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Newsgroups: comp.theory.cell-automata Path: sparky!uunet!munnari.oz.au!manuel!sserve!pdact!dbell From: dbell@pdact.pd.necisa.oz.au (David I. Bell) Subject: Spaceships in Conway's Life (Part 3a) Organization: NEC Information Systems Australia, Canberra Date: Fri, 11 Sep 1992 01:11:05 GMT Message-ID: <1992Sep11.011105.18848@pdact.pd.necisa.oz.au> Keywords: life Sender: news@pdact.pd.necisa.oz.au (News Holder) Lines: 819 Spaceships in Conway's Life (Part 3) by David I. Bell dbell@pdact.pd.necisa.oz.au 11 Sep 1992 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........................................................... 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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. --------------------- [Continued in next mail article] ----------------------