## Systematic survey of small patterns

For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.
NickGotts
Posts: 82
Joined: November 10th, 2011, 6:20 pm

### Re: Systematic survey of small patterns

Having generated and run 21,186,740 10-cell single-cluster patterns (out of an estimated 90,000,000), I have the first interesting result to report: a methuselah lasting 17,425 steps - thus beating bunnies10 by just 2 steps:

Code: Select all

x = 6, y = 6, rule = B3/S23
bo$2obo$4b2o$o2bo$o$o!  Step 30 of the pattern is almost the same as step 28 of bunnies10, but lacks a glider which the latter has just thrown off. It thus ends with a population of 1,744, including 40 gliders, as opposed to bunnies10's 1749 and 41 gliders. Since it's hardly a radical improvement, I suggest calling it bunnies10a. Its MCPS is 12, its bounding box 8*5. Just to note what interesting results I could and could not find. Since my main aim is to find all 10-cell infinite growth patterns, I'm just looking at patterns that my algorithm does not judge to be quiescent at 16,384 steps. It's a conservative algorithm, since it just checks that the pattern contains nothing other than still lifes, oscillators, gliders and *WSSs, and requires the gliders and *WSSs to be a certain distance away from the still lifes, oscillators, and anything travelling in other directions. The version I'm using at present checks that the still lifes and oscillators are unchanged after 12 steps - so if an oscillator has a period not divisible by 12, the pattern is reported as non-quiescent. All such patterns reported so far either are or include a pentadecathlon, or are variants of bunnies10; but all the latter except bunnies10 itself and now bunnies10a, resolve in less than 17,423 steps. I'll cross-post part of this message to the methuselahs thread. Moosey Posts: 2525 Joined: January 27th, 2019, 5:54 pm Location: A house, or perhaps the OCA board. Contact: ### Re: Systematic survey of small patterns Just so that nobody ends up confused if they look at this thread and not the other, On that thread, I wrote: NickGotts wrote:Here's a 10-cell methuselah lasting 17,425 steps - thus beating bunnies10 by just 2 steps: Code: Select all x = 6, y = 6, rule = B3/S23 bo$2obo$4b2o$o2bo$o$o!

Step 30 of the pattern is almost the same as step 28 of bunnies10, but lacks a glider which the latter has just thrown off. It thus ends with a population of 1,744, including 40 gliders, as opposed to bunnies10's 1749 and 41 gliders. Since it's hardly a radical improvement, I suggest calling it bunnies10a. Its MCPS is 12, its bounding box 8*5.

A bit of background is available in the "Systematic survey of small patterns" thread.
I think you must have miscounted its lifespan— that IS bunnies 10
Last edited by Moosey on June 4th, 2019, 5:39 pm, edited 1 time in total.
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NickGotts
Posts: 82
Joined: November 10th, 2011, 6:20 pm

### Re: Systematic survey of small patterns

Just a note on why I estimate 90,000,000 10-cell single-cluster patterns. The numbers of single-cell patterns with successive numbers of cells are (using my algorithm for generating them) as follows, with ratios between the counts for successive numbers of cells, to three decimal places, added in parentheses:
o 1 cell: 0
o 2 cells: 0
o 3 cells: 10
o 4 cells: 66 (4:3 ratio 6.6)
o 5 cells: 551 (5:4 ratio 8.348)
o 6 cells: 5,777 (6:5 ratio 10.485)
o 7 cells: 61,898 (7:6 ratio 10.715)
o 8 cells: 692,809 (8:7 ratio 11.193)
o 9 cells: 7,870,790 (9:8 ratio 11.361)

As can be seen, the successive ratios increase, but the rate of increase shows a tendency to fall (although this is not quite monotonic). Multiplying 7,870,790 by 11.361 gives 89,420,045 (rounding down). So I expect the 10-cell count to be slightly above 90,000,000.

NickGotts
Posts: 82
Joined: November 10th, 2011, 6:20 pm

### Re: Systematic survey of small patterns

I managed to miscopy bunnies10a, posting bunnies10 instead. Thanks to moosey for catching the error on the Long-lived methuselahs thread. Here's the real bunnies10a:

Code: Select all

x = 8, y = 5, rule = B3/S23
2bo$ob2obo$6bo$o4b2o$7bo!

A for awesome
Posts: 1906
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1
Contact:

### Re: Systematic survey of small patterns

NickGotts wrote:Here's the real bunnies10a:

Code: Select all

x = 8, y = 5, rule = B3/S23
2bo$ob2obo$6bo$o4b2o$7bo!
Trivial variant with a smaller bounding box:

Code: Select all

x = 7, y = 5, rule = B3/S23
bo$3obo$5bo$o3b2o$6bo!
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

NickGotts
Posts: 82
Joined: November 10th, 2011, 6:20 pm

### Re: Systematic survey of small patterns

A for Awesome wrote:
Trivial variant with a smaller bounding box:
Code: Select all / Show in Viewer
x = 7, y = 5, rule = B3/S23
bo$3obo$5bo$o3b2o$6bo!
Thanks! Now you point it out, I find* it had actually turned up in my search a little earlier, only I didn't notice it, i.e., I failed to check exactly how long it lasted! With its smaller bounding box, it should clearly be the preferred variant.

*My generating algorithm generates all the patterns with a given distribution across columns together, and yours is a 2,2,1,0,2,2,1, which comes before 2,0,2,1,0,2,2,1 in the ordering I use, so I looked back to find that distribution.

NickGotts
Posts: 82
Joined: November 10th, 2011, 6:20 pm

### Re: Systematic survey of small patterns

This is not something that has yet turned up in my survey of single-cluster 10-cell patterns, but something I noticed should do so later. As far as oscillator periods are concerned, we know that 3 cells are required for period 2, 7 for period 3, and 8 for period 15. Turns out 10 are needed for period 8:

Code: Select all

x = 6, y = 6, rule = B3/S23
bo$2obo$4bo$bo$2bob2o$4bo! I'd be surprised if no-one's noticed this before - probably just not thought worth noting, as my focus on minimal-cell starting pattern problems (mainly a result of their connection to sparse life) is unusual. There may be period 4, 6 or 12 oscillators resulting from 9 or 10 cells - my searches would not turn them up as non-quiescent, as the test for quiescence I've used in these cases runs a pattern forward 12 steps for reasons of convenience (the factor of 4 makes it easy to identify gliders and *WSSs, the factor of 3 prevents pulsars giving a false positive for non-quiescence - at 8 cells and above, they become fairly common). There are no period n oscillators for any n that is not a factor of 12 produced from initial patterns with less than 10 cells. (A quibble here: if you count the whole final pattern, excluding gliders and *WSSs, as a single oscillator consisting of multiple independent parts, you could say there are period 6 oscillators starting from 8 cells (final patterns including a blinker and a pulsar), and period 30 oscillators starting from 9 cells (blinker plus pentadecathlon - the only 7-cell patterns producing a pulsar produce nothing else, the only 8-cell patterns producing a pentadecathlon produce nothing else). Sarp Posts: 203 Joined: March 1st, 2015, 1:28 pm ### Re: Systematic survey of small patterns NickGotts wrote: ... the only 8-cell patterns producing a pentadecathlon produce nothing else). Are you sure? Code: Select all x = 8, y = 6, rule = B3/S23 7bo$6b2o$2o$o$o$bo!


2718281828
Posts: 738
Joined: August 8th, 2017, 5:38 pm

### Re: Systematic survey of small patterns

Sarp wrote:
NickGotts wrote: ... the only 8-cell patterns producing a pentadecathlon produce nothing else).
Are you sure?

Code: Select all

x = 8, y = 6, rule = B3/S23
7bo$6b2o$2o$o$o$bo!  I think he is. The only 8-cell pattern which produces only(!) the PD. NickGotts Posts: 82 Joined: November 10th, 2011, 6:20 pm ### Re: Systematic survey of small patterns Sarp wrote: Are you sure? Code: Select all / Show in Viewer x = 8, y = 6, rule = B3/S23 7bo$6b2o$2o$o$o$bo!
You're right of course - and in an earlier post, I described the 8-cell patterns that produce a pentadecathlon plus debris as well as those which produce a pentadecathlon alone - but managed to post the wrong rle (which 2718281828 quotes in the post after yours)! So apologies and thanks, here is the rle I should have posted on 12th December 2018:

Code: Select all

x = 65, y = 39, rule = Life
22bo$31bobo8bo$21b3o8bo8b3o$32bo3$32bo$21b3o8bo8b3o$31bobo8bo$22bo21$
2b3o27b3o28bo$bo2bo27bo2bo26b3o2$63bo3$2o28b2o$o29bo29b2o$60bo! (I've discovered why I'm prone to post the wrong rle. The editor I use most of the time to look at rle files is a version of emacs which doesn't copy a selection to the clipboard when you hit Ctrl-C, as most of the software I use does. So if I'm not careful, I think I've copied the rle I want to include in a post, when I haven't.) NickGotts Posts: 82 Joined: November 10th, 2011, 6:20 pm ### Re: Systematic survey of small patterns Here are two more variants of bunnies10a (10-cell single-cluster patterns that last 17425 steps). The one on the left has a bounding box 7*5, and an MCPS of 11, which according to the usual criteria for methuselahs makes it the preferred variant. Code: Select all x = 27, y = 5, rule = B3/S23 bo19bo$3o17b2obo$3bobo19bo$bo2b2o15b2ob2o$6bo19bo! My search has also (as it should) found Paul Callahan's 10-cell single-cluster pattern producing a block-laying switch engine, together with 5 trivial variants: Code: Select all x = 50008, y = 50007, rule = B3/S23 bo$2obo$bobo$3bo$5bobo$5bo9995$10001bo$10000b2obo$10001bobo$10003bo$10005bo$10005bobo9995$20001bo$20000b2obo$20003bo$20001bobo$20005bobo$
20005bo9995$30001bo$30000b2obo$30003bo$30001bobo$30005bo$30005bobo
9995$40001bo2$40000b2obo$40001bobo$40003bo$40005bobo$40005bo9994$50001bo2$50000b2obo$50001bobo$50003bo$50005bo$50005bobo!

Macbi
Posts: 699
Joined: March 29th, 2009, 4:58 am

### Re: Systematic survey of small patterns

Nice progress!
NickGotts wrote:My search has also (as it should) found Paul Callahan's 10-cell single-cluster pattern producing a block-laying switch engine, together with 5 trivial variants:

Code: Select all

x = 50008, y = 50007, rule = B3/S23
bo$2obo$bobo$3bo$5bobo$5bo9995$10001bo$10000b2obo$10001bobo$10003bo$
10005bo$10005bobo9995$20001bo$20000b2obo$20003bo$20001bobo$20005bobo$20005bo9995$30001bo$30000b2obo$30003bo$30001bobo$30005bo$30005bobo 9995$40001bo2$40000b2obo$40001bobo$40003bo$40005bobo$40005bo9994$
50001bo2$50000b2obo$50001bobo$50003bo$50005bo$50005bobo! For those of us whose monitors are smaller than a parsec: Code: Select all x = 487, y = 479, rule = B3/S23 bo$2obo$3bo$bobo$5bobo$5bo93$96bo2$95b2obo$96bobo$98bo$100bo$100bobo
88$195bo$194b2obo$195bobo$197bo$199bo$199bobo93$290bo2$289b2obo$290bob o$292bo$294bobo$294bo79$385bo$384b2obo$385bobo$387bo$389bobo$389bo93$480bo$479b2obo$482bo$480bobo$484bo$484bobo!