From: David I. Bell (dbell@pdact.pd.necisa.oz.au)
Date: Fri Oct 23 1992 - 02:55:50 UTC
----------------- [Continued from previous mail article] --------------------
The method of tweaking the disconnection clouds to re-ignite the new blinker
row can produce many different puffer trains with many different periods.
(Probably an infinite number of essentially different puffer trains can be
constructed.) But the products of each puffer are different and hard to
control, and even worse, there is no control over what the period is. For
example, creating a puffer train with period 2000 by randomly tweaking
(multiple times probably) the disconnection cloud is not obvious. At any
rate it would be very time-consuming to search for.
As part of constructing a new type of sawtooth pattern using a blinker puffer,
Dean Hickerson found the following useful reaction. By adding a spaceship to
a blinker puffer, a glider can be produced by the disconnection cloud. The
produced glider travels backwards and outwards, to the upper right. This
reaction is shown below.
[Creating a glider from disconnection cloud of blinker fuse]
........................OOOOOO.
........................O.....O
........................O......
.........................O....O
...........................OO..
...............................
...............................
...............................
...............................
...............................
...............................
...............................
..OOOOOO.......................
..O.....O......................
..O............................
...O....O......................
.....OO........................
...............OO.O..OO........
..OOOOO.......O.O.OOOOO........
.OOOOOOO..OOO.O................
OO.OOO.O......O.O.OOOOO........
.OO............OO.O..OO........
...............................
...............................
...............................
OOOOO..........................
O....O.........................
O..............................
.O...O.........................
...O...........................
Much earlier, and for another purpose, Dean Hickerson had found useful
glider turning reactions using spaceships. Some of these are shown below.
In the first one, a forwards glider is converted into another forwards
glider, but traveling towards the upper left. This is a 90 degree right
handed turn. The upper HWSS does not hinder the glider coming in from a
long distance.
[Forwards glider converted into other forwards direction]
...........OO..
.........O....O
........O......
........O.....O
........OOOOOO.
...............
...............
...............
...............
...............
......O........
....OO.........
.....OO........
...............
...OO..........
.O....O........
O..............
O.....O........
OOOOOO.........
In the following, the forwards glider is converted into a backwards glider,
traveling towards the lower right. This is a 90 degree left turn. The
bottom right MWSS is only needed to delete a block which is created during
the reaction. Once again, the upper HWSS does not hinder the glider coming
in from a long distance.
[Forwards glider converted into backwards glider]
..................OOOO..
.................OOOOOO.
................OO.OOOO.
.................OO.....
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
......O.................
....OO..................
.....OO.................
........................
...OO...............OOO.
.O....O............OOOOO
O.................OO.OOO
O.....O............OO...
OOOOOO..................
In the following, a backwards traveling glider is converted into a forwards
moving glider heading towards the lower left. This is a 90 degree right
handed turn. The top MWSS is only needed to delete a block which is created
during the reaction.
[Backwards glider converted into forwards glider]
..............................................OOO.
.............................................OOOOO
............................................OO.OOO
.............................................OO...
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
..................................................
......O.O.........................................
.......OO.........................................
.......O..........................................
..O..O............................................
.O................................................
.O...O............................................
.OOOO.............................................
..................................................
..................................................
..........OO......................................
.........OO.OOOO..................................
..........OOOOOO..................................
...........OOOO...................................
Combinations of these reactions (and their mirror reflections) can be used
to make a glider travel in any desired direction.
After seeing all of the above reactions, I thought of a method of using them
along with the blinker puffers to create puffer trains with arbitrarily
large periods. How this works is as follows.
Take a blinker puffer with the fuse lit. Add two spaceships on either side
of it as shown above to turn the disconnection cloud into two gliders flying
symmetrically outwards and backwards from the glider puffer. Use two
copies of the last reaction above to turn those gliders into forwards and
outwards traveling gliders. Then use two copies of the first reaction
to turn the gliders into inwards and forwards traveling gliders. If the
reactions are properly placed, then the two final gliders will converge
onto the end of the new blinker row, and then re-ignite the blinker row
using the method shown previously.
The quickest way to use these reactions in this manner produces the following
period 1136 spaceship. This was built independently by myself and by Dean
Hickerson, once I announced the construction method. In this object, the
third glider turning reactions only needs two spaceships because the parallel
reactions run into each other and thereby suppress the creation of the blocks
that would need the other two spaceships to clean up.
[Period 1136 spaceship using blinker puffer and glider turning reactions]
$$$$$$$$$$$..OOOOOO.........
$$$$$$$$$$$..O.....O........
$$$$$$$$$$$..O$....
$$$$$$$$$$$...O....O........
$$$$$$$$$$$.....OO$
$$$$......OO$$$$$$$.........
$$$$....O....O$$$$$$$.......
$$$$...O$$$$$$$$...
$$$$...O.....O$$$$$$$.......
$$$$...OOOOOO$$$$$$$........
$$$$$$$$$$$$.......
$$.......OO.......OO$$$$$$$$.........
$$.....O....O....OOOO$$$$$$$$........
$$....O.........OO.OO$$$$$$$$........
$$....O.....O....OO$$$$$$$$...OOOOOO.
$$....OOOOOO$$$$$$$$$O.....O
$$$$$$$$$$$$O......
$$$$$$$$$$$$.O....O
$$$$$$$$$$$$...OO..
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
OOOOO$$$$$$$$$$$$..
O....O$$$$$$$$$$$$.
O$$$$$$$$$$$$......
.O...O$$$$$$$$$$$$.
...O$$$$$$$$$$$$...
..OOO$$$$$$$$$$$$..
.OOOO.......OO.O..OO$$$$$$$$$$.......
OO.OO......O.O.OOOOO$$$$$$$$$$.......
.OO....OOO.O$$$$$$$$$$$.....
$.O.O.OOOOO$$$$$$$$$$.......
$..OO.O..OO$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
..OOOOOO$$$$$$$$$$$.........
..O.....O$$$$$$$$$$$........
..O$$$$$$$$$$$$....
...O....O$$$$$$$$$$$........
.....OO$$$$$$$$$$$$
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$.......
$$$$$$$$$$$$...OO..
$$$$$$$$$$$$.O....O
$$$$$$$$$$$$O......
$$....OOOOOO$$$$$$$$$O.....O
$$....O.....O....OO$$$$$$$$...OOOOOO.
$$....O.........OO.OO$$$$$$$$........
$$.....O....O....OOOO$$$$$$$$........
$$.......OO.......OO$$$$$$$$.........
$$$$$$$$$$$$.......
$$$$...OOOOOO$$$$$$$........
$$$$...O.....O$$$$$$$.......
$$$$...O$$$$$$$$...
$$$$....O....O$$$$$$$.......
$$$$......OO$$$$$$$.........
$$$$$$$$$$$.....OO$
$$$$$$$$$$$...O....O........
$$$$$$$$$$$..O$....
$$$$$$$$$$$..O.....O........
$$$$$$$$$$$..OOOOOO.........
The interesting thing about this construction is that the glider's initial
backward-and-outward path is parallel to its final forward-and-inward path.
This means that these path lengths can be increased by moving the turning
reactions further away from the blinker puffer, and the gliders will still
hit the end of the blinker row. This allows a puffer to be built with an
arbitrarily large period by moving the turning reactions sufficiently far
away.
Because of the timing constraints on the phase of blinker ignition, it turns
out that the allowable changes are as follows. Move the LWSSs and all the
spaceships behind them 9N units backwards, and 3N units outwards from the
center line. This will increase the period by 96. This will also leave
enough room to allow the creation of the two blocks that one of the turning
reactions leaves, turning the construction into a puffer. Adding two more
spaceships to the final result to delete those blocks will turn the puffer
back into a spaceship.
The following shows the above repositioning for the case N=1. The spaceships
have been moved 9 units backwards and 3 units outwards, increasing the
period by 96. This results in a double-block puffer of period 1232. The
block deleting spaceships have not been added.
[Period 1232 puffer train leaving behind two blocks]
$$$$$$$$$$$$.OOOOOO.........
$$$$$$$$$$$$.O.....O........
$$$$$$$$$$$$.O$....
$$$$$$$$$$$$..O....O........
$$$$$$$$$$$$....OO$
$$$$$.....OO$$$$$$$.........
$$$$$...O....O$$$$$$$.......
$$$$$..O$$$$$$$$...
$$$$$..O.....O$$$$$$$.......
$$$$$..OOOOOO$$$$$$$........
$$$$$$$$$$$$$......
$$$$.....OO$$$$$$$$.........
$$$$....OOOO$$$$$$$$........
$$$$...OO.OO$$$$$$$$........
$$.......OO$.....OO$$$$$$$$...OOOOOO.
$$.....O....O$$$$$$$$$........O.....O
$$....O$$$$$$$$$$....O......
$$....O.....O$$$$$$$$$.........O....O
$$....OOOOOO$$$$$$$$$$..OO..
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
OOOOO$$$$$$$$$$$$$.
O....O$$$$$$$$$$$$$
O$$$$$$$$$$$$$.....
.O...O$$$$$$$$$$$$$
...O$$$$$$$$$$$$$..
..OOO$$$$$$$$$$$$$.
.OOOO.......OO.O..OO$$$$$$$$$$$......
OO.OO......O.O.OOOOO$$$$$$$$$$$......
.OO....OOO.O$$$$$$$$$$$$....
$.O.O.OOOOO$$$$$$$$$$$......
$..OO.O..OO$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
..OOOOOO$$$$$$$$$$$$........
..O.....O$$$$$$$$$$$$.......
..O$$$$$$$$$$$$$...
...O....O$$$$$$$$$$$$.......
.....OO$$$$$$$$$$$$.........
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$$$$$$$$$$$$......
$$....OOOOOO$$$$$$$$$$..OO..
$$....O.....O$$$$$$$$$.........O....O
$$....O$$$$$$$$$$....O......
$$.....O....O$$$$$$$$$........O.....O
$$.......OO$.....OO$$$$$$$$...OOOOOO.
$$$$...OO.OO$$$$$$$$........
$$$$....OOOO$$$$$$$$........
$$$$.....OO$$$$$$$$.........
$$$$$$$$$$$$$......
$$$$$..OOOOOO$$$$$$$........
$$$$$..O.....O$$$$$$$.......
$$$$$..O$$$$$$$$...
$$$$$...O....O$$$$$$$.......
$$$$$.....OO$$$$$$$.........
$$$$$$$$$$$$....OO$
$$$$$$$$$$$$..O....O........
$$$$$$$$$$$$.O$....
$$$$$$$$$$$$.O.....O........
$$$$$$$$$$$$.OOOOOO.........
So far this allows for the construction of spaceships of any period which
is 1136 + 96N. This is a great improvement over the period-doubling methods.
But this is not all that can be done.
If the LWSSs and their neighboring HWSSs are moved 3N cells forwards and N
cells inwards towards the center line, and the back four HWSSs are moved N
units forwards and N units outwards, then the period is increased by 8.
For this repositioning to work, the spaceships must be kept far enough away
from the center line for the reactions to work, so that this cannot be used
indefinitely by itself. But since the first repositioning method moves the
spaceships outwards, by using the first repositioning method first, the
second repositioning can then be applied as many times as is necessary.
This second repositioning method has the side effect that the final glider
paths no longer hit the last blinker in the blinker row. Each increment of
the period by 8 results in the gliders being shifted forwards to impact on
the next blinker forwards. The blinker ignition will still work, but some
blinkers at the end can be left unignited, and therefore will be added to the
puffer's output. Besides the block deleting spaceships, another spaceship can
be added to delete these leftover blinkers to turn the puffer back into a
spaceship.
Here are two examples of combining these two repositionings. In this first
one, the second repositioning method is applied to the period 1232 puffer
train above to produce a period 1240 puffer train. In this particular case,
since the final gliders impact the second to last blinker, no blinkers are
left by the puffer.
[Period 1240 puffer producing pair of blocks]
$$$$$$$$$$$$OOOOOO.........
$$$$$$$$$$$$O.....O........
$$$$$$$$$$$$O$....
$$$$$$$$$$$$.O....O........
$$$$$$$$$$$$...OO$
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$..OO$$$$$$$$.
$$$$$O....O$$$$$$$.........
$$$$.........O$$$$$$$$.....
$$$$.........O.....O$$$$$$$.........
$$$$.........OOOOOO$$$$$$$$
$$$$$$$$$$$$$.....
$$$$..OO$$$$$$$$$.
$$$$.OOOO$$$$$$$$...OOOOOO.
$$.......OO$.OO.OO$$$$$$$$...O.....O
$$.....O....O$OO$$$$$$$$.....O......
$$....O$$$$$$$$$$....O....O
$$....O.....O$$$$$$$$$$OO..
$$....OOOOOO$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
OOOOO$$$$$$$$$$$$$
O....O$$$$$$$$$$$$.........
O$$$$$$$$$$$$$....
.O...O$$$$$$$$$$$$.........
...O$$$$$$$$$$$$$.
..OOO$$$$$$$$$$$$$
.OOOO.......OO.O..OO$$$$$$$$$$$.....
OO.OO......O.O.OOOOO$$$$$$$$$$$.....
.OO....OOO.O$$$$$$$$$$$$...
$.O.O.OOOOO$$$$$$$$$$$.....
$..OO.O..OO$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
..OOOOOO$$$$$$$$$$$$.......
..O.....O$$$$$$$$$$$$......
..O$$$$$$$$$$$$$..
...O....O$$$$$$$$$$$$......
.....OO$$$$$$$$$$$$........
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$....OOOOOO$$$$$$$$$$.....
$$....O.....O$$$$$$$$$$OO..
$$....O$$$$$$$$$$....O....O
$$.....O....O$OO$$$$$$$$.....O......
$$.......OO$.OO.OO$$$$$$$$...O.....O
$$$$.OOOO$$$$$$$$...OOOOOO.
$$$$..OO$$$$$$$$$.
$$$$$$$$$$$$$.....
$$$$.........OOOOOO$$$$$$$$
$$$$.........O.....O$$$$$$$.........
$$$$.........O$$$$$$$$.....
$$$$$O....O$$$$$$$.........
$$$$$..OO$$$$$$$$.
$$$$$$$$$$$$$.....
$$$$$$$$$$$$$.....
$$$$$$$$$$$$...OO$
$$$$$$$$$$$$.O....O........
$$$$$$$$$$$$O$....
$$$$$$$$$$$$O.....O........
$$$$$$$$$$$$OOOOOO.........
Here is another example. This is a period 1368 puffer train which leaves
four blinkers. This is made from the period 1136 spaceship by two iterations
of the first positioning, (increasing the period by 192), followed by five
iterations of the second positioning (increasing the period by 40). As in
the period 1136 spaceship, the reactions are close enough to the center line
to suppress the creating of the two blocks.
[Period 1368 4-blinker puffer train]
$$$$$$$$$$$$.....OOOOOO.........
$$$$$$$$$$$$.....O.....O........
$$$$$$$$$$$$.....O$....
$$$$$$$$$$$$......O....O........
$$$$$$$$$$$$........OO$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$...OOOOOO.
$$$$.........OO$$$$$$$$..O.....O
$$$$.......O....O$$$$$$$$O......
$$$$......O$$$$$$$$.......O....O
$$$$......O.....O$$$$$$$$...OO..
$$$$......OOOOOO$$$$$$$$........
$$$$$$$$$$$$$$
$$$.........OO$$$$$$$$$.........
$$.......OO.........OOOO$$$$$$$$$........
$$.....O....O......OO.OO$$$$$$$$$........
$$....O$...OO$$$$$$$$$$
$$....O.....O$$$$$$$$$$.........
$$....OOOOOO$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
.....OO$$$$$$$$$$$$$...
...O....O$$$$$$$$$$$$$.
..O$$$$$$$$$$$$$.......
..O.....O$$$$$$$$$$$$$.
..OOOOOO$$$$$$$$$$$$$..
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$..OO.O..OO$$$$$$$$$$$$
$.O.O.OOOOO$$$$$$$$$$$$
.OO....OOO.O$$$$$$$$$$$$........
OO.OO......O.O.OOOOO$$$$$$$$$$$$
.OOOO.......OO.O..OO$$$$$$$$$$$$
..OOO$$$$$$$$$$$$$.....
...O$$$$$$$$$$$$$......
.O...O$$$$$$$$$$$$$....
O$$$$$$$$$$$$$.........
O....O$$$$$$$$$$$$$....
OOOOO$$$$$$$$$$$$$.....
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$....OOOOOO$$$$$$$$$$$
$$....O.....O$$$$$$$$$$.........
$$....O$...OO$$$$$$$$$$
$$.....O....O......OO.OO$$$$$$$$$........
$$.......OO.........OOOO$$$$$$$$$........
$$$.........OO$$$$$$$$$.........
$$$$$$$$$$$$$$
$$$$......OOOOOO$$$$$$$$........
$$$$......O.....O$$$$$$$$...OO..
$$$$......O$$$$$$$$.......O....O
$$$$.......O....O$$$$$$$$O......
$$$$.........OO$$$$$$$$..O.....O
$$$$$$$$$$$$$...OOOOOO.
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$$$
$$$$$$$$$$$$........OO$
$$$$$$$$$$$$......O....O........
$$$$$$$$$$$$.....O$....
$$$$$$$$$$$$.....O.....O........
$$$$$$$$$$$$.....OOOOOO.........
So as can be seen above, you can make a puffer train (and spaceship) of any
sufficiently large period which is a multiple of 8.
Can we do better than this? The answer is yes, if we switch to the larger
blinker puffer which can produce spaceships of periods 4 (mod 8). I found a
method for creating arbitrarily large periods using this other puffer engine.
The following is a period 1044 puffer train which leaves a row of 9 blinkers
and two more isolated blinkers.
[Period 1044 puffer train leaving 11 blinkers]
$$$$$$$$....OOOOOO.........
$$$$$$$$....O.....O........
$$$$$$$$....O$....
$$$$$$$$.....O....O........
$$$$$$$$.......OO$
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$..OOOOOO.
$$$$$$$$$..O.....O
$$$$$$$$$..O......
$$$$$$$$$...O....O
$$$$$$$$$.....OO..
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$.......O..O$$$$........
$$$$......O$$$$$..
$$$$......O...O$$$$........
$$$$......OOOO$$$$.........
$$$$$$$$$.........
$....O$$$$$$$$....
$..O...O$$$$$$$$..
$.O$O......OOO$$$$$$.......
$.O....O...O...O...OOOOO$$$$$$......
$.OOOOO...O.......OO.OOO$$$$$$......
$.........O....O...OO$$$$$$.........
$.........OOOOO$$$$$$$.....
$O$$$$$$$$........
.OO.O...OO.OO$$$$$$$$......
O..OOOOO.O$$$$$$$$.........
O..O.OO.OO$$$$$$$$.........
OO...OOO.OO......OO.O..OO$$$$$$$....
..O......OO.....O.O.OOOOO$$$$$$$....
.........OO.OOO.O$$$$$$$$..
..O......OO.....O.O.OOOOO$$$$$$$....
OO...OOO.OO......OO.O..OO$$$$$$$....
O..O.OO.OO$$$$$$$$.........
O..OOOOO.O$$$$$$$$.........
.OO.O...OO.OO$$$$$$$$......
$O$$$$$$$$........
$.........OOOOO$$$$$$$.....
$.........O....O...OO$$$$$$.........
$.OOOOO...O.......OO.OOO$$$$$$......
$.O....O...O...O...OOOOO$$$$$$......
$.O$O......OOO$$$$$$.......
$..O...O$$$$$$$$..
$....O$$$$$$$$....
$$$$$$$$$.........
$$$$......OOOO$$$$.........
$$$$......O...O$$$$........
$$$$......O$$$$$..
$$$$.......O..O$$$$........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.....OO..
$$$$$$$$$...O....O
$$$$$$$$$..O......
$$$$$$$$$..O.....O
$$$$$$$$$..OOOOOO.
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$$.........
$$$$$$$$.......OO$
$$$$$$$$.....O....O........
$$$$$$$$....O$....
$$$$$$$$....O.....O........
$$$$$$$$....OOOOOO.........
This basic result can be modified to produce other puffer trains. The four
LWSSs in the back can be moved 6N units backwards and 6N units outwards.
This increases the period by 120N, and leave 3N more blinkers in the row.
Since the period increase is a multiple of 8, each of these new puffers will
also have a period which is 4 (mod 8).
For example, here is the next puffer in this series, which has period 1164,
and leaves 14 blinkers.
[Period 1164 puffer train]
$$$$$$$$$OOOOOO.........
$$$$$$$$$O.....O........
$$$$$$$$$O$....
$$$$$$$$$.O....O........
$$$$$$$$$...OO$
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$........OOOOOO.
$$$$$$$$$........O.....O
$$$$$$$$$........O......
$$$$$$$$$.........O....O
$$$$$$$$$$.OO..
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$.......O..O$$$$$....
$$$$......O$$$$$........
$$$$......O...O$$$$$....
$$$$......OOOO$$$$$.....
$$$$$$$$$$.....
$....O$$$$$$$$$
$..O...O$$$$$$$$........
$.O$O......OOO$$$$$$$...
$.O....O...O...O...OOOOO$$$$$$$..
$.OOOOO...O.......OO.OOO$$$$$$$..
$.........O....O...OO$$$$$$$.....
$.........OOOOO$$$$$$$$.
$O$$$$$$$$$....
.OO.O...OO.OO$$$$$$$$$..
O..OOOOO.O$$$$$$$$$.....
O..O.OO.OO$$$$$$$$$.....
OO...OOO.OO......OO.O..OO$$$$$$$$
..O......OO.....O.O.OOOOO$$$$$$$$
.........OO.OOO.O$$$$$$$$........
..O......OO.....O.O.OOOOO$$$$$$$$
OO...OOO.OO......OO.O..OO$$$$$$$$
O..O.OO.OO$$$$$$$$$.....
O..OOOOO.O$$$$$$$$$.....
.OO.O...OO.OO$$$$$$$$$..
$O$$$$$$$$$....
$.........OOOOO$$$$$$$$.
$.........O....O...OO$$$$$$$.....
$.OOOOO...O.......OO.OOO$$$$$$$..
$.O....O...O...O...OOOOO$$$$$$$..
$.O$O......OOO$$$$$$$...
$..O...O$$$$$$$$........
$....O$$$$$$$$$
$$$$$$$$$$.....
$$$$......OOOO$$$$$.....
$$$$......O...O$$$$$....
$$$$......O$$$$$........
$$$$.......O..O$$$$$....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.OO..
$$$$$$$$$.........O....O
$$$$$$$$$........O......
$$$$$$$$$........O.....O
$$$$$$$$$........OOOOOO.
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$$.....
$$$$$$$$$...OO$
$$$$$$$$$.O....O........
$$$$$$$$$O$....
$$$$$$$$$O.....O........
$$$$$$$$$OOOOOO.........
Dean Hickerson recently found another glider perturbing reaction which does
not change the direction of a glider, but which does delay it and shift its
path. This reaction is shown below. The spaceship on the right is only
needed to delete an extraneous block produced by the reaction.
[Forwards glider being delayed and path-shifted]
........OO.....$$$$$$.......
.......OO.OOO..$$$$$$.......
........OOOOO..$$$$$$.......
.........OOO...$$$$$$.......
...............$$$$$$.......
...............$$$$$$.......
...............$$$$$$...OOO.
...............$$$$$$..OOOOO
...............$$$$$$.OO.OOO
......O........$$$$$$..OO...
....OO.........$$$$$$.......
.....OO........$$$$$$.......
...............$$$$$$.......
...OO..........$$$$$$.......
.O....O........$$$$$$.......
O..............$$$$$$.......
O.....O........$$$$$$.......
OOOOOO.........$$$$$$.......
This reaction can be used to modify the period of these new spaceships,
allowing more of the periods which are 4 (mod 8) to be constructed.
The following puffer train is made from the period 1044 puffer train by
adding one (doubled) copy of the above reaction to it. This produces a
puffer train of period 1732. It produces 16 blinkers and two blocks.
[Period 1732 puffer train]
$$$$$$$$$$$$$$$........OOOOOO.........
$$$$$$$$$$$$$$$........O.....O........
$$$$$$$$$$$$$$$........O$....
$$$$$$$$$$$$$$$.........O$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$..O$$$$$$$$$
$$$$$$$$.O$$$$$$$$$.
$$$$$$$$.O.....O$$$$$$$$.....
$$$$$$$$.OOOOOO$$$$$$$$......
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$......OOOOOO.
$$$$$$$$$$$$$$$$......O.....O
$$$$$$$$$$$$$$$$......O......
$$$$$$$$$$$$$$$$.......O.....
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$.........OO$$$$$$$$..
$$$$$$$$........OO.OOO$$$$$$$.........
$$$$$$$$.........OOOOO$$$$$$$.........
$$$$$$$$$OOO$$$$$$$$
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$......OOOO$$$$$$$$$$$$...
$$$$......O...O$$$$$$$$$$$$..
$$$$......O$$$$$$$$$$$$......
$$$$.......O$$$$$$$$$$$$.....
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$.OOOOO$$$$$$$$$$$$$$$.......
$.O....O$.OO$$$$$$$$$$$$$$...
$.O.......OOOOO...OO.OOO$$$$$$$$$$$$$$
$..O......O....O...OOOOO$$$$$$$$$$$$$$
$.OOOO....O.........OOO$$$$$$$$$$$$$$.
........O.O.O.......O$$$$$$$$$$$$$$$..
..O.....OOO.O$$$$$$$$$$$$$$$$
OO......OO$$$$$$$$$$$$$$$$...
O..OOO$....O$$$$$$$$$$$$$$$..
O....OOO.........O.O.O.O.OO$$$$$$$$$$$$$$......
.OOO..OOO.......OOOO...OOOO$$$$$$$$$$$$$$......
........O..OOO.O$$$$$$$$$$$$$$$.......
.OOO..OOO.......OOOO...OOOO$$$$$$$$$$$$$$......
O....OOO.........O.O.O.O.OO$$$$$$$$$$$$$$......
O..OOO$....O$$$$$$$$$$$$$$$..
OO......OO$$$$$$$$$$$$$$$$...
..O.....OOO.O$$$$$$$$$$$$$$$$
........O.O.O.......O$$$$$$$$$$$$$$$..
$.OOOO....O.........OOO$$$$$$$$$$$$$$.
$..O......O....O...OOOOO$$$$$$$$$$$$$$
$.O.......OOOOO...OO.OOO$$$$$$$$$$$$$$
$.O....O$.OO$$$$$$$$$$$$$$...
$.OOOOO$$$$$$$$$$$$$$$.......
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$.......O$$$$$$$$$$$$.....
$$$$......O$$$$$$$$$$$$......
$$$$......O...O$$$$$$$$$$$$..
$$$$......OOOO$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$OOO$$$$$$$$
$$$$$$$$.........OOOOO$$$$$$$.........
$$$$$$$$........OO.OOO$$$$$$$.........
$$$$$$$$.........OO$$$$$$$$..
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$.......O.....
$$$$$$$$$$$$$$$$......O......
$$$$$$$$$$$$$$$$......O.....O
$$$$$$$$$$$$$$$$......OOOOOO.
$$$$$$$$$$$$$$$$$...
$$$$$$$$.OOOOOO$$$$$$$$......
$$$$$$$$.O.....O$$$$$$$$.....
$$$$$$$$.O$$$$$$$$$.
$$$$$$$$..O$$$$$$$$$
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$$$...
$$$$$$$$$$$$$$$.........O$...
$$$$$$$$$$$$$$$........O$....
$$$$$$$$$$$$$$$........O.....O........
$$$$$$$$$$$$$$$........OOOOOO.........
By adding one copy of the delaying reaction, the period is increased by 688.
Since 688 is a multiple of 8, and the GCD of 688 and 120 is 8, by adding
enough copies of this reaction (up to 14 of them) to the period 120 increasing
method above, spaceships of all sufficiently large periods which are 4 (mod 8)
can be built.
So overall, this means that spaceships can be built with any sufficiently
large period which is a multiple of 4. Improving the coverage of periods to
include nonmultiples of 4 will have to be based on truly new knowledge, since
such spaceships cannot use any of the standard spaceships as components.
The glider reactions shown above are certainly not the best possible ones for
building large period spaceships. There are probably much better reactions
available (smaller and faster), both for igniting the blinker puffer's debris
clouds, and for turning and shifting gliders. In addition, it should be
possible to ignite the blinker row using only one glider with help from
perturbations from an accompanying spaceship. This would suppress the need
for one side of the glider reactions to make the spaceships. Unlike most of
the results from these articles which require complicated search programs
to find, improvements on making these large period spaceships could be found
by anyone with access to a reasonable Life program.
I want to conclude this article with a few objects which are not exactly
spaceships, but which are not puffer trains either. They occupy a middle
ground between the two. Under a loose definition of spaceship which is
"an object which moves into new areas without leaving any permanent output",
these objects are spaceships. But they are strange ones, for they never
repeat themselves since the number of ON cells keeps increasing. But they
still have what is called a "period", which is the number of generations
before a similar-looking object reappears, except for another unit added in
the middle. These can be thought of as spaceships which "grow".
The first example of such a loosely defined spaceship is the following old
result. A puffer at the front produces a row of blinkers, which are all
consumed at the back by a smokeless reaction. But the eating reaction is
slower than the generating reaction, so that the number of blinkers grows.
This has a "period" of 156. After each period, the spaceship has one extra
blinker.
[A "blinker spaceship" which grows arbitrarily large]
...........O..O...............
..........O...................
..........O...O...............
..OO......OOOO................
.OOOO.........................
OO.OO.........................
.OO.....OO.OOO................
.......O.....OO.......O....OOO
......OO.......O......O....O.O
.......O.....OO.......O....OOO
.OO.....OO.OOO................
OO.OO.........................
.OOOO.........................
..OO......OOOO................
..........O...O...............
..........O...................
...........O..O...............
This second more recent example by Dean Hickerson has a "period" of 72.
Here two rows of blinkers are produced by a puffer. A glider at the back
travels between the two rows, consuming all of the blinkers. Once again,
the eating reaction is (much) slower than than the generating reaction,
so that the spaceship grows. After each "period" there are 14 new blinkers.
[A "blinker and glider spaceship" which grows arbitrarily large]
.O....................
O.....................
O....O................
OOOOO.................
......................
......................
......................
.OO...................
OO.OOO.O..............
.OOOOOOO..OOO.OOO.OOO.
..OOOOO...............
......................
.....OO.............O.
...O...............OO.
..O.........O......O.O
..O.....O..O..........
..OOOOOO...O.....O....
...........OOOOOO.....
......................
......................
......................
......................
..........OO..OO......
.........OO.OO.O......
..........OOOOO.......
...........OOO........
............O.........
..........O...O.......
.........O............
.........O....O.......
.........OOOOO........
The last example is a "PI ship", built by me. This is the first finite
pattern in which the PI-heptomino's almost-spaceship behavior has been tamed.
Here a puffer train produces two parallel rows of blocks. Between the rows
of blocks is generation 1 of the PI-heptomino. The PI-heptomino reacts with
each pair of blocks and deletes them, moving forwards with a period of 30
and a speed of 3c/10. Since this is slower than the rate of block creation
by the puffer, the spaceship slowly gets larger. This has a "period" of 180.
[A "PI and blocks spaceship" which grows arbitrarily large]
$$.....OOOO$$$$$$$$........
$$....OOOOOO$$$$$$$$.......
$$...OO.OOOO$.......OOOOOO$$$$$$....
$$....OO$$.O.....O$$$$$$...
$$$$.......O$$$$$$.........
$.....O$$$..O....O$$$$$$...
$...O...O$$$..OO$$$$$$.....
$..O$.........OO$$$$$$$$...
$..O....O........OO...OOO$$$$$$$$...
$..OOOOO........O..OOO...OO$$$$$$$$.
$$.....O.O.......OO.....OO$$$$$$$...
$$......O...O.....OO....O.O$$$$$$$..
OOOO$$.......O....O......O.O$$$$$$$.
O...O.........O$$O........O$$$$$$........OO..
O$.OO$$$$$$$$$........OO.OO
.O..O..OO.....OOO$$$$$$$$$......OOOO
......OOO......OOO$$$$$$$$$......OO.
.O..O..OO.....OOO$.O$$$$$$$$........
O$.OO$$$$$$$$$$...
O...O.........O$$$$$$$$$$..
OOOO$........O.O$$$.........OOO$$$$$
$$$......OO$$......O.O$$$$$
$$$.....OO.OOO$.........O...OO$$$$$.
$...OO$$.OOOOO$.......O...O$$$$$....
$..OO.OO$$OOO......O$O$$$$$.........
$...OOOO$$.......O...O........O....O$$$$$....
$....OO$$.......O$...OOOOO$$$$$.....
$$$$...O....O$$$$$$........
$$$$...OOOOO$$$$$$.........
$$$$$$$$$$$.......
$$$$$$$$$$$.......
$$$.OOOO$$$$$$$$..
$$$.O...O$O$$$$$$$
$$$.O.......OO....O.O$$$$$$.........
$$$..O..O..OOOO...O..OO........OO$......OO$$$.........
$$$.......OO...O.....O.OO......OO$......OO$$$.........
$$$..O..O..OOOO...O..OO$$$$$$.......
$$$.O.......OO....O.O$$$$$$.........
$$$.O...O$O$$$$$$$
$$$.OOOO$$$$$$$$..
$$$$$$$$$$$.......
$$$$$$$$$$$.......
$$$$....OOOOO$$$$$$........
$$$$....O....O$$$$$$.......
$$$$....O$$$.O$$$$
$$$$.....O...O$$.....OO$$$$
$$$$$$$....O$$$$..
$$$$.....O...O$$.....OO$$$$
$$$$....O$$$.O$$$$
$$$$....O....O$$$$$$.......
$$$$....OOOOO$$$$$$........
$$$$$$$$$$$.......
$$$$$$$$$$$.......
$$$.OOOO$$$$$$$$..
$$$.O...O$O$$$$$$$
$$$.O.......OO....O.O$$$$$$.........
$$$..O..O..OOOO...O..OO$$$$$$.......
$$$.......OO...O.....O.OO......OO$......OO$$$.........
$$$..O..O..OOOO...O..OO........OO$......OO$$$.........
$$$.O.......OO....O.O$$$$$$.........
$$$.O...O$O$$$$$$$
$$$.OOOO$$$$$$$$..
$$$$$$$$$$$.......
$$$$$$$$$$$.......
$$$$...OOOOO$$$$$$.........
$$$$...O....O$$$$$$........
$....OO$$.......O$...OOOOO$$$$$.....
$...OOOO$$.......O...O........O....O$$$$$....
$..OO.OO$$OOO......O$O$$$$$.........
$...OO$$.OOOOO$.......O...O$$$$$....
$$$.....OO.OOO$.........O...OO$$$$$.
$$$......OO$$......O.O$$$$$
OOOO$........O.O$$$.........OOO$$$$$
O...O.........O$$$$$$$$$$..
O$.OO$$$$$$$$$$...
.O..O..OO.....OOO$.O$$$$$$$$........
......OOO......OOO$$$$$$$$$......OO.
.O..O..OO.....OOO$$$$$$$$$......OOOO
O$.OO$$$$$$$$$........OO.OO
O...O.........O$$O........O$$$$$$........OO..
OOOO$$.......O....O......O.O$$$$$$$.
$$......O...O.....OO....O.O$$$$$$$..
$$.....O.O.......OO.....OO$$$$$$$...
$..OOOOO........O..OOO...OO$$$$$$$$.
$..O....O........OO...OOO$$$$$$$$...
$..O$.........OO$$$$$$$$...
$...O...O$$$..OO$$$$$$.....
$.....O$$$..O....O$$$$$$...
$$$$.......O$$$$$$.........
$$....OO$$.O.....O$$$$$$...
$$...OO.OOOO$.......OOOOOO$$$$$$....
$$....OOOOOO$$$$$$$$.......
$$.....OOOO$$$$$$$$........
This concludes my series of six articles on spaceships. I wish to thank
several people for their help with information for these articles, and for
reviewing them. These people include Richard Schroeppel, Bill Gosper,
Allan Wechsler, Hartmut Holzwart, and Nina Bell. I especially want to thank
Dean Hickerson for his huge help in supplying historical information and
results, reviewing these articles thoroughly, and for starting off the search
for the new spaceships.
As you have seen, Life certainly has more spaceship-related results than
anyone dreamed of in its early days. I hope you have found this series
of articles entertaining.
David I. Bell
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