HighLife - An Interesting Variant of Life (part 1/3)
David I. Bell
by David I. Bell
db...@canb.auug.org.au
7 May 1994
INTRODUCTION
------------
In February 1994, Nathan Thompson reported several interesting objects
that he found in a cellular automaton closely related to Conway's Life.
Since then, a small group of people have made further discoveries in this
automaton. This article documents their discoveries.
It is assumed here that you know many of the concepts and terms used in
Conway's Game of Life since many of them are also applicable here. These
are terms such as blinker, eater, glider, oscillator, spaceship, glider gun,
puffer train, breeder, wick, and the speed of light.
The rule for this universe is specified within our group as "B36/S23", and
has been named "HighLife" by John Conway. This notation describes the
number of live neighbors required for birth, and the number of live
neighbors required for survival. Here birth requires 3 or 6 live neighbors,
whereas survival requires 2 or 3 live neighbors. (The rule for Conway's
Life is specified by "B3/S23".)
The only difference between these rules is the extra birth which occurs
for six live cells. Since this case doesn't occur extremely often, you might
expect that HighLife and Life should appear similar to each other and
contain many of the same objects. This turns out to be the case. However,
that small rule change does allow for some surprises, as you will see below.
HighLife can be run under X11 if you have the "xlife" program. The "R"
command is used to change the rule, with the arguments "23/36" to select
the HighLife rule.
Where known, each figure is labeled with the initials of the discoverer or
creator of the objects in the figure. The initials stand for the following
people:
DB David Bell
HE Hugh Everett
AH Alan Hensel
DH Dean Hickerson
NT Nathan Thompson
Because of their size, some objects are shown as a "rle" pattern instead of
as a picture. (The name "rle" stands for run length encoding, and is our
standard storage format for large objects.) It is not difficult to write a
program to convert the rle pattern into other formats.
In an rle pattern, the first line gives the width (x) and height (y) of the
pattern and optionally a rule string (defaulting to the Life rule). The
remaining lines describe the pattern, where 'b' represents a blank cell,
'o' represents a live cell, '$' represents an end of line, and '!' marks the
end of the Life pattern. Numeric repeat counts can precede these values.
The pattern contains newlines so that no line is longer than 70 characters.
Newlines cannot interrupt a repeat count or the following character.
As a simple example, a glider in HighLife can be described as:
x = 3, y = 3, rule = B36/S23
bo$2bo$3o!
SIMPLE STUFF
------------
Highlife has many of the same objects as Life. There are blinkers, blocks,
beehives, boats, and loaves. But there are no ships, and no natural traffic
lights or honey farms. The ship self destructs, and the predecessors to the
traffic lights and honey farms self-destruct in spectacular manners.
To replace them there occurs an occasional object composed of four boats
as shown in figure 1. Besides this, there are several other common results
which you can discover on your own.
..... .OO.OO.
..... O.O.O.O
..... .O...O.
.OOO. .......
O...O .......
.O.O. .......
..O.. .O...O.
..... O.O.O.O
..... .OO.OO.
[Figure 1: Predecessor and resulting still life.]
The glider survives in HighLife, as do the orthogonal spaceships LWSS,
MWSS, and HWSS. Almost all of the other known spaceships from Life fail.
Most of the common period 2 oscillators from Life survive in HighLife.
Some of the oscillators from Life with higher periods also survive in
HighLife. As might be expected, there are some oscillators in HighLife
which do not occur in Life.
The "eater" from Life still works in HighLife, as does some of the "eating"
capability of a block. But sometimes the block needs help to prevent a new
spark being created which would destroy it.
All of the known glider guns and puffer trains from Life fail in Highlife.
The glider kickback reaction in Life fails in HighLife, and no substitute
for it is yet known. Such a reaction (if it exists) must be assisted by
eaters or other objects.
THE REPLICATOR
--------------
The reason that HighLife has been investigated so much is because of the
object known as the "replicator". This amazing object starts with only six
live cells as shown in figure 2.
.OOO
O...
O...
O...
[Figure 2: The basis of the replicator. (NT)]
This is the basis of a period 12 linear self-replicator. After 3 generations,
it turns into the symmetrical object shown in figure 3.
..OOO
.O..O
O...O
O..O.
OOO..
[Figure 3: One of the phases of the replicator. (NT)]
Starting from the above object, figure 4 shows the results after several
multiples of 12 generations.
..... ......... ............. ................. ..OOO................
..... ......... ............. ................. .O..O................
..... ......... ............. ..OOO............ O...O................
..... ......... ............. .O..O............ O..O.................
..... ......... ..OOO........ O...O............ OOO..................
..... ......... .O..O........ O..O............. .....................
..... ..OOO.... O...O........ OOO...OOO........ .....................
..... .O..O.... O..O......... .....O..O........ .....................
..OOO O...O.... OOO.......... ....O...O........ .....................
.O..O O..O..... ............. ....O..O......... .....................
O...O OOO...OOO ............. ....OOO...OOO.... .....................
O..O. .....O..O ............. .........O..O.... .....................
OOO.. ....O...O ..........OOO ........O...O.... .....................
..... ....O..O. .........O..O ........O..O..... .....................
..... ....OOO.. ........O...O ........OOO...OOO .....................
..... ......... ........O..O. .............O..O .....................
..... ......... ........OOO.. ............O...O ..................OOO
..... ......... ............. ............O..O. .................O..O
..... ......... ............. ............OOO.. ................O...O
..... ......... ............. ................. ................O..O.
..... ......... ............. ................. ................OOO..
[Figure 4: Generations 0, 12, 24, 36, and 48 of the replicator. (NT)]
As you can see, the replicator expands along a diagonal line creating
copies of itself. Where two copies of the replicator meet, they cleanly
destroy each other. Every 12 * 2^N generations there are exactly two copies
of the replicator, with their centers separated by 4 * 2^N cells. At
generation 12 * (2^N - 1), there are 2^N copies of the replicator with
their centers evenly spaced 4 cells apart. The two ends of the replicator
line expand at a speed of c/6.
The replicator is a natural "sawtooth" pattern. A sawtooth is any pattern
whose population is unbounded but which does not tend to infinity. That
is, the population alternates between a fixed low value and an ever
increasing high value. (In Life, the only known sawtooth patterns are
very elaborate structures.)
Since the replicator is so small, it appears naturally from many reactions.
Because of this, the population of a random soup of cells of a reasonable
size (say 1000 by 1000) will usually grow without bound once a replicator
appears on the boundary. (The population of random soups in Life are also
expected to grow without bound if they are large enough, but only with a
vastly larger starting size.)
Dean Hickerson has partially solved the problem of designing sets of
replicators which react against other objects in any specified periodic
manner to create oscillators or puffer trains. (Many of these results are
in this article.) But his method currently can only be applied to a subset
of such problems.
THE BOMBER
----------
There is a replicator-based spaceship in HighLife which occurs naturally.
This is simply formed by putting a blinker in the path of the replicator.
Figure 5 shows the starting object.
.OOO......
O.........
O.........
O........O
.........O
.........O
[Figure 5: Beginning of spaceship. (NT)]
Figure 6 shows the completed spaceship. It has a period of 48 and travels
at a speed of c/6 to the upper left. The blinker reacts with one of the
spawned replicators such that it destroys itself and the spawned replicator.
But it also leaves another blinker on the other side of the spaceship
which allows the reaction to be repeated. So this is actually a glide-
reflective spaceship with a semi-period of 24.
..OOO..........
.O..O..........
O...O..........
O..O...........
OOO............
...............
............OOO
............O.O
............OOO
.....O.........
.....O.........
.....O.........
[Figure 6: Completed period 48 c/6 "bomber" spaceship. (NT)]
The traffic-lights predecessor seen at the right (using Life terminology)
self-destructs with a ring of sparks. This reaction is called a "bomb".
Since this spaceship continually creates bombs on alternate sides, the
spaceship is named the "bomber".
In one sense the spaceship can be described as "pulling" the blinker along
behind itself. This blinker pulling behavior also occurs in other replicator
based situations. In those cases, the result is not a true spaceship, but is
instead a "growing" spaceship. An example of this is shown in figure 7.
..OOO...............
.O..O...............
O...O...............
O..O................
OOO.................
....................
....................
....................
....................
....................
....................
....................
....................
.................OOO
[Figure 7: Replicator occasionally "pulling" a blinker. (DH)]
Here the blinker is pulled along only occasionally for a while as a
replicator reaches it and is transformed into a bomber. The bomber
continues until it runs into another replicator, at which point both
replicators self-destruct leaving only the blinker. The blinker then
waits until another replicator reaches it to repeat the process. So
the blinker's progress is sporadic. Meanwhile, the front replicator
travels at a constant speed, causing the whole object to grow in size.
Figure 8 shows two replicators playing "tug of war" with a blinker.
Here the blinker gets pulled arbitrarily far in both directions.
For N > 6, in generation 2^N the blinker is pulled about 2^(N-3)/3 units
diagonally (to the upper left if N is odd, to the lower right if N is even).
.OOO.......................
O..........................
O..........................
O..........................
...........................
...........................
...........................
.............O.............
.............O.............
.............O.............
...........................
...........................
...........................
...........................
.................O.........
................OO.........
...............O.O.........
..............OOO..........
...........................
...........................
...........................
...........................
...........................
........................OOO
.......................O.O.
.......................OO..
.......................O...
[Figure 8: A tug of war between two replicators. (DH)]
Arrangements similar to the one in figure 8 can appear chaotic, with the
blinker staying on one side for a long time before crossing to the other side.
An example of this is shown in figure 9. Achim Flammenkamp and Dean Hickerson
wrote programs which simulated this tug of war. Up to generation 201119907,
the blinker changes sides at generations 33430, 33695, 35350, 35519, 45430,
45983, 46102, 203519, and 12686902. It is not known whether the blinker
changes sides infinitely often.
.OOO..............................
O.................................
O.................................
O.................................
..................................
..................................
..................................
.............O....................
.............O....................
.............O....................
..................................
..................................
..................................
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..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
..................................
.................................O
................................OO
...............................O.O
..............................OOO.
[Figure 9: An unpredictable tug of war between replicators. (DH)]
This glider pulling behavior is very common when replicators impact against
other objects. Each replicator reacts against the object and changes it in
some way. Once a blinker randomly appears at the right spot, the replicator
then pulls it away and no longer reacts with the object. But if the object
is symmetrical along the replicator's long axis, then the lone blinker cannot
be formed. Then the replicator must either "burn through" the object, or
else pull along some other type of object. It is possible that the replicator
could be destroyed, but this is thought to be very unlikely.
It is possible for a set of replicators to pull a blinker in the normal manner
an arbitrarily large number of times, and then fail. An example of this is
shown in figure 10. Here the blinker is pulled twice before being destroyed.
..OOO.........................
.O..O.........................
O...O.........................
O..O..........................
OOO...........................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............OOO.............
.............O..O.............
............O...O.............
............O..O..............
............OOO...............
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
.............................O
.............................O
.............................O
[Figure 10: Replicators pulling a blinker only twice. (HE)]
PUFFER TRAINS
-------------
There are three known methods of building diagonal puffer trains in HighLife.
As might be expected, all three methods are based on the properties of the
replicator (or the bomber which is derived from the replicator.)
The first type of diagonal puffer train is based on bombers. The bomber has
proven to be very useful because the bombs occur on the edges of the spaceship
beyond the path of the rest of the spaceship, and so can be used to perturb
other objects as the spaceship travels. In particular, two copies of the
bomber can fly side by side so that the two bombs interact to form interesting
things such as blinkers, blocks, and gliders. Figure 11 shows two such
reactions which produce gliders.
...........OOO. ............OO.......
..........O...O .............OO......
.........O....O ..........O.OO.......
........O..O..O ..........OOO........
........O....O. ...........O........O
........O...O.. ....................O
.........OOO... ....................O
............... .....................
............... ...O.................
............... ..OO.................
..............O .O.O.................
..............O OOO..................
..............O .....................
............... .....................
..OOO.......... .....................
.O..O.......... .....................
O...O.......... .....................
O..O........... .....O...............
OOO............ .....O...............
.........OOO... .....O...............
[Figure 11: Pairs of bombers producing gliders. (DB, DH)]
The left reaction produces a glider traveling down and left every 48
generations (i.e., at right angles to the bomber's path). The right reaction
produces a glider traveling down and right every 48 generations (i.e., in
the opposite direction to the bomber.)
The second type of diagonal puffer train is constructed by using a bomber and
a replicator, and reacting them together. Figure 12 shows a period 96 rake
made using this method. Many other objects can be constructed using similar
reactions.
..OOO...............
.....O..............
O.O...O.............
O......O............
O.......O...........
.O......O...........
..O...O.O...........
...O............OOO.
....OOO............O
...................O
..........O........O
..........O.........
..........O.........
[Figure 12: A period 96 rake. (DH)]
In general, in any construction of this sort the period can be increased
by a factor of 2^N by moving the replicator farther in front of the bomber.
So we can, for example, make rakes with periods of 192, 384, etc.
The third type of diagonal puffer train is constructed by reacting replicators
against some debris in such a way that more debris is created such that a
periodicity appears. These type of puffers demonstrate that replicators
don't have to burn through or pull blinkers when they react with objects.
An example of this type of puffer train is shown in figure 13. Here a set
of replicators reacts against a beacon. Every 96 generations a pair of
blinkers and a biloaf are formed.
..OOO...................................
.O..O...................................
O...O...................................
O..O....................................
OOO...OOO...............................
.....O..O...............................
....O...O...............................
....O..O................................
....OOO.................................
........................................
........................................
........................................
..............OOO.......................
.............O..O.......................
............O...O.......................
............O..O........................
............OOO...OOO...................
.................O..O...................
................O...O...................
................O..O....................
................OOO.....................
........................................
........................................
........................................
..........................OOO...........
.........................O..O...........
........................O...O...........
........................O..O............
........................OOO.............
........................................
........................................
........................................
..................................OOO...
.................................O..O...
................................O...O...
................................O..O....
................................OOO...OO
......................................OO
....................................OO..
....................................OO..
[Figure 13: Period 96 puffer producing two blinkers and a biloaf. (DH)]
It is possible to create a similar puffer which produces exactly the same
output as the one in figure 13, but with increasing time intervals between
the puffer output. Besides getting slower, the puffer also grows in size.
This is shown in figure 14.
..OOO...............................................
.O..O...............................................
O...O...............................................
O..O................................................
OOO...OOO...........................................
.....O..O...........................................
....O...O...........................................
....O..O............................................
....OOO...OOO.......................................
.........O..O.......................................
........O...O.......................................
........O..O........................................
........OOO.........................................
....................................................
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....................................................
....................................................
....................................................
..........................OOO.......................
.........................O..O.......................
........................O...O.......................
........................O..O........................
........................OOO.........................
....................................................
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....................................................
....................................................
..................................................OO
..................................................OO
................................................OO..
................................................OO..
[Figure 14: Aperiodic puffer producing two blinkers and a biloaf. (DH)]
Dean Hickerson found that you can create a puffer with any cyclic pattern of
pairs of blinkers and biloaves if and only if the value 5L + 3B is a power
of two, where L is the number of biloaves and B is the number of pairs of
blinkers. (Figure 13 shows a puffer where both B and L are 1.) The rle
pattern below shows a case where L is 1 and B is 9. So this puffer makes
one biloaf and 9 blinker pairs every 384 generations.
x = 134, y = 134, rule = B36/S23
2b3o$bo2bo$o3bo$o2bo$3o8$14b3o$13bo2bo$12bo3bo$12bo2bo$12b3o8$26b3o$
25bo2bo$24bo3bo$24bo2bo$24b3o8$38b3o$37bo2bo$36bo3bo$36bo2bo$36b3o8$
50b3o$49bo2bo$48bo3bo$48bo2bo$48b3o8$62b3o$61bo2bo$60bo3bo$60bo2bo$60b
3o8$74b3o$73bo2bo$72bo3bo$72bo2bo$72b3o8$86b3o$85bo2bo$84bo3bo$84bo2bo
$84b3o8$98b3o$97bo2bo$96bo3bo$96bo2bo$96b3o8$110b3o$109bo2bo$108bo3bo$
108bo2bo$108b3o8$122b3o$121bo2bo$120bo3bo$120bo2bo$120b3o3b3o$125bo2bo
$124bo3bo$124bo2bo$124b3o2$132b2o$132b2o$130b2o$130b2o!
Here is another puffer by Dean which is asymmetric. It creates 4 blocks,
a boat, and a glider every 768 generations. Unlike the previous puffers,
the output from this one is sparse because for much of the cycle a glider
just travels up to the replicators.
x = 282, y = 288, rule = B36/S23
b3o$o$o$o12$18bo$18bo$18bo$15b3o18$37b3o$36bo$36bo$36bo5bo$42bo$42bo$
39b3o22$65b3o$64bo$64bo$64bo12$82bo$82bo$82bo$79b3o14$97b3o$96bo$96bo$
96bo12$114bo$114bo$114bo$111b3o171$280bo$279b2o$279bobo!
The first orthogonal puffer train in HighLife was found by Alan W. Hensel
using a search program. It is a new type of object called a "line puffer"
which has a very complicated exhaust with a very large period. (Line puffers
were first found in Life very recently.) The middle components can be
repeated in an obvious manner to make arbitrarily wide line puffers. Since
the spacing between the components can be varied by one cell, from some point
onwards a line puffer of any width can be constructed. Unfortunately some
of these line puffers get destroyed by the exhaust at some point. Alan has
since made a new line puffer which is more robust, but some widths still get
destroyed by the exhaust. The original line puffer is given below.
x = 161, y = 17, rule = B36/S23
58b3o7b3o8b3o8b3o7b3o$57bo3bo5bo3bo6bo3bo6bo3bo5bo3bo$56bo5bo3bo5bo4bo
5bo4bo5bo3bo5bo$34b3o3b3o13bo5bo3bo5bo4bo5bo4bo5bo3bo5bo13b3o3b3o$29bo
3bobbobbo3bo5b3o3boob3obooboob3oboobboob3oboobboob3obooboob3oboo3b3o5b
o3bobbobbo3bo$28b3obo3bobo5bo3bobbo3bobooboobobobooboobobbobooboobobbo
booboobobobooboobo3bobbo3bo5bobo3bob3o$20bo5boobobbob3obo3bobbobo3bobo
bo3bo3bobo3bo3bobbo3bo3bobbo3bo3bobo3bo3bobobo3bobobbo3bob3obobboboo5b
o$10bo8b3o4bo7bobo3b3obbobobb6o3boboo3booboboo4booboboo4booboboo3boobo
3b6obbobobb3o3bobo7bo4b3o8bo$6bobooboobo3b5o5bobobobob3obobboobo5bobb
4obobbobobbobobbobbobbobobbobbobbobobbobobbob4obbo5boboobbob3obobobobo
5b5o3bobooboobo$5boo7booboo3booboobob3obobb5obbo3bo4boobbo45bobboo4bo
3bobb5obbob3obobooboo3booboo7boo$6boobobobobobobbobbob3obobooboboobbob
obob3o7bobbobbooboobbobboobboobbobboobboobbobbooboobbobbo7b3obobobobb
ooboboobob3obobbobboboboboboboo$4b3o7bobobbobobbobb3o4bo7bo3bo5bobobo
47bobobo5bo3bo7bo4b3obbobbobobbobo7b3o$6bo9booboboboo4bo4bo6boboobo4b
ooboobb45obbooboo4boboobo6bo4bo4booboboboo9bo$b3o10bobobb3o7bob3ob6o3b
oob3o10bo39bo10b3oboo3b6ob3obo7b3obbobo10b3o$bo12b3obobbo6bobo5boboobo
bo5bobobobobbo43bobbobobobo5boboboobo5bobo6bobbob3o12bo$14b3o3bo12bobo
9bo3bo9bo41bo9bo3bo9bobo12bo3b3o$oo11boo14bobo97bobo14boo11boo!
There are no known simple orthogonal puffer trains in HighLife. For these
to be found, a small "engine" needs to be found which works similarly to the
Schick engine or B-heptomino in Life. Very little searching has been done
so far to find one.
OTHER SPACESHIPS
----------------
Like the Life universe, HighLife has complicated "space-dust" spaceships.
Figure 15 shows the first one of these found, which is a period 2 "bird"
spaceship. A smaller spaceship can be constructed from this one by clipping
off one of the wings and making it symmetrical. Or, a larger spaceship can
be made by inserting more central components in an obvious way.
.........OOO........OOO........OOO.........
.....OOO.OOO.OOO...O...O...OOO.OOO.OOO.....
.....O...O.O...O..O.....O..O...O.O...O.....
....OO.O.....O..OOO.....OOO..O.....O.OO....
..............O.O.O.OOO.O.O.O..............
..OOO.O.........O.O.O.O.O.O.........O.OOO..
.OO.............O.O.....O.O.............OO.
.O................OO...OO................O.
OO................OO...OO................OO
[Figure 15: A period 2 spaceship traveling up. (AH)]
Figure 16 shows the period 3 c/3 "turtle" spaceship from Life which also
works in HighLife. None of the other known period 3 spaceships from Life
work in HighLife.
.OOO.......O
.OO..O.OO.OO
...OOO....O.
.O..O.O...O.
O....O....O.
O....O....O.
.O..O.O...O.
...OOO....O.
.OO..O.OO.OO
.OOO.......O
[Figure 16: The "turtle" period 3 c/3 spaceship from Life traveling left. (DH)]
Figure 17 shows a period 3 c/3 spaceship that works in HighLife but not
in Life.
.......O...O.......
......O.O.O.O......
......OOO.OOO......
OOOOO.O.....O.OOOOO
OO..OO...O...OO..OO
..OOO....O....OOO..
..O.............O..
[Figure 17: Period 3 c/3 spaceship traveling up. (DB)]
Harold McIntosh has run an analysis of the differences between period 2
c/2 spaceships in Life and HighLife, using de Bruijn diagrams. Both short
but wide and long but thin ships can be calculated. (The adjectives refer
to the appearance of ships with respect to their direction of movement.)
In both Life and HighLife, ships of both types appear in quickly increasing
numbers once some minimum sizes in width and height are reached.
He found that the numbers in HighLife increase more rapidly but with a larger
size threshold for freestanding objects, with the consequence that while it
was barely possible to find Dean Hickerson's wicktrailer as the smallest
instance amongst Life spaceships of width 8, the data for Highlife already
defied analysis beyond width 7.
The smallest components for wide but short spaceships coincide between the
two kinds of Life, but thereafter the components appear to diverge. He was
able to calculate Bruijn diagrams up to a height of 10 for Life, but only up
a height of 6 for HighLife, indicating that HighLife probably has more of
these kinds of spaceships than Life does.
OSCILLATORS
-----------
Many of the oscillators in Life also work in HighLife. Almost all of the
period 2 oscillators still work. Figures 18 through 21 show a representative
sample of the oscillators with periods above 2 which still work in HighLife.
(The stable parts of the rightmost one in figure 19 and the third one in
figure 21 required some slight modifications to work in HighLife.)
..................................................OO...........
...................................................O...........
......................................OO.......OO.O......OOOO..
......................................O.......O..O.......O..O..
....................................O.O.......O........OOO..OOO
.....O...........O..................OO..........O......O......O
.O.OO.O.OO.......OOO...O.OOO.....OO..........O..O......O......O
O.O...O.OO......O.O...O.O.......O.O.........O.OO.......OOO..OOO
.O....O......OOO.O...OOO........O...........O............O..O..
.....OO................O.......OO..........OO............OOOO..
[Figure 18: Some period 3 oscillators taken from Life.]
...........O......................................................
...........O..................................OO.OO..OOOO..OO.OO..
...OOO...OOO.......OO...OO..................O..O.O..OOOOOO..O.O..O
.O.OOO...OO.........O.O.O....OO........OO...OO....OOOOOOOOOO....OO
O.O.O....OO..OOO....OO.OO....OO.OO..OO.OO.......OO..........OO....
O..O.......OOO......O.O.O.....O..O..O..O.......O..O........O..O...
.OO........OOO.....OO...OO....OO.OOOO.OO........OO..........OO....
[Figure 19: Some period 4 oscillators taken from Life.]
...............................................................OOO.....
............................................................OO.OO......
............................................................OO.OO......
.............................................................O.........
...............................OO............OO........................
.OO.........O..................OO............OO......OO................
..O.........OOO......................................OO.OO...O.........
..O.O..........O.............OOOO..........OOOO.......O..O..OO.OOOO.OO.
...O.O........O..........OO.O....O.....OO.O....O......OO.OOOO.OO..O..O.
.....OO.OO....O..O.......OO.OO...O.....OO.O..O.O..............O...OO.OO
........OO.......O..........O..O.O.OO.....O..O.O.OO..................OO
..OO...........O.O..........O.O..O.OO.....O.O..O.OO....................
...O...........OO..OO........OOOO..........OOOO...............O........
OOO................O.O.....................................OO.OO.......
O....................O.......OO............OO..............OO.OO.......
.....................OO......OO............OO.............OOO..........
[Figure 20: More period 4 oscillators taken from Life.]
............................OO.......OO......................
...........................O..O.....O..O.....................
..........................O.O.O.....O.O.O....................
.........................O.OO.OO...OO.OO.O...................
.............O.....O.....O...............O...................
.............OO...OO......OOO.OO.....OOOO....................
.............OO...OO........O.OO...OO.O.........O............
..............O...O.............................OOO........OO
................................OO.................O.......O.
...............OOO...............O................O......O.O.
..............O...O..............O................O...O..OO..
.................................OO....................O.....
....................................................O..O.....
...OO.........O...O.........O.OO...OO.O..............OOO.....
.O....O........OOO........OOOO.....OO.OOO....................
.O....O..................O...............O.......OO..........
.O....O.......O...O......O.OO.OO...OO.OO.O......O.O.....OO...
..O..O.......OO...OO......O.O.O.....O.O.O.......O.......O....
O.O..O.O.....OO...OO.......O..O.....O..O.......OO........OOO.
OO....OO.....O.....O........OO.......OO....................O.
[Figure 21: Periods 5, 6, 6, and 8 oscillators taken from Life.]
Dean Hickerson has found many oscillators in HighLife using a "random
torus" searching program that he wrote. There are a wide range of periods.
These oscillators are shown in figures 22 and 23. The period 9 oscillator
in figure 22 also works if the one cell gap in the row of cells is increased
to two cells, and that result also works in Life.
................................OO..........................................
................................OO..........................................
............................................................................
..............................OOOO...................OO...................OO
.................OO..........O....O...................O...................O.
................O.O.........O.O.OO.O.OO...............O.O...............O.O.
....OO.....OO...O...........O.O....O.OO................OO...............OO..
...O..O....O..O.OO.......OO.O....O.O........................................
.....O.O....OOO..........OO.O.OO.O.O......................OOOOOO.OOOOOO.....
.O.O...O.........OOO.........O....O.........................................
O...O.O.......OO.O..O.........OOOO........OOO..........OO...............OO..
O.O............O...OO.....................OOO.........O.O...............O.O.
.O..O........O.O..............OO..............OOO.....O...................O.
..OO.........OO...............OO..............OOO....OO...................OO
[Figure 22: Oscillators with periods 3, 4, 4, 7, and 9. (DH)]
..........................OOO.............O.................................
.........................O.OO.O.........OOO...............O...........O.....
........................O......O.......O..O...............O.O.......O.O.....
..........OO..O.O.........O..O..O......O.O.............O..OO.........OO..O..
............O.O.O......OO......OO.....OOO.............OOO...............OOO.
..........OO...O.......OO......OO............OOO.....OO.OO.............OO.OO
.......................O..O..O..............O.O.......OOO...............OOO.
O..O.......O...OO.......O......O...........O..O........O..OO.........OO..O..
O...O.....O.O.O..........O.OO.O............OOO............O.O.......O.O.....
OO.OO.....O.O..OO..........OOO.............O..............O...........O.....
[Figure 23: Oscillators with periods 10, 12, 14, 20, and 28. (DH)]
Figure 24 shows a period 24 oscillator formed from a "shuttle" and two eaters
which stabilize it. By itself, the shuttle creates a flipped copy of itself
(with some junk) in 12 generations, but the junk then destroys it. Adding
eaters as shown inhibits the junk and turns it into an oscillator. The
shuttle occurs naturally but usually self-destructs. It was discovered in
a symmetrical random soup experiment as a set of four shuttles stabilizing
each other.
........OO............
........O.O.O.O.......
........OO............
..OO..............OO..
.O.O..............O.O.
.O..................O.
OO..................OO
[Figure 24: Period 24 oscillator. (DH)]
Two copies of the shuttle can react with a still life to form more
complicated period 24 oscillators. Two of these oscillators are shown in
figure 25.
..................................... ............OO........................
..............O..............O....... .......O.O.O.O........................
..........O.OOOO.........O.OOOO...... ............OO......................OO
........OO.O.O.OO......OO.O.O.OO..... ..OO..............OO................O.
..........O.OOOO.........O.OOOO...... .O.O.............O..O.............O.O.
..OO..........O....O.........O...OO.. .O................OO..............OO..
.O.O..............O.O............O.O. OO......................OO............
.O...............O..O..............O. ........................O.O.O.O.......
OO................OO...............OO ........................OO............
[Figure 25: Period 24 oscillators using shuttles and a still life. (DH)]
As might be expected, the replicator can be turned into a large period
oscillator. This is done by inhibiting the growth of the ends of the
replicator. The first such oscillator found is shown in figure 26.
This uses four replicators positioned so that they react with each other
with a period of 48. (The two pieces in the middle are sparks which are
created and die away again.) Larger periods may be constructed by moving
the replicators further apart.
.....OOO.............OOO.....
.....O..O...........O..O.....
.....O...O.........O...O.....
......O..O.........O..O......
.......OOO.........OOO.......
.............................
.............................
.............................
.............................
.............................
.............................
...O.....................O...
..O.O...................O.O..
OO.........................OO
..O.O...................O.O..
...O.....................O...
.............................
.............................
.............................
.............................
.............................
.............................
.......OOO.........OOO.......
......O..O.........O..O......
.....O...O.........O...O.....
.....O..O...........O..O.....
.....OOO.............OOO.....
[Figure 26: Period 48 oscillator constructed from four replicators. (NT)]
A much smaller period 48 oscillator can be made using eaters to constrain
a replicator. Figure 27 shows this.
.O......................
.OOO....................
....O...................
...OO...................
........................
........................
...........OOO..........
..........O..O..........
.........O...O..........
.........O..O...........
.........OOO............
........................
........................
........................
........................
........................
........................
........................
........................
....................OO..
....................O.O.
......................O.
......................OO
[Figure 27: Period 48 replicator based oscillator. (DH)]
Higher period oscillators with periods which are multiples of 48 can be made
by moving the eaters further apart.
Period 96 replicator based oscillators can be built in an alternative
manner, as shown in figure 28. Here blocks are used instead of eaters.
Blocks cannot be used for a period 48 oscillator because the reaction
is too slow.
OO.........................
OO.........................
...........................
...........................
...........................
...........................
...........................
...........................
...........................
....................O......
...................OO......
..................O.O......
.................OOO.......
...........................
.......................OOO.
......................O.O..
......................OO...
......................O....
...........................
...........................
...........................
...........................
...........................
...........................
...........................
.........................OO
.........................OO
[Figure 28: Period 96 replicator based oscillator. (HE)]
A period 72 replicator oscillator can be made by starting with several
replicators in the oscillator, as shown in figure 29.
OO..................................
OO..................................
....................................
.............OOO....................
............O..O....................
...........O...O....................
...........O..O.....................
...........OOO......................
....................................
....................................
....................................
....................................
....................................
....................................
....................................
.........................OOO........
........................O..O........
.......................O...O........
.......................O..O.........
.......................OOO...OOO....
............................O..O....
...........................O...O....
...........................O..O.....
...........................OOO...OOO
................................O..O
...............................O...O
...............................O..O.
...............................OOO..
....................................
....................................
....................................
...............................OO...
...............................OO...
[Figure 29: Period 72 replicator based oscillator. (DH)]
If a replicator is perturbed by period 4 "mold" oscillators, then period 24
oscillators can be built. The smallest of these is shown in figure 30,
and a larger one is shown in figure 31.
..........OO.............
.........O..O............
..........O.O............
........OO.O.............
........OOO..............
........OO...............
.........................
.........................
...OOO...................
.O.OOO.....OOO...........
O.O.O.....O..O...........
O..O.....O...O...........
.OO......O..O.........OO.
.........OOO.........O..O
....................O.O.O
...................OOO.O.
...................OOO...
.........................
.........................
...............OO........
..............OOO........
.............O.OO........
............O.O..........
............O..O.........
.............OO..........
[Figure 30: Small period 24 replicator based oscillator. (DH)]
..........OO...............................
.........O..O..............................
..........O.O..............................
........OO.O...............................
........OOO................................
........OO.................................
...........................................
...........................................
...OOO.....................................
.O.OOO.....OOO.............................
O.O.O.....O..O.............................
O..O.....O...O.............................
.OO......O..O..............................
.........OOO...............................
...........................................
...........................................
...........................................
...................OOO.....................
..................O..O.....................
.................O...O.....................
.................O..O......................
.................OOO...OOO.................
......................O..O.................
.....................O...O.................
.....................O..O..................
.....................OOO...................
...........................................
...........................................
...........................................
...............................OOO.........
..............................O..O......OO.
.............................O...O.....O..O
.............................O..O.....O.O.O
.............................OOO.....OOO.O.
.....................................OOO...
...........................................
...........................................
.................................OO........
................................OOO........
...............................O.OO........
..............................O.O..........
..............................O..O.........
...............................OO..........
[Figure 31: Larger period 24 replicator based oscillator. (DH)]
------------------------------ End of part 1 ------------------------------