Newsgroups: comp.theory.cell-automata From: dbell@pdact.pd.necisa.oz.au (David I. Bell) Subject: Spaceships in Conway's Life (Addendum 1a) Organization: NEC Information Systems Australia, Canberra Date: Tue, 02 Mar 1993 04:57:14 GMT Message-ID: <1993Mar02.045714.24332@pdact.pd.necisa.oz.au> Lines: 971 Spaceships in Conway's Life (Addendum 1) by David I. Bell dbell@pdact.pd.necisa.oz.au 2 Mar 1993 This is an addendum to my series of articles concerning Conway's Game of Life. In this article, I will show the new results about spaceships that have been discovered since my series of articles was first posted in August through October of 1992. There have been new discoveries made in many of the categories of spaceships which I had reported on in my articles. The order of presentation in this article will be based on the period of the spaceship. So I will begin with the period 2 spaceships, and end with the large period spaceships. The vast majority of results in this article were found by Hartmut Holzwart. These are mostly period 2 and period 4 spaceships. (For period 4 spaceships, almost all the results are his.) I have contributed by finding some period 2 and period 3 spaceships, and working on large period spaceships. Dean Hickerson made a small but very significant contribution to these new results. So now I will begin with the new results for period 2 c/2 spaceships. Hartmut Holzwart found the following spaceship last September. One of his repeatable tagalongs from my article 2 has been slightly modified to allow two of them to connect together. This allows a spaceship to be constructed which is as long and as wide as desired, and is quite dense. [Dense period 2 spaceship (speed c/2)] ..............O................................... ...........OOOO................................... ...........OO..................................... .........O...OO................................... .........OOOO.O................................... ......O.O......................................... .....OOOOOO....................................... ....OO.....O...................................... ...OO....O........................................ ....OO...OO....................................... ..........OOO..................................... ....OO......O..................................... ...OO.O........................................... ..OO.............................................. .OO..O............................................ OO................................................ .OOOOO............................................ .................................................. .OOOOO............................................ OO................................................ .OO..O............................................ ..OO.............................................. ...OO.O........................................... ....OO............................................ .................................................. ....OO.............O.........O.........O.........O ...OO..OO........OOO.......OOO.......OOO.......OOO ..OO..OOO.......OO........OO........OO........OO.. .OOO.O..........O.........O.........O.........O... ..OO..OOO.....O.O.......O.O.......O.O.......O.O... ...OO..OO...OOO.......OOO.......OOO.......OOO..... ....OO.....OO.O......OO.O......OO.O......OO.O..... ...........O.O.OO....O.O.OO....O.O.OO....O.O.OO... ....OO.....OO..OO....OO..OO....OO..OO....OO..OO... ...OO..OO..O.....OO..O.....OO..O.....OO..O.....OO. ..OO..OOO..OO..OO.O..OO..OO.O..OO..OO.O..OO..OO.O. .OOO.O.....OO.O......OO.O......OO.O......OO.O..... ..OO..OOO..OO..OO.O..OO..OO.O..OO..OO.O..OO..OO.O. ...OO..OO..O.....OO..O.....OO..O.....OO..O.....OO. ....OO.....OO..OO....OO..OO....OO..OO....OO..OO... ...........O.O.OO....O.O.OO....O.O.OO....O.O.OO... ....OO.....OO.O......OO.O......OO.O......OO.O..... ...OO..OO...OOO.......OOO.......OOO.......OOO..... ..OO..OOO.....O.O.......O.O.......O.O.......O.O... .OOO.O..........O.........O.........O.........O... ..OO..OOO.....O.O.......O.O.......O.O.......O.O... ...OO..OO...OOO.......OOO.......OOO.......OOO..... ....OO.....OO.O......OO.O......OO.O......OO.O..... ...........O.O.OO....O.O.OO....O.O.OO....O.O.OO... ....OO.....OO..OO....OO..OO....OO..OO....OO..OO... ...OO..OO..O.....OO..O.....OO..O.....OO..O.....OO. ..OO..OOO..OO..OO.O..OO..OO.O..OO..OO.O..OO..OO.O. .OOO.O.....OO.O......OO.O......OO.O......OO.O..... ..OO..OOO..OO..OO.O..OO..OO.O..OO..OO.O..OO..OO.O. ...OO..OO..O.....OO..O.....OO..O.....OO..O.....OO. ....OO.....OO..OO....OO..OO....OO..OO....OO..OO... ...........O.O.OO....O.O.OO....O.O.OO....O.O.OO... ....OO.....OO.O......OO.O......OO.O......OO.O..... ...OO..OO...OOO.......OOO.......OOO.......OOO..... ..OO..OOO.....O.O.......O.O.......O.O.......O.O... .OOO.O..........O.........O.........O.........O... ..OO..OOO.....O.O.......O.O.......O.O.......O.O... ...OO..OO...OOO.......OOO.......OOO.......OOO..... ....OO.....OO.O......OO.O......OO.O......OO.O..... ...........O.O.OO....O.O.OO....O.O.OO....O.O.OO... ....OO.....OO..OO....OO..OO....OO..OO....OO..OO... ...OO..OO..O.....OO..O.....OO..O.....OO..O.....OO. ..OO..OOO..OO..OO.O..OO..OO.O..OO..OO.O..OO..OO.O. .OOO.O.....OO.O......OO.O......OO.O......OO.O..... ..OO..OOO..OO..OO.O..OO..OO.O..OO..OO.O..OO..OO.O. ...OO..OO..O.....OO..O.....OO..O.....OO..O.....OO. ....OO.....OO..OO....OO..OO....OO..OO....OO..OO... ...........O.O.OO....O.O.OO....O.O.OO....O.O.OO... ....OO.....OO.O......OO.O......OO.O......OO.O..... ...OO..OO...OOO.......OOO.......OOO.......OOO..... ..OO..OOO.....O.O.......O.O.......O.O.......O.O... .OOO.O..........O.........O.........O.........O... ..OO..OOO.......OO........OO........OO........OO.. ...OO..OO........OOO.......OOO.......OOO.......OOO ....OO.............O.........O.........O.........O .................................................. ....OO............................................ ...OO.O........................................... ..OO.............................................. .OO..O............................................ OO................................................ .OOOOO............................................ .................................................. .OOOOO............................................ OO................................................ .OO..O............................................ ..OO.............................................. ...OO.O........................................... ....OO......O..................................... ..........OOO..................................... ....OO...OO....................................... ...OO....O........................................ ....OO.....O...................................... .....OOOOOO....................................... ......O.O......................................... .........OOOO.O................................... .........O...OO................................... ...........OO..................................... ...........OOOO................................... ..............O................................... Hartmut also found two new "arm" components for period 2 spaceships. These arms are derived from the "glancing head" component, and have slopes of 5/17 and 5/13. The following spaceship shows two copies of both of these components. [Period 2 spaceship with arm components (speed c/2)] .............................O.O ............................O..O ...........................OO... ..........................O...O. .........................OOO.O.. ........................O....... .......................O..OOOO.. .......................O..O..... .......................OOO.OOO.. .........................OO..... ............................O... ..........................O..... ..........................OO.... ........................OOO..... .......................O...O.... ......................OO........ .....................O...O...... ....................OOO.O....... ...................O............ ..................O..OOOO....... ..................O..O.......... ..................OOO.OOO....... ....................OO.......... .......................O........ .....................O.......... .....................OO......... ...................OOO.......... ..................O...O......... .................OO............. ................O...O........... ...............OOO.O............ ..............O................. .............O..OOOO............ .............O..O............... .............OOO.OOO............ ...............OO............... ..................O............. ...............O.O.............. ..............OO................ ...............OO............... ................OOO............. .................O.............. ...................O............ ..............O.O.O............. .............O..O............... ............OO.................. ...........O...O................ ..........OOO.O................. .........O...................... ........O..OOOO................. ........O..O.................... ........OOO.OOO................. ..........OO.................... .............O.................. ..........O.O................... .........OO..................... ..........OO.................... ...........OOO.................. ............O................... ..............O................. .........O.O.O.................. ........O..O.................... .......OO....................... ......O......................... .....OOOOOO..................... ..OO.......O.................... .O...OOO.O...................... O...O....O...................... O.....O....O.................... OOO...OOOOO..................... ................................ OOO...OOOOO..................... O.....O....O.................... O...O....O...................... .O...OOO.O...................... ..OO.......O.................... .....OOOOOO..................... ......O......................... .......OO....................... ........O..O.................... .........O.O.................... Hartmut also found the following spaceship, which has repeatable glide- reflective wicks trailing from the ends. Between the two main components is a large repeatable component which is only a slightly modified version of Dean Hickerson's long period 2 spaceship. The central component was found using Hartmut's cylindrical search feature of the life search program. [Period 2 spaceship with glide-reflective wick and repeatable component (speed c/2)] ...............................................O........ ...................O.............O..........OO.O........ ................OOOO..........OOOO..........O........... ................OO............OO............O........... ............O.O...........O.O...........O.O............. ..........OO...OOOOO....OO...OOOOO....OO...OOO.......O.. ........OO...O........OO...O........OO...O.........OOO.. ........OOO.O..OOO.O..OOO.O..OOO.O..OOO.O..OOO....OO.... ......O........OO...O........OO...O........OO.O...O..... ......OOOOOOO....OO...OOOOO....OO...OOOOO....OO.....O... ...O.O.............O.O...........O.O..........OOOOOO.... ..OOOOOOO.O............OO............OO........O.O...... .OO.......O............OOOO..........OOOO.........OOOO.O OO....OO..................O.............O.........O...OO .OO....O.O..........................................OO.. ....................................................OOOO .OO....O.O.............................................O OO....OO.......O.O...................................... .OO.......O..O.O........................................ ..OOOOOOO.O.O.O...O..................................... ...O.O........OOO.O..................................... ......OOO.OO............................................ ......O...OO.OO......................................... ........OOO............................................. ........OOO............................................. ........................................................ ........OOO............................................. ........OOO............................................. ......O...OO.OO......................................... ......OOO.OO............................................ ...O.O........OOO.O..................................... ..OOOOOOO.O.O.O...O..................................... .OO.......O..O.O........................................ OO....OO.......O.O...................................... .OO....O.O.............................................. ........................................................ .OO....O.O.............................................. OO....OO.......O.O...................................... .OO.......O..O.O........................................ ..OOOOOOO.O.O.O...O..................................... ...O.O........OOO.O..................................... ......OOO.OO............................................ ......O...OO.OO......................................... ........OOO............................................. ........OOO............................................. ........................................................ ........OOO............................................. ........OOO............................................. ......O...OO.OO......................................... ......OOO.OO............................................ ...O.O........OOO.O..................................... ..OOOOOOO.O.O.O...O..................................... .OO.......O..O.O........................................ OO....OO.......O.O...................................... .OO....O.O.............................................O ....................................................OOOO .OO....O.O..........................................OO.. OO....OO..................O.............O.........O...OO .OO.......O............OOOO..........OOOO.........OOOO.O ..OOOOOOO.O............OO............OO........O.O...... ...O.O.............O.O...........O.O..........OOOOOO.... ......OOOOOOO....OO...OOOOO....OO...OOOOO....OO.....O... ......O........OO...O........OO...O........OO.O...O..... ........OOO.O..OOO.O..OOO.O..OOO.O..OOO.O..OOO....OO.... ........OO...O........OO...O........OO...O.........OOO.. ..........OO...OOOOO....OO...OOOOO....OO...OOO.......O.. ............O.O...........O.O...........O.O............. ................OO............OO............O........... ................OOOO..........OOOO..........O........... ...................O.............O..........OO.O........ ...............................................O........ Hartmut's cylindrical search also found several other repeatable components related to the one in the above spaceship. Most of these join each other in the same manner in one spot. [Period 2 spaceship demonstrating several repeating components (speed c/2)] .............O......... ...........OO.......... ........OOOO.O......... ........OO............. ......O...OO.O......... ......OOOO.O.O.OO...... ...O.O......OO......... ..OOOOOO.OO.OO.O....O.. .OO.....OO...O...OOOO.. OO....OO....O....OO.... .OO....OOOO............ ....................... .OO....OOOO............ OO....OO....O....OO.... .OO.....OO...O...OOOO.. ..OOOOOO.OO.OO.O....O.. ...O.O......OO......... ......OOOO.O.O.OO...... ......O...OO.O......... ........OOO............ ........OOO............ ....................... ........OOO............ ........OOO......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.........O .OO.....OO...O.....OOOO ..OOOOOO.OO.OO.....OO.. ...O.O......OO.O.O..... ......OOOO.O.O.OO.OO.O. ......O...OO..O..OO.O.O ........OOO..OO....OO.. ........OOO...O.OOO.O.. ....................... ........OOO...O.OOO.O.. ........OOO..OO....OO.. ......O...OO..O..OO.O.O ......OOOO.O.O.OO.OO.O. ...O.O......OO.O.O..... ..OOOOOO.OO.OO.....OO.. .OO.....OO...O.....OOOO OO....OO....O.........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.. ........OOO......OO.... ........OOO............ ....................... ........OOO............ ........OOO............ ......O...OO.O......... ......OOOO.O.O...O.O... ...O.O......OO.OO..O... ..OOOOOO.OO.OO.OO...... .OO.....OO...O....O.... OO....OO....O..OO...... .OO....OOOO....OOO..... ....................... .OO....OOOO....OOO..... OO....OO....O..OO...... .OO.....OO...O....O.... ..OOOOOO.OO.OO.OO...... ...O.O......OO.OO..O... ......OOOO.O.O...O.O... ......O...OO.O......... ........OOO............ ........OOO............ ....................... ........OOO............ ........OOO............ ......O...OO.O......... ......OOOO.O.O......... ...O.O......OO.OO..O... ..OOOOOO.OO.OO.OO..O... .OO.....OO...O....O.... OO....OO....O..OO...... .OO....OOOO....OOO..... ....................... .OO....OOOO....OOO..... OO....OO....O..OO...... .OO.....OO...O....O.... ..OOOOOO.OO.OO.OO..O... ...O.O......OO.OO..O... ......OOOO.O.O......... ......O...OO.O......... ........OOO............ ........OOO............ ....................... ........OOO............ ........OOO........O.O. ......O...OO.O....O..O. ......OOOO.O.O.O.OO.... ...O.O......OO......... ..OOOOOO.OO.OO.OOOOOO.O .OO.....OO...O...O...OO OO....OO....O......OOO. .OO....OOOO........OOO. ....................... .OO....OOOO........OOO. OO....OO....O......OOO. .OO.....OO...O...O...OO ..OOOOOO.OO.OO.OOOOOO.O ...O.O......OO......... ......OOOO.O.O.O.OO.... ......O...OO.O....O..O. ........OOO........O.O. ........OOO............ ....................... ........OOO............ ........OOO............ ......O...OO.O......... ......OOOO.O.O.OO...... ...O.O......OO......... ..OOOOOO.OO.OO.O....O.. .OO.....OO...O...OOOO.. OO....OO....O....OO.... .OO....OOOO............ ....................... .OO....OOOO............ OO....OO....O......O.O. .OO.....OO...O....O..O. ..OOOOOO.OO.OO.O.OO.... ...O.O......OO......... ......OOOO.O.O.OOOOOO.O ......O...OO.O...O...OO ........OOO........OOO. ........OOO........OOO. ....................... ........OOO........OOO. ........OOO........OOO. ......O...OO.O...O...OO ......OOOO.O.O.OOOOOO.O ...O.O......OO......... ..OOOOOO.OO.OO.O.OO.... .OO.....OO...O....O..O. OO....OO....O......O.O. .OO....OOOO............ ....................... .OO....OOOO............ OO....OO....O....OO.... .OO.....OO...O...OOOO.. ..OOOOOO.OO.OO.O....O.. ...O.O......OO......... ......OOOO.O.O.OO...... ......O...OO.O......... ........OO............. ........OOOO.O......... ...........OO.......... .............O......... Hartmut looked for period 2 spaceships with even symmetry, and found that there are no spaceships with a width less than 28. The smallest minimum width spaceship which has even symmetry is the following. Removing one of the two middle rows will produce a spaceship with odd symmetry (but not the one of minimum width). [Period 2 spaceship with even symmetry of minimum width 28 (speed c/2)] .........O.O ........O..O .......OO... ......O....O .....OOOOO.O ..OO........ .O...OOO.... O...O....... O.....O..... O..O..O..... .O.O..OO.O.. .........O.. OOO......... OO.......... OO.......... OOO......... .........O.. .O.O..OO.O.. O..O..O..... O.....O..... O...O....... .O...OOO.... ..OO........ .....OOOOO.O ......O....O .......OO... ........O..O .........O.O Hartmut found several modifications and tagalongs for the above spaceship. Two of these are shown below. [Period 2 spaceship with large tagalongs (speed c/2)] .........OO.................O..... ........O...............OO.OO..O.. .......OO..O...........O.....O.O.. ......O....O..........OO..OO...... .....OOOOO...........O............ ..OO.....OO....O....OOO.OO.O.....O .O...OOO..O..OOO...O.....O....OOOO O...O.....OO.OO...OOO....OO...OO.. O.....O...O...O..O.....OO...O..... O..O..O.....OOO.OOO....O..OO...... .O.O..OO.O..OO.....O...O.O........ .........O.....OOO.........OOO.... OOO............OO.....OOO..O..O... OO...............O....OO...OO..... OO...............O....OO...OO..... OOO............OO.....OOO..O..O... .........O.....OOO.........OOO.... .O.O..OO.O..OO.....O...O.O........ O..O..O.....OOO.OOO....O..OO...... O.....O...O...O..O.....OO...O..... O...O.....OO.OO...OOO....OO...OO.. .O...OOO..O..OOO...O.....O....OOOO ..OO.....OO....O....OOO.OO.O.....O .....OOOOO...........O............ ......O....O..........OO..OO...... .......OO..O...........O.....O.O.. ........O...............OO.OO..O.. .........OO.................O..... .........OO.................O........... ........O...............OO.OO..O........ .......OO..O...........O.....O.O........ ......O....O..........OO..OO............ .....OOOOO...........O.................O ..OO.....OO....O....OOO.OO.O.....O.O.OO. .O...OOO..O..OOO...O.....O....OOOO.O.O.O O...O.....OO.OO...OOO....OO...OO..O..... O.....O...O...O..O.....OO...O.....OOO... O..O..O.....OOO.OOO....O..OO......O..... .O.O..OO.O..OO.....O...O.O........O.O... .........O.....OOO.........OOO.......... OOO............OO.....OOO..O..O..OOO.... OO...............O....OO...OO....OO..... OO...............O....OO...OO....OO..... OOO............OO.....OOO..O..O..OOO.... .........O.....OOO.........OOO.......... .O.O..OO.O..OO.....O...O.O........O.O... O..O..O.....OOO.OOO....O..OO......O..... O.....O...O...O..O.....OO...O.....OOO... O...O.....OO.OO...OOO....OO...OO..O..... .O...OOO..O..OOO...O.....O....OOOO.O.O.O ..OO.....OO....O....OOO.OO.O.....O.O.OO. .....OOOOO...........O.................O ......O....O..........OO..OO............ .......OO..O...........O.....O.O........ ........O...............OO.OO..O........ .........OO.................O........... The second spaceship above differs by the addition of a short wide tagalong which attaches to the feet of the previous tagalong. This tagalong can also be used in other spaceships where the feet appear, to join the two spaceships. For example, an odd-symmetrical version of the tagalong is shown below which joins two old spaceships. [Period 2 spaceships joined by bridging tagalong (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............O ......OOOO.O.O.O....O.O.OO. ......O...OO.O...OOOO.O.O.O ........OO.......OO..O..... ........OOOO.O.......OOO... ...........OO........O..... .............O.......O.O... ........................... ....................OOO.... ....................OO..... ....................OOO.... ........................... .............O.......O.O... ...........OO........O..... ........OOOO.O.......OOO... ........OO.......OO..O..... ......O...OO.O...OOOO.O.O.O ......OOOO.O.O.O....O.O.OO. ...O.O......OO............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............. In one of Hartmut's results there appeared an L-shaped repeating component which is very small and simple. I was asked to "tame" it by finding components which terminated the two ends. Amazingly, the component that I found for the front of the L components connects to them on both of its sides. The following spaceship shows the front component, two arms made of the L components, and the back component that I found. Also shown is one of a number of "feet" which can be attached to the L component. [Period 2 spaceship with simple "L" arms (speed c/2)] ............................O .........................OOOO .........................OO.. .......................O...OO .......................OOOO.O ....................O.O...... ...................OOOOOO.... ..................OO.....O... .................OO....O..... ..................O.O..OO.... ........................OOO.. ..................OO......O.. .................O........... .................OOO......... ...............O............. ...............OOO........... .............O............... .............OOO............. ...........O................. ...........OOO............... .........O................... .........OOO................. .......O.....OO.............. .......OOO...OOOO............ .....O..........O............ .....OOO..................... ...O......................... ...OOO....................... ..O.......................... .OOO.O....................... .O........................... .OOOOO....................... ........O.O.................. .O.O...O..O.................. OO....OO..................... .OO......O................... ..OOOOO.O.................... ...O.O....................... ......OOO.................... ......O...................... ........OOO.................. ........O.................... ..........OOO................ ..........O.................. ............OOO.............. ............O................ ..............OOO............ ..............O.............. ...............OO......O..... .....................OOO..... ...............O.O..OO....... ..............OO....O........ ...............OO.....O...... ................OOOOOO....... .................O.O......... ....................OOOO.O... ....................O...OO... ......................OO..... ......................OOOO... .........................O... I did not find a front component for a single arm of the L components. But I did find that the L component can be connected to the standard component that connects to the feet of many period 2 spaceships. This is shown below, where the L arm is connected to the back foot of the wicktrailer on a base spaceship. [Period 2 spaceship with L arm component connection (speed c/2)] ............................O..................... .............O...........OOOO..................... ...........OO..........OOO.O...................... ........OOOO.O........OO...O...................... ........OO.......OO...O....OO..................... ......O...OO.O...OOOO.O.....OOO.O................. ......OOOO.O.O.O....OOOO......O.O................. ...O.O......OO.........O........OO................ ..OOOOOO.OO.OO.OO..............OO................. .OO.....OO...O................O..O................ OO....OO....O................O.................... .OO....OOOO..................O.................... .............................OOO.................. .OO....OOOO...................O................... OO....OO....O.................OOOO................ .OO.....OO...O...............O....O............... ..OOOOOO.OO.OO.OO.............O.O................. ...O.O......OO................O...O............... ......OOOO.O.O.O....O..........OOO................ ......O...OO.O...OOOO...........O................. ........OO.......OO..............OOO.............. ........OOOO.O....................O............... ...........OO......................OOO............ .............O......................O............. .....................................OOO.......... ......................................O........... .......................................O.......O.. .......................................O....OO.O.. ......................................OOO...O..... ......................................O.....O..... ......................................O...O....... .......................................O...OOO.... ........................................OO........ ...........................................OOOOO.O ............................................O....O .............................................OO... ..............................................O..O ...............................................O.O I found another very messy connection of the L component to the feet of period 2 spaceships. This is illustrated below. The almost-symmetrical component in the middle is interesting, for it appears that it might be made larger somehow. But this has not been done yet. [Messy connection of L component with base period 2 spaceship (speed c/2)] .........................O ......................OOOO ......................OO.. ....................O...OO ....................OOOO.O .................O.O...... ................OOOOOO.... ...............OO.....O... ..............OO....O..... ...............O.O..OO.... .....................OOO.. ...............OO......O.. ..............O........... ..............OOO......... ............O............. ............OOO........... ..........O............... ..........OOO............. ........O................. ........OOO............... .......O.................. ......OOO.O............... ......O................... ......OOOOO............... .............O.O.......... ......O.O...O..O.......... .....OO....OO............. ......OO.................. .......OOOOOOOO........... ........O.O....O.......... ..........OO.O............ ..........O..O............ ...............O.......... ..........OOOOO........... .........OO............... ........OO..OOO........... .......OOO.O...O.......... ........OO..OO............ .........OO....O.......... ..........OOOO.O.......... ..............O........... ........OOOO.O.O.......... .......OO...OO............ ......OO..OO...O.......... .....OOO..OOOO.O.......... ......O......O.O.......... .......O.................. .......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.................. I found another "barber pole" component for period 2 spaceships. Like the previous barber pole component, this one appears the same in both phases, only shifted. The effect is not as obvious in this example, however, since the component is so large. The component can have feet attached to it at many places, as is shown below. [Period 2 spaceship with "barber pole" repeating component (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.. ..................... .................O.O. ................OO..O ................O.O.. ................OO... ................O.... ...............OOO... ...............O..O.. ..............O..O... ..............OO..... ..................... ..............O.O.... .............OO..O... .............O.O..... .............OO...... .............O....... ............OOO...... ............O..O..... ...........O..O...... ...........OO........ ..................... ...........O.O....... ..........OO..O...... ..........O.O...OO... ..........OO....OOOO. ..........O........O. .........OOO......... .........O..O........ ........O..O......... ........OO........... ..................... ........O.O.......... .......OO..O......... .......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............. That was a summary of the new developments for period 2 spaceships. Now I will move on to period 3 c/3 spaceships. There are not many new results to report for period 3 spaceships. All of these results were found by me. Newsgroups: comp.theory.cell-automata From: dbell@pdact.pd.necisa.oz.au (David I. Bell) Subject: Spaceships in Conway's Life (Addendum 1b) Organization: NEC Information Systems Australia, Canberra Date: Tue, 02 Mar 1993 04:58:02 GMT Message-ID: <1993Mar02.045802.24394@pdact.pd.necisa.oz.au> Lines: 1063 Ever since I found the period 9 tail to period 3 c/3 spaceships, I have wanted to turn that tail into a small component compatible with the rest of Dean Hickerson's period 3 grammar. At the beginning of 1993, I finally succeeded. The resulting spaceship is shown below. [Period 9 tail connecting to period 3 components (speed c/3)] .......O...... ......O.O..... ......O.O..... ......O....... .....OO....... .....O.O...... .............. ....OOO....... ....OO........ .............. ......OOO..... .....O.OO..... ....O......... ....O......... ...OOO........ ..O........... ..O..O........ ..O..OO....... .....O........ ....OO........ .....OO....... ..O.....O..... ..O..O.O...... .O.O..O....... OOO........... ....OOO.O..... ...OOO..O..... .OO........... .OOO.......... .............. .O.O.......... .O.O.......... .O..OO........ ..O.OO........ ..O........... ....O......... ..OOO......... ..OOO......... ......OO...... ......OOO..... .............. ......OOO..... ...O..O....... ..O.O.O....... .O..O.O.O..... ..O...O....... ...OO......... ..OO...O...... .OOOOO........ .O...OOO...... ...O.......... ...OO......... .............. ..O..O........ ..O..OO....... .OO...O....... .O....O....... .OO.OOO....... ..O...O....... ..OO...OO..... ...O.OOO.....O ....OO.O....OO ......O...O.O. ......OOOO..O. ............O. ............O. ............O. ......OOOO..O. ......O...O.O. ....OO.O....OO ...O.OOO.....O ..OO...OO..... ..O...O....... .OO.OOO....... .O....O....... .OO...O....... ..O..OO....... ..O..O........ .............. ...OO......... ...O.......... .O...OOO...... .OOOOO........ ..OO...O...... ...OO......... ..O...O....... .O..O.O.O..... ..O.O.O....... ...O..O....... ......OOO..... .............. ......OOO..... ......OO...... ..OOO......... ..OOO......... ....O......... ..O........... ..O.OO........ .O..OO........ .O.O.......... .O.O.......... .............. .OOO.......... .OO........... ...OOO..O..... ....OOO.O..... OOO........... .O.O..O....... ..O..O.O...... ..O.....O..... .....OO....... ....OO........ .....O........ ..O..OO....... ..O..O........ ..O........... ...OOO........ ....O......... ....O......... .....O.OO..... ......OOO..... .............. ....OO........ ....OOO....... .............. .....O.O...... .....OO....... ......O....... ......O.O..... ......O.O..... .......O...... This is a large solution to the problem, however. It only connects to Dean's grammar at about 13 cells from either end. The rest of it is made up of new components. I then looked for a smaller solution. The following shows two things. Firstly, a shorter arm is demonstrated which connects to the p9 component. This is seen at the top and bottom. The component has the same two-bit spark as is produced by Dean's "turtle" spaceship. Secondly, between two copies of the period 9 component is a large symmetrical component. Using this you can make a wide spaceship with as many period 9 components in it as you want. [Period 3 spaceship with several period 9 components (speed c/3)] ...O......... ..O.O........ .OO.......... ...O......... .OO.......... .OOO......... ............. ..O.O........ ..OOO........ ............. .....OOO..... .....OOO..... ............. ....OO....... ....OO....... ......O...... ............. ...OOO....... ...O.OO...... ......O...... ..OOO.O...... .O....O...... .OO.OOO...... OO.O..O...... ....OO....... .O..OO....... .O..O........ .O.OO........ ..O.OO.O..... .....O.O..... ............. .....OO...... ....O..O..... ...OOO....... .......O..... ......O...... ..O.......... .OO.O........ .O..OO....... OO...O....... O....O....... OO.OOO....... .O...O....... .OO...OO..... ..O.OOO.....O ...OO.O....OO .....O...O.O. .....OOOO..O. ...........O. ...........O. ...........O. .....OOOO..O. .....O...O.O. ...OO.O....OO ..O.OOO.....O .OO...OO..... .O...O....... OO.OOO....... O....O....... OO...O....... .O..OO....... .OO.O........ ..O.......... ......O...... .......O..... ...OOO....... ....O..O..... .....OO...... ............. .....O.O..... ..O.OO.O..... .O.OO........ .O..O........ .O..OO....... ...OO........ OO..OO....... .OO.......... ..OOOO....... ......O...... ..OOO........ OO....O...... OO..O.O...... O..O......... OO.OOO....... ....OO....... ..O...OO..... ..O.OOOO..... ..O.......... ...O......... ....OOO...... ...OO.O...... ..O.......... .O..O........ ..O.OOO...... ............. ..OOO........ ...OOO.O..... .....O....... .....O....... .....OO...... ............. .....OO...... .....O....... .....O....... ...OOO.O..... ..OOO........ ............. ..O.OOO...... .O..O........ ..O.......... ...OO.O...... ....OOO...... ...O......... ..O.......... ..O.OOOO..... ..O...OO..... ....OO....... OO.OOO....... O..O......... OO..O.O...... OO....O...... ..OOO........ ......O...... ..OOOO....... .OO.......... OO..OO....... ...OO........ .O..OO....... .O..O........ .O.OO........ ..O.OO.O..... .....O.O..... ............. .....OO...... ....O..O..... ...OOO....... .......O..... ......O...... ..O.......... .OO.O........ .O..OO....... OO...O....... O....O....... OO.OOO....... .O...O....... .OO...OO..... ..O.OOO.....O ...OO.O....OO .....O...O.O. .....OOOO..O. ...........O. ...........O. ...........O. .....OOOO..O. .....O...O.O. ...OO.O....OO ..O.OOO.....O .OO...OO..... .O...O....... OO.OOO....... O....O....... OO...O....... .O..OO....... .OO.O........ ..O.......... ......O...... .......O..... ...OOO....... ....O..O..... .....OO...... ............. .....O.O..... ..O.OO.O..... .O.OO........ .O..O........ .O..OO....... ....OO....... OO.O..O...... .OO.OOO...... .O....O...... ..OOO.O...... ......O...... ...O.OO...... ...OOO....... ............. ......O...... ....OO....... ....OO....... ............. .....OOO..... .....OOO..... ............. ..OOO........ ..O.O........ ............. .OOO......... .OO.......... ...O......... .OO.......... ..O.O........ ...O......... The structure of the middle 3 rows in the above spaceship intrigued me, and I then looked for more components that joined in that same manner. The following shows the results. At the top is the smallest terminating component that contains that connection (about 20 rows). Below that is a symmetrical component which includes a middle piece which is also in some period 2 spaceships, but which here acts as period 3 (about 26 rows). Below that is another symmetrical component (about 40 rows). And at the bottom is another copy of the termination component. [Period 3 spaceship with components connected in common manner (speed c/3)] ....O.. ...O.O. ..OO... ...O... ..O.O.. ..O.... .O..... .O.O... .O.O... ..OOOO. ....O.O ....O.O ...O... ...OOO. .O..... .OOO... ..OOO.O ....O.. ....O.. ....OO. ....... ....OO. ....O.. ....O.. ..OOO.O .OOO... ....... .O.OOO. O..O... .O.O... ..O..OO .....OO ....OO. ...O.O. ..O.... ...O.O. ....OO. .....OO ..O..OO .O.O... O..O... .O.OOO. ....... .OOO... ..OOO.O ....O.. ....O.. ....OO. ....... ....OO. ....O.. ....O.. ..OOO.O .OOO... ....... .O.OOO. O..O... .O.O... ..O..OO .....OO ....OO. ...O.O. ...O... ..O..O. ..OOO.O .O..O.. ..OOOOO ..O..O. ...OOOO .OO.... ...OOOO ..O..O. ..OOOOO .O..O.. ..OOO.O ..O..O. ...O... ...O.O. ....OO. .....OO ..O..OO .O.O... O..O... .O.OOO. ....... .OOO... ..OOO.O ....O.. ....O.. ....OO. ....... ....OO. ....O.. ....O.. ..OOO.O .OOO... .O..... ...OOO. ...O... ....O.O ....O.O ..OOOO. .O.O... .O.O... .O..... ..O.... ..O.O.. ...O... ..OO... ...O.O. ....O.. A few months ago Hartmut Holzwart did a little searching for cylindrical components for period 3 spaceships. He found an interesting partial result, but could not finish it. Recently I looked at it and found a way to tame it. Here is the finished object. It is a dense c/3 spaceship, which can be made as wide and as long as desired. Hartmut had found both of the repeatable components in the center. The thin stabilizing pieces on the sides of the front are just the terminating component of the spaceship shown above. At least one column of the repeating tagalongs is required to stabilize the front of the spaceship. So here is a case where the first column of tagalongs aren't really tagalongs, but extra copies of the same component are! [Dense period 3 spaceship (speed c/3)] ....O .............................................................. ...O.O.............................................................. ..OO................................................................ ...O................................................................ ..O.O............................................................... ..O................................................................. .O.................................................................. .O.O................................................................ .O.O................................................................ ..OOOO.............................................................. ....O.O............................................................. ....O.O............................................................. ...O................................................................ ...OOO.............................................................. .O.................................................................. .OOO................................................................ ..OOO.O............................................................. ....O............................................................... ....O............................................................... ....OO.............................................................. .................................................................... ....OO.....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O. ....O..O..O...O..OO...O...O..OO...O...O..OO...O...O..OO...O...O..OO. ....O.OO.O.....O.....O.....O.....O.....O.....O.....O.....O.....O.... .....OOO..O......O.O..O......O.O..O......O.O..O......O.O..O......O.O .....OO.....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O .......O...O...O.......O...O.......O...O.......O...O.......O...O.... .....OO....O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O ......O.O..O...........O...........O...........O...........O........ .......O....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO .................................................................... .......O....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO ......O.O..O...........O...........O...........O...........O........ .....OO....O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O .......O...O...O.......O...O.......O...O.......O...O.......O...O.... .....OO.....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O .....OOO..O......O.O..O......O.O..O......O.O..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..OO...O...O..OO...O...O..OO. ....OO.....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O. .................................................................... ....OO.....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O. ....O..O..O...O..OO...O...O..OO...O...O..OO...O...O..OO...O...O..OO. ....O.OO.O.....O.....O.....O.....O.....O.....O.....O.....O.....O.... .....OOO..O......O.O..O......O.O..O......O.O..O......O.O..O......O.O .....OO.....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O .......O...O...O.......O...O.......O...O.......O...O.......O...O.... .....OO....O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O ......O.O..O...........O...........O...........O...........O........ .......O....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO .................................................................... .......O....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO ......O.O..O...........O...........O...........O...........O........ .....OO....O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O .......O...O...O.......O...O.......O...O.......O...O.......O...O.... .....OO.....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O .....OOO..O......O.O..O......O.O..O......O.O..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..OO...O...O..OO...O...O..OO. ....OO.....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O. .................................................................... ....OO.....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O....OO.O.O.O. ....O..O..O...O..OO...O...O..OO...O...O..OO...O...O..OO...O...O..OO. ....O.OO.O.....O.....O.....O.....O.....O.....O.....O.....O.....O.... .....OOO..O......O.O..O......O.O..O......O.O..O......O.O..O......O.O .....OO.....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O .......O...O...O.......O...O.......O...O.......O...O.......O...O.... .....OO....O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O ......O.O..O...........O...........O...........O...........O........ .......O....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO .................................................................... .......O....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO....OOOOOOOO ......O.O..O...........O...........O...........O...........O........ .....OO....O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O...O.OOO...O .......O...O...O.......O...O.......O...O.......O...O.......O...O.... .....OO.....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O....OO..OO.O .....OOO..O......O.O..O......O.O..O......O.O..O......O.O..O......O.O ....O.OO.O.....