3/S23
64b2o$64b2o3$61b2o$61b2o8b2o$12b2o57bo$12b2o50b2o6bo$64b2o5b2o2$15b2o$
5b2o8b2o$6bo46b3o$5bo6b2o40bo$5b2o5b2o40b3o4$22b3o$23bo$21b3o4$53b2o$
53b2o2$56b2o$56b2o2$23b2o$23b2o28b2o$45bo7b2o$20b2o21b3o$20b2o20bo$2b
4o36b2o$2o4b2o$2o5bo15b2o5bobo$2b2obobo15b2o5b2o$7bo23bo$3bo3bo$3bo4bo
$5b3o3bo$5b2o4bo$11b2o16bo$13bo15bobo5b2o3b2o$13b3o13b2o8b3o$38bo3bo$
39bobo$16bo23bo$15bob5o$14b2o5bo$14b2o3bo2bo$22bo17b2o$16b2obo2bo17b2o
$19bo2bo$20b2o$20b2o5bo4bo$25b2ob4ob2o$27bo4bo!
```