I don't think anyone else has transcribed it for LifeViewer, so here goes. (I'm a little surprised to see the date here, because I had only been in Aberystwyth a little over a month at that point and was writing CGOL infinite-growth searches for Nick Gotts. I must have been very busy.)

**Subject: Experiments with a somewhat "Life-like" hexagonal CA (long)**

By Paul Callahan (3 Dec 97)

**Introduction**

Occasionally the question comes up of whether there is a 2-state CA that runs on a hexagonal grid and works something like Conway's Game of Life. I've never seen any detailed discussion of a particular rule, though I had heard rumors that there is one with gliders. I decided to look for such a rule, partly to satisfy my own curiosity and partly to test the generality of some software tools I've been working on. (After working on this for a while, I found out that some rules had been found by Robert Andreen with some similar properties).

After settling on one rule, which I'll describe below, I was able to duplicate many of the properties associated with Life, including the ones needed to embed digital logic. A number of growth-rate results and oscillator-construction methods should also transfer to this rule. In fact, I was surprised at the variety of interesting patterns and reactions, some of which actually lead to simpler constructions than in Life. The rule is tamer than Life in that small patterns usually stabilize very fast.

I'm fairly certain that there is no totalistic rule on a hex grid that supports the variety of interesting reactions found in Life, because if there were it would be better known. In any case, I checked briefly and found nothing promising. In particular B3/S23 (birth on three live neighbors, survival on 2 or 3, like Life) settles down very fast and doesn't seem likely to have spaceships. Birth on 2 neighbors is useful, because an axis-parallel front of 3 neighbors cannot occur in a hex grid. However, always having birth on 2 neighbors seems likely to cause uncontrolled growth.

**The Rule**

With that in mind, I started playing around with rules based on different-shaped neighborhoods, but insisting on symmetry. Taking symmetries into account, there are three ways each of getting 2, 3, and 4 neighbors, for a total of 13 different neighborhoods. I tried a lot of variations, but the one I settled on only distinguishes between different ways of getting two neighbors. For these, one can use the organic chemists' naming convention for benzene-ring disubstitutions: ortho, meta, and para. Rich Schroeppel and Achim Flammenkamp suggested this, and it may be a useful mnemonic for some people. It has the disadvantage that there is no standard extension to subsets of 3 or 4 neighbors. Fortunately, we don't need this for the rule used here. So let's call these neighborhoods 2o, 2m, and 2p as follows:

Code: Select all

```
O O O . O .
2o: . x . 2m: . x O 2p: . x .
. . . . . O
```

That is, we always have survival but not birth on 3 or 4 neighbors. On two neighbors we have birth but not survival if the neighbors are adjacent (2o), and survival but not birth if there is one empty neighbor cell in between (2m). If they are opposite each other (2p), there is no birth or survival (which is also the case for 0, 1, 5, or 6 neighbors).

I settled on these rules in order to get a glider. It uses the only really small mechanism I'm aware of. I consider it crucial that gliders show up spontaneously from random starting states.

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$3bo$o2bo$3bo$3bo!
```

(Robert Andreen, and probably others as well, have looked at rules that support this glider.)

**Variations**

You can vary the rule for 2p. If you have birth but not survival, the same glider works (other factors made this variation less interesting to me). If you have birth and survival on 2p and death on 4 (this can be relaxed somewhat) you get two distinct sparky gliders that propagate at the same speed but higher periods (6 and 8 ).

The following discussion, however, will be limited to B2o/S2m34. This rule is interesting because it doesn't seem to grow explosively but it does have a number of natural high-period oscillators. It also has a fairly common (i.e. observable in nature) infinite-growth pattern.

**Initial finds**

Patterns usually settle down much faster than in Life. The kinds of things that remain are either little and isolated, or clumped into honey-comb patterns. The latter are very stable, and seem to act as anchors for oscillators (mostly low-period blinkers on the fringes, but interesting high-period sparkers as well). There are a number of reactions--similar to the "kickback" in Life--in which two gliders collide to produce a new glider on a different path. Unlike Life, one glider can also destroy another without being affected.

The first thing I really liked about this rule was the following period-48 spaceship (it flips after 24 moves and moves down one position).

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
obo4b$b2o4b$b3obob$2b2o2bo$3bo3b!
```

Infinite growth is an easier question to answer here than in Life, because there is a stalk that leaves a trail of double honey-combs behind it. It shows up often enough to notice--maybe as often as the replicator in HighLife--but not so much as to dominate the dynamics of random patterns. This one is propagating down.

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
ob2o7b$b2obo6b$2ob2o6b$2b2obo5b$2bob2o5b$3b2obo4b$3bob2o4b$3b3obo3b$6b2o3b$
6bobo2b$6b3o2b$8b3o!
```

Note: gliders cannot propagate down (270 degrees) but are limited to 0, 60, 120, 180, 240 degrees. Clearly, this affects the kinds of constructions we can build out of spaceship/glider interactions.

**Some natural oscillators and a period-15 spaceship**

There were a lot of oscillators that appeared in random testing, so I ran a program to test 10x10 (rhombic) random starting states for new ones. I'll summarize all the periods later. One very useful early find has period 42. This is its smallest state.

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
b2o3b$bobo2b$4o2b$o3bob$b5o$3bo2b$3bo2b!
```

High-period oscillators in this rule have a different feel from Life oscillators. One of the weirdest ones is period 66 and seems to extend and retract a tentacle (which is big and temporarily looks like a permanent, stable part of the oscillator).

Compact form:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2b2ob$3obob$obobob$b2ob2ob$bob2obobob$5b2ob3ob$7b2ob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
4bob$2bo2bob2ob$3o3b2obob$2bo4bobob$2bo4bob2ob$3b2ob4obobob$5b2obob2ob3ob$
12b2ob$13bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2b2ob$b3ob$3bob$4ob$2ob!
```

Finally, there is another spaceship that comes up in random searching, though it is much less common than the period-48. It's symmetric and moves in glider directions one position every 15 steps:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bobo5b$2b3o4b$3obo4b$2bobo4b$3b2o4b$9b$4b2o3b$4bobo2b$3b3obob$6b3o$6bobo!
```

**Larger constructions: A Glider Gun**

Three of the period-42 oscillators can be combined to make a glider gun. I found this by first pairing up oscillators and testing the results. This didn't yield a glider gun, but it yielded a pair with a larger spark, to which I added the third:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
12b2ob$12bob$12bo4b2ob$2bo9bo5b2obob$b2obo10bo2bob2ob$2ob3o13b2ob2ob$
2bobobo13bobob$2b2ob4o3b2o6bob2ob$2bob2obo5b3o4b3obob$13bobo6bob$12b4ob$
12bo3bob$13b5ob$15bob$15bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bo$2ob!
```

**Representation of Patterns**

So far, pictures have sufficed to give an idea of this rule, but I hope at least a few people will try to run these patterns on their own CA programs. The format for electronic distribution is unfortunately less standardized than Life or other 2-state CAs on a cartesian grid. The obvious way to emulate a hexagonal grid is to use the Moore neighborhood omitting two opposite diagonal corners, which is what I did. But we have to agree on which corners. I chose the northeast and southwest corners for removal. So the embedding is given by the following

transformation:

Code: Select all

```
ABC A B C
DEF -> D E F
GHI G H I
```

[Of historical interest only. I stopped writing in Perl many moons ago.]

Code: Select all

```
#!/usr/local/bin/perl
$x=0; $y=0;
findpat:
while(<>) {
last findpat if (/x *= *[0-9]+ *, *y *= *[0-9]+/);
}
main:
while(<>) {
while ( /([0-9]*)([bo\$])(.*)/ ) {
if ($1 eq "") { $count = 1; }
else { $count = $1; }
if ($2 eq "b") { $x+=$count; }
elsif ($2 eq "\$") { $x=0; $y+=$count; }
else {
for ($i=0; $i<$count; $i++) {
print $x++, " ", $y, "\n";
}
}
$_ = $3;
}
last main if (/\!/);
}
```

**Glider Reflections**

There are many ways of reflecting gliders both from stationary oscillators and moving spaceships. I won't try to list the 180-degree reflections exhaustively, since there are so many. I will include some of them in the larger patterns. So far, I have found one way of turning a glider 60 degrees and four ways of turning one 120 degrees. I haven't tried all the high-period oscillators, so there may be others as well. It's especially interesting that the period-48 spaceship can reflect gliders, because gliders move fast enough that they can circulate around a flotilla of moving objects in this way. Here are the three non-180-degree spaceship

reflections:

Glider is reflected 60 degrees, traveling up:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
14bob$14bob$14bo2bob$4o10bob$o2bo11bob$b3ob$b2ob!
```

Glider is reflected 120 degrees, traveling to the right:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$ob$bob$2bob$3b2ob$b$b$b2ob$2bob$3bob$3bobob$4bobob$2b3obob$b4ob2ob$4bobob$
4bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
8bob$6bob$7bob$8bob$9b2ob$b$b$b$2ob$bob$2bob$2bob$2bob$b5ob$4bob$4bob!
```

In the above three examples, the spaceship is traveling down. Thus, the two distinct 120-degree reflections are not interchangeable.

Here are two more 120 degree reflections with stationary oscillators. In both cases, the reflected glider travels to the right.

Period-20:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$ob$bob$2bob$3b2ob$b$b$b$b$4b3ob$6b2ob$6bobob$7b2ob$7bobob$8b2ob$8bob$9bob$
9bob$7b4ob$9bobob$10bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$ob$bob$2bob$3b2ob$b$b$b$4bob$5b2ob$4b4ob$6b3ob$6bobob$7b4ob$7bo3bob$7b5ob$
10bob$11bob!
```

Finally, these can be used to send gliders in a loop. First, here's one based on period-42 reflectors:

Code: Select all

```
x = 89, y = 89, rule = B2o/S2m34H
29bo$30b2o$5bo22b3ob3o$4bobo23bobo2bo$5b4o21bob4o7bobo$4b2o3bo20b3obob
o5bob2obo$6b4o23b2o8b2ob2o$5b2obobo22bo9bobobo$7b5o32b3obo$10bo30bo3bo
b2o$38b4o3b3o$47bo$2b2o5bo19bo$2bob2o4bo19bo$b2obo5bo17bo2bo$2ob3o4bo
21bo$2bobobo3bo21bo$2b2ob2o$2bob2obo$5bo8$12b4o$16bo41bo11bo$61bo9bobo
$14bo46bo7b5o$61bo8bo3bo$60b2o9b4o$70b2obobo$72b3obo$75b2o$75bo2$72bo$
73bo$4b2o68bo$2b3ob3o66b2o$4bobobo72bo$3b2ob4o70b2obo$5b2obo71bob2o$5b
ob2o70b4ob2o$7bo72bobobo$12b2o66b3ob3o$14bo68b2o$15bo$16bo2$13bo$12b2o
$12bob3o$13bobob2o$14b4o9b2o$14bo3bo8bo$15b5o7bo46bo$15bobo9bo$18bo11b
o41bo$73b4o8$83bo$81bob2obo$82b2ob2o$56bo21bo3bobobo$56bo21bo4b3ob2o$
57bo2bo17bo5bob2o$58bo19bo4b2obo$59bo19bo5b2o$41bo$41b3o3b4o$40b2obo3b
o30bo$40bob3o32b5o$41bobobo9bo22bobob2o$41b2ob2o8b2o23b4o$41bob2obo5bo
bob3o20bo3b2o$43bobo7b4obo21b4o$53bo2bobo23bobo$54b3ob3o22bo$57b2o$59b
o!
```

Code: Select all

```
x = 270, y = 51, rule = B2o/S2m34H
17bo$17bobobo$17bo3b2o$20b2o$20bo$3bobo14b2o$3b4o13bob2o$4bo2bo12bo2bo
$5bobo15bo$5b4o$5bo2$o$obobo13bo$o3b2o12b3o$3b2o13bo2bo$3bo13b3ob2o$3b
2o15b2o$3bob2o14b2o$3bo2bo$6bo74bo$81bo$81bo2bo$81bo$82bo5$31b3o7b3o$
30b2o2bo5b3o2b2o219bo$31bo11bo222bobo$31b3o10bob2o220bo$31bobo10bo220b
2o$34bo232bobo$268b2o$268b2o$268bo5$12bo3$16b4o$15b2o$15b5o2$20bo$20bo
bo!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bob$ob4ob$b2ob3o4bob$2bo3bo2bob$2bo3b2o2bob$11bob$3bo3b2o3b2ob$4bo3bob$4b2ob3ob$
4bob4ob$6bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$2b3ob$3ob2ob$2bo2b2ob$3bob2o5bob$10bob$4bob2o3bob$4bo2b2o3bob$3b3ob2o4b2ob$
6b3ob$7bob!
```

**Glider duplication**

While doing collision enumerations to find reflections, I stumbled across a very surprising reaction between a glider and a spaceship:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
ob$3bob$3bob$3bob$2b2ob$b$b$b2ob$b2ob$bobob$2o2bob$b6ob$3bobob$4bob!
```

Code: Select all

```
x = 270, y = 51, rule = B2o/S2m34H
17bo$17bobobo$17bo3b2o$20b2o$20bo$3bobo14b2o$3b4o13bob2o$4bo2bo12bo2bo
$5bobo15bo$5b4o$5bo$50bo3bo$o50b2o2bo$obobo13bo33bobobo$o3b2o12b3o30b
3o2b2o$3b2o13bo2bo32b3o$3bo13b3ob2o$3b2o15b2o$3bob2o14b2o$3bo2bo$6bo
74bo$81bo$81bo2bo$81bo$60bo21bo$60bo5bo$62bo2bo$63b3o$63b2o$31b3o7b3o
19b3o$30b2o2bo5b3o2b2o16bo2bo199bo$31bo11bo21bob2o197bobo$31b3o10bob2o
20bo199bo$31bobo10bo220b2o$34bo232bobo$268b2o$268b2o$268bo5$12bo3$16b
4o$15b2o$15b5o2$20bo$20bobo!
```

Period-42:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
12bobob$4bo8b2obob$11b3ob2ob$8bo3bobobob$5b4o3b4obob$14bob3ob$17bob$2ob3ob$
ob6ob$bobob2ob$3ob2obob$3b2ob$4bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bo7bob$5bo2b4ob$5bo3b2ob3ob$5bo5b2obob$bo2b2o7b2ob$obo12b2ob$3bo13b3ob$
bobo14bob$2bobo13b3ob$3b2o13bob$3bobob$4b2ob$6bob$6bob$7bobob$6b4ob$9bob$10bob!
```

Here's a period-210 gun built by inserting a p42 glider duplication into a period-210 glider relay consisting of two 180-degree reflectors:

Code: Select all

```
x = 69, y = 26, rule = B2o/S2m34H
49bo$50bo$48b5o$49bo3bo$50b4o$50bobobo$51bob3o$50b2obo$52bo10b2o$2bo
50b2o6b3ob3o$b2o45b4o2bo2bo4bobobo$2b2o44bo9bo4b3obo$3ob2o50bo2bo3bo2b
3o$bob4o53bobo2b2o$2bobobo53bob4o$2b2ob3o58bo$2bob2o$56bo$54bobo$53b4o
$54bobobo$55b6o$54b2o3bo$56b4o$56bobo$58bo!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
4bob$3bobob$4b4ob$5bob$5bob$6bob$4b3ob$5bobob$5b3o5b2ob$2o5b2o4bob$bo6bo14bob$
b2o18b3ob$2bo20bob$23b2ob$22b2obob$2bo18b2ob$2bo17b2obob$2bo2bo14bob3ob$
2bo18b2ob$3bo8b2o8bob$14bob$15bob$16bob$10b2o2bob$9b3ob$8b2obo11bo3b2ob$
8bob3o11b4obob$9bobo11b3obobob$9b2ob2o3b3o6b2ob2ob$9bob2o6b3o6b2obob$
11bo7bobo7bob$20b5ob$19b2o3bob$21b5ob$21bobob$24bob!
```

**Spaceship factories**

Spaceships are a lot more interesting when they can be produced on demand. Just as in Life, there are ways to synthesize these from gliders. Here's a three-glider synthesis of the period-48 spaceship:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
18bob$19bob$19bobob$2o10b4o4bobob$2bo9bo8bob$3bob$4bo11bob$2bob!
```

Code: Select all

```
x = 36, y = 93, rule = B2o/S2m34H
10bo$11bo$11b4o$11bobo$14bo$14bo$15b3o$8bo6bobo$2bo5bo7b3o$3o6bo15bo$
2bo21b3o$b3o21bob3o$4b2o12bo7bobo$6b2o9bo7b3ob2o$5b2ob2obo14bob2o$7b2o
b2o14b3obo$7bobo2b2o14bo$9b3o$11bo2$30bo$17bo13b2o$5bo9bo13b3ob2o$4b5o
7bo6b4o2bobobo$5bobobo7bo5bo6b2ob3o$6b4o8b2o10bob2o$6bo3bo16bo$6b5o13b
2o$9bobo12bobo$10bo14b4o$25bo3bo$25b5o$28bo$29bo6$15b4o$19bo2$17bo22$b
o$obo3bo$b4ob2o$2bo2b2ob3o$2bo4b2obo$9b3o13b2o$11b2o11b2ob3o$22b3obobo
$23bo2b2ob2o$17bo5b4ob2o$8b2o7bo6b5o$8bo8bo2bo5bo$17bo$18bo$25b2o$26bo
bo$17b3o5bo3b2obo$16b2o13b2o$11bo3b2o3bo3bo4b3ob2o$11bo2b2ob3o4b2o2b2o
bobo$10b5ob2o8bo3b2ob2o$13bo11bob2obob2obo$14bo8b3obo5bo$24bobobo$25b
4o$25bo3bo$26b5o$28bo$29bo!
```

One use of these spaceship factories would be a "standing breeder" like the one built in Life by Dean Hickerson. That is, it should be possible to generate a set of spaceship streams through which a glider stream can be sent on a zig-zag path. If this path intersects a stream of glider duplication flotillas, then each such reaction point would be a source of linear growth, resulting in quadratic growth overall. The standing breeder could also be used to realize other growth rates, since it converts any f(n)-growth stream of gliders into an O(n f(n)) breeder. I haven't built such a pattern--so far, anyway. There is a simpler way to get quadratic growth, which I'll explain in the next section.

**Quadratic Growth: The Stalk Breeder**

The stalk is common enough that it seemed likely that there was a simple way to synthesize it. As with the period-48 spaceship, there are several. Here's one using 3 gliders:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
ob$3bob$3bob$3bob$2b2ob$b$b$b$b$b$5bob$4bobob$5bobob$7bob$8bo4bob$13bob$
13bo2bob$13bob$14bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
10bob$10bob$2bo8bo2bob$3b2o7bob$13bob$b$b$2ob$2bob$3bob$4bob$2bob!
```

That's all there is to this pattern, though it takes up a lot of space:

Code: Select all

```
x = 413, y = 328, rule = B2o/S2m34H
396b2o$397bo$398bo$398bobo$399bobo$397b3obo$396b4ob2o$399bobo$15bo383b
o2$15b2o$17bo$17bo2bo$16bobobo$16b3obo$16bobobo$16bobo$19bo6$391bo$
392b2o$394bo$393bobo$394bo31$2o$o$o$bobo$b5o$3bo$4bo22$273bo2$274b2o$
274bo$274bobo$274bobo$275b3o$276bobo$276bo23$131bo$131bobo$133bo$132b
2o$132bo42$214bo$214bo$206bo8bo2bo$207b2o7bo$217bo3$204b2o$206bo$207bo
$208bo$206bo4$373bo2$373b2o$375bo$374bobo$375bobo$375b3o$375bobo$378bo
11$372b3o$373bob2o12bo$372b2o2bo$373b4o12b2o$376bo14bo$377bo12bobo$
391bobo$391b3o$391bobo$230b4o160bo$231bo2bo$232b3o$219b2o13b2o$221bo$
222bo$220bobo$219b5o$222bo$222bo4$365b2o$366bo$367bo19b2o$367bo20bo$
367bo20bo$366b5o16bobo$369bo17bobo$369bo18bob3o$262bo125b2ob4o$390bobo
$240bo21b2o129bo$264bo$241b2o20bobo112bo$241bo22bobo110bobo$239bo2bo
21b3o112bo$240bobobo19bobo109b2o2bo$241bob3o21bo109b3o$242bobobo131bob
2o11bo$245bobo118bo11bo13b4o$245bo120bobo24bob2o$367bo25b4o$269bo97bo
2b2o25bo$269bo99b3o$266bo2bo98b2obo$267bo104bo$266b4o$268bobo$253b4o
12bobo2bo$253bo2bo18bo$254b3o19bo2bo97b2o$254b2o23bo98bo$278b4o97bo$
278bobo98bo$278bobo98bo$217bobo158b5o$218b2o161bo$216bob3o160bo$203b2o
11bo2b2o58bo$204b2o14bo$204bobo72b2o$204bo2b2o72bo$203b6o71bobo$205bob
o73bobo$207bo73b3o$281bobo$284bo$408b2o$410bo$411bo$409bobo$408b5o$
411bo$411bo2$266bo$267b2o$269bo$269bo$247b3o19bo$248bob2o15b3o$228b2o
17b2o2bo15bobo$230bo17b4o18bo$231bo19bo$229bobo20bo$228b5o$231bo$231bo
$248b2o$249bo$248bobo$247bo2bo2bo$240bo7bo2b4o$240bobo6bo2bo2bo$239bo
2bo10bo10b2o$239bo14bo10bo$239b4o5bo16bobo$240bo2bo7bo11bo2bo2bo$240bo
10bo11b4o2bo$251bo11bo2bo2bo$250b2o14bo$266bo7$264b3o$265bob2o$264b2o
2bo$265b4o$268bo$269bo9$250bo$251b2o$251b3o$250bo$251bo2bo$251b6o$254b
o$255bo!
```

**Other guns**

The period-20 oscillator actually yields a compact period-20 gun. Two combined produce a spark that is almost a glider: all that is needed to fix it is the 3-cell eater:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
8b2ob$8bob$b$5b6ob$5bob$2bo5bob$b5ob$2bob2o8bob$3b2o8b3o4bob$2b2o9b2ob2o2bob$
3o12b2ob4ob$2bo14bo2bobob$b3o17bob$4bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
5bob$6b3ob$7bob$7b3ob$3bo2b2ob$2b2ob2ob$obob2ob$b3ob2ob$4b2obo11bob$6b4o7b3ob$
8bo10bob$9bo9b2ob$18b2obob$17b2ob$16b2ob2ob$5bo8b3ob2ob$6bo9b3obob$6bobo7bo2bob$
7bobob$8bo5b2ob$7bo5b3ob$13b3ob$8bo5bobob$8b2o5b2ob$8bobo4bobob$15b3ob$18bob$
18bob$19bobob$18b4ob$21bob$22bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
11b2ob$2o11bobob$2bobo8b3ob$2b3o8bobob$2bobo9b4ob$3b3o8bo2bob$2b2o2bo7b5ob$
4b3o8b2ob$4b2obo8bob$4bob2ob$2bo2bobob$b3o2b2ob$o5bob$b$b$b$b$b$b$b$b$18bo2bob$
19b3ob$20bob3ob$20bobobobob$20b7ob$21bo2bob$25bob!
```

Code: Select all

```
x = 86, y = 49, rule = B2o/S2m34H
o$o10bo$bo9bobo$9b5o$10bo3bo$11b4o$11bobobo$11b5o6bo$13bo7b2obo$20b2ob
2o$18bo3bobo$19bo2b2obo$18b3obob2o$20b2ob2o$24bo2$7bo$7bo18bo$8bo2bo
15b2o$9bo17b2o$10bo16bob2o$26b4obo$29bobo$29bob3o$29b3o$31bo$15bo9b2o$
15b2obo6bo$13b3ob4o4bo$15bobobo5bo20bo31b2o$14b2ob3o8bo16bobo31b2obo$
16b2obo26bobo30bob2o$16b2o2b2o26bo31b2ob2o$20b2o27bo31bobo$22bo58bob2o
$81b3obo$23bo14b2o43bo$23b3o12bo$22b2obo13bob2o$22bob5o10b3obo$23bobo
2bo4bo5bobobo$23b2obo2bo4b2o4b2ob2o$23bob3o4b3obo5b2obo$26bo4b2obobo5b
obo$33b4o$33bo3bo$33b5o$36bobo$36bo!
```

**Miscellaneous oscillators**

Many different oscillators turned up in a random survey of 10x10 (rhombic) starting states on an otherwise empty infinite grid. Most of these consisted of an oscillating part stabilized by an anchor organized into honey-combs. Like the period-43 mentioned in the preceding section, these often have a low-period fringe that multiplies the total oscillation by 2, 3, or 5, without affecting the interesting part, which I call the principal oscillation.

I'm going to list oscillators here by principal oscillation. Sometimes it may be possible to eliminate the discrepancy between overall period and principal oscillation, but I haven't tried to. It is also likely that many of the stabilizing anchors below can be reduced in size. This is not an exhaustive list of known oscillators, and I've left out common low-period patterns that are easy to generate. Oscillators that have appeared already in this article will not be repeated.

Period 7:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
3bob$2b2ob$b2ob2obob$3bob3ob$ob2obo2b2ob$b2obo3bob$2ob4o2bob$2b2o3bobob$
2bobob2o2bob$2b3ob2ob3ob$5b2ob2ob$7b2ob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
ob$ob$b2ob$2b3ob$b2o2b2obob$bo4b3ob$2bo5bob$9bob$8b2ob$8bob$9bobob$8b4ob$11bob$
12bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2b2ob$3obob$bob2ob$b3obob$4b2ob$4bob$3b3ob$3bo2bob$2b3obob$2bobo2bob$3b2ob2ob$
5b2ob$6bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$b2ob$2b2obob$3obobob$2bobo2b2ob$2b6ob3ob$2bo3b2o2bob$8b3ob$10bob$11bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2obob$o2b2ob$2b2obob$3ob2ob$bob2obob$2b2ob2ob$4b2ob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
b3ob$2obob$o3b2ob$2ob2ob$2b2ob2ob$2bob2ob$4bob!
```

Period 17:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
2bob$2bob$6ob$2bob2ob$3b2ob$b$3b2o2bob$2b2ob2ob$2bo2bobob$3b2ob2ob$5b2ob2ob$
7b2ob$7bob!
```

Period 18:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bobob$2b3ob$b2obob$2ob2ob3obob$ob2obobob4ob$b2ob5ob2ob$3b2obob3obob$3bobo4bob!
```

Period 19:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bo3bob$2o2b3obob$ob2obob3ob$b2ob5obob$bob2o2bob2ob$3bobob3ob2ob$7b2ob3ob$
8b3o2bob$10bob!
```

Period 22:

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bob$2bob$5ob$2bobob$2b5ob$3bobob2ob$6b4ob$6b2ob$b$10bob$11b2ob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
bob$bobob$5ob$bobob2ob$bo2b2obob$3b2o2bob$5b5ob$7bobob$5b6ob$2b2obobobobob$
4bob2obob$8b3ob$8bo2bob$11bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
3bob$3b2ob$2b2ob2ob$3o4b3ob$bobo3bobob$2b4o2b3ob$4o2b3ob$2o2b4ob$b3obobob$
2bo2bob$b$6bob!
```

Code: Select all

```
x = 10, y = 10, rule = B2o/S2m34H
3bob$3b3ob$b3obob$obob5ob$b5o2bob$2bobobo2bob$b6ob$4bob$4bob!
```

**General Comments and Conclusion**

After working with this rule for several weeks, I was surprised at the variety of interesting objects and reactions that existed in it. It's a fairly simple rule, though admittedly more complicated than Conway's Life. However, like Life, it is far simpler than the phenomena it supports. I think it's interesting, among other things, for demonstrating that there is a 2-state hexagonal CA with at least some Life-like properties. I had been skeptical of this, since I had never seen a discussion of one.

There are some properties missing from this rule that make Life interesting. Unlike Life, patterns die down very rapidly unless they stabilize into honey-comb structures. The result of a collision between gliders and stable objects is similarly tame, with the most common results being an eater reaction or mutual annihilation. On the other hand, this can actually be beneficial for designing patterns to realize some logical function or growth rate.

Unlike Life, this rule does not seem to support finite still-life patterns. The honeycomb structure is stable and can present an arbitrarily long stable boundary, but corners always result in cell-producing neighborhoods that usually result in low-period oscillations. The stability of the honey-comb lattice is also very unlike Life, in which a small change to a dense lattice almost always propagates very fast, destroying it. It may be worth doing some systematic tests on random mutations of a lattice background.