Infinite growth patterns easy to memorize and enter by hand

[pcallahan]

This is a very subjective topic, but I started thinking about it after writing this posting. What kinds of infinite growth patterns are so easy to remember that you would never need to look them up, and are small enough to enter cell by cell without your audience getting bored?

Of course, there are tons of patterns large and small available online that are just a click away, so you sort of have to imagine yourself with a time machine to get the most benefit. You step out of your device sometime in the 70s or 80s to find a Life enthusiast busy with whatever program they use (and very likely wrote themselves).

"I am a future human," you intone, "From... the future. But wait! Here's the good part:"

And you proceed to show them cool stuff they have never seen before. Sadly, you are hampered by the lack of internet, lack of copy-paste, and don't want to look silly pulling folded pieces of paper out of your pocket. Remember, you represent... the future... so this had better be good. You decide on infinite growth patterns that you know by heart and can draw easily.

Easy to remember is not the same as small patterns, though there is some intersection. Start with the 5x5 infinite growth pattern (known result). That's small and not hard to memorize.

Code: Select all

x = 5, y = 5, rule = B3/S23
3obo$o$3b2o$b2obo$obobo!

At this orientation, it looks to me like a minimalist landscape with a mountain at the bottom and something in the sky, perhaps the sun in the upper right. That sideways L could be clouds (YMMV). Pretty easy to remember one way or another, but a little hard to explain.

Most people have more experience memorizing symbols, so what if we try to keep a symbolic representation as small as possible. The smallest patterns don't have the smallest RLE necessarily. I can't guarantee this is minimal, but here is an infinite growth pattern with an 11-character RLE string that I found by enumeration. 11-characters is enough to encode very long lines, but this is a compromise in terms of encoding length and what you can enter.

Code: Select all

x = 22, y = 2, rule = B3/S23
bo$8o3b11o!

I do not know if anyone else has done a search for short RLE. This is "new to me" as of this morning.

Still, the string "bo$8o3b11o!" is not very intuitive. Enter 7-segment display integers, encoded as 5x3 bitmaps. Dean Hickerson explored these some years back and found that 154299 was an infinite growth pattern in this simple "font" (and turns out to be pretty easy to enter by hand, but be sure to remember the lowest segment is lit on those 9s).

Code: Select all

x = 21, y = 5, rule = B3/S23
ob3obobob3ob3ob3o$obo3bobo3bobobobobo$ob3ob3ob3ob3ob3o$o3bo3bobo5bo3bo
$ob3o3bob3ob3ob3o!

There is a smaller number, 140732, that works and somehow eluded DRH.

Code: Select all

x = 21, y = 5, rule = B3/S23
obobob3ob3ob3ob3o$obobobobo3bo3bo3bo$ob3obobo3bob3ob3o$o3bobobo3bo3bob
o$o3bob3o3bob3ob3o!

I believe it was first mentioned on this blog.

But what about other things you can display with 7-segments? You can do hexadecimal, for one thing, if you don't mind mixing lowercase b and d with A, C, E, F. There were some old calculators that did this.

You can go down to 5 digits this way, and here are the examples of infinite growth I found. I did not find any with 4 digits, unfortunately:

20Ab1

Code: Select all

x = 19, y = 5, rule = B3/S23
3ob3ob3obo5bo$2bobobobobobo5bo$3obobob3ob3o3bo$o3bobobobobobo3bo$3ob3o
bobob3o3bo!

99bC8

Code: Select all

x = 19, y = 5, rule = B3/S23
3ob3obo3b3ob3o$obobobobo3bo3bobo$3ob3ob3obo3b3o$2bo3bobobobo3bobo$3ob
3ob3ob3ob3o!

b0bCd

Code: Select all

x = 19, y = 5, rule = B3/S23
o3b3obo3b3o3bo$o3bobobo3bo5bo$3obobob3obo3b3o$obobobobobobo3bobo$3ob3o
b3ob3ob3o!

F7663

Code: Select all

x = 19, y = 5, rule = B3/S23
3ob3ob3ob3ob3o$o5bobo3bo5bo$3o3bob3ob3ob3o$o5bobobobobo3bo$o5bob3ob3ob
3o!

My favorite is b0bCd. It's easy to remember and write. And it has a name: Bob.

How much further can we reduce the number of display digits? Here is a pattern that works with 3 7-segment display cells:

Code: Select all

x = 11, y = 5, rule = B3/S23
4bobob3o$4bobo3bo$3ob3ob3o$2bo5bo$3ob3obo!

The only recognizable character is the rightmost, which can generously be called a question mark. So this one is small, but not really easy to memorize as a string.

Finally, here are some that encode an extended character set. These have all been used before on 7 segment displays. My favorite is probably _39-. That is easy to remember. AoCu is sort of cool, but would be much better if Ao was a chemical element (which it is not... Darn!).

5?28

Code: Select all

x = 15, y = 5, rule = B3/S23
3ob3ob3ob3o$o5bo3bobobo$3ob3ob3ob3o$2bobo3bo3bobo$3obo3b3ob3o!

96u4

Code: Select all

x = 15, y = 5, rule = B3/S23
3ob3o5bobo$obobo7bobo$3ob3obobob3o$2bobobobobo3bo$3ob3ob3o3bo!

AoCu

Code: Select all

x = 15, y = 5, rule = B3/S23
3o5b3o$obo5bo$3ob3obo3bobo$obobobobo3bobo$obob3ob3ob3o!

_39-

Code: Select all

x = 15, y = 5, rule = B3/S23
4b3ob3o$6bobobo$4b3ob3ob3o$6bo3bo$3ob3ob3o!

[pcallahan]

For completeness, here are some 5-character examples that create glider-producing switch engines.

027n0

Code: Select all

x = 19, y = 5, rule = B3/S23
3ob3ob3o5b3o$obo3bo3bo5bobo$obob3o3bob3obobo$obobo5bobobobobo$3ob3o3bobobob3o!

15nCn

Code: Select all

x = 19, y = 5, rule = B3/S23
2bob3o5b3o$2bobo7bo$2bob3ob3obo3b3o$2bo3bobobobo3bobo$2bob3obobob3obobo!

2UJyo

Code: Select all

x = 19, y = 5, rule = B3/S23
3obobo3bobobo$2bobobo3bobobo$3obobobobob3ob3o$o3bobobobo3bobobo$3ob3ob3ob3ob3o!

519tH

Code: Select all

x = 19, y = 5, rule = B3/S23
3o3bob3obo3bobo$o5bobobobo3bobo$3o3bob3ob3ob3o$2bo3bo3bobo3bobo$3o3bob3ob3obobo!

[pcallahan]

Finally, a vanity pattern.

Code: Select all

x = 19, y = 5, rule = B3/S23
3ob5obobo3b3o$obobobobobobo3bo$3ob3obobobo3bobo$o3bobobobobo3bo$o3bobo
b7ob3o!

It is my name PAULC with 3 bit flips to turn it into an infinite growth pattern. My enumeration found two others, but they are less legible and this one produces infinite gliders as a bonus.

OK, I'll stop now... Basically, I dusted off my old ANSI C code (circa 1996) and remembered enough to apply the crude hammer of detecting infinite growth patterns. Today everything looks like a nail. I wrote Python script to generate the 7 segment display characters. I will clean it up and post it if and only if some expresses interest. There is nothing very special about it but it could save some work.

[Freywa]

Finally, a vanity pattern.

Code: Select all

x = 19, y = 5, rule = B3/S23
3ob5obobo3b3o$obobobobobobo3bo$3ob3obobobo3bobo$o3bobobobobo3bo$o3bobo
b7ob3o!

Looks like Paul E to me.

[pcallahan]

Looks like Paul E to me.

The dot is at least disconnected. I find the havoc of connecting the U and L to be more of a problem.

I worked with a Paul E once. Maybe it's his pattern!

[pcallahan]

If my previous vanity pattern doesn't look enough like PAULC, an alternative is to add a spark row below the pattern that does not touch it. Out of 2^19 possibilities, there are 4 results and two functionally distinct patterns.

Code: Select all

x = 19, y = 7, rule = B3/S23
3ob3obobobo3b3o$obobobobobobo3bo$3ob3obobobo3bo$o3bobobobobo3bo$o3bobo
b3ob3ob3o2$4b4obo3bob3o!

An extra cell can be added in either the leftmost position or to the right without affecting the pattern, but the above is minimal. It is the block-producing form of the switch engine.

This one has more live cells but makes the glider-producing form of the switch engine.

Code: Select all

x = 19, y = 7, rule = B3/S23
3ob3obobobo3b3o$obobobobobobo3bo$3ob3obobobo3bo$o3bobobobobo3bo$o3bobo
b3ob3ob3o2$2o2b2ob7o2b2o!

It is not hard to remember where to place the 2-cell segments. Each is lined up with letters P, A, or C.

[dani]

As for the 11-character rle, there may be a string smaller than that...e.g. a row of n live cells where n is less than 9 digits.

[Hunting]

Off-topic: Can we use this "extended charset" system to other object? Like Pentadecathlon, or even... Breeder?

(Off-forum: Can we use this system to other rules?)

[pcallahan]

Off-topic: Can we use this "extended charset" system to other object? Like Pentadecathlon, or even... Breeder?

(Off-forum: Can we use this system to other rules?)

I imagine some other objects can be produced, though I haven't looked. In most cases there would be a lot of extra junk. If there are some that produce clean gliders, we can collide them into complex objects. We might need to leave spaces.

As for the 11-character rle, there may be a string smaller than that...e.g. a row of n live cells where n is less than 9 digits.

True. I do not know how many contiguous straight line cell patterns have been examined. Is there a point where they become predictable? However, a line of over a million cells cannot be entered by hand feasibly. I was unable to find shorter RLE with runs going up to (I think) 100.

I don't want to claim a definitive result, but I think "bo$8o3b11o!" may be the shortest RLE for infinite growth that can be entered by hand in under a minute. (I might look a little harder, or anyone else can.)

[Moosey]

What if we find a square wave in the form 123456789... n?
We could just remember it as 1-through-n

[pcallahan]

What if we find a square wave in the form 123456789... n?
We could just remember it as 1-through-n

Yes, the most general form of the question is finding interesting patterns with low Kolmogorov complexity. However, there are practical limits on how many cells you will want to enter by hand.

[Moosey]

What if we find a square wave in the form 123456789... n?
We could just remember it as 1-through-n

Yes, the most general form of the question is finding interesting patterns with low Kolmogorov complexity. However, there are practical limits on how many cells you will want to enter by hand.

True. I tested up to 13 before I got tired and there are no infinite growths.

[dvgrn]

Off-topic: Can we use this "extended charset" system to other object? Like Pentadecathlon, or even... Breeder?

Have you looked at Golly's Patterns/Syntheses/life-integer-constructions.rle, and Scripts/Lua/life-integergun30.lua? If we can make a pentadecathlon even without an extended charset, then the same trick can probably be done a little smaller with more character options:

Code: Select all

x = 85, y = 5, rule = B3/S23
obobobobob3obobob3obob3ob3ob3ob3ob3obobobob3ob3obobobobobobob3obob3ob
3obo$obobobobobo3bobobobobobobobobo3bo3bobobobobobo3bobobobobobobobobo
3bobo3bo3bobo$ob3ob3ob3ob3ob3obobobobobob3ob3obobob3obob3ob3obob3ob3ob
o3bobo3bo3bobo$o3bo3bo3bo3bobobobobobobobo3bo3bobobo3bobobo5bobo3bo3bo
bo3bobo3bo3bobo$o3bo3bob3o3bob3obob3ob3ob3ob3ob3o3bobob3ob3obo3bo3bobo
3bobo3bo3bobo!

And Dean's method of building a Gosper glider gun with slow pairs of gliders generalizes easily to building a breeder, or whatever else you want. That's also pretty certain to be slightly more efficient with more characters to choose from.

Last time around the exploration went in the opposite direction: what's the _smallest_ number of different characters that allows for universal construction?

[muzik]

How much further can we reduce the number of display digits? Here is a pattern that works with 3 7-segment display cells:

Code: Select all

x = 11, y = 5, rule = B3/S23
4bobob3o$4bobo3bo$3ob3ob3o$2bo5bo$3ob3obo!

The only recognizable character is the rightmost, which can generously be called a question mark. So this one is small, but not really easy to memorize as a string.

Bit late, but I'd think this looks a bit more like a glottal stop.

[pcallahan]

For reference, here is the extended 7-segment display character set I'm using. You can run the pattern, but it just fizzles.

-0123456789?ACEFGHJLPU_bcdnoqrtuy

Code: Select all

x = 131, y = 5, rule = B3/S23
4b3o3bob3ob3obobob3ob3ob3ob3ob3ob3ob3ob3ob3ob3ob3obobo3bobo3b3obobo5bo
9bo9b3o5bo7bobo$4bobo3bo3bo3bobobobo3bo5bobobobobo3bobobobo3bo3bo3bo3b
obo3bobo3bobobobo5bo9bo9bobo5bo7bobo$3obobo3bob3ob3ob3ob3ob3o3bob3ob3o
b3ob3obo3b3ob3obobob3obobobo3b3obobo5b3ob3ob3ob3ob3ob3ob3ob3obobob3o$
4bobo3bobo5bo3bo3bobobo3bobobo3bobo3bobobo3bo3bo3bobobobobobobo3bo3bob
o5bobobo3bobobobobobo3bobo3bo3bobo3bo$4b3o3bob3ob3o3bob3ob3o3bob3ob3ob
o3bobob3ob3obo3b3obobob3ob3obo3b3ob3ob3ob3ob3obobob3o3bobo3b3ob3ob3o!

Some letters appear in both upper and lower case, which is problematic for memorization but increases the odds of finding things. I just finished enumerating all 5-character strings with infinite growth. There are 277, barring bugs. It is conceivable that one of these conceals a higher period oscillator, like p14 tumbler, but a switch engine seems the most likely given their population (3342-6055 reported by my testing code).

EL_?5 (block-laying) has the least population at generation 30000:

Code: Select all

x = 19, y = 5, rule = B3/S23
3obo7b3ob3o$o3bo9bobo$3obo7b3ob3o$o3bo7bo5bo$3ob3ob3obo3b3o!

coU_F (glider-shooting) has the greatest population at generation 30000:

Code: Select all

x = 19, y = 5, rule = B3/S23
8bobo5b3o$8bobo5bo$3ob3obobo5b3o$o3bobobobo5bo$3ob3ob3ob3obo!

And here they all are. I am deliberately not including RLE for each, both to save space and to persuade readers to convince themselves they are quite easy to draw by hand, e.g. in Golly (added after comments: fixed width of 4 for the character including space in between, so there is a lot of space before 1).

--?bG -EAd8 -EP10 -Eb10 -GdLP -_F2P -d4cc 027n0 07A6F 0ELrn 0Fccd 0b0G3 0b4JG 0uA8_ 15nCn 1?FGy 1Cu3t 1FLEo 1G9CP 1q_d2 20Ab1 2C0uA 2P8_d 2UJyo 2ccU- 2nF3_ 2q16? 2r5AP 2ud9J 2yEyF 3-C63 32yCr 38EP? 3APUu 3A_0C 3JPcn 3bcq4 3cnLn 3ry93 42cn9 44yqt 47P9c 4JLbE 4L9Ct 4c0Pd 4n?Eq 4o?H? 4oP3t 519tH 5806y 5EG7n 5c6P? 5r?yL 5tEoq 5u7?6 5u_0c 62Fnu 69-G- 6CEuU 6Cd1c 6GU7J 6Ju0E 6U3EG 6bc4u 6bry- 6bryH 6ot8t 730yJ 733qy 7Jb?r 7UC0o 7co_0 7nFd0 8-_Jo 814yb 84U6t 8J-d6 8J-dL 8J-dq 8UC6- 8bA3- 8oP84 8qJJH 8rb-n 8u-q_ 9-rJ- 900-F 93J91 99b?C 99bC8 9L5nq 9rLFd 9tyu2 ?4520 ?4rAU ?_39- ?qnPE ?tG8J ?yroC ?yrqL A1t1o A6qu6 A76d? AGFJJ An6G? C6LJ5 CLGn_ Cqu77 CrGn5 E127L E21Jy EL_?5 EUt?1 En?Ur Et8uF F6duy F7663 F86bc FL3nu FUF7H F_F2P FcuCq Fr4Eq G6t-5 GL4Gb GP?Hb G_8nE Gn-U_ Gn1-H Gru?6 H1t1o H2ccL H4o_- HEL5q H_39- HoL1q Hq6o5 HuFrU J4Pu2 J8JHb J9GG8 JA1tu JC6LA JFn-0 J_t3P JouPr JqPu2 Jyc3n L-4dP L-4dn L-4o4 L-4oc L00HJ L076A L65y3 L?G59 LE?U2 LF?Gc LPHou LoFU4 LtLrr P0PCq P2cFG P52C4 PC988 PE44L PE_4L PGLdc PUu6q PbG3n Po5Gu Po642 PtH72 U-Fnn UC8LF UEE6u UF-qb UGctF UHn_F UJo20 UL2cn Un13b Un13r Ur0J2 _2?b0 _39-A _43U0 _5cEu _8Au0 _9nE2 _JJ6J _L30- _P-u8 b0bCd b573H b7HUr bC4J- bJFG1 b_8q5 btPd? c?HUr c?JE7 cC470 cUP84 cbHF2 coU_F cr776 d2rLu d65Jq dFr6q duCb0 n-2GP n0tun n664J nG109 nJ4n4 nUG_u nUP84 noCL4 o51t2 oP190 oP19G oP1y0 ocH6L od_JE ou4?J ou?J9 q-21y q0Jq6 q76d? qHnGo qUHUr qy?_o r5AG8 r9HUr r9P-9 rCrG5 rGb7E rU5yF rcLFu rnG?0 tL3FG tLb6d t_39- t_At7 t_oJ1 tn2dC tr0?t tuHUr u9y5- uF2n4 uG96t uJLLy uL0u6 uLdE1 uU4AE uUqAE ubP84 ud0EG uyJF- y-G8U y-dGA y0br8 y1t1o yFqrc yH_0o yLGF? yLGFF yLGFP yLU16 yLU1H yLU1U yLU74 yLU7J yLU7t yrtUq yu1qF

[dvgrn]

I am deliberately not including RLE for each, both to save space and to persuade readers to convince themselves they are quite easy to draw by hand, e.g. in Golly...

Fixed-width font, eh? That's a difference from Eric Angelini's and Dean Hickerson's use of the digits-only version of these 5x3 numbers.

I checked the fixed-width theory by trying H1t10, which turns out to be an unusually dramatic one, producing an MWSS immediately, a beehive on dock via a couple of successive Herschels entering the same area, and a switch engine created by the last Herschel's second natural glider, which manages an impressive long-distance strike without which the reaction would have quickly fizzled out:

Code: Select all

x = 19, y = 5, rule = B3/S23
obo3bobo5bo$obo3bobo5bo$3o3bob3o3bob3o$obo3bobo5bobobo$obo3bob3o3bob3o!
#C [[  STOP 1350 X -23 Y 0 Z 3 STEP 5 T 935 PAUSE 2 ]]

Non-fixed-width search would open up a few alternate possibilities -- maybe try a four-character search and see if infinite growth turns up after all?

[pcallahan]

Fixed-width font, eh? That's a difference from Eric Angelini's and Dean Hickerson's use of the digits-only version of these 5x3 numbers.

Yes, fixed width is easier to implement, and I would argue it is more in the spirit of 7-segment displays. It does make the spacing of 1 seem very off. I don't plan to investigate this much further, but it might be worth doing variable width.

[pcallahan]

Non-fixed-width search would open up a few alternate possibilities -- maybe try a four-character search and see if infinite growth turns up after all?

BTW, I posted four-character results above: 5?28, 96u4, AoCu, _39-

Fortunately none of them depend on fixed/variable font choice. I went up to 5 hoping to find something memorable or funny. I still like the hexadecimal one b0bCd

[AforAmpere]

It would be interesting to find an infinite growth pattern with numbers sequentially like this:

Code: Select all

x = 259, y = 5, rule = B3/S23
3obob3ob3obobob3ob3ob3ob3ob3obob3obobobob3obob3obobobobob3obob3obob3ob
ob3obob3ob3ob3ob3obob3ob3ob3ob3ob3obobob3ob3ob3ob3ob3ob3ob3ob3ob3ob3ob
3ob3ob3obob3ob3ob3ob3ob3obobob3ob3ob3ob3ob3ob3ob3ob3ob3ob3obobob3o$obo
bo3bo3bobobobo3bo5bobobobobobobobobobobo3bobo3bobobobobobo3bobo3bo3bob
obobobobobo3bobobo3bobo3bo3bo3bo3bo3bobobo3bobo5bobo5bo3bo3bobobo3bobo
bo3bobobo3bobo3bo3bo3bo3bo3bobobo3bobo5bobo5bo3bo3bobobo3bobobobobobob
o$obobob3ob3ob3ob3ob3o3bob3ob3obobobobobobob3obob3obob3obob3obob3obo3b
obob3obob3ob3obobob3obob3ob3ob3ob3ob3ob3ob3ob3ob3ob3ob3o3bob3ob3ob3ob
3ob3obobob3obob3ob3ob3ob3ob3ob3ob3ob3ob3ob3ob3o3bob3ob3ob3ob3ob3obobo$
obobobo5bo3bo3bobobo3bobobo3bobobobobobobobo3bo3bobo3bobo3bobobobobo3b
obobobobo3bobo3bobobo3bobo3bo3bo5bobo5bobo5bobo3bobobo5bobo3bobobo5bo
3bobobo3bobo3bobo5bo3bo3bo3bo3bo3bo3bobobo3bo3bo3bobobo3bo3bo3bobobo$
3obob3ob3o3bob3ob3o3bob3ob3obob3obobobob3obob3obo3bobob3obob3obo3bobob
3obob3ob3ob3ob3obob3ob3ob3ob3ob3o3bob3ob3ob3ob3ob3o3bob3ob3ob3ob3ob3ob
3ob3obob3ob3ob3ob3ob3o3bob3ob3ob3ob3ob3o3bob3ob3ob3ob3o3bob3o!

It would probably be huge.

[Moosey]

It would be interesting to find an infinite growth pattern with numbers sequentially like this:

Code: Select all

 a bunch of numbers

It would probably be huge.

I posted something along those lines a few posts up, but with increasing lengths of lines in a "square wave".

[AforAmpere]

I posted something along those lines a few posts up, but with increasing lengths of lines in a "square wave".

Yeah, it was kind of inspired by that. I feel like one exists for both.
EDIT, you are at a ridiculous 57 posts per day.

[Moosey]

I posted something along those lines a few posts up, but with increasing lengths of lines in a "square wave".

Yeah, it was kind of inspired by that. I feel like one exists for both.
EDIT, you are at a ridiculous 57 posts per day.

Well of course I am. I joined yesterday.

[pcallahan]

It would be interesting to find an infinite growth pattern with numbers sequentially ...
It would probably be huge.

Switch engines show up pretty often if you adjust your notion of "often." The single 5x5 example would suggest one in 33 million or so, but depending on what you're enumerating, it may be 1 in 100000 or better. The small clusters like 5x5 tend to dissipate in the first few generations. So maybe there is a more stable statistic on number of stabilized switch engines appearing from the distinct patterns you get after a few generations (of course if your sample space is still life patterns you won't get any).

You can already get infinite growth from a few 6 digit numbers by themselves. Assuming the switch engine emerges in less than 1000 generations, you could find the solution just by checking the tail ends of sequences rather than the sequences themselves. I would guess you do not need to go beyond six digits. Fewer might work. The pattern would be very cumbersome though.

[Macbi]

Based on catagolue.appspot.com/statistics, a switch engine shows up once in every 85,000 soups of size 16 by 16.

[Saka]

--(I posted in the wrong thread, post deleted and moved here)
--(Wrong thread again, whoops )
THCTLH and UTHCTLON in a small 3x3 font both make a Glider-Producing SE

Code: Select all

x = 23, y = 3, rule = B3/S23
3obobob3ob3obo3bobo$bo2b3obo4bo2bo3b3o$bo2bobob3o2bo2b3obobo!

Code: Select all

x = 31, y = 3, rule = B3/S23
obob3obobob3ob3obo3b3ob3o$obo2bo2b3obo4bo2bo3bobobobo$3o2bo2bobob3o2bo
2b3ob3obobo!

(Notice that "THCTL" is contained in both patterns)

EDIT:
TUTHCTON

Code: Select all

x = 31, y = 3, rule = B3/S23
3obobob3obobob3ob3ob3ob3o$bo2bobo2bo2b3obo4bo2bobobobo$bo2b3o2bo2bobob
3o2bo2b3obobo!