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This is the first of a series of articles on recent results in Conway's Game of Life. There are many aspects of Life that are interesting and have recent developments, such as glider guns, spaceships, puffer trains, large period oscillators, and the construction of objects with desired properties.
I am restricting this set of articles to spaceships because of two reasons. Firstly, since the earliest days of Life around 1970, no truly new spaceships had been discovered until just about three years ago, and therefore this area is very new and exciting. And secondly, I have had a part in developing this area, and therefore know first hand about some of the results. But I will touch on several of the other areas as I go.
Almost all of the new spaceship results I will be describing were found by one of three people. Possibly others have independently discovered some of these things, and if so, we certainly should share credit with them. But until I know otherwise, the credit for the discoveries belongs to the people I mention. These people are Dean Hickerson, David Bell (myself), and Hartmut Holzwart (listed in order of involvement).
Most of the new spaceships were found by computer search. The size and complexity of most of the discoveries are such that they would not be expected to turn up by running random patterns, or by manually trying a set of possibilities. However, some of the "fine tuning" of the results are the work of human thinking and manipulating pieces of the found results.
Firstly, I should define some terminology. A spaceship is a finite pattern of live cells in Life that after a certain number of generations reappears, but translated in some direction by a nonzero distance. Translation is measured by the shortest king-wise connected path between two cells. So two cells adjacent to each other are one cell apart, whether or not the cells are adjacent orthogonally or diagonally. The process of translating after a certain number of generations is called moving or traveling.
The number of generations before the pattern reappears (but translated) is called the period of the spaceship. (This is analogous to the period of stationary oscillators.) Many spaceships appear to be the same pattern after a number of generations, but on closer examination are not really the same. Instead of being merely translated, they are also reflected (or flipped) as in a mirror. After twice that number of generations, the spaceship does reappear in its original form, with just a translation. The period always measures this full number of generations. Spaceships which show their mirror image after half their period are called glide- reflection spaceships.
Spaceships can travel orthogonally (straight left-right or up-down), or diagonally. All known diagonal spaceships travel the same distance left- right as they do up-down. There is no reason why a spaceship could not travel (for example) two units across and one unit up for its translation. However no example of such a spaceship is known. Theoretically, we could build ships with any translation by using a universal computer/constructor. But they would be very large and slow.
The translated distance divided by the period is the speed of the spaceship. Since the maximum speed of propagation of a signal in Life is one cell per generation, this speed is known as c (the speed of light). This speed is also the fastest possible growth of any finite object for a finite number of generations (think of a long line of ON cells). However, growth (or even just movement) of a finite object for an infinite time cannot occur at this speed. It was proven in the early days of Life that the maximum possible speed of any spaceship is c/2 for orthogonal spaceships, and c/4 for purely diagonal spaceships. More generally, if a spaceship moves A cells across and B cells up, then its period must be at least 2 * (A + B). This means (for example) that an orthogonal spaceship might move by one cell in two generations, or two cells in four generations, or three cells in six generations, and so on. But no spaceship (for example) can move by one cell in one generation, or two cells in three generations, or three cells in four generations.
Whenever I list spaceships or puffers, I will follow some standard rules. Since most known spaceships move orthogonally, that movement will not be explicitly mentioned. They will always be drawn so as to move from right to left. The few diagonal spaceships will be marked as such, and will move in the indicated direction.
To begin with and to be complete, I will list the original spaceships. These are the glider, the lightweight spaceship (LWSS), middleweight spaceship (MWSS), and the heavyweight spaceship (HWSS). These were all found very soon after Life was invented by Conway. They are:
Glider (diagonal to lower left, glide-reflective, period 4, speed c/4):
.O. O.. OOO
LWSS (glide-reflective, period 4, speed c/2):
.O..O O.... O...O OOOO.
MWSS (glide-reflective, period 4, speed c/2):
...O.. .O...O O..... O....O OOOOO.
HWSS (glide-reflective, period 4, speed c/2):
...OO.. .O....O O...... O.....O OOOOOO.
The glider is the most basic spaceship. Since it is so small, it appears spontaneously from random objects. Collisions between gliders produce many useful things. For example, two gliders can produce another glider, and three gliders can produce a LWSS or MWSS or a HWSS. Gliders can be produced indefinitely by many kinds of "glider guns." The first glider gun was found by Bill Gosper in 1970, and was the first example of a Life object whose population grew arbitrarily large.
The three orthogonal spaceships are very useful because of the presence of their "sparks." Sparks are cells at the edge of an object which die off, and which can perturb other objects without destroying the object which generates the spark. The commonest use of sparks from these spaceships is to perturb an "engine" to produce puffer trains, or to modify stationary objects when the spaceships pass by.
The three orthogonal spaceships above show an obvious pattern, and the casual Life-player might wonder why the pattern cannot be continued to make even larger spaceships. This will not work directly because the sparks do not die off anymore for larger ships, but wreck the ship.
The following for example, doesn't work because the spark at the top is actually a blinker, and doesn't die. Without the blinker, this object is known as an OWSS (overweight spaceship). This name is also given to even larger objects following the same pattern.
Non-working OWSS:
...OOO.. .O.....O O....... O......O OOOOOOO.
It was discovered rather early that an OWSS can survive if it is "escorted" by other smaller spaceships. The escorting spaceships are positioned to prevent the formation of the deadly non-sparks. The following is an extreme example of this. This shows the largest overweight spaceship which can be safely escorted by only two other spaceships.
OWSS escorted by two HWSS (period 4, speed c/2):
....OOOO....... ...OOOOOO...... ..OO.OOOO...... ...OO.......... ............... ...........OO.. .O............O O.............. O.............O OOOOOOOOOOOOOO. ............... ............... ....OOOO....... ...OOOOOO...... ..OO.OOOO...... ...OO..........
There are many other combinations of spaceships that will support one or more OWSS. Some OWSS can be supported by a convoy of two spaceships on each side. But for very long OWSS, no convoy of small spaceships can work to stabilize it. However, it is possible to support an arbitrarily long OWSS by nesting different sized OWSS side by side to stabilize them all, with final small spaceships on the outside.
The next topic related to spaceships is called a tagalong. A tagalong is something which follows behind one or more spaceships, and needs the spaceships in order to survive. Generally, tagalongs are attached to the sparks of a spaceship so that they don't destroy the spaceship. But some tagalongs attach right to the base ship itself, and some even need some modification of the base ships in order to work. A tagalong along with its base ship can be considered as a large spaceship.
One of the first tagalongs was found a couple of years into the history of Life. This is the Schick engine, found by Paul Schick in 1972. It uses the sparks from two adjacent LWSS to keep the engine going.
Schick engine tagalong (period 12, speed c/2):
OOOO..... O...O.... O........ .O..O..OO ......OOO .O..O..OO O........ O...O.... OOOO.....
This tagalong is very useful, because it splits into two separate parts. The back part dies off on its own. But it can be perturbed by adjacent LWSS, MWSS, or HWSS in many ways to produce output which remains. It then becomes a puffer train. For example, adding one LWSS gives:
Simple puffer train producing pairs of beehives (period 24, speed c/2):
...........OO. ..........OOOO .........OO.OO ..........OO.. OOOO.......... O...O......... O............. .O..O..OO..... ......OOO..... .O..O..OO..... O............. O...O......... OOOO..........
If a second LWSS is also placed at the bottom of the above object symmetrically to the top LWSS, then the puffer produces a stream of blinkers with period 12.
A tagalong can be changed into a puffer engine by perturbations as in the above example. The reverse can also happen. A puffer engine can be tamed and turned into a tagalong. This can generally be done by using passing spaceships to destroy all of the puffer output. Such an object will then be a large spaceship. I will give two examples.
The first example uses a well known puffer engine, the B heptomino. The first two puffers in Life were found by Bill Gosper and are both based on the B heptomino. The one given here is the second one he found, where a single engine is perturbed by two LWSS to become a very dirty puffer. It takes over 4600 generations for this puffer to settle down. It then produces a large collection of objects with period 140.
Dirty puffer train based on B heptomino (eventual period 140, speed c/2):
..... .OO.. OO.OO .OOOO ..OO. ..... ..... ....O ...OO ..OO. ...OO ..... ..... ..... ..... .OO.. OO.OO .OOOO ..OO.
By adding another LWSS, this dirty puffer is tamed and becomes a spaceship with period 20. This object has a rather large spark which can then be perturbed with other spaceships to produce simple useful outputs such as gliders.
Period 20 spaceship based on B heptomino (period 20, speed c/2):
..........O..O........ .OO......O............ OO.OO....O...O........ .OOOO....OOOO......... ..OO.................. ...................... ...........O.......... ....O...O....O........ ...OO....O...O...OO.OO ..OO.....O........O..O ...OO.............OOO. ............O..O...... .............OO....... ...................... ...................... .OO................... OO.OO................. .OOOO................. ..OO..................
The second example is based on a puffer discovered by Bob Wainwright in 1984. The period of this puffer is only eight, which is the minimum puffer period known. (There are several other different period 8 puffers known.) The one shown here produces a row of blinkers.
Period 8 puffer train producing blinkers (speed c/2):
...O..... .O...O... O........ O....O... OOOOO.... ......... ......... ......... .OO...... OO.OOO... .OOOO.... ..OO..... ......... .....OO.. ...O....O ..O...... ..O.....O ..OOOOOO.
By using a passing HWSS, the blinkers can be deleted, producing a true spaceship which can be made as large as desired by moving the trailing HWSS back.
Period 8 spaceship (speed c/2):
...O............. .O...O........... O................ O....O.....OO.... OOOOO.....OO.OOOO ...........OOOOOO ............OOOO. ................. .OO.............. OO.OOO........... .OOOO...OOO.OOO.. ..OO............. ................. .....OO.......... ...O....O........ ..O.............. ..O.....O........ ..OOOOOO.........
By using different puffers, and deleting the output in many many different ways, a very large number of spaceships can be produced. But they all have a few things in common. All of these orthogonal spaceships use the LWSS, MWSS, or HWSS as essential components. Therefore they must all have a speed of c/2, and a period which is a multiple of four. The question arises as to whether there are some spaceships which move at different speeds, have other periods, or don't make use of the normal spaceships. The answer is YES, and is why this area has been exciting in the last few years.
Before proceeding to these new things, I will recap two tantalizing early results which showed that possibly such ships might be found.
This first one was noticed by many people, and is the pi heptomino. The following is generation 1 of the pi heptomino. It reappears 30 generations later, with a translation of 9, for a very obscure speed of 3c/10.
Generation 1 of pi heptomino, a non-working spaceship:
..O .OO O.. .OO ..O
Unfortunately, it also produces a large amount of other debris which then quickly destroys the object. That debris can be controlled by carefully placed blocks in its path, or by placing many copies of the pi heptomino side by side. But all such constructions must be carried out to infinity to work forever. No one has figured out how to make a finite object based on this work.
The second tantalizing result was discovered by Charles Corderman in 1971 while doing an exhaustive search of polyominoes. A small object was discovered which translated itself diagonally by 8 cells, with some other debris remaining. The debris doesn't interfere immediately with the object, and so it translates itself again. Only after many such translations does the debris catch up to the engine and destroy it. Corderman found that by perturbing the debris in various ways, or by placing multiple copies of the engine alongside each other, the engine can survive forever. However, in none of these cases was it a spaceship. It was instead a puffer, and left debris behind. The following is one of the simplest of these puffers, and leaves just blocks behind.
Switch Engine puffer train (diagonal to upper left, glide-reflective, period 288, speed c/12):
.O.O........................ O........................... .O..O....................... ...OOO...................... ............................ ..........................OO ..........................O.
The above object is one of the smallest known objects whose population grows arbitrarily large. (All of these are based on the switch engine and have only 11 ON cells.) The only known diagonal puffer trains are all based on the switch engine. But no one succeeded in taming the debris completely to create a spaceship until Dean Hickerson did this in April, 1991. He did this by placing a large number of switch engines together so as to eliminate all debris. His original ship required 13 switch engines, but his smaller one given here only uses 10 of them. The number of them needed can possibly be reduced even further, but this isn't known.
The spaceship is just a little too wide to be seen in an 80 column width, so the description below is compressed in the following manner. Each dollar sign represents 10 empty cells. Just use an editor to replace each dollar sign by 10 periods, and you will be able to reconstruct the picture of the full object.
Cordership (diagonal to upper right, symmetrical, period 96, speed c/12):
$$$.....OO$$$$$ $$$.....O...OO$$$$...... $$$....O......O.......OOO$$$..... $$$..OOOOO...O.....OOO$$$........ $$$.....O...O......OOO$$$........ $$$..O..O$.O$$$......... $$....OO.......OO$$$$$.. $$....OO$$$$$$. $$$$$$$$....... $$$$$........O$$........ $$$$$........O$$........ $$$$$........O$$........ $$$$$......OO$$......... $$$$$.....OOO$$......... $......OO$$$........OO$$......... $......OO$$$$$$......... $$$$$$$$....... $$$$$$$$....... $$$$$$$$....... $$$$$........O$$........ $$$$$.......O$$......... $$$$$........O$$........ ........OO$$$$$$$....... ........OO$$$......O$$$$ $$$$.....O.O$$$......... $$$$....OO.OO$$$........ $$$$.......OO$$$........ $$$$.OO$$$$.... $$$$...O..OOO$......O.O.......OOO......... $$$$.OO.O$$.O.....OOO$.. OO$$$$.O..O$$.....OOO$.. OO...O.......O$$.........O$$.........O$... ....O.O.....OOO$$.........OO$$$$. ...O..O..OOOO.OO$$$$$$$. .........O..OOO$$$O$$$$. ......O..OO..O$$.........OO.OO$$$......O.. ....O.O.OO$$$....O$$$.........O.. .....O$$$$.O$$$......O.. $$$$.....O$..O.OO$$OO... $$$$.........O.O......O.OOO$........OOO... $$$$$.O....O.O....O$........OO... $$$$........O...O.O......OO$$.... $$$$$OO..O..O...O$$..... $$$$$.O...OO.O$$........ $$$$$.......O.O$$....... $$.......O$$.........O.O$$....O.. $$......OOO$$$$$....O.O. $$.....OO.OO$$$$$..O..O. $$......OOO$$$$$........ $$.......O$$$$$......... $$.....O.O$$$$$.....O..O $$....OOOO$$$$$...OOO.OO $$....O$$$$$.....O..OO.. $$$$$$$$O..O... $$....OO.OO$$$$$..O.O... $$...O.....O$$$$$....... $$....O...O$$$$$........ $$.......O...O.......O$$$$....... $$$O.O.....OOO$$$$...... $$.........O..O..OOOO.OO$$$$..... $$$.....O..OOO$$$$...... $$$..O..OO..O$$$.........OO...... $$$O.O.OO$$$$...OO...... $$$.O$$$$$..... $$$$$$$$....... $$$$$$$$....... $$$$$$$$....... $$$$$$$$....... $$$$$$$$....... $$$$$$$.OO$.... $$$$$$$.OO$.... $$$$$...O$$$... $$$$$..OOO$$$.. $$$$$.OO.OO$$$. $$$$$..OOO$$$.. $$$$$...O$$$... $$$$$.O.O$$$... $$$$$OOOO.........OO$$.. $$$$$O$..OO$$.. $$$$$$$$....... $$$$$OO.OO$$$.. $$$$.........O.....O$$$. $$$$$O...O$$$.. $$$$$...O$$$... $$$$$$$$....... $$$$$.....OO$$$ $$$$$.....OO$$$
The Cordership has lots of debris that dies away. Gliders can hit this debris and do interesting things. In particular, gliders can be turned by 90 or 180 degrees by hitting the debris appropriately. Dean Hickerson has used this fact to create some interesting constructions, such as reflecting gliders back and forth forever between two Corderships that are slowly pulling apart.
The Corderships form one of several new classes of spaceships, ending a period of about 20 years when no new classes were found. However, it was not the first of these new classes. The next part of this series will begin exploring these new classes in earnest, starting with the c/2 period 2 spaceships.
This is the second in a series of articles concerning Conway's Game of Life. In this article, I will describe the early history of the discovery of the new classes of spaceships, and then survey the results for all the known period 2 spaceships. All period 2 spaceships must be orthogonal, and must travel at the speed of c/2. This follows from the speed restrictions mentioned in my previous article.
In July of 1989 Dean Hickerson started writing a Life search program for his Apple IIe. It was written in 6502 assembly language and Applesoft BASIC. Dean's Life search program looks for patterns that repeat themselves after a small number of generations, with or without translations. (Translations are used when you want to find spaceships instead of oscillators.)
During a search, the program recursively attempts to set cells ON or OFF, and for each cell that is set uses transition and implication rules in order to detect contradictions in the current state of cells. This allows the program to quickly stop and backtrack over impossible situations, and thus reduce enormously the size of the search. This program was described in the xlife 3.0 distribution.
Search programs typically attempt to set the cells in a column-by-column order. Since a cell's fate in N generations is usually dependent on the state of all cells within a distance of N, contradictions cannot usually be detected until the state of the next N columns has been specified. This means that looking for oscillators or spaceships is much harder for larger periods. The current useful limit on search programs of this type is for period 5. Some searches in small areas have been done for periods 6 and 7.
Another effect of the order of setting cells column-by-column is that objects which have many rows are harder to find than those with fewer rows. (The number of columns can be very large and has little effect on the search.) The current search programs can usefully look for objects which have about 10 rows (20 rows if symmetry is used), but this limit can be raised somewhat for low-period searches, such as up to 14 rows for period 2.
Since the useful number of rows being searched is limited, many of the spaceships which are found are of one of two types. They are either wide and short or are thin and long. These terms refer to the dimensions of the spaceship with respect to the direction that it travels. Therefore a thin and long spaceship looks like an "arrow," whereas a wide and short spaceship looks like a "wave." However, for many smaller space ships where both dimensions fit within the row limit and are of comparable size, they can look like "blobs."
Using his program, and looking for short, wide spaceships, Dean Hickerson found the first period 2 spaceships on July 28, 1989. These were the first examples of a new class of spaceship. He does not remember which specific spaceship was found first, but the following ship was among the first ones found, and is the smallest known period 2 spaceship.
Smallest known period 2 spaceship (speed c/2):
.....O.O ....O..O ...OO... ..O..... .OOOO... O....O.. O..O.... O..O.... .O...... ..OOOO.O ...O...O ....O... ....O.O. ........ ...OOO.. ...OO... ...OOO.. ........ ....O.O. ....O... ...O...O ..OOOO.O .O...... O..O.... O..O.... O....O.. .OOOO... ..O..... ...OO... ....O..O .....O.O
Within a few hours of finding the first period 2 ship, Dean had discovered a grammar for constructing an infinite number of different short, wide, period 2 spaceships. A grammar is an "alphabet" of "components," along with rules for the possible sequences of connections between components. Components are simply the identifiable pieces of a ship which reappear over and over in different ships in different combinations. There are three components in the above spaceship, as will be seen below.
The complete grammar describes the components, the allowed sequences of the components, and the manner in which the components are joined together. The following are the components of Dean's first grammar.
[A] [A'] [B] [B'] [C] [C']
.....O.O .......O X X X X
....O..O ....OOOO ... .O.. ..... .O...
...OO... ....OO.. OOO .O.O OOO.. .O.O.
..O..... ..O..... OO. O... OO... O....
.OOOO... ..OOOO.. OOO .O.O OO... O...O
O....O.. .O...... ... .O.. OOOOO .O..O
O..O.... OOO..O.. X X ..O.. .O.O.
O..O.... .OOO.... X X
.O...... ..O.....
..OOOO.O ...OOO.O
...O...O ...O..OO
....O... ....OOO.
....O.O. ......O.
X X
[D] [D'] [E] [E'] [F] [F']
X X X X X X
... .O.. ...O.O ....O ...O .O.O
OOO .O.O ..O..O ..OOO .OOO O..O
OO. O... ..O... .OOO. OOO. O...
OO. O... .O..O. O..O. .OOO O..O
OOO .O.O OOO... O.... ...O .O.O
... .O.. .OOO.. O..O. X X
X X ...O.. .O.O.
X X
[G] [G']
X X
.O.O.. ...O...
O..O.. .OOO...
O..... OOO....
O..... .OO....
.O..OO ..OO..O
..OOOO ...OO.O
...... .......
..OOOO ...OO.O
.O..OO ..OO..O
O..... .OO....
O..... OOO....
O..O.. .OOO...
.O.O.. ...O...
X X
The components occur in pairs identified by the same letter, but with or without a quote mark (e.g., A and A'). Such pairs are related in that they are the two phases of the same section of a period 2 spaceship. So that, for example, if a period 2 spaceship contains component B in generation 0, then in generation 1 it must contain component B' in the same position within the ship.
Besides the listed components, there are other components which are their mirror images. The mirroring is done by flipping the component across a horizontal line. The mirrored component names are the same as for the original components, except that they end with a trailing dash. So that, for example, the mirror image of component A' is A'-. Mirror images for symmetrical components such as B would be duplicates and so are not used.
The components that make up a spaceship are strung together like beads on a string, stacked one above the other. A sequence of components specifies the order of components, arranged from top to bottom. When doing this, the components must be correctly aligned horizontally. The 'X' characters in the component diagrams specify the proper alignment. When two components are stacked together, they must be placed adjacent to each other so that the 'X' characters are in the same column. The rows containing the 'X' characters are not part of the components, and should be removed.
The following shows an example of stacking two components correctly.
Section B F' of a period 2 spaceship:
X .... OOO. OO.. OOO. .... .O.O O..O O... O..O .O.O X
The final part of the grammar specifies the allowed sequences of components. Only certain sequences of components can be used to make a valid spaceship. The following rules specify these allowed sequences.
The sequence must begin with A or A'.
The sequence must end with A- or A'-.
Each pair of adjacent symbols must appear in one line of the following table, with the first symbol found before the vertical bar, and the second symbol found after the vertical bar.
A C' E- E' F' G | B C D
A' C E E'- F G' | B' C' D'
B C- D | A- C'- E E'- F' G
B' C'- D' | A'- C- E- E' F G'
The simplest example of a sequence which follows these rules is A B A-, which represents the c/2 period 2 ship given above. Another example of a spaceship created using these rules is A D E B' A'-, which represents the following spaceship.
One of many period 2 spaceships constructed by the above grammar (speed c/ 2):
.......O.O ......O..O .....OO... ....O..... ...OOOO... ..O....O.. ..O..O.... ..O..O.... ...O...... ....OOOO.O .....O...O ......O... ......O.O. .......... .....OOO.. .....OO... .....OO... .....OOO.. .......... ......O.O. .....O..O. .....O.... ....O..O.. ...OOO.... ....OOO... ......O... ....O..... ....O.O... ...O...... ....O.O... ....O..... ......O... ....OOO... ...O..OO.. ...OOO.O.. ..O....... .OOO...... OOO..O.... .O........ ..OOOO.... ..O....... ....OO.... ....OOOO.. .......O..
Small tagalongs were soon found for several of these components. (These tagalongs can also be attached to many of the other period 2 spaceships.) The one for component C was found by Robert Wainwright, and the one for component A was found by Bill Gosper.
C component with tagalong (left), A component with tagalong (right):
X ........O.
......... .....O.O.O
OOO...... ....O..O..
OO....... ...OO.....
OO....O.. ..O.......
OOOOO.O.O .OOOO.....
.O.....O. O....O....
X O..O......
O..O......
.O........
..OOOO.O..
...O...O..
....O.....
....O.O...
X
Dean Hickerson also looked for thin, long period 2 ships in those first weeks. These spaceships are much harder to find, and so he only found one basic ship before moving on to search for other things. This spaceship is shown below.
First long period 2 spaceship (speed c/2):
.............O....... ...........OO........ ........OOOO.O....... ........OO.......OO.. ......O...OO.O...OOOO ......OOOO.O.O.O....O ...O.O......OO....... ..OOOOOO.OO.OO.OO.... .OO.....OO...O....... OO....OO....O........ .OO....OOOO.......... ..................... .OO....OOOO.......... OO....OO....O........ .OO.....OO...O....... ..OOOOOO.OO.OO.OO.... ...O.O......OO....... ......OOOO.O.O.O....O ......O...OO.O...OOOO ........OO.......OO.. ........OOOO.O....... ...........OO........ .............O.......
At the end of September, 1989, Dean discovered an extensible tagalong for the above ship that is now named a wicktrailer. This tagalong allows the ship to be made as long as desired. Such a tagalong can be described as having a period, which is the number of generations it takes before a unit of the tagalong reappears in the same place.
Period 2 ship with wicktrailer (period 20 extensible tagalong) (speed c/2):
............................O.........O.........O.. .............O...........OOOO......OOOO......OOOO.. ...........OO..........OOO.O.....OOO.O.....OOO.O... ........OOOO.O........OO...O....OO...O....OO...O... ........OO.......OO...O....OO...O....OO...O....OO.. ......O...OO.O...OOOO.O.....OOO.O.....OOO.O.....OOO ......OOOO.O.O.O....OOOO......OOOO......OOOO......O ...O.O......OO.........O.........O.........O....... ..OOOOOO.OO.OO.OO.................................. .OO.....OO...O..................................... OO....OO....O...................................... .OO....OOOO........................................ ................................................... .OO....OOOO........................................ OO....OO....O...................................... .OO.....OO...O..................................... ..OOOOOO.OO.OO.OO.................................. ...O.O......OO..................................... ......OOOO.O.O.O....O.............................. ......O...OO.O...OOOO.............................. ........OO.......OO................................ ........OOOO.O..................................... ...........OO...................................... .............O.....................................
The tagalong can be attached to the other "foot" of this ship in a similar manner to produce a symmetrical ship of any length, or with two unequal length wicks. The tagalong can also be attached to the "foot" of the other period 2 ships on component A'. Doing this allows the construction of a ship which is as wide and as long as desired.
There is a hidden motivation in looking for new ships and their tagalongs. Besides the elegance of new ships, Dean was hunting for new puffer trains. Puffer trains allow for the construction of interesting large Life objects, including devices which perform logic operations or numeric calculations. There are only a small number of basic types of puffer trains, and new methods for making puffer trains would be useful. For every new spaceship which is found, there is the possibility that it contains a new set of "sparks" that has not been seen before. Such a new set of sparks might allow a new type of puffer engine to work, or might be useful to catalyze a reaction in a new manner. So far this has not been successful.
Another way that a new puffer train might be constructed is by perturbing an extensible tagalong such as the one in the above ship. If the perturbation results in a reaction that travels at a speed less than or equal to the ship (c/2 in this case), and that reaction leaves debris behind, then a puffer train results. Unfortunately, this process has so far not worked with any extensible tagalong. The reaction either breaks off, or travels faster than the ship and ends up destroying the ship. For example, deleting one cell from the end of the wicktrailer, or adding one cell to it almost always results in one of either two reactions which travel at 6c/7 or 11c/12 and catch up to the ship and destroy it.
After Dean found the above results, he went looking for other things, and so the class of period 2 spaceships wasn't further developed for a while. Meanwhile, I had heard of Dean's search program, read his notes about it, and wrote my own program in C, and started making my own discoveries. I sent a copy of the program to Hartmut Holzwart. He made some modifications to it (such as speedups, new symmetries, and different search orders), and then also started making many new discoveries. But it wasn't until June of this year that we started looking for period 2 spaceships.
On June 7, 1992, I was experimenting with a new search feature, and found three new components for period 2 ships. The most interesting component is a repeatable diagonal component which bends the ship backwards with a slope of 3/6. The other two larger components just provide a connection to an already known ship component, and terminate the end of the new component. The following shows a ship using all three components, with the diagonal one repeated twice.
Period 2 spaceship with repeatable "barber pole" component (speed c/2):
........................O ......................OOO .....................OO.. .....................O... ...................O.O... .................OOO..... ................OO..O.... ................O.O...... ................OOO...... ................O.OOO.... ...............OO........ ..............OO.O....... .................O.O..... .............O........... .............OO.O........ .............O.O......... ............OO........... ...........OO.O.......... ..............O.......... ..........O.............. ..........OO.O........... ..........O.O............ .........OO.............. ........OO.O............. ...........O............. .......O................. .......OO.O.............. .......O.O............... ......OO................. ...OO.OOO................ ...O....O................ ...O..................... ....O..O................. .....OO.................. ......OO................. ...OOO.O................. ..O...OO.O.O............. .O...O..O..O............. .O..O.O.O................ .O........O.............. ..O..OO...O.............. ...OO.OOOOOO............. ....OO...O.O............. ....O.O.................. ...O..................... ....O..O................. ....O.O.................. ......................... ...OOO................... ...OO.................... ...OOO................... ......................... ....O.O.................. ....O.................... ...O...O................. ..OOOO.O................. .O....................... O..O..................... O..O..................... O....O................... .OOOO.................... ..O...................... ...OO.................... ....O..O................. .....O.O.................
The diagonal component can obviously be repeated arbitrarily often to make an "arm" as long as desired. The same arm can also be constructed on the other side of the ship to make it symmetrical, creating a bow shaped ship.
One interesting thing about a long string of these diagonal components is that both phases appear identical to each other, but shifted. This makes the arm act somewhat like a "barber pole" with a pattern appearing to move up (or down) the arm as the ship moves.
A few days later I found a second repeatable diagonal component. This component is actually a tagalong since it just attaches to a "foot" of an intact period 2 ship. The slope of this tagalong is 8/13. This tagalong is called the "glancing head" because it resembles a head with two eyes looking sideways. The following shows two copies of this tagalong, one attached to the base ship and the second to the first tagalong. Again, this arm can be made as long as desired.
Period 2 spaceship with repeatable "glancing head" tagalong (speed c/2):
.......................O .....................OOO ....................OO.. ....................O... ..................O.O... ................OOO..... ...............OO..O.... ..............O..O...... ..............O..O...... ..............O......... ...............O..O..... ................OO...... .................OO..... ...............O.O...... .............OOO.O...... ............OO.......... ............O........... ..........O.O........... ........OOO............. .......OO..O............ ......O..O.............. ......O..O.............. ......O................. .......O..O............. ........OO.............. .........OO............. .......O.O.............. ....OOOO.O.............. ....OO.................. ..O..................... ..OOOO.................. .O...................... OOO..O.................. .OOO.................... ..O..................... ...OOO.O................ ...O..OO................ ....OOO................. ......O................. ....O................... ....O.O................. ...O.................... ....O.O................. ....O................... ......O................. ....OOO................. ...O..OO................ ...OOO.O................ ..O..................... .OOO.................... OOO..O.................. .O...................... ..OOOO.................. ..O..................... ....OO.................. ....OOOO................ .......O................
By using a piece of Dean Hickerson's wicktrailer, one foot can be turned into two feet. This allows the construction of a "binary tree" spaceship, which branches an arbitrary number of times. Here the wicktrailer tagalong can be attached to any foot, and two copies of the glancing head tagalong can be attached to the two feet of the wicktrailer. The following shows a simple example of connecting these tagalongs together to make a branching spaceship.
Period 2 binary tree spaceship (speed c/2):
.......................................O.. .....................................OOO.. ....................................OO.... ....................................O..... ..................................O.O..... ................................OOO....... ...............................OO..O...... ..............................O..O........ ..............................O..O........ ..............................O........... ...............................O..O....... ................................OO........ .................................OO....... ...............................O.O........ ............................OOOO.O........ .......O..................OOO.O........... ....OOOO.................OO...O........... ....OO...................O....OO.......... ..O....................O.O.....OOO.O...... ..OOOO...............OOO.........O.O...... .O..................OO..O..........OO..... OOO..O.............O..O...........OO...... .OOO...............O..O..........O..O..... ..O................O............O......... ...OOO.O............O..O........O..O...... ...O..OO.............OO.........O..O...... ....OOO...............OO.........OO..O.... ......O.............O.O...........OOO..... ....O.............OOO.O.............O.O... ....O.O..........OO...................O... ...O.............O....................OO.. ....O.O........O.O.....................OOO ....O........OOO.........................O ......O.....OO..O......................... ....OOO....O..O........................... ...O..OO...O..O........................... ...OOO.O...O.............................. ..O.........O..O.......................... .OOO.........OO........................... OOO..O........OO.......................... .O..........O.O........................... ..OOOO....OOO.O................O.......... ..O......OO..................OOO.......... ....OO...O..................OO............ ....OOOO.O..................O............. .......OOOO.O.............O.O............. ..........O.O...........OOO............... ............OO.........OO..O.............. ...........OO.........O..O................ ..........O..O........O..O................ .........O............O................... .........O..O..........O..O............... .........O..O...........OO................ ..........OO..O..........OO............... ...........OOO.........O.O................ .............O.O.....OOO.O................ ...............O....OO.................... ...............OO...O..................... ................OOO.O..................... ..................OOOO.O.................. .....................O.O.................. .......................OO................. ......................OO.................. .....................O..O................. ....................O..................... ....................O..O.................. ....................O..O.................. .....................OO..O................ ......................OOO................. ........................O.O............... ..........................O............... ..........................OO.............. ...........................OOO............ .............................O............
Since each branch can be extended to be as long as necessary by using more glancing head tagalongs, enough room can be created to build a fully populated binary tree ship, containing 2^N nodes.
The wicktrailer is even more versatile than is shown above. A longer section of the wicktrailer has even more feet. There is room for the glancing head tagalong to be attached to each one of those feet, and for more glancing head tagalongs to be attached to those tagalongs. Counting each section of wicktrailer as a node, this means that not only can full binary trees be constructed, but full N-ary trees can also be constructed, for any N. The following illustrates the construction details that allow such a spaceship to be made.
Period 2 N-ary tree spaceship details (speed c/2):
.............................O.........O.........O.......... ...........................OOO.......OOO.......OOO.......... ..........................OO........OO........OO............ ..........................O.........O.........O............. ........................O.O.......O.O.......O.O............. ......................OOO.......OOO.......OOO............... .....................OO..O.....OO..O.....OO..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............... .....................O.O.......O.O.......O.O................ ..................OOOO.O....OOOO.O....OOOO.O....O........... ................OOO.O.....OOO.O.....OOO.O.....OOO........... ...............OO...O....OO...O....OO...O....OO............. ...............O....OO...O....OO...O....OO...O.............. .............O.O.....OOO.O.....OOO.O.....OOO.O.............. ...........OOO.........OOOO.O....OOOO.O....OOOO.O........... ..........OO..O...........O.O.......O.O.......O.O........... .........O..O...............OO........OO........OO.......... .........O..O..............OO........OO........OO........... .........O................O..O......O..O......O..O.......... ..........O..O...........O.........O.........O.............. ...........OO............O..O......O..O......O..O........... ............OO...........O..O......O..O......O..O........... ..........O.O.............OO..O.....OO..O.....OO..O......... .......OOOO.O..............OOO.......OOO.......OOO.......... ....OOOO.O...................O.O.......O.O.......O.O........ ....OO...O.....................O.........O.........O........ ..O......OO....................OO........OO........OO....... ..OOOO....OOO.O.................OOO.......OOO.......OOO..... .O..........O.O...................O.........O.........O..... OOO..O........OO............................................ .OOO.........OO............................................. ..O.........O..O............................................ ...OOO.O...O................................................ ...O..OO...O..O............................................. ....OOO....O..O............................................. ......O.....OO..O...........O.........O.........O........... ....O........OOO.........OOOO......OOOO......OOOO........... ....O.O........O.O.....OOO.O.....OOO.O.....OOO.O............ ...O.............O....OO...O....OO...O....OO...O............ ....O.O..........OO...O....OO...O....OO...O....OO........... ....O.............OOO.O.....OOO.O.....OOO.O.....OOO......... ......O.............OOOO.O....OOOO.O....OOOO.O....O......... ....OOO................O.O.......O.O.......O.O.............. ...O..OO.................OO........OO........OO............. ...OOO.O................OO........OO........OO.............. ..O....................O..O......O..O......O..O............. .OOO..................O.........O.........O................. OOO..O................O..O......O..O......O..O.............. .O....................O..O......O..O......O..O.............. ..OOOO.................OO..O.....OO..O.....OO..O............ ..O.....................OOO.......OOO.......OOO............. ....OO....................O.O.......O.O.......O.O........... ....OOOO....................O.........O.........O........... .......O....................OO........OO........OO.......... .............................OOO.O.....OOO.O.....OOO.O...... ...............................O.O.......O.O.......O.O...... .................................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...... ...............................OO..O.....OO..O.....OO..O.... ................................OOO.......OOO.......OOO..... ..................................O.O.......O.O.......O.O... ....................................O.........O.........O... ....................................OO........OO........OO.. .....................................OOO.......OOO.......OOO .......................................O.........O.........O
A side effect of this construction is that for any sized square, no matter how large, there exists a period 2 spaceship whose cells occupy at least some constant percentage of the area of the square (around 10%).
Later, another repeating diagonal tagalong was discovered, which also allows N-ary tree ships to be built. Here the slope is 4/17, which is shallower than the previous tagalong. This tagalong has the name "staring head," and looks similar to the main component of the base ship. The following illustrates this tagalong and how it may be connected.
Period 2 spaceship demonstrating "staring head" tagalong (speed c/2):
..........O.........O............. .......OOOO......OOOO......O...... ....OOOO.O.....OOO.O.....OOO...... ....OO...O....OO...O....OO........ ..O......OO...O....OO...O......... ..OOOO....OOO.O.....OOO.O......... .O..........OOOO.O....OOOO.O...... OOO..O.........O.O.......O.O...... .OOO.............OO........OO..... ..O.............OO........OO...... ...OOO.O.......O..O......O..O..... ...O..OO......O.........O......... ....OOO.......O.........O......... ......O.......OOO.......OOO....... ....O..........O..O......O..O..... ....O.O........OOO.......OOO...... ...O..........O.........O......... ....O.O......OOO.......OOO........ ....O.......OOO..O....OOO..O...... ......O......O.........O.......... ....OOO.......OOOO......OOOO...... ...O..OO......O.........O......... ...OOO.O........OO........OO...... ..O.............OOOO.O....OOOO.O.. .OOO...............O.O.......O.O.. OOO..O...............OO........OO. .O..................OO........OO.. ..OOOO.............O..O......O..O. ..O...............O.........O..... ....OO............O.........O..... ....OOOO.O........OOO.......OOO... .......O.O.........O..O......O..O. .........OO........OOO.......OOO.. ........OO........O.........O..... .......O..O......OOO.......OOO.... ......O.........OOO..O....OOO..O.. ......O..........O.........O...... ......OOO.........OOOO......OOOO.. .......O..O.......O.........O..... .......OOO..........OO........OO.. ......O.............OOOO......OOOO .....OOO...............O.........O ....OOO..O........................ .....O............................ ......OOOO........................ ......O........................... ........OO........................ ........OOOO...................... ...........O......................
Using his modified version of my program, Hartmut Holzwart recently did a search for large long period 2 spaceships. He found several dozen such ships with many possible variations. In this article I will give only a representative sample of these spaceships.
Following are three variations on relatively small period 2 ships. These are related to the first long spaceship found by Dean Hickerson.
Several similar small period 2 spaceships (speed c/2):
............ ............. ............O ............ ............. .........OOOO .........O.O .........O.O. .........OO.. ........O..O ........O..O. .......O...OO .......OO... .......OO.... .......OOOO.O ......O..... ......O...... ......O...... .....OOOOOO. .....OOOOOO.O .....OOOOOO.O ..OO.......O ..OO.......OO ..OO.......OO .O...OOO.O.. .O...OOO.OO.. .O...OOO.OO.. O...O....O.. O...O....OO.. O...O....OO.. O.....O....O O.....O...OO. O.....O...OO. OOO...OOOOO. OOO...OOOO.O. OOO...OOOO.O. ............ ............. ............. OOO...OOOOO. OOO...OOOO.O. OOO...OOOO.O. O.....O....O O.....O...OO. O.....O...OO. O...O....O.. O...O....OO.. O...O....OO.. .O...OOO.O.. .O...OOO.OO.. .O...OOO.OO.. ..OO.......O ..OO.......OO ..OO.......OO .....OOOOOO. .....OOOOOO.O .....OOOOOO.O ......O..... ......O...... ......O...... .......OO... .......OO.... .......OOOO.O ........O..O ........O..O. .......O...OO .........O.O .........O.O. .........OO.. ............ ............. .........OOOO ............ ............. ............O
A section of the wicktrailer can be attached to the two center feet of the middle ship above, creating feet which are separated by a one cell gap. Hartmut found that a repeatable tagalong can be attached to that pair of feet. This tagalong is unusual in that unlike most other tagalongs which work by having cells turned on by the base ship, this tagalong works by having some cells turned off by the base ship. The following shows the base ship, the wicktrailer, and two copies of the repeatable tagalong.
Period 2 spaceship with repeatable tagalong (speed c/2):
............................O.........O .........O.O..............OOO.......OOO ........O..O.............OO........OO.. .......OO................O.........O... ......O........O.......O.O.......O.O... .....OOOOOO.OOOO.....OOO.......OOO..... ..OO.......OO.O.....OO.O......OO.O..... .O...OOO.OO...O.....O.O.OO....O.O.OO... O...O....OO...OO....OO..OO....OO..OO... O.....O...OO...OOO..O.....OO..O.....OO. OOO...OOOO.O.....O..OO..OO.O..OO..OO.O. ....................OO.O......OO.O..... OOO...OOOO.O.....O..OO..OO.O..OO..OO.O. O.....O...OO...OOO..O.....OO..O.....OO. O...O....OO...OO....OO..OO....OO..OO... .O...OOO.OO...O.....O.O.OO....O.O.OO... ..OO.......OO.O.....OO.O......OO.O..... .....OOOOOO.OOOO.....OOO.......OOO..... ......O........O.......O.O.......O.O... .......OO................O.........O... ........O..O.............OO........OO.. .........O.O..............OOO.......OOO ............................O.........O
By using two feet at the ends of two separate spaceships, this tagalong can be used to connect the two ships into one larger ship. Since the tagalong has feet of its own, two copies of it can be used to attach another copy of the tagalong to them too.
The following spaceships are related, and are both extensible. Here extensible means that they contain recognizable components which can be repeated to make the ships wider. (This will be explained in more detail a little later.) The first spaceship here is important because it demonstrates a new end component for grammar-based ships.
Two similar extensible period 2 spaceships (speed c/2):
............O.O ............O.O.. ...........O..O ...........O..O.. ..........OO... ..........OO..... .........O....O .........O....... ........OOOOO.O ........OOOOOO... .....OO........ .....OO.......O.. ....O...OOO.... ....O...OOO.O.... ...O...O....... ...O...O....O.... ...O.....O..... ...O.....O....O.. ...OOO...O..... ...OOO...OOOOO... .........OO.O.. ................. ...OOO......O.. ...OOO.....OOOOO. ..O............ ..O........O....O .O...O......... .O...O.....O..O.. O..O........... O..O........O...O O....O......... O....O.......O... OOOOO.......... OOOOO.........OOO ............... ................. OOOOO.......... OOOOO.........OOO O....O......... O....O.......O... O..O........... O..O........O...O .O...O......... .O...O.....O..O.. ..O............ ..O........O....O ...OOO......O.. ...OOO.....OOOOO. .........OO.O.. ................. ...OOO...O..... ...OOO...OOOOO... ...O.....O..... ...O.....O....O.. ...O...O....... ...O...O....O.... ....O...OOO.... ....O...OOO.O.... .....OO........ .....OO.......O.. ........OOOOO.O ........OOOOOO... .........O....O .........O....... ..........OO... ..........OO..... ...........O..O ...........O..O.. ............O.O ............O.O..
The following shows a tagalong for a slightly modified base ship. Here the tagalong works in the normal manner, by having new cells created by the base ship. This tagalong is not repeatable, however.
Period 2 spaceship with tagalong (speed c/2):
............O.............. .........O.O.O............. ........O..O............... .......OO.....O...........O ......O.......OOO.......OOO .....OOOOOO....OOO.....OO.. ..OO.......O....O......O... .O...OOO.O.......OOO.O.O... O...O....O.......O..OO..... O.....O....O......OO...O... OOO...OOOOO........OOOO.... ........................... OOO...OOOOO........OOOO.... O.....O....O......OO...O... O...O....O.......O..OO..... .O...OOO.O.......OOO.O.O... ..OO.......O....O......O... .....OOOOOO....OOO.....OO.. ......O.......OOO.......OOO .......OO.....O...........O ........O..O............... .........O.O.O............. ............O..............
The following spaceship is one of the more complicated ships found by Hartmut Holzwart. It has a curious pattern of several long rows of ON cells within it. Possibly this hints at a pattern that could reappear at a larger scale in even larger ships. This spaceship is also extensible.
One of the more complicated period 2 spaceships (c/2):
..............O.O......... ...........O.O..O......... ..........O..O...........O .........OO....O...O.O.OO. ........O....O...OOO.O.O.O .......OOOOO.O...O..O..... ....OO...........OOO.OO... ...O...OOOOOOOO...O....O.. ..O...O............OOO.... ..O.....OOOOOOOOO..O...O.. ..OOO...O............OOO.. ..........OOO.O.O....OOOO. ..OOO.....O..OO.O.......O. .O.....O...OO............. O...OO.O....OOOOO......... O...O..................... O...OO.O....OOOOO......... .O.....O...OO............. ..OOO.....O..OO.O.......O. ..........OOO.O.O....OOOO. ..OOO...O............OOO.. ..O.....OOOOOOOOO..O...O.. ..O...O............OOO.... ...O...OOOOOOOO...O....O.. ....OO...........OOO.OO... .......OOOOO.O...O..O..... ........O....O...OOO.O.O.O .........OO....O...O.O.OO. ..........O..O...........O ...........O.O..O......... ..............O.O.........
Not all of the spaceships that Hartmut found are extensible. (I have mostly selected those that were.) The following is one of those which is not known to be extensible.
Another complicated period 2 (non-extensible) spaceship (speed c/2):
...........O.O...........O.O ..........O..O.........OOOOO .........OO...........O...O. ........O...O........OO...O. .......OOO.O........O...O... ....OO.............OOOO...O. ...O...OOOOO......O....OOO.. ..O...O...........OOO....... ..O.....OOOO...O.O.....OOO.. ..OOO...O.....OOOOOOOOO...O. ..........O..OO........OO... ..OOO.....OO.OO...OO.O..O.O. .O.....O...OO......O..OO.O.. O...OO.O...O........OO.O.O.. O...O.....O..O.............. O...OO.O...O........OO.O.O.. .O.....O...OO......O..OO.O.. ..OOO.....OO.OO...OO.O..O.O. ..........O..OO........OO... ..OOO...O.....OOOOOOOOO...O. ..O.....OOOO...O.O.....OOO.. ..O...O...........OOO....... ...O...OOOOO......O....OOO.. ....OO.............OOOO...O. .......OOO.O........O...O... ........O...O........OO...O. .........OO...........O...O. ..........O..O.........OOOOO ...........O.O...........O.O
I will now explain how many of the above ships are extensible. This could be done by extending Dean's grammar to include all the new components and how they attach to each other. But since the attachments of these new components are so obvious, an example seems easiest. The following spaceship demonstrates the new components and how they may be connected. All the components are separated from each other by a gap of one cell. You should recognize several of these from the collection of spaceships above.
Example components to make Hartmut's ships extensible (speed c/2):
.......O...... ....OOOO...... ....OO........ ..O........... ..OOOO........ .O............ OOO..O........ .OOO.......... ..O........... ...OOO.O...... ...O..OO...... ....OOO....... ......O....... ....O......... ....O.O....... ...O.......... ....O.O....... ....O......... ......O....... ....OOO....... ...O..OO...... ...OOO.O...... ..O........... .OOO.......... OOO..O........ .O............ ..OOOO........ ..O........... ....OO........ ....OOOO.O.... .......O.O.... .........OO... ........OO.... .......O..O... ......O....... ......O....... ......OOO..... .............. ......OOO..... .....O........ ....O...O..... ...O..O....... ...O....O..... ..O.OOOO...... .OOO.......... ..O.OOOO...... ...O....O..... ...O..O....... ....O...O..... .....O........ ......OOO..... .............. ......OOO..... .....O........ ....O...O..... ...O..O....... ...O....O..... ...OOOOO...... .............. ...OOOOO...... ...O....O..... ...O..O....... ....O...O..... .....O........ ......OOO..... .............. ......OOO..... .....O........ ....O...O..... ...O..O....... ...O....O..... ...OOOOO...... .............. .....OOOOO.... .....O....O... .....O..O..... ......O...O... .......O...... ........OOO... .............. ........OOO... .......O.....O ......O...OO.O ......O...O... ......O...OO.O .......O.....O ........OOO... .............. ........OOO... ........O..... ........O..... .........O..O. ..........OO.. ...........OO. .........O.O.. ......OOOO.O.. ......OO...... ....O......... ....OOOO...... ...O.......... ..OOO..O...... ...OOO........ ....O......... .....OOO.O.... .....O..OO.... ......OOO..... ........O..... ......O....... ......O.O..... .....O........ ......O.O..... ......O....... ........O..... ......OOO..... .....O..OO.... .....OOO.O.... ....O......... ...OOO........ ..OOO..O...... ...O.......... ....OOOO...... ....O......... ......OO...... ......OOOO.... .........O....
Below the standard base period 2 spaceship, there is a small component that I found which attaches to the foot of a ship, and provides a bridge to the new components. Below that is a symmetrical component with a leading "plus sign" of five cells. This component featured in many of Hartmut's first period 2 ships. But it turns out that the plus sign is optional and can be removed, splitting the large component into two smaller components. This is shown next. Following that are the same smaller components shifted with respect to each other by two cells. Finally, there is a "jellyfish" component, then the bridge and another base ship.
To summerize, all the components with three cells in a row along the edge can join together, and the components with five cells in a row along the edge can join up together, possibly shifted.
As an example of extending one of Hartmut's spaceships, we can take the complicated spaceship above with the several rows of ON cells. By adding more of the same kind of components already present in the ship, we can create the following new spaceship.
Example of extending a period 2 spaceship (speed c/2):
..............O.O......... ...........O.O..O......... ..........O..O...........O .........OO....O...O.O.OO. ........O....O...OOO.O.O.O .......OOOOO.O...O..O..... ....OO...........OOO.OO... ...O...OOOOOOOO...O....O.. ..O...O............OOO.... ..O.....OOOOOOOOO..O...O.. ..OOO...O............OOO.. ..........OOO.O.O....OOOO. ..OOO.....O..OO.O.......O. .O.....O...OO............. O...OO.O....OOOOO......... O...O..................... O...OO.O....OOOOO......... .O.....O...OO............. ..OOO.......OO..O......... .............OO........... ..OOO.........OO.O........ .O.....O.......OO......... O...OO.O.................. O...O..........OO......... O...OO.O......OO.O........ .O.....O.....OO........... ..OOO.......OO..O......... ...........OO............. ..OOO.......OOOOO......... .O.....O.................. O...OO.O....OOOOO......... O...O......OO............. O...OO.O....OO..O......... .O.....O.....OO........... ..OOO.........OO.O........ ...............OO......... ..OOO..................... .O.....O.......OO......... O...OO.O......OO.O........ O...O........OO........... O...OO.O....OO..O......... .O.....O...OO............. ..OOO.......OOOOO......... .......................... ..OOO.......OOOOO......... .O.....O...OO............. O...OO.O....OO..O......... O...O........OO........... O...OO.O......OO.O........ .O.....O.......OO......... ..OOO..................... ...............OO......... ..OOO.........OO.O........ .O.....O.....OO........... O...OO.O....OO..O......... O...O......OO............. O...OO.O....OOOOO......... .O.....O.................. ..OOO.......OOOOO......... ...........OO............. ..OOO.......OO..O......... .O.....O.....OO........... O...OO.O......OO.O........ O...O..........OO......... O...OO.O.................. .O.....O.......OO......... ..OOO.........OO.O........ .............OO........... ..OOO.......OO..O......... .O.....O...OO............. O...OO.O....OOOOO......... O...O..................... O...OO.O....OOOOO......... .O.....O...OO............. ..OOO.....O..OO.O.......O. ..........OOO.O.O....OOOO. ..OOO...O............OOO.. ..O.....OOOOOOOOO..O...O.. ..O...O............OOO.... ...O...OOOOOOOO...O....O.. ....OO...........OOO.OO... .......OOOOO.O...O..O..... ........O....O...OOO.O.O.O .........OO....O...O.O.OO. ..........O..O...........O ...........O.O..O......... ..............O.O.........
These new components can be combined with the branching ship components to form "loopy" structures. So we can construct spaceships with arbitrarily large holes inside them. The following is a simple example of such a spaceship.
Period 2 spaceship with hole (speed c/2):
..........................O.............. .......................OOOO.............. .....................OOO.O............... ....................OO...O............... ....................O....OO.............. ..................O.O.....OOO.O.......... ................OOO.........O.O.......... ...............OO..O..........OO......... ..............O..O...........OO.......... ..............O..O..........O..O......... ..............O............O............. ...............O..O........O..O.......... ................OO.........O..O.......... .................OO.........OO..O........ ...............O.O...........OOO......... .............OOO.O.............O.O....... ............OO...................O....... ............O....................OO...... ..........O.O.....................OOO.O.. ........OOO.........................O.O.. .......OO..O..........................OO. ......O..O...........................OO.. ......O..O..........................O..O. ......O............................O..... .......O..O........................O..... ........OO.........................OOO... .........OO.............................. .......O.O.........................OOO... ....OOOO.O........................O.....O ....OO...........................O...OO.O ..O..............................O...O... ..OOOO...........................O...OO.O .O................................O.....O OOO..O.............................OOO... .OOO..................................... ..O................................OOO... ...OOO.O..........................O.....O ...O..OO.........................O...OO.O ....OOO..........................O...O... ......O..........................O...OO.O ....O.............................O.....O ....O.O............................OOO... ...O..................................... ....O.O............................OOO... ....O.............................O.....O ......O..........................O...OO.O ....OOO..........................O...O... ...O..OO.........................O...OO.O ...OOO.O..........................O.....O ..O................................OOO... .OOO..................................... OOO..O.............................OOO... .O................................O.....O ..OOOO...........................O...OO.O ..O..............................O...O... ....OO...........................O...OO.O ....OOOO.O........................O.....O .......O.O.........................OOO... .........OO.............................. ........OO.........................OOO... .......O..O........................O..... ......O............................O..... ......O..O..........................O..O. ......O..O...........................OO.. .......OO..O..........................OO. ........OOO.........................O.O.. ..........O.O.....................OOO.O.. ............O....................OO...... ............OO...................O....... .............OOO.O.............O.O....... ...............O.O...........OOO......... .................OO.........OO..O........ ................OO.........O..O.......... ...............O..O........O..O.......... ..............O............O............. ..............O..O..........O..O......... ..............O..O...........OO.......... ...............OO..O..........OO......... ................OOO.........O.O.......... ..................O.O.....OOO.O.......... ....................O....OO.............. ....................OO...O............... .....................OOO.O............... .......................OOOO.............. ..........................O..............
Recently I found that the "barber pole" diagonal tagalong can be used as a bridge between two spaceships. This is shown below.
Barber pole component connecting two period 2 spaceships (speed c/2):
.......O................. ....OOOO................. ....OO................... ..O...................... ..OOOO................... .O....................... OOO..O................... .OOO..................... ..O...................... ...OOO.O................. ...O..OO................. ....OOO.................. ......O.................. ....O.................... ....O.O.................. ...O..................... ....O.O.................. ....O.................... ......O.................. ....OOO.................. ...O..OO................. ...OOO.O................. ..O...................... .OOO..................... OOO..O................... .O....................... ..OOOO................... ..O...................... ....OO................... ....OOOO.O............... .......O.O............... .........OO.............. ........OO............... .......O..O.............. ......O.................. ......O....O............. ......OO.OOO............. .........OO.............. ..........O.O............ ..........OO.O........... ..........O.............. ..............O.......... ...........OO.O.......... ............OO........... .............O.O......... .............OO.O........ .............O........... .................O....... ..............OO.O....... ...............OO........ ................O.O...... ................OO.O..... ................O........ ....................O.... .................OO.O.... ..................OO..... ...................O.O... ...................OO.O.. ...................O..... .......................O. ....................OO.O. ....................O.... .....................O.O. .....................O... ......................OOO ......................... ....................O.OOO ...................OO..OO ...................O.O... ...................OO.... ...................O..... ..................OOO.... ..................O...... ..................O...... ...................O..O.. ....................OO... .....................OO.. ...................O.O... ................OOOO.O... ................OO....... ..............O.......... ..............OOOO....... .............O........... ............OOO..O....... .............OOO......... ..............O.......... ...............OOO.O..... ...............O..OO..... ................OOO...... ..................O...... ................O........ ................O.O...... ...............O......... ................O.O...... ................O........ ..................O...... ................OOO...... ...............O..OO..... ...............OOO.O..... ..............O.......... .............OOO......... ............OOO..O....... .............O........... ..............OOOO....... ..............O.......... ................OO....... ................OOOO..... ...................O.....
Finally, I have discovered two different repeatable tagalongs for period 2 spaceships, and a small tagalong for them which makes the ship a period 6 spaceship. This is the only method known for making a period 6 spaceship.
Period 2 spaceship with repeatable tagalongs and period 6 tagalong (speed c/2):
.....O.O....................................................... ....O..O....................................................... ...OO.......................................................... ..O............................................................ .OOOO.......................................................... O....O......................................................... O..O........................................................... O..O........................................................... .O............................................................. ..OOOO.O....................................................... ...O...O....................................................... ....O.......................................................... ....O.O........................................................ ............................................................... ...OOO......................................................... ...OO.......................................................... ...OOO......................................................... ............................................................... ....O.O........................................................ ....O.......................................................... ...O...O....................................................... ..OOOO.O....................................................... .O............................................................. O..O........................................................... O..O........................................................... O....O......................................................... .OOOO.......................................................... ..O............................................................ ...OO.......................................................... ....O..O.O....O.O....O.O....O.O.....O....O.O.....O....O.O...... .....O.O..O..O..O.OO.O..O..O..O.OO.O.O..O..O.OO.O.O..O..O.OO.O. ........O...OO........O...OO.......O...OO.......O...OO.......OO .........O......O......O......O.....OOO....O.....OOO....O...... ..........OOOOO.O.......OOOOO.O.......OOOO.O.......OOOO.O....OO ..................OO............OO...........OO...........OO.O. ..........OOOOO.O.......OOOOO.O.......OOOO.O.......OOOO.O...... .........O......O......O......O.....OOO....O.....OOO....O...... ........O...OO........O...OO.......O...OO.......O...OO......... .....O.O..O..O..O.OO.O..O..O..O.OO.O.O..O..O.OO.O.O..O..O.OO... ....O..O.O....O.O....O.O....O.O.....O....O.O.....O....O.O...... ...OO.......................................................... ..O............................................................ .OOOO.......................................................... O....O......................................................... O..O........................................................... O..O........................................................... .O............................................................. ..OOOO.O....................................................... ...O...O....................................................... ....O.......................................................... ....O.O........................................................ ............................................................... ...OOO......................................................... ...OO.......................................................... ...OOO......................................................... ............................................................... ....O.O........................................................ ....O.......................................................... ...O...O....................................................... ..OOOO.O....................................................... .O............................................................. O..O........................................................... O..O........................................................... O....O......................................................... .OOOO.......................................................... ..O............................................................ ...OO.......................................................... ....O..O....................................................... .....O.O.......................................................
In the above, the two types of period 2 tagalongs are repeated twice. The two pre-blocks at the right form the period 6 tagalong. The pre-blocks can be removed to make a normal period 2 spaceship. The final three sets of two-bit sparks can also be removed and the ship will still work.
There are a few more known period 2 components that I have not mentioned in this article. The number of possible components is probably infinite. As we search larger and larger areas, we find more and more components, so that cataloging them becomes impossible. Instead of trying to catalog everything, we can search directly for objects which meet certain conditions, such as spaceships which have sparks of a desired kind. An example of this is given by the first "barber pole" spaceship in this article. The component at the front which connects the diagonal component to the rest of the ship is larger than the others, and is messy. It was searched for explicitly in order to connect the diagonal component to a known spaceship.
Considering the long time interval of 20 years between the classic spaceships and these new classes, you might be surprised at the number of new spaceships that have been found in the last few years. But I am not done yet! The next article in this series will survey the period 3 c/3 spaceships. These were the first spaceships found which travel at a "non-standard" speed.
This is the third in a series of articles concerning Conway's Game of Life. In this article, I will survey the results for all the known period 3 spaceships, and give some applications of them. All period 3 spaceships must be orthogonal, and must travel at the speed of c/3. This follows from the speed restrictions mentioned in my first article.
When Dean Hickerson started looking for spaceships using his search program, the first spaceships he found were of period 2. But he soon also tried looking for period 3 spaceships. In August 1989 he found a grammar for constructing an infinite number of short wide c/3 period 3 spaceships. (This grammar is used similarly to the grammar for period 2 spaceships that was in my previous article.) These period 3 spaceships were the first orthogonal spaceships found which didn't travel at the "normal" speed of c/2.
In Dean's grammar, the components are labeled using letters, or letters followed by either a single quote or a double quote (e.g., A, A', and A"). Any three components with the same letter are related, and represent the same section of a period 3 spaceship in three successive generations. Therefore, if component A appears in generation 0 of a spaceship, then component A' must appear in the same location in generation 1, and component A" must appear in the same location in generation 2.
A component name followed by a dash represents the mirror image of a component. The mirroring is done by reflecting the component across a horizontal line. For example, component B"- is the mirror image of component B".
The components in Dean's grammar are the following:
[A] [A'] [A"] [B] [B'] [B"] [C] [C'] [C"]
..O. ..O. ..O ..O. ..O. ..O ...O... ....... ...O...
.O.O .O.O OO. .O.O .O.O OO. ..O.... .OOO... ..O.O..
.O.O OO.. OO. .O.O OO.. OO. .OO.... .OOO... ..O....
.O.. ..O. ..O .O.. .O.. ..O ...O.O. ...OO.. ..O...O
.... OO.. O.. OO.. O.O. O.. .OOO..O .O.O.OO ....O.O
OO.. OOO. O.O X X X .OO..O. .O.O... .OO..O.
X X X ....... O...... .O.....
OOO.... OOO.... OO.....
.O..... O...... .O.....
X X X
[D] [D'] [D"] [E] [E'] [E"] [F] [F'] [F"]
X X X X X X X X X
..O.O ...... ..O.. .O.O .... .O.O .OO.. .O.O ..O..
..OO. ...OO. ...OO O.O. O..O ..O. ..O.. ..O. ....O
..... ..O..O .O... OOO. O.O. OO.. ..O.O .OO. ..OO.
OO.OO .OOO.. .O.OO .... O... .O.. ..OO. .... ..OO.
OO... O..O.. O..OO .O.O .O.. .OOO .OO.. O..O .....
O.... O..O.. O..O. X X X OO... O... OO...
OO.O. .OO... .OOO. X X X
.O.O. ....O. .O...
X X X
[G] [G'] [G"] [H] [H'] [H"] [I] [I'] [I"]
X X X X X X X X X
O..O O... O.O ..OO.. ...O.O ...O.. .O.O. .O... .OOO..
OO.. OOO. OO. ...O.. ....O. .....O ..... O.... .O....
.O.. .... O.. ...O.O ...OO. ...OO. OOO.. O.O.. OO....
.OOO .O.O .O. ...OO. ...... ...OO. OO... O.O.. .O....
X X X .OOO.. ..O..O ...... ..... .OOO. ......
O.O... .O.... OOO... ..OOO ..O.O .....O
O.O... OO.... OOO... .O.OO .O..O .OOOO.
O..... .O.... ...... .O... OO... .OO...
.O.... .OOO.. .O.O.. .O... ..... ......
X X X ..... OO... .O.O..
OO... OOO.. .O.O..
X X X
[J] [J'] [J"] [K] [K'] [K"]
X X X X X X
..O..... ..OO.... .O.O.... .O.O. .O... .OOO..
..OOO... ..OO.... ....O... ..... O.... .O....
..O..... ...O.... ...O.... OOO.. O.O.. OO....
........ ..O..... ..OOO... OO... O.O.. .O....
OO...O.. .O..OO.. .O.OO... ..... .OOO. ......
OOOOO... O...OOOO OO..O..O ..OOO ..O.O .....O
OO...O.O .O....O. ........ .O.OO .O..O .OOOO.
..O..OO. ...O.OOO .O...OOO O.... OO... .OO...
OO.O.... .OO..... .O.OOO.. O.... ..... .OO...
.O...... .OO..O.. .O..O... ..... O.... .O....
..OOO... ...OO... ..OOO... OOO.. OOO.. OO....
........ ........ ........ .O... O.... .O....
..OOO... ...OO... ..OOO... X X X
.O...... .OO..O.. .O..O...
OO.O.... .OO..... .O.OOO..
..O..OO. ...O.OOO .O...OOO
OO...O.O .O....O. ........
OOOOO... O...OOOO OO..O..O
OO...O.. .O..OO.. .O.OO...
........ ..O..... ..OOO...
..O..... ...O.... ...O....
..OOO... ..OO.... ....O...
..O..... ..OO.... .O.O....
X X X
The components are strung together by stacking them above each other, similarly to the way that period 2 components are stacked. (The X's indicate the horizontal alignment of components, and should be removed.)
The rules which give the allowed sequences of components to make a valid spaceship are the following.
The sequence must begin with A, A', A", B, B', B", C, C', or C".
The sequence must end with A-, A'-, A"-, B-, B'-, B"-, C-, C'-, or C"-.
Each pair of adjacent symbols must appear in one line of the following table, with the first symbol found before the vertical bar, and the second symbol found after the vertical bar.
A | D E
A' | D' E'
A" | D" E"
I | D
I' | D'
I" | D"
D- | A- I-
D'- | A'- I'-
D"- | A"- I"-
E- | A-
E'- | A'-
E"- | A"-
B F- H- | E- G'- H"- I K
B' F'- H'- | E'- G"- H- I' K'
B" F"- H"- | E"- G- H'- I" K"
E G' H" I- K- | B- F H
E' G" H I'- K'- | B'- F' H'
E" G H' I"- K"- | B"- F" H"
C K | J
C' K' | J'
C" K" | J"
J | C- K-
J' | C'- K'-
J" | C"- K"-
D' G- | F-
D" G'- | F'-
D G"- | F"-
F | D'- G
F' | D"- G'
F" | D- G"
The simplest spaceship which can be constructed by these rules is A E B-, which is shown below. This spaceship has 25 ON cells in every generation. There is no known period 3 spaceship which has fewer ON cells than 25. (Any such spaceship must be spread out very thinly.)
Smallest known period 3 spaceship (speed c/3):
..O.. .O.O. .O.O. .O... ..... OO... .O.O. O.O.. OOO.. ..... .O.O. .OO.. ..O.. ..O.O ..O.O ...O.
Another example spaceship created using these rules is C J C-, which represents the following symmetrical spaceship.
One of many period 3 spaceships constructed by the above grammar (speed c/3):
...O.... ..O..... .OO..... ...O.O.. .OOO..O. .OO..O.. ........ OOO..... .O...... ..O..... ..OOO... ..O..... ........ OO...O.. OOOOO... OO...O.O ..O..OO. OO.O.... .O...... ..OOO... ........ ..OOO... .O...... OO.O.... ..O..OO. OO...O.O OOOOO... OO...O.. ........ ..O..... ..OOO... ..O..... .O...... OOO..... ........ .OO..O.. .OOO..O. ...O.O.. .OO..... ..O..... ...O....
Dean Hickerson also tried looking for long and thin period 3 spaceships. He found two basic spaceships, which are given below. They have the same front ends. Dean has named the leftmost spaceship the "turtle."
The "turtle" and another related period 3 spaceship (speed c/3):
.OOO.......O .........O...
.OO..O.OO.OO ........O.O..
...OOO....O. ........O....
.O..O.O...O. .OOO....O...O
O....O....O. .OO..O....O.O
O....O....O. ...OOO.OO.OO.
.O..O.O...O. .O..O.O.OOO..
...OOO....O. O....O.....O.
.OO..O.OO.OO O....O.....O.
.OOO.......O .O..O.O.OOO..
...OOO.OO.OO.
.OO..O....O.O
.OOO....O...O
........O....
........O.O..
.........O...
For many people, the turtle spaceship is the most aesthetically pleasing new spaceship that has been found so far. Besides being pretty, it has a feature which makes it useful. In generation 2, it produces a two bit spark at the back. This spark can be used in several ways.
The first use of the spark is that it makes a good attachment point for tagalongs. Dean quickly found a repeatable tagalong for the turtle ship. This tagalong can be said to have a period of 28, which is the number of generations for it to reappear in the same location. This tagalong is not self-terminating, however. A different kind of tagalong is required to terminate the repeating one. The following shows the base ship, four copies of the repeating tagalong, and a small terminating tagalong.
Period 3 spaceship with repeatable tagalong and ending tagalong (speed c/3):
...................O........O........O.........O........... .................OO........O.O......O.O......OO............ .OOO.......O.....OO.......OO........O.O......OO.........O.O .OO..O.OO.OO.......O........O.......O..........O.......OOOO ...OOO....O......O...O....OO............O....O...O....OO... .O..O.O...O...O.O...OOOO..OOOOOO...OO..O..O.O...OOOO..OO... O....O....O..O.OO...O.O..O.O...O.O..O.OO...OO...O.O..O..... O....O....O..O.OO...O.O..O.O...O.O..O.OO...OO...O.O..O..... .O..O.O...O...O.O...OOOO..OOOOOO...OO..O..O.O...OOOO..OO... ...OOO....O......O...O....OO............O....O...O....OO... .OO..O.OO.OO.......O........O.......O..........O.......OOOO .OOO.......O.....OO.......OO........O.O......OO.........O.O .................OO........O.O......O.O......OO............ ...................O........O........O.........O...........The same repeating tagalong can also be attached in a different way to the base ship, as shown below. Also shown is an alternate terminating tagalong.
Period 3 spaceship with repeatable tagalong and another ending (speed c/3):
..................O.........O........O.........O.O .................O.O......OO........O..OOO...O..O. .OOO.......O.....O.O......OO.......OO....O.O.O..OO .OO..O.OO.OO.....O..........O..........O.O........ ...OOO....O..........O....O...O....OO.OO.......... .O..O.O...O...O.OO..O..O.O...OOOO..OOO............ O....O....O..O...O.OO...OO...O.O..O.OO............ O....O....O..O...O.OO...OO...O.O..O.OO............ .O..O.O...O...O.OO..O..O.O...OOOO..OOO............ ...OOO....O..........O....O...O....OO.OO.......... .OO..O.OO.OO.....O..........O..........O.O........ .OOO.......O.....O.O......OO.......OO....O.O.O..OO .................O.O......OO........O..OOO...O..O. ..................O.........O........O.........O.O
The tails of the terminating tagalong above have alternate forms which also work, and which are simple permutations of the positions of the cells in the final few columns. The following diagrams show these permutations for the bottom 5 rows and rightmost 9 columns.
Alternate endings for tail of final component:
......... ......... ......O.O O.....O.O O...O..OO O...O..O. O.O.O..O. O.O.O..O. O.O.O..OO O...O..OO O.....O.O O........ ......... ......... .........
The second use of the spark from the turtle spaceship (and the sparks from other c/3 spaceships) is that they can interact with faster spaceships to produce various reactions (which is not possible for period 2 spaceships). For example, a LWSS catches up to a period 3 spaceship with a relative speed of c/6, and can then interact with it. The following shows a simple case of this, where the spark from the turtle ship destroys the pursuing LWSS, and so saves the turtle.
Period 3 spaceship uses its spark to destroy a pursuing LWSS:
.OOO.......O......... .OO..O.OO.OO......... ...OOO....O......O..O .O..O.O...O.....O.... O....O....O.....O...O O....O....O.....OOOO. .O..O.O...O.......... ...OOO....O.......... .OO..O.OO.OO......... .OOO.......O.........
The following demonstrates two useful reactions found by Dean Hickerson. Here two turtle spaceships are pursued by a salvo of five LWSSs and a MWSS. The three lower spaceships collide with the lower turtle spaceship to produce a glider. This glider then collides with the debris created by the collision of the upper spaceships with the upper turtle spaceship to produce a MWSS which travels in the reverse direction.
Salvo of c/2 spaceships hits c/3 ships to produce backwards MWSS (speed c/ 3):
.OOO.......O$$$$$. .OO..O.OO.OO$$$$$$. ...OOO....O$.........OO$$$$. .O..O.O...O$........OOOO$$$$ O....O....O........OOOO.....OO.OO$$$$ O....O....O........O...O.....OO$$$$.. .O..O.O...O........O$$$$........OOOO. ...OOO....O.........O..O$$$$....O...O .OO..O.OO.OO$$$$$......O.... .OOO.......O$$$$$.......O..O $$$$$$$... $$$$$$$... $$$$$$$... $$....OOO.......O$$$........ $$....OO..O.OO.OO$$$........ $$......OOO....O$$$......... $$....O..O.O...O.....OO$$$.. $$...O....O....O....OO.OOO...OOOO$$.. $$...O....O....O.....OOOOO...O...O$$. $$....O..O.O...O......OOO....O$$..... $$......OOO....O$....O..O......OOOO$. $$....OO..O.OO.OO$$...O...O$ $$....OOO.......O$$...O$.... $$$$$.........O..O$
Dean found the above reaction in order to construct a Life pattern which had a population growth like log(t), according to a method suggested by Bill Gosper. This is done by shooting one salvo of c/2 ships at the pair of c/3 ships, and waiting until the returning MWSS arrives. When it does, then shoot out a glider somewhere, and send another salvo of the c/2 ships. Since the period 3 ships quintuple their distance from the salvo gun every cycle, the number of gliders grows at a rate which keeps dividing by 5, and this causes the population to grow like log(t). However, after finding the reaction shown above, Dean found other simpler ways to create this behavior, and so this construction has not been completed. Later in this article is another simpler reaction which produces the same result.
After finding the period 3 spaceships given above and their tagalongs, Dean Hickerson went on to other things, and (like the period 2 spaceships), nobody looked for further period 3 spaceships for a while.
In March, 1992 I started looking for period 3 spaceships and their tagalongs, eventually doing an exhaustive search for long thin symmetrical spaceships in an area up to 18 cells wide and 79 cells long. I found several new spaceships, and many tagalongs. I have been the only person doing a search for period 3 spaceships since Dean's earliest searches. Therefore the rest of the spaceships and tagalongs in this article were found by me, and so I will omit mentioning the discoverer of each spaceship.
The first new spaceship found was the following.
Small symmetrical period 3 spaceship (speed c/3):
.....O....... ....O.O....O. ....O.O...O.O ....O.....OO. ..........O.. ...OO......O. ....O.OO...O. ...OO.OOOOO.. .O..OOO...... OO........... .O..OOO...... ...OO.OOOOO.. ....O.OO...O. ...OO......O. ..........O.. ....O.....OO. ....O.O...O.O ....O.O....O. .....O.......
The following spaceship was the first new spaceship found using a new search feature which allows searching in a large number of rows, but limits the number of ON cells in any column to a specified number of adjacent rows. This ship has the same number of ON cells as the smallest known period 3 spaceship, and thus ties in minimum size (but it has more ON cells in the other phases).
Second smallest known period 3 spaceship (speed c/3):
.....O.. ....O.O. ...OO... ....O.O. ....OO.. .......O ...O..O. ..O..... .OO..... .O.O.... OO...... ........ .O...... .O.O.... .O.O.... ..O.....
The spaceship above has two nice properties. First, it is more spread out than any other period 3 spaceship. This makes it the best candidate for being constructed from a set of glider collisions. If it could be constructed, then a spaceship gun for it could be built. So far none of the new spaceships have been able to be constructed from gliders. (David Buckingham is an expert at constructing objects from gliders. He despairs of constructing most of the new spaceships because they are what he calls "space dust." Only a small number of the new spaceships are simple enough to be constructed from gliders using known techniques. David says he does have a method that would allow the above spaceship to be constructed, but he hasn't completed the construction.)
The second nice property of the spaceship is that it contains a different set of sparks that can have new tagalongs attached to them. The following shows a repeatable tagalong for this ship. Surprisingly, this tagalong was not found by a search program, but was found manually. This was possible because most of the tagalong resembles a component of the base ship.
Period 3 spaceship with small repeatable tagalong (speed c/3):
............O.. ...........O.O. ..........OO... .....O.....O.O. ....O.O....OO.. ...OO.........O ....O.O...O..O. ....OO...O..... .......O.O..... ...O..O...O..O. ..O...........O .OO........OO.. .O.O.......O.O. OO........OO... ...........O.O. .O..........O.. .O.O........... .O.O........... ..O............
Since the tagalong has a pair of the same sparks as the base spaceship, it can be attached to itself an arbitrary number of times. Because there is always a choice of two sets of sparks that another copy of the tagalong can attach to, you can make a ship that weaves back and forth as desired.
This tagalong is even more versatile than is described above. A section of it can be repeatedly attached to itself to make a arbitrarily long "arm." This allows a binary tree spaceship to be built for period 3 spaceships (therefore sharing this capability with period 2 spaceships). An example of such a spaceship is shown below. One of its phases is very striking in appearance (one of the most unlikely-looking spaceships known).
Period 3 "binary tree" spaceship (speed c/3):
...........................O............... ..........................O.O.............. .........................OO................ ..........................O.O.............. ..........................OO............... .............................O............. .........................O..O.............. ........................O.................. .......................OO.................. ..................O.....O.O................ .................O.O....OO................. ................OO.........O............... .................O.O...O..O................ .................OO...O.................... ....................O.O.................... ................O..O...O..O................ ...............O...........O............... ..............OO........OO................. ...............O.O......O.O................ ...............OO......OO.................. ..................O.....O.................. ..............O..O.......O..O...........O.. .............O...............O.........O.O. ............OO............OO..........OO... ..O..........O.O..........O.O..........O.O. .O.O.........OO..........OO............OO.. .O.O............O.........O.O.............O .O..........O..O...........O..........O..O. ...........O.........................O..... OO........OO........................OO..... .O.O.......O.O.......................O.O... .OO........OO........................OO.... ..O...........O.........................O.. ...O..O...O..O......................O..O... .......O.O.........................O....... ....OO...O........................OO....... ....O.O...O..O...............O.....O.O..... ...OO.........O.............O.O....OO...... ....O.O....OO..............OO.........O.... .....O.....O.O..............O.O...O..O..... ..........OO................OO...O......... ...........O...................O.O......... ............O..O...........O..O...O..O..... ................O.........O...........O.... .............OO..........OO........OO...... .............O.O..........O.O......O.O..... ............OO............OO......OO....... .............O...............O.....O....... ..............O..O.......O..O.......O..O... ..................O.....O...............O.. ...............OO......OO............OO.... ...............O.O......O.O..........O.O... ..............OO........OO..........OO..... ...............O...........O.........O..... ................O..O...O..O...........O..O. ....................O.O...................O .................OO...O................OO.. .................O.O...O..O............O.O. ................OO.........O..........OO... .................O.O....OO.............O.O. ..................O.....O.O.............O.. .......................OO.................. ........................O.................. .........................O..O.............. .............................O............. ..........................OO............... ..........................O.O.............. .........................OO................ ..........................O.O.............. ...........................O...............
The base ship can be extended in the same manner as is done to the tagalong to make it wider. One interesting thing about the above spaceship is that many perturbations to it will simply break off and not destroy the rest of the spaceship. The reason for this is that the repeating component has a self-repairing feature. If the dangling cell next to the edge cell of an arm is removed (or equivalently, the end component is removed leaving another similar component at the end), then the ship will regenerate the missing cell.
The following shows a slightly extended base spaceship with the dangling cell removed which will regenerate in generation 3.
Regenerating period 3 spaceship (speed c/3):
..O....... .O.O...... .O.O...... .O........ .......... OO........ .O.O...... .OO....... ..O....... ...O..O... .......O.. ....OO.... ....O.O... ...OO..... ....O..... .....O..O. .........O ......OO.. ......O.O. .....OO... ......O... .......O..
Other early tagalongs found were the following. These are the same except for where they attach to the base ship. They have sparks at the back which allow them to be repeatedly attached to each other. These tagalongs can be mixed with the one just above.
Two period 3 ships with related repeatable tagalongs (speed c/3):
..............O............ ..............O............ .............O.O........... .............O.O........... .....O.......O.O.......OO.O .....O......OO.........O.O. ....O.O......O........O...O ....O.O.......O........OO.O ...OO............O.......O. ...OO.......OO........O.... ....O.O...O.OO..O..O.O.O... ....O.O.....OOOOOO...OO.... ....OO...O...O.OO...OO..... ....OO...OO..O...O.O....... .......O.O...O.OO...OO..... .......O.OO..O...O.O....... ...O..O...O.OO..O..O.O.O... ...O..O.....OOOOOO...OO.... ..O..............O.......O. ..O.........OO........O.... .OO..........O........O...O .OO...........O........OO.O .O.O.........O.O.......OO.O .O.O........OO.........O.O. OO...........O.O........... OO...........O.O........... ..............O............ ..............O............ .O......................... .O......................... .O.O....................... .O.O....................... .O.O....................... .O.O....................... ..O........................ ..O........................
These tagalongs are similar to the back ends of the original repeating tagalongs that Dean Hickerson found for his turtle spaceship. This means that all these tagalongs can also be attached to that spaceship.
Another early tagalong is the following, which I called the "fly." It turns out that many tagalongs look similar to this one, but this was the first one found with this appearance. It can be repeatedly attached to itself, as shown here. The front part of this tagalong resembles the front of the turtle spaceship, but is wider.
Period 3 spaceship with repeatable "fly" tagalong (speed c/3):
..........................O.O...O........................... .........................OO.O.O..O.......................... ...........OOO........O.........O........................... ...........OO..O.OO...O..OOOO............................... .....O.......OOOO..O.O..OO....OO............................ ....O.O....O..O...OOO.....OOO............................... ...OO.....O....O..OO..OO..O..O.............................. ....O.O...O....O..OOO.O.O....OO............................. ....OO...OO....O..OOOO.....O........................OO...O.O .......O.OO....O..OOOO.....O...........O............OOO.O..O ...O..O...O....O..OOO.O.O....OO.......O..........OO....OO.O. ..O.......O....O..OO..OO..O..O.......OO..O.OO...OOOOOOO.OO.. .OO........O..O...OOO.....OOO............OO..O.O........O... .O.O.........OOOO..O.O..OO....OO.....OO.O...O...O....OO..... OO.........OO..O.OO...O..OOOO........OOOO....O......OO....O. ...........OOO........O.........O...O..OO....O..OO....OO.O.. .O.......................OO.O.O..O..O.OOO....O..OOO...OO.... .O.O......................O.O...O...O.OOO....O..OOO...OO.... .O.O................................O..OO....O..OO....OO.O.. ..O..................................OOOO....O......OO....O. .....................................OO.O...O...O....OO..... .........................................OO..O.O........O... .....................................OO..O.OO...OOOOOOO.OO.. ......................................O..........OO....OO.O. .......................................O............OOO.O..O ....................................................OO...O.O
Here are two short, wide period 3 spaceships which show more components that could be added to Dean's grammar. These ships have additional sparks that are useful for perturbing following spaceships and for attaching tagalongs to. The spaceship on the right is the only known period 3 spaceship with a spark on the edge. This ship was found by an explicit search starting with that spark. But the edge spark has not been too useful so far.
Two more period 3 spaceships with useful sparks (speed c/3):
...O.... .....O.O...
..O.O... .....O.....
..O.O... ..OOO......
..O..... ..OO.O.....
........ .O..OO.....
.OO..... ..OOOO.....
..O.O... .....O.....
.O.O.... ..O.O..O...
.OOO.... OO.OO.O....
........ OOO........
..O.O... .....O.....
..OO.... .OO...OOO..
...O.... .OOO..O....
...O.O.. ........O..
...OO... ....OOOO...
.OOO.... .....O.....
O.O..... .....OO....
O..O.... ......O.OOO
O.OO.... .......O...
...O.... ..........O
O..O.... ......O....
.OO..... ....OOOOO.O
..O...O. ...O...OO.O
...OO..O ..O...O....
...OO... ...O...O...
....O... .....O.....
...OO... ...OOO.....
..O...O. ..O..O.....
.OO.OO.. ......OO...
..O..... ..OOO.OO...
..OO.... ..OO...OO..
...O.... ......O.O..
....O... ....OOO....
....OOO....
....OO.....
......O....
....OO.....
.....O.O...
......O....
I found a salvo of two LWSSs that could hit the back of the first ship above and which generates a glider. Dean Hickerson then extended that to a salvo of four LWSSs that could hit two period 3 ships to generate a backwards- traveling MWSS. This is shown below.
Four LWSSs hit two period 3 ships to produce backwards MWSS:
....O......................................................... ..OO.......................................................... ..OO.......................................................... ....O......................................................... ..O........................................................... ..O.O......................................................... ..O.O......................................................... ...O.......................................................... .OO........................................................... ..O........................................................... ..OOO......................................................... ...O.......................................................... .....O........................................................ ...OO......................................................... ...OO......................................................... .............................................................. OOO........................................................... OO..OO........................................................ ..O........................................................... .OO........................................................... .OO........................................................... .O..O......................................................... .....O........................................................ ..OOOOO....................................................... ......OO...................................................... ......OO..OO.................................................. ...OOOO..OO.OO................................................ ..O...O...OOOO............................................OO.. ..OOO......OO............................................OO.OO ..O..O....................................................OOOO ...O.......................................................OO. ...O.O........................................................ ....O......................................................... .............................................................. .............................................................. .............................................................. .............................................................. .............................................................. .............................................................. .............................................................. .............................................................. .............................................................. .......................O...................................... ......................O.O..................................... ......................O....................................... .....................O..O..............O..O................... .....................OOO......OO......O....................... .....................O...O...OOOO.....O...O................... ......................OOOO..OO.OO.....OOOO.................... .........................OO..OO............................... .........................OO................................... .....................OOOOO.................................... ........................O..................................... ....................O..O...................................... ....................OO........................................ ....................OO........................................ .....................O........................................ ...................OO..OO..................................... ...................OOO........................................ .............................................................. ......................OO...................................... ......................OO...................................... ........................O..................................... ......................O....................................... .....................OOO...................................... .....................O........................................ ....................OO........................................ ......................O....................................... .....................O.O...................................... .....................O.O...................................... .....................O........................................ .......................O...................................... .....................OO....................................... .....................OO....................................... .......................O......................................
This pattern is similar to the one found earlier by Dean, but uses two fewer spaceships. It has another important advantage over the earlier pattern. The returning MWSS is on a different path than the incoming LWSSs. This means that multiple salvos of LWSSs can be heading for the period 3 ships, at the same time as multiple returning MWSSs can be heading backwards.
In August 1992, the above reaction was used by Dean Hickerson to implement an example of a "sawtooth" pattern (his seventh). A sawtooth is a pattern whose population is unbounded, but which doesn't tend to infinity (the graph of the population looks like a zig-zag pattern, with fixed lower points and increasing higher points). In this sawtooth, a salvo gun tries to shoot salvos of the four LWSSs at the receding period 3 spaceships, but 5 of the salvos are inhibited if a MWSS has just arrived. So a stream of salvos grows towards the period 3 ships and is reflected back towards the gun. When the first MWSS reaches the gun, the salvos are turned off, and stay off until the last returned MWSS arrives. Since the period 3 spaceships are receding, each cycle requires more salvos before being turned off, and thus the population keeps reaching higher maximums. But the population always returns to the same low number of ON cells when the salvos are used up.
The following shows a weaving wicktrailer tagalong that looks reminiscent of the wicktrailer for period 2 spaceships. It can be said to have a period of 34. The three apparently different cycles are actually the same pattern, just in the 3 different phases.
Period 3 spaceship with wicktrailer (speed c/3):
..O................O.....................OO..........O............... .O.......O.O......OO.........OOO....................OO............... OO..O.O..O......OOOOO.......O..........OO..O......OOOOO.OOO.......O.O ....OO.OO......O......O...OO....O.....O..........O........O.O...O..O. OO.O..OOO.....O..O..OO....OOO...O.O..OO....OOO...O.O..OO....O.O.O..OO OOOO...OO...O..O.O...O..........O......O...OO....O.....O..OOO........ OOOO...OO...O.........OO..O......OOOOO.......O..........O............ OO.O..OOO..........................OO.........OOO.................... ....OO.OO...............OO..........O................................ OO..O.O..O........................................................... .O.......O.O......................................................... ..O..................................................................
The wicktrailer can be attached to some other spaceships if it is slightly modified at the front. Here is a wicktrailer attached to Dean's second long spaceship, along with another related tagalong. Compare the top tagalong with the spaceship above to see the slight modification required.
Period 3 spaceship with two independent tagalongs (speed c/3):
.........O........O.................. ........O.O......OO.................. ........O......OOOOO.OOO............. .OOO....O...O.O........O.O........... .OO..O....O.O.O.O..OO....O.O.O..OO... ...OOO.OO.OO...O....O..OOO...O..O.... .O..O.O.OOO..........O.........O.O... O....O.....O......................... O....O.....O......................... .O..O.O.OOO..........O...O.O......... ...OOO.OO.OO...O....O.O.OO.O......... .OO..O....O.O.O.O..OO................ .OOO....O...O.O......OOOOOO.O.....O.. ........O......OOOOO...O........O.O.O ........O.O......OO......O..O.O.O.OO. .........O........O......O........... ..........................OO.........
Only one cycle of the wicktrailer is shown above because there is no room for more cycles without removing the bottom tagalong. The bottom tagalong does not appear to be extensible.
Sometimes a tagalong is not really a tagalong, but becomes a required part of a spaceship. An example of this is shown below. Here there is a large tagalong for the turtle spaceship, with what appears to be two copies of the wicktrailer attached to the back of the tagalong. But if either of these wicktrailers is removed, then the tagalong will self destruct. So this is really a single large tagalong.
Period 3 spaceship with a tagalong with required wicktrailers (speed c/3):
...............................O............... ..................O...........OO............... ..O..............O..........OOOOO.OOO.......... .O.O......OO....OO..O.OO...O........O.O........ .O...OO..OOO........OO..O.O..O..OO....O.O.O..OO .O.O...O..O.....OO.O...O...O.O...O..OOO...O..O. ...O..O..OOO....OOOO....O.........O.........O.O OO..OOO..OOO.OO..OOO....O...................... OO..OOO..OOO.OO..OOO....O...................... ...O..O..OOO....OOOO....O.........O.........O.O .O.O...O..O.....OO.O...O...O.O...O..OOO...O..O. .O...OO..OOO........OO..O.O..O..OO....O.O.O..OO .O.O......OO....OO..O.OO...O........O.O........ ..O..............O..........OOOOO.OOO.......... ..................O...........OO............... ...............................O...............
There are many more tagalongs for Dean's two symmetrical spaceships. Here is a small selection of them. You can see several components used repeatedly. Also of interest is that the front and back parts of the turtle ship appear in various widths.
Several period 3 spaceships with tagalongs (speed c/3):
.....................OO..... .....................OOO.O.. ...................O....OOO. .................OOOOOOO.OO. .OOO.......O....O........OO. .OO..O.OO.OO....O.OO.OO.OO.. ...OOO....O....O..OO..OO...O .O..O.O...O....OOO.O.OO...O. O....O....O..O...O..OO...... O....O....O..O...O..OO...... .O..O.O...O....OOO.O.OO...O. ...OOO....O....O..OO..OO...O .OO..O.OO.OO....O.OO.OO.OO.. .OOO.......O....O........OO. .................OOOOOOO.OO. ...................O....OOO. .....................OOO.O.. .....................OO.....
.................O...........OOO...O........... ................O.O.........O.....O..O......... ................O...O.OO..OO....O.O...O........ .OOO.......O....O.O..O....OOO...O...O.....O.OO. .OO..O.OO.OO......O..OO...OO....O.O.O..OO.O.O.O ...OOO....O....OO..OO.O..........O...O......O.. .O..O.O...O....OO..OOOO........................ O....O....O..O..O..OOOO...O.................... O....O....O..O..O..OOOO...O.................... .O..O.O...O....OO..OOOO........................ ...OOO....O....OO..OO.O..........O...O......O.. .OO..O.OO.OO......O..OO...OO....O.O.O..OO.O.O.O .OOO.......O....O.O..O....OOO...O...O.....O.OO. ................O...O.OO..OO....O.O...O........ ................O.O.........O.....O..O......... .................O...........OOO...O...........
..............................OO..O.... .................O............OO..O.OOO ................O.O..............OO.... ................O...O.OO..OOOOOOO...O.. .OOO.......O....O.O..O....O..O....O.... .OO..O.OO.OO......O..OO...OOOO.OOOOO... ...OOO....O....OO..OO.O....OO.OO..OO... .O..O.O...O....OO..OOOO....O.....OOO... O....O....O..O..O..OOOO....O.....O..... O....O....O..O..O..OOOO....O.....O..... .O..O.O...O....OO..OOOO....O.....OOO... ...OOO....O....OO..OO.O....OO.OO..OO... .OO..O.OO.OO......O..OO...OOOO.OOOOO... .OOO.......O....O.O..O....O..O....O.... ................O...O.OO..OOOOOOO...O.. ................O.O..............OO.... .................O............OO..O.OOO ..............................OO..O....
..............................OO..O...........................O......... .................O............OO..O..................OO......O.O........ ................O.O..............OO.........................O........O.O ................O...O.OO..OOOOOOO......OOO.........OO..O...O..OO.O.O..O. .OOO.......O....O.O..O....O..O....O....OO..O.OO...O......OOOOO.....O..OO .OO..O.OO.OO......O..OO...OOOO.OOOOO.....OOOO..O.OO....OO......O........ ...OOO....O....OO..OO.O....OO.OO..OO...O..O...OOO..O...OO...OO.......... .O..O.O...O....OO..OOOO....O.....OOOO.O....O..OO.........O..OO.......... O....O....O..O..O..OOOO....O.....O...OO....O..OOO....................... O....O....O..O..O..OOOO....O.....O...OO....O..OOO....................... .O..O.O...O....OO..OOOO....O.....OOOO.O....O..OO.........O..OO.......... ...OOO....O....OO..OO.O....OO.OO..OO...O..O...OOO..O...OO...OO.......... .OO..O.OO.OO......O..OO...OOOO.OOOOO.....OOOO..O.OO....OO......O........ .OOO.......O....O.O..O....O..O....O....OO..O.OO...O......OOOOO.....O..OO ................O...O.OO..OOOOOOO......OOO.........OO..O...O..OO.O.O..O. ................O.O..............OO.........................O........O.O .................O............OO..O..................OO......O.O........ ..............................OO..O...........................O.........
.....................O..............................O...O.O.O............... .........O..........OO..OO.........OO...O......OO.OO..OOOO.OO............... ........O.O......O.OO.O.........O.OOO..O.O.....OO.O.O.OO....O.O............. ........O......OOO.OO...O.O....O...O...O.O....O....O......OO..OO.O.......O.O .OOO....O...O.O.O..OO.....O..O.O..OO...O.....OO..O.O......OOO..O...O....OOOO .OO..O....O.O.O....O....OOO.O....OOO..OO.O.O..O..OO............OO.OO...OO... ...OOO.OO.OO...OO.OOO...OOO.O.....OO.....O.........O..............OO..O.O... .O..O.O.OOO........OO.........OOO.OO.O....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.OOO........OO.........OOO.OO.O....OO.......O.O..........O.O...O..... ...OOO.OO.OO...OO.OOO...OOO.O.....OO.....O.........O..............OO..O.O... .OO..O....O.O.O....O....OOO.O....OOO..OO.O.O..O..OO............OO.OO...OO... .OOO....O...O.O.O..OO.....O..O.O..OO...O.....OO..O.O......OOO..O...O....OOOO ........O......OOO.OO...O.O....O...O...O.O....O....O......OO..OO.O.......O.O ........O.O......O.OO.O.........O.OOO..O.O.....OO.O.O.OO....O.O............. .........O..........OO..OO.........OO...O......OO.OO..OOOO.OO............... .....................O..............................O...O.O.O...............
Here are more period 3 spaceships found by the exhaustive search. The first two are rather small, whereas the right one is much larger. I call the one the left the "dart," and the middle one the "brain" because of their appearances.
Three more period 3 spaceships (speed c/3):
........O. .OO........ ........O.O......................
.......O.O O..O.....OO ........O...OOO..................
......OO.. OOO...OOO.. ......OO.....OO..................
.........O O..O.OOOO.. ......OO...O......O........OO....
.....O...O .OOO...O... .........O.OO..OO.O.......OOO....
..O..O.... .O..OOO.... .....O...O......O..OO.O.O..O.....
.O.O..OOOO ...O....OO. ..O..O......O...OOOO...OO..OO...O
O..O...... ...OOOOO.O. .O.O..OOOOOO........O..O..OO..OOO
.O.O..OOOO ........... O..O..................O...OO..O..
..O..O.... ...OOOOO.O. .O.O..OOOOOO........O..O..OO..OOO
.....O...O ...O....OO. ..O..O......O...OOOO...OO..OO...O
.........O .O..OOO.... .....O...O......O..OO.O.O..O.....
......OO.. .OOO...O... .........O.OO..OO.O.......OOO....
.......O.O O..O.OOOO.. ......OO...O......O........OO....
........O. OOO...OOO.. ......OO.....OO..................
O..O.....OO ........O...OOO..................
.OO........ ........O.O......................
The large spaceship on the right is important for the following reason: Whenever a spaceship is found which appears to be composed of components which are only loosely interacting (as in the above spaceship), then a good thing to try is to remove the components from the back end, and see what happens to the spaceship. In almost all cases, the ship will be destroyed. However, occasionally this doesn't occur, and something interesting will happen. (The best result is that a puffer engine might be found). In the case of the above spaceship, removing the tail component creates a period 9 spaceship, as shown below.
Period 9 spaceship (speed c/3):
........O.O.................. ........O...OOO.............. ......OO.....OO.............. ......OO...O......O........OO .........O.OO..OO.O.......OOO .....O...O......O..OO.O.O..O. ..O..O......O...OOOO...OO..OO .O.O..OOOOOO........O..O..OO. O..O..................O...OO. .O.O..OOOOOO........O..O..OO. ..O..O......O...OOOO...OO..OO .....O...O......O..OO.O.O..O. .........O.OO..OO.O.......OOO ......OO...O......O........OO ......OO.....OO.............. ........O...OOO.............. ........O.O..................
The spaceship creates a blinker at the back which unfortunately reacts with the tail again, and gets eaten. Along with some tagalongs which have the same tail end as this ship, this is the only method known to make a period 9 spaceship.
Following are two other small tagalongs which turn the period 9 ship back into a period 3 spaceship. Here the rightmost 3 columns from the spaceship above have been extracted, with the new tagalongs appended.
Alternative small tagalongs to make the period 9 spaceship into period 3:
.......O. ........O. ......O.O .......O.O .OO..OO.. .OO....O.. OOO...O.. OOO...O..O .O....OO. .O....OOO. .OO..O.O. .OO..O.... OO..OOO.. OO..OOO... OO..O.... OO..O..... OO..OOO.. OO..OOO... .OO..O.O. .OO..O.... .O....OO. .O....OOO. OOO...O.. OOO...O..O .OO..OO.. .OO....O.. ......O.O .......O.O .......O. ........O.
The large spark in the period 9 spaceship is useful. Running a LWSS into it can produce various debris. One of these reactions produces a loaf. When the loaf is properly hit with other LWSSs, it can be pulled backwards. This mechanism was found by Dean Hickerson, and is shown below.
Period 9 spaceship turns LW