HighLife - An Interesting Variant of Life (part 3/3)
David I. Bell
by David I. Bell
db...@canb.auug.org.au
7 May 1994
GLIDER TURNING
--------------
Many glider turning reactions have been found in HighLife. Such reactions
exist for the period 24 shuttle, the bomber, the replicator, and standard
spaceships.
Pairs of period 24 shuttles can turn gliders by 90 degrees in several ways.
Figures 45 and 46 show two such glider turning reactions. The sparks used
for both of these turning reactions are identical to the ones used in Life.
In the reaction in figure 46, the glider can be slid one cell to the right
and the turning reaction still works.
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O.O....................
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.......................
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.......................
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......OOO..............
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[Figure 45: First glider turning reaction using two period 24 shuttles. (DH)]
...............O...................
.............O.O...................
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.................O...O...O.........
................O..O...OO.O........
.........O...........O...O.........
.........O...O...........O.....OO..
........O.OO...O..O............O.O.
.........O...O...O...............O.
..OO.....O.......................OO
.O.O...............................
.O.................................
OO.................................
[Figure 46: Second glider turning reaction using two period 24 shuttles. (DB)]
The bomber is able to turn gliders in various directions. Figure 47 shows
two reactions which turn a glider coming from the side to the back.
..........OOO.... .O.........OOO....
.............O... ..O...........O...
.O......O.O...O.. OOO......O.O...O..
..O............O. ................O.
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.........O......O ..........O......O
....O.O...O...O.O .....O.O...O...O.O
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....O.......O.O.. .....O.......O.O..
.OO.............. ..OO..............
......O...O.O.... .......O...O.O....
.......O......... ........O.........
........OOO...... .........OOO......
[Figure 47: Two side to back glider turning reactions with bombers. (DB)]
Figure 48 shows a glider coming in from the side being reflected by 180
degrees.
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...............OOO....
..............O.OOO...
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..............OO...OO.
...............OO..OOO
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....O...O.........O...
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....O...O.............
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.O.O..................
..OO..................
[Figure 48: Glider reflection reaction with bomber. (DB)]
Figure 49 shows a glider coming in from the back being turned by a
symmetrical pair of bombers.
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............OOO.........
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....................OO.O
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...O.......O............
..........O.............
..........O.............
.................OOO....
[Figure 49: Back to side glider turner using two bombers. (DB)]
Figure 50 shows a glider coming in from the front of a pair of bombers being
turned 90 degrees to the side. This uses the interaction between two bombs
to create the new glider. The incoming glider creates the second bomb to
create the glider.
........OOO........................
...........O.......................
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....OOO.......................O..O.
...O.O.O......................O...O
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....OOO................OOO.........
.....O.............................
...................................
...................................
...................................
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..........................OOO......
..........................O.O......
..........................OOO......
[Figure 50: Glider from front of two bombers turned 90 degrees. (DB)]
Figure 51 shows a more useful glider reflection reaction. Here a bomber
and a replicator interact with a period of 96 to form a bomb. The bomb
is used to turn a glider by 180 degrees. (The glider which appears near
the top is only transient and is part of the period 96 reaction.) A glider
cannot be turned by the normal bomb produced by just a bomber in this way
because it reacts with the bomber too early.
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O.O......OO........OOO.OO..OOO
..........O.......O.OOO.OOO.O.
.................OOO..OO.OOO..
...............O.OOO...OO.O...
...............O...OO..OOO....
...............O....OOO.O.....
.....................OOO......
......................O.......
..............................
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..............................
.OOO..........................
O...O.........................
O...O.........................
O...O.........................
.OOO..........................
.....OOO......................
.......O......................
......O.......................
[Figure 51: Bomber and replicator turn glider from back 180 degrees. (DB)]
The timing of this reaction is such that two of the period 96 spaceships
can be placed back to back so as to continually turn a glider back and
forth between them in a growing glider loop. Furthermore, the two paths
of the glider don't overlap, so multiple gliders can be traveling in the
glider loop at one time.
Figure 52 shows a glider coming in from the side of a bomber and replicator
being turned 90 degrees so as to travel in the same direction as the bomber,
thus overtaking it. The second bomber at the left is used to clean up some
debris. This reaction is rather slow and takes about 90 generations.
..............................................OOO...
.............................................O...O..
.............................................O....O.
.............................................O..O..O
..........................................OOO.O....O
.........................................O...O.O...O
.........................................O....O.OOO.
.........................................O..O..O....
..........................................O....O....
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...........................................O.O...O..
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............................O...O......O.......O.O..
...............................O........O...........
...............................O.........O...O.O....
....................................................
...........................................O.O......
............................................O.......
..........................................O.........
..........................................O.........
[Figure 52: Glider from side turned 90 degrees to front. (DB)]
Free replicators can turn gliders by 90 or 180 degrees. These reactions
destroy the replicator, and so it must be regenerated from another replicator
in order for the glider turning reaction to be repeated. This is easily
accomplished by removing one of the end stabilizers from the period 48 or 96
replicator oscillators.
For example, figure 53 shows a reaction in which a replicator reflects a
glider by 180 degrees. The incoming and outgoing paths of the glider do not
overlap.
....O..
...OOO.
..OO.OO
.OO.OO.
OO.OO..
.OOO...
..O....
....OO.
....O.O
....O..
[Figure 53: Replicator reflecting glider by 180 degrees. (DB)]
By using most of two copies of a period 48 replicator oscillator and two
gliders, this reaction can be used to make a glider loop which stabilizes
the oscillators. This is shown in figure 54. The ends can be separated
and more gliders inserted into the loop to make larger period 48 oscillators.
The glider loop can also be constructed using the period 96 oscillator.
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.................................................OOO
...................................................O
[Figure 54: Period 48 oscillator with gliders and replicators. (DB)]
Figure 55 shows a reaction in which a replicator turns a glider by 90
degrees. The outgoing glider travels to the upper right. This reaction
can be used with four copies of the period 96 replicator oscillator to
construct a larger oscillator in which four gliders travel in a square loop.
As before, the loop can be made larger and more gliders can be inserted.
The reaction is too slow to be used for the period 48 replicator oscillator.
...OOO
..O..O
.O...O
.O..O.
.OOO..
......
.O....
OO....
O.O...
[Figure 55: Replicator turning glider by 90 degrees. (DB)]
Figure 56 shows a reaction in which a glider hits a replicator and forms
two gliders. One glider is reflected by 180 degrees and the other glider is
turned by 90 degrees to travel to the lower right. The obvious attempt to
make a glider gun from this reaction fails.
..O.....
.O......
.OOO....
........
...O....
..OOO...
.OO.OO..
..OO.OO.
...OO.OO
....OOO.
.....O..
[Figure 56: Replicator duplicating a glider. (DB)]
A bound replicator can be used to turn gliders by 90 or 180 degrees in six
different known ways. Four of these reactions are basically the same, in
which a perturbation by a glider at a particular point of the replicator
causes a glider to be created. (This is also the basic reaction which powers
the period 96 glider gun in figure 32.) The other two reactions work
differently. The reactions differ from the previous ones in that the glider's
presence is optional and is not needed to stabilize the replicator.
Figure 57 shows two closely related glider turning reactions, in which a
glider coming from the lower right is turned 90 degrees to the upper right.
Figure 58 shows the third 90 degree turning reaction.
OO.................................. OO..................................
OO.................................. OO..................................
.................................... ....................................
................O................... ................O...................
...............OOO.................. ...............OOO..................
..............OOO.O................. ..............OOO.O.................
.............OO..OOO................ .............OO..OOO................
............OO...OO.O............... ............OO...OO.O...............
...........OOO..OO.OOO.............. ...........OOO..OO.OOO..............
............O.OOO.OOO.O............. ............O.OOO.OOO.O.............
.............OOO.OO..OOO............ .............OOO.OO..OOO............
..............O.OO...OO.O........... ..............O.OO...OO.O...........
...............OOO..OO.OOO.......... ...............OOO..OO.OOO..........
................O.OOO.OOO.O......... ................O.OOO.OOO.O.........
.................OOO.OO..OOO........ .................OOO.OO..OOO........
..................O.OO...OO.O....... ..................O.OO...OO.O.......
...................OOO..OO.OOO...... ...................OOO..OO.OOO......
....................O.OOO.OOO.O...O. ....................O.OOO.OOO.O.....
.....................OOO.OO..OOO.OO. .....................OOO.OO..OOO.OOO
......................O.OO...OO..O.O ......................O.OO...OO..O..
.......................OOO..OO...... .......................OOO..OO....O.
........................O.OOO....... ........................O.OOO.......
.........................OOO........ .........................OOO........
..........................O......... ..........................O.........
.................................... ....................................
.........................OO......... .........................OO.........
.........................OO......... .........................OO.........
[Figure 57: Replicators turning gliders by 90 degrees. (DB)]
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.........................O.O.........
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[Figure 58: Replicator turning a glider by 90 degrees. (DB)]
Figure 59 shows two of the three ways that a replicator can reflect a
glider by 180 degrees.
OO.................................. OO................................
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.................................... ..................................
................O................... ................O.................
...............OOO.................. ...............OOO................
..............OOO.O................. ..............OOO.O...............
.............OO..OOO................ .............OO..OOO..............
............OO...OO.O............... ............OO...OO.O.............
...........OOO..OO.OOO.............. ...........OOO..OO.OOO............
............O.OOO.OOO.O............. ............O.OOO.OOO.O...........
.............OOO.OO..OOO............ .............OOO.OO..OOO..........
..............O.OO...OO.O........... ..............O.OO...OO.O.........
...............OOO..OO.OOO.......... ...............OOO..OO.OOO........
................O.OOO.OOO.O......... ................O.OOO.OOO.O.....O.
.................OOO.OO..OOO........ .................OOO.OO..OOO...O..
..................O.OO...OO.O....... ..................O.OO...OO.O..OOO
...................OOO..OO.OOO...... ...................OOO..OO.OOO....
....................O.OOO.OOO.O..... ....................O.OOO.OOO.O...
.....................OOO.OO..OOO.O.O .....................OOO.OO..OOO..
......................O.OO...OO..OO. ......................O.OO...OO...
.......................OOO..OO....O. .......................OOO..OO....
........................O.OOO....... ........................O.OOO.....
.........................OOO........ .........................OOO......
..........................O......... ..........................O.......
.................................... ..................................
.........................OO......... .........................OO.......
.........................OO......... .........................OO.......
[Figure 59: Replicators reflecting gliders by 180 degrees. (DB)]
Figure 60 shows the third way of using a replicator to reflect a glider
by 180 degrees, in a construction which forms a glider relay. Here two
replicator oscillators are positioned to reflect a glider back and forth
between them. The replicators can be separated further to create any
period which is a multiple of 96. The incoming and outgoing paths for the
glider do not overlap, so that for larger separations, multiple gliders
can be put into the loop.
.........O.............................................................
........OOO............................................................
.......OOO.O...........................................................
......OO..OOO..........................................................
.....OO...OO.O.........................................................
.OO.OOO..OO.OOO........................................................
.OO..O.OOO.OOO.O.......................................................
......OOO.OO..OOO...........................OO.........................
.......O.OO...OO.O..........................OO.........................
........OOO..OO.OOO....................................................
.........O.OOO.OOO.O........................O..........................
..........OOO.OO..OOO......................OOO.........................
...........O.OO...OO.O....................OOO.O........................
............OOO..OO.OOO..................OO..OOO.......................
.............O.OOO.OOO.O...........OOO..OO...OO.O......................
..............OOO.OO..OOO............O.OOO..OO.OOO.....................
...............O.OO...OO............O...O.OOO.OOO.O....................
................OOO..OO..................OOO.OO..OOO...................
.................O.OOO....................O.OO...OO.O..................
..................OOO......................OOO..OO.OOO.................
...................O........................O.OOO.OOO.O................
.............................................OOO.OO..OOO...............
..............................................O.OO...OO.O..............
...............................................OOO..OO.OOO.............
................................................O.OOO.OOO.O............
.................................................OOO.OO..OOO...........
..................................................O.OO...OO............
...................................................OOO..OO.............
....................................................O.OOO..............
.....................................................OOO...............
..........................OO..........................O................
..........................OO...........................................
.....................................................................OO
.....................................................................OO
[Figure 60: A period 96 glider relay. (DB)]
A glider loop can also be constructed by the second reflection shown in
figure 59, but such a glider loop has the disadvantage that the two
glider paths overlap so that multiple gliders cannot be in the loop.
The first reflection reaction in figure 59 cannot be used to make a glider
relay at all because of the phase difference introduced by the reaction.
Standard spaceships can also be used to turn or reflect a glider (while
destroying the spaceship). There are no such reactions using the LWSS.
Figure 61 shows three such reactions using the MWSS. The first reaction
turns the glider 90 degrees to the upper right. The second reaction turns
the glider 90 degrees to the lower right. The third reaction reflects the
glider by 180 degrees.
.....O .......... .......O..
...O.O .......... .......O.O
....OO .........O .......OO.
...... .OOOOO.OO. .OOOOO....
.OOO.. O....O..OO O....O....
OOOOO. .....O.... .....O....
OOO.OO O...O..... O...O.....
...OO. ..O....... ..O.......
[Figure 61: Three glider turning reactions using a MWSS. (DB)]
Figure 62 shows four glider turning reactions using the HWSS. The first
two reactions turn the glider 90 degrees to the upper right. The third
reaction turns the glider 90 degrees to the lower right. The fourth reaction
reflects the glider by 180 degrees.
......O ........... ........... ...........
....O.O ........... ........... ...........
.....OO ........O.O ..........O ...........
....... .OOOOOO..OO .OOOOOO.OO. .OOOOOO...O
.OOOO.. O.....O..O. O.....O..OO O.....O.OO.
OOOOOO. ......O.... ......O.... ......O..OO
OOOO.OO O....O..... O....O..... O....O.....
....OO. ..OO....... ..OO....... ..OO.......
[Figure 62: Four glider turning reactions using a HWSS. (DB)]
By using multiple spaceships, there are probably some glider turning
reactions which do not destroy any of the spaceships. This would be done
by using the sparks from one spaceship to turn a glider into some junk,
and then using the sparks from another spaceship or two to turn that junk
back into a glider. No searching has yet been done for such reactions.
In Life, there are reactions between a glider and a boat or a toad which
turn the glider by 90 degrees (and destroying the boat or toad). Both of
those reactions fail in Highlife. But a different reaction allows a toad
to turn a glider by 90 degrees as shown in figure 63. The turned glider
travels to the lower left.
.......OO.
..O......O
O.O...O...
.OO....OO.
[Figure 63: Toad turning a glider by 90 degrees. (DB)]
WICKS
-----
Dean Hickerson found some infinite wicks of various periods in HighLife
using his toroidal Life program. Sections of each of these are shown in
figures 64 through 73. The ones in figures 64 and 65 also work in normal
Life. There are no known ways to make finite oscillators out of these
wicks.
OOOOOO..OOOOOO..OOOOOO..OOOOOO..OOOOOO..OOOOOO..OOOOOO..OOOOOO
O....O..O....O..O....O..O....O..O....O..O....O..O....O..O....O
[Figure 64: Period 6 wick. (DH)]
OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO
.O...O...O...O...O...O...O...O...O...O...O...O...O...O...O...O...O
OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO..OO
[Figure 65: Period 8 wick. (DH)]
OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO
OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO
OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO
OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO..OO....OO
[Figure 66: Period 10 wick. (DH)]
OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO.....
OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO.....
OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO.....
....................................................................
....................................................................
....................................................................
.....OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO
.....OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO
.....OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO
[Figure 67: Period 12 wick. (DH)]
.....OO......OO......OO......OO......OO......OO......OO......OO.....
.OO......OO......OO......OO......OO......OO......OO......OO......OO.
O..O....O..O....O..O....O..O....O..O....O..O....O..O....O..O....O..O
O..O....O..O....O..O....O..O....O..O....O..O....O..O....O..O....O..O
O..O....O..O....O..O....O..O....O..O....O..O....O..O....O..O....O..O
.OO......OO......OO......OO......OO......OO......OO......OO......OO.
.....OO......OO......OO......OO......OO......OO......OO......OO.....
[Figure 68: Period 17 wick. (DH)]
OO.....OO.....OO.....OO.....OO.....OO.....OO.....OO.....OO.....OO.
O.O....O.O....O.O....O.O....O.O....O.O....O.O....O.O....O.O....O.O
OO.....OO.....OO.....OO.....OO.....OO.....OO.....OO.....OO.....OO.
[Figure 69: Period 26 wick. (DH)]
OOOOOO....OOOOOO....OOOOOO....OOOOOO....OOOOOO....OOOOOO....OOOOOO
[Figure 70: Period 28 wick. (DH)]
OO....OO............OO....OO............OO....OO............OO....OO
OOO..OOO............OOO..OOO............OOO..OOO............OOO..OOO
OO....OO............OO....OO............OO....OO............OO....OO
[Figure 71: Period 32 wick. (DH)]
..OO.O..O...OO........OO.O..O...OO........OO.O..O...OO........OO.O..O...OO
..O..O..O..O..........O..O..O..O..........O..O..O..O..........O..O..O..O..
OO...O..O.OO........OO...O..O.OO........OO...O..O.OO........OO...O..O.OO..
[Figure 72: Period 38 wick. (DH)]
..O.O.......O.O.......O.O.......O.O.......O.O.......O.O.......O.O..
.O.O.O.....O.O.O.....O.O.O.....O.O.O.....O.O.O.....O.O.O.....O.O.O.
OO...OO...OO...OO...OO...OO...OO...OO...OO...OO...OO...OO...OO...OO
.O.O.O.....O.O.O.....O.O.O.....O.O.O.....O.O.O.....O.O.O.....O.O.O.
..OOO.......OOO.......OOO.......OOO.......OOO.......OOO.......OOO..
[Figure 73: Period 54 wick. (DH)]
OTHER REACTIONS
---------------
Figure 74 shows a reaction in which two bombs react to form two bombers
plus a little bit of junk.
......OOO
......O.O
OOO...OOO
O.O......
OOO......
[Figure 74: Two bombs react to form two bombers. (DB)]
Three gliders can construct a replicator as shown in figure 75. This reaction
for making a replicator was used in the first replicator puffer shown earlier.
.O.............
..O..O.........
OOO..O.O.......
.....OO........
...............
...............
...............
............OOO
............O..
.............O.
[Figure 75: Three gliders can create a replicator. (DH)]
In Life, only two gliders are needed to form an eater. But three gliders
are required in HighLife to make an eater. Figure 76 shows two methods of
making an eater. These reactions are very slow and require 110 and 119
generations respectively in order to finish.
........... .O............
O.O........ ..O...........
.OO.......O OOO...........
.O......OO. ..............
.........OO ..............
........... ..............
.........O. ......OOO..OOO
.........OO ........O..O..
........O.O .......O....O.
[Figure 76: Two ways that three gliders can create an eater. (DH)]
Figures 77 and 78 show the synthesis of period 2 oscillators with only one
active cell. (Oscillators with only one active cell are impossible in Life.)
............O....O.
.............OO.OO.
O.O..OO.....OO..O.O
.OO.OO.............
.O....O............
[Figure 77: Four gliders form a period 2 oscillator. (DH)]
O........................
.OO......................
OO.......................
.........................
.........................
.........................
...O.O...O.....OO...O....
....OO.OO.......OO.OO....
....O...OO.....O...O.O...
.........................
.........................
.........................
.......................OO
......................OO.
........................O
[Figure 78: Six gliders form a period 2 oscillator. (DH)]
Figure 79 shows a synthesis of the period 7 oscillator using 3 gliders.
..............O.
.............O..
.............OOO
..O.............
O.O.............
.OO.............
................
.O..............
.OO.............
O.O.............
[Figure 79: Three gliders form a period 7 oscillator. (DH)]
Figures 80 and 81 show two different ways of forming the period 10 oscillator.
The symmetrical pair of period 10 oscillators formed in figure 81 commonly
appears in random soup experiments.
.........O
...O...OO.
.O.O....OO
..OO......
..........
...OO.....
..O.O.....
....O.....
[Figure 80: Three gliders form a period 10 oscillator. (DH)]
..O.....................................
O.O.....................................
.OO.....................................
........................................
........................................
........................................
........................................
........................................
........................................
........................................
........................................
...............O.O......................
................OO......................
................O..................O.O..
...................................OO...
....................................O...
......................................O.
.....................................OO.
.....................................O.O
[Figure 81: Four gliders create a pair of period 10 oscillators. (DH)]
Figure 82 shows a synthesis of the period 20 oscillator using 12 gliders.
........................O.....................
......................O.O.....................
.......................OO.....................
..............................................
..........O..............O.................O.O
...........O.............O.O...............OO.
.........OOO.............OO.................O.
..............................................
..............................................
..............................................
..............................................
..............................................
..............................................
...................................O..........
..................................O...........
..................................OOO.........
......................................OOO.....
......O...............................O.......
.......O...............................O......
.....OOO......................................
.........OOO..................................
...........O..................................
..........O...................................
..............................................
..............................................
..............................................
..............................................
..............................................
..............................................
.O.................OO.............OOO.........
.OO...............O.O.............O...........
O.O.................O..............O..........
..............................................
.....................OO.......................
.....................O.O......................
.....................O........................
[Figure 82: Twelve gliders form period 20 oscillator. (DH)]
Figure 83 shows a synthesis of the period 12 oscillator using 16 gliders.
...............O...................
................O..................
..............OOO..................
......................O............
.....................O.............
.....................OOO...........
...................................
..................O................
.................O.................
.................OOO...............
...................................
.....O..........O..................
...O.O...........O.................
....OO.........OOO.................
................................O..
.........O...........O..........O.O
.......O.O...........O.O........OO.
........OO..OO.......OO..OO........
.OO........O.O...........O.O.......
O.O..........O...........O.........
..O................................
.................OOO.........OO....
.................O...........O.O...
..................O..........O.....
...................................
...............OOO.................
.................O.................
................O..................
...................................
...........OOO.....................
.............O.....................
............O......................
..................OOO..............
..................O................
...................O...............
[Figure 83: Sixteen gliders form period 12 oscillator. (DH)]
Figure 84 shows an intriguing object. It reappears in generation 78
(along with some junk) rotated by 90 degrees. Perturbations of its reaction
also reproduce it in different generations. This might be tamed someday to
form glider guns or puffers, as the B-heptomino in Life has been.
OO..
O.OO
.OOO
..O.
[Figure 84: Object which reappears in generation 78. (DH)]
An infinite row of these objects makes a moving wick which moves upwards
at a speed of c/15. A section of this wick is shown in figure 85.
Alternatively, this can be viewed as a spaceship on a cylindrical universe
of width 8.
OO......OO......OO......OO......OO......OO......OO......OO..
O.OO....O.OO....O.OO....O.OO....O.OO....O.OO....O.OO....O.OO
.OOO.....OOO.....OOO.....OOO.....OOO.....OOO.....OOO.....OOO
..O.......O.......O.......O.......O.......O.......O.......O.
[Figure 85: A section of an infinite period 90 c/15 spaceship. (DH)]
Long rows of cells are not nearly as chaotic in HighLife as in Life. For
example, a row of 149 cells results in a pretty display of blinkers. Such
a display can occur for arbitrarily large line lengths.
OTHER COMMENTS
--------------
To me, HighLife appears slightly less "vigorous" than normal Life for
random soups. Objects don't seem to last as long or grow as far before
settling down. But, this comment only applies until a replicator appears!
Paul Callahan has investigated other modifications to the Life rule that
also give rise to the replicator. The minimal rule change is the "Ship"
rule, which says that the normal Life rule applies EXCEPT in the following
two neighborhoods:
OO. .OO
O.O O.O
.OO OO.
Under normal Life rules, the central cell stays dead. But under the Ship
rule, the central cell is born. So the full generality of the B36/S23 rule
is not needed to make the replicator work. The six cell birth case only
needs to apply in the above situations.
Under the Ship rule, the period 24 glider gun fails, but the period 30
glider run from Life can be made to work with the addition of some clocks
near the blocks. Furthermore, the pentadecathlon works, as do some of the
orthogonal puffers from Life. The "kickback" reaction also works.
Paul Callahan has also pointed out that the replicator is an embedding
of a 1-D automaton into HighLife, where the new state of a cell is the XOR
of its two neighbors. This follows since a replicator next to empty
space expands into it, but two replicators next to each other cancel each
other out.
By making a read/write head on one end of a replicator, and allowing the
replicator to grow indefinitely on the other end, an arbitrarily large
amount of information can be stored in the replicator stream. Since it
runs using XOR logic, all of the information will eventually be presented
to the read head (but in a very complicated manner!). Paul suggests that
such a construction might allow a nice proof of the universality of
HighLife.
Paul Callahan has also investigated the results of aiming a glider gun at a
replicator. Assuming that the glider gun survives the occasional backwards
glider (by protecting it with some debris), the results are quite complicated
and unpredictable. This occurs because the glider stream cannot tunnel
through the replicator, and the replicator cannot escape the glider stream.
Paul has run one of his experiments over 145000 generations without observing
periodicity.
Finally, John Conway (who discovered Life) has remarked that:
"It seems to me that "B36/S23" is really the game I should have found,
since it's so rich in nice things."
That statement may or may not be true. (For example, I think the Life rule
is more "realistic" and "natural" in having births only for three neighbors.)
But you will probably have to agree that HighLife has many interesting things
and is worth investigating further.
I would like to thank all of the people who contributed results for this
article. I would also like to thank the following people who reviewed my
article to make it much better and more accurate:
John Conway
Achim Flammenkamp
Bill Gosper
Dean Hickerson
Harold McIntosh
Richard Schroeppel
Nathan Thompson
If anyone makes any exciting discoveries in HighLife (or in Life) and they
want to send them to me, I would be happy to receive them. I would send
them on to people who are interested in such things and possibly use them
in future articles. Proper credit would be given, of course.
David I. Bell
db...@canb.auug.org.au
Paul Callahan
(sorry I didn't notice it when the article was being reviewed):
db...@pdact.pd.necisa.oz.au (David I. Bell) writes:
...
>Paul Callahan has investigated other modifications to the Life rule that
>also give rise to the replicator. The minimal rule change is the "Ship"
>rule, which says that the normal Life rule applies EXCEPT in the following
>two neighborhoods:
> OO. .OO
> O.O O.O
> .OO OO.
>Under normal Life rules, the central cell stays dead. But under the Ship
>rule, the central cell is born.
...
>Under the Ship rule, the period 24 glider gun fails, but the period 30
^^^^^^^^^
>glider run from Life can be made to work with the addition of some clocks
>near the blocks.
Actually, the period 30 "queen bee" shuttle seems irreparable, as it
contains several instances of the Ship neighborhood duriod its cycle.
It's the period 46 b-heptomino shuttle that can be stabilized with clocks.
The following shows the only generation in which the exception matters.
.o............................o..
..oo..........................o.o
oo...........................o.o.
..o............................o.
....oo...........................
...oooo.......oo.................
..oXo.o......o..oo...........oo..
..oo.ooo....oooooo...........oo..
.....oo.o.....oooo...............
.....oo.o........................
.................................
.....oo.o........................
.....oo.o.....oooo...............
....o..o....oooooo...............
.....oo......o..oo...............
.....oo.......oo.................
The spot marked 'X' is a dead cell which will stay dead in standard
Life, but which will become alive under the Ship rule. The presence
of the clock (upper right) suppresses a spark that interacts with cell
'X' when it is alive, damaging the block stabilizer. In standard
Life, the absence of the spark has no effect, so this pattern works
under both rules.
The gun formed by combining parallel period 46 shuttles also works
under both rules, as does the interaction to produce a lightweight
spaceship by colliding a glider with the shuttle's spark. Hence, many
constructions continue to work under the Ship rule, provided the clock
stabilizers are added. I haven't done an exhaustive survey of which
patterns can be patched in this way. If anyone else wants to check,
I'm interested in their observations.
--
Paul Callahan
call...@biffvm.cs.jhu.edu