Spaceship
A spaceship (also referred to as a glider[1], or less commonly a fish[2], and commonly shortened to "ship") is a finite pattern that returns to its initial state after a number of generations (known as its period) but in a different location.
Spaceship speed
- Main article: Speed
The speed of a spaceship is the number of cells that the pattern moves during its period, divided by the period length. This is expressed in terms of c (the metaphorical "speed of light") which is one cell per generation; thus, a spaceship with a period of five that moves two cells to the left during its period travels at the speed of 2c/5.
Until the construction of Gemini in May 2010, all known spaceships in Life traveled either orthogonally (only horizontal or vertical displacement) or diagonally (equal horizontal and vertical displacement); other oblique spaceships have been constructed since then, e.g. waterbear and the Parallel HBK. It is known that Life has spaceships that travel in all rational directions at arbitrarily slow speeds (see universal constructor).
Spaceships traveling in other directions and at different speeds have also been constructed in other two-dimensional cellular automata.[3]
The maximum speed for Life is c/2 orthogonal and c/4 diagonal[4] (and continually, (2, 1)c/6 for slope 2, (3, 1)c/8 for slope 3, etc.); however, certain agar crawlers such as lightspeed wire can break this speed in their respective medium. Spaceships in the traditional Moore neighbourhood of range 1 have a maximum speed of c, although Larger than Life neighbourhoods can increase this limit depending on the interactable distance between cells. For Life, no spaceship can move at speed (m,n)c/x where (m+n)/x > 1/2; for other non-strobing outer-totalistic and non-totalistic rules, the limit is at most 1, and in strobing rules, 3/2.[5] Certain range-1 non-isotropic rules can harbour c diagonal spaceships, giving a limit of 2.
Spaceship types
- Main article: Types of spaceships
Spaceships are most commonly sorted by their speed and direction, and sometimes their period. They can be separated into three fundamental categories:
Elementary spaceships
Elementary spaceships are the smallest classification of spaceships. This class consists of naturally-occurring ships, as well as those found by direct computer search, such as the weekender.
Despite their generally small size, only a few of them have been known to have emerged from soups.
Constructing guns for elementary spaceships is usually difficult, as only a few have known glider syntheses (with those that do generally requiring a lot of gliders in awkward positions), with larger elementary spaceships tending to contain high concentrations of space dust.
The following table lists the smallest known cases for the lowest possible periods for each speed:
| Speed | Slope | Direction | Smallest known | Minimum # of cells |
|---|---|---|---|---|
| c/2 | 0 | orthogonal | 64P2H1V0 | 64 |
| c/3 | 0 | orthogonal | 25P3H1V0.1[n 1] | 25 |
| c/4 | 0 | orthogonal | 37P4H1V0 | 37 |
| c/5 | 0 | orthogonal | spider | 58 |
| 2c/5 | 0 | orthogonal | 30P5H2V0 | 30 |
| c/6 | 0 | orthogonal | 56P6H1V0 | 56 |
| c/7 | 0 | orthogonal | loafer | 20 |
| 2c/7 | 0 | orthogonal | weekender | 36 |
| 3c/7 | 0 | orthogonal | 232P7H3V0 | 232 |
| c/10 | 0 | orthogonal | copperhead | 28 |
| c/4 | 1 | diagonal | glider | 5 |
| c/5 | 1 | diagonal | 58P5H1V1 | 58 |
| c/6 | 1 | diagonal | 77P6H1V1 | 77 |
| c/7 | 1 | diagonal | lobster | 83 |
| c/8 | 1 | diagonal | walrus | 68 |
| (2,1)c/6 | 2 | knightwise | Sir Robin | 282 |
- ↑ 25P3H1V0.2 has the same population of 25, but has a more expansive bounding box
The following is an incomplete list of some higher-period ships which travel at the same velocities:[6]
Engineered spaceships
Engineered spaceships are defined as spaceships consisting of small interacting components.[8] Some engineered spaceships are composed of hundreds of thousands or even millions of active cells; these are also sometimes referred to as caterpillars after the first engineered macro-spaceship. Other simpler mechanisms like the Corderships might need a relatively small number of components, down to a minimum of just two for the 2-engine Cordership. Such spaceships have occurred naturally in other rules, but not yet in Conway Life.
Engineered spaceships have fixed speeds, because their mechanisms depend on supporting a specific active reaction that travels at one particular speed. Some rely on "crawlers", reactions in which a pattern reacts with another pattern, producing both the patterns in different positions as in the pi crawler. Others, like the Corderships and pufferfish spaceships, rely on stabilised puffer engines, and are often referred to as "Corderoids". The process of stabilising a moving engine is referred to as corderisation.
| Speed | Direction | Smallest known | Minimum # of cells | Classification |
|---|---|---|---|---|
| 17c/45 | orthogonal | caterpillar | 11,880,063 | Crawler-based |
| 31c/240 | orthogonal | silverfish | 210,108 | Crawler-based |
| (23,5)c/79 | slope 23/5 | waterbear | 197,896 | Crawler-based |
| (34,7)c/156 | slope 34/7 | Unnamed (34,7)c/156 spaceship | 25,013,268 | Crawler-based |
| (6,3)c/1024 | knightwise | HBK caterpillar | 329071 | Crawler-based |
| c/2 | orthogonal | pufferfish spaceship | 235 | Stabilized puffer engine |
| c/12 | diagonal | 2-engine Cordership | 100 | Stabilized puffer engine |
Traveling signal loops
A traveling signal loop is a subcategory somewhat similar to an engineered spaceship, but with key differences. The usual reason for constructing a traveling signal loop is to extract an extra output glider at some point in each cycle, producing an engineless rake. Suppressing a rake's output glider will trivially produce a high-period spaceship.
One main difference between engineered spaceships and traveling signal loops is that the "specific active reactions that travel at one particular speed" are spaceships themselves, and don't need any additional support. The signals traveling in the signal loops can generally be removed without having any effect on the fleet of spaceships supporting the loop.
| Speed | Direction | Examples | Classification |
|---|---|---|---|
| 2c/7 | orthogonal | weekender distaff | High-period signal loop |
| c/3 | orthogonal | c/3 orthogonal rake | High-period signal loop |
| c/4 | orthogonal | p388 c/4 forward rake, David Bell's adjustable-period backrake | High-period signal loop |
| c/5 | orthogonal | c/5 orthogonal rake | High-period signal loop |
| 2c/5 | orthogonal | 2c/5 orthogonal rake | High-period signal loop |
| c/4 | diagonal | c/4 diagonal rake | High-period signal loop |
| c/5 | diagonal | c/5 diagonal rake | High-period signal loop |
| c/12 | diagonal | c/12 diagonal rake | High-period signal loop |
Adjustable spaceships
Adjustable spaceships (formerly known as engineerable spaceships) are the third extant class of spaceships. On average smaller than the engineered spaceships in terms of population, but much larger in bounding box, their magic comes from having (to an extent) adjustable features, usually speed. With some almost always trivial modifications, these spaceships can be made to travel at different velocities and even directions. Rather than being searched for, programs exist that explicitly construct the spaceships. Adjustable spaceships can be based on variable-speed reactions such as the half-bakery and glider reaction or the freezing/reanimation cycle of the Caterloopillar, or by reading instruction tapes as in the Gemini. Families of adjustable spaceships include the Geminoids, Demonoids, Orthogonoids, loopships, half-baked knightships and Caterloopillars.
| Speed | Direction | Smallest known | Minimum # of cells | |
|---|---|---|---|---|
| 31c/240 | orthogonal | centipede caterloopillar | 361,070 | |
| c/8 | orthogonal | Original caterloopillar | 232,815 | |
| All speeds slower than c/4 | orthogonal | other caterloopillars | Varies | |
| 16c/217251 | orthogonal | Orthogonoid | 467,746 | |
| c/16384 | orthogonal | Hashlife-friendly Orthogonoid | 467,585 | |
| c/32768 | orthogonal | Double-wide Orthogonoid | 467,346 | |
| 1000130c/20003511 | orthogonal | loopship | 267,672 | |
| (200,7)c/81976184 | slope 200/7 | Stable Storage Spaceship | 633,388 | |
| (5120,1024)c/16849793 | slope 5 (ibiswise) | Gemini | 846,278 | |
| (3,1)c/3948264 | slope 3 (camelwise) | camelship | 239,822 | |
| (4096,2048)c/17783745 | slope 2 (knightwise) | Gemini 2 | <=872,252 | |
| (6,3)c/245912 | slope 2 (knightwise) | Parallel HBK | 132,945 | |
| (6,3)c/2621440 | slope 2 (knightwise) | Half-baked knightship | 1,049,395 | |
| 65c/438852 | diagonal | 0hd Demonoid | 27,250 | |
| 65c/818356 | diagonal | 10hd Demonoid | 47,701 | |
| 79c/1183842 | diagonal | single-channel Demonoid | 60,672 | |
| c/16384 | diagonal | Scorbie's Demonoid | 72,207 | |
| c/512 | diagonal | Hashlife-friendly Demonoid | 105,369 | |
| 1746011c/8916896 | diagonal | Speed Demonoid | 1,021,124 | |
Other classifications
- Main article: Spaceship types
Although spaceships are most commonly categorized by their speed and direction, other categorizations have been applied to spaceships based on their appearances, components, or other properties. One such categorization is the symmetry of spaceships: spaceships can be bilaterally symmetric (e.g. copperhead), exhibit glide symmetry (e.g. glider), or simply be asymmetric (e.g. loafer). Other somewhat subjective categorizations have also been made, such as greyships, spaceships filled with large amounts of static, live cells in the form of an agar, or smoking ships, which produce large sparks, a notable example being the Schick engine. A spaceship may also support other components which would not function as spaceships on their own. Given a freestanding spaceship, such additional components are often referred to as tagalongs; however they can be attached to any side of a spaceship, such as pushalong 1 and sidecar. Unstable spaceships immersed in a sustaining cloud are known as flotillae. A well known example is that of the overweight spaceships, which are unstable alone but may be 'escorted' by two or more smaller spaceships.
History
1970s
The four smallest spaceships in life, the glider, lightweight spaceship, middleweight spaceship and heavyweight spaceship, were all found by hand in 1970. For almost twenty years spaceship development was limited to adding tagalongs to the known c/2 spaceships, such as the Schick engine.
1980s
Significant advances in spaceship technology came in 1989, when Dean Hickerson began using automated searches based on a depth-first backtracking algorithm. These searches found orthogonal spaceships with speeds of c/3 and c/4, new c/2 ships, and the first spaceship other than the glider to travel at the speed of c/4 diagonally, dubbed the big glider.
1990s
Hickerson continued to find new spaceship speeds, the first of this decade being 2c/5 orthogonal, plus several ways to combine switch engines to create the first c/12 diagonal spaceships, named Corderships in honour of Charles Corderman.
The next spaceship speed to be discovered was that of the orthogonal c/5 snail, found by Tim Coe in 1996, with a program he had designed using breadth-first searching, and which could split tasks between multiple CPUs.[9] In the following year, David Bell found the much smaller c/5 spider using lifesrc, a program based on Hickerson's search algorithm.[10]
In March of 1998 David Eppstein created gfind, a breadth-first program that uses a depth-first search to limit the size of the search queue.[11]
2000s
Eppstein put his search program to good use in 2000, discovering the first spaceship that travels at the speed of 2c/7 orthogonally, the weekender. A search by Paul Tooke using the same program found the first c/6 orthogonal spaceship, the dragon, later that year. Also in 2000, Jason Summers found the first c/5 diagonal spaceship using David Bell's lifesrc program.
In 2004 Gabriel Nivasch, with the help of Jason Summers and David Bell, finished construction on the caterpillar, the first known orthogonal 17c/45 spaceship, which made use of the 17c/45 reaction.
2010s
In May 2010 Andrew J. Wade created a universal constructor-based spaceship, Gemini, which travels at a speed of (5120,1024)c/33699586.[12] This was the first explicitly constructed spaceship in Life to travel in an oblique direction, and also yielded the first explicit method of constructing arbitrarily slow spaceships.
In August 2011, Matthias Merzenich discovered lobster, the first c/7 diagonal spaceship.
In February 2013, the first c/7 orthogonal spaceship, loafer, was discovered by Josh Ball.
2014 provided a handful of new engineered spaceships, using various new technologies. In July, several half-bakery-based knightships were constructed with a new technique not requiring universal-constructor circuitry. These produced spaceships that were both much slower and much smaller than the Gemini variants. In September, Dave Greene and Chris Cain completed two 31c/240 orthogonal spaceships, along the same general lines as the original Caterpillar but using a number of new mechanisms. Finally, in December, an oblique caterpillar dubbed the Waterbear was completed by Brett Berger, traveling at (23,5)c/79. Richard Schank discovered pufferfish, a c/2 puffer, and Ivan Fomichev found a c/2 fuse for its exhaust and combined two pufferfish with fuses to assemble the first wholly high-period c/2 spaceship.
In December 2015, Chris Cain completed a diagonal self-constructing spaceship -- a "0hd Demonoid" -- based on Geminoid technology, adapted from a larger 10hd version constructed in November in collaboration with Dave Greene.
In March 2016, zdr discovered copperhead, an extremely small c/10 spaceship. A pseudo-tagalong for this spaceship, alongside many other c/10 technologies, were constructed within two months after the discovery of the ship.
In April 2016, Michael Simkin finished the adjustable caterloopillar project, making it possible to build spaceships of arbitrary orthogonal speeds slower than c/4.
In June 2016, Tim Coe found a large elementary 3c/7 orthogonal spaceship, the spaghetti monster.
In March 2018, Adam P. Goucher and Tomas Rokicki found the first elementary knightship, Sir Robin, which translates itself by (2,1) every six generations.
2020s
Over the course of December 2020 and the first half of 2021, three speeds received versatile new technology, including the first intermediate-period 2c/7 orthogonal ships (based on the doo-dah by John Winston Garth), several smaller 3c/7 orthogonal ships by Dylan Chen allowing higher-period ships to be constructed, and higher-period (2,1)c/6 knightships based on the sprayer by Adam P. Goucher and Dylan Chen, all aided by the new ikpx2 spaceship search program.
In 2023, the first elementary c/8 diagonal spaceship was found: the walrus.
The first known cases for unsimplified speeds are unknown.
In other rules
.png) A map of outer-totalistic rules. Birth conditions iterate in binary over the x axis (the right half has B0), survival in the y axis (the bottom half has S8). Outside of the bottom-left quadrant (where only the all-off agar is considered space), it is symmetrical due to the duality of black/white reversal. Green rules have known explicit examples of spaceships within them, red ones have been proven not to support any. (See file description for more details.)  A map of outer-totalistic rules. Birth conditions iterate in a Gray code over the x axis, survival in the y axis. Green rules have known explicit examples of spaceships within them, red ones have been proven not to support any. (See file description for more details.) |
Many Life-like cellular automata afford spaceships, as do their generalizations; this includes both outer-totalistic and isotropic non-totalistic rules, as well as non-isotropic, Generations and Larger than Life rules.
Various segments of the Life-like rulespace cannot contain spaceships, however. For instance, assuming B0 is not active:
- In any rule with B1c, any pattern expands in all directions.
- For totalistic rules, this is rounded up to B1.
- In any rule with all of S012acek3aijn4a (all survival conditions with a subset of the on cells in 4a), the trailing edge of a pattern cannot die.
- For totalistic rules, this is rounded up to S01234.
- In any rule with B23/S0, the trailing edge of a pattern cannot die.
- In any rule with none of B1ce2ac3i, no pattern can escape its initial bounding box.
- In any rule with none of B1ce2ae3a, no pattern can escape its initial bounding diamond.
- For totalistic rules, these are both rounded up to B123.
- With B3 and none of B1245/S012345, with a pattern's bounding diamond replaced with that of its bounding box, it remains inescapable.[14]
- It appears that in some rules patterns cannot escape the bounding box or the bounding diamond despite having the B3a or B2e condition. This is likely the case for LongLife and Assimilation.
In strobing rules with (not B1c) XOR S7c (such that one member of the pair of rules they emulate alternating between has B1c), a spaceship may move with velocity vector (x,y)cp only if x+y ≤ 32*p.[5]
In rules with both of B3ai and none of B12ace, a spaceship may move at (x,y)cp only if 2*max(x,y)+min(x,y) ≤ p.
In the subset without either S4w or S5a (such as Life), this criterion restricts to 2*(x+y) ≤ p; the diagonal speed limit is (1,1)c/4, not (1,1)c/3, and the knightwise speed limit is (2,1)c/6, not (2,1)c/5 and so on. All patterns at the maximum speed in any direction travel at c/2 in the taxicab metric.[4]
The orthogonal limit without any of B1ec2a is c/2.
While assuming B0 is active, we can treat them as either strobing rules or black/white reversals of rules. Over the outer-totalistics, this yields the following spaceshipless rulespaces:[15]
- Rules containing all of the transitions in at least one of {B0/S7,B08/S56,B05678/S6,B045678} without any of the transitions in at least one of {S01234,B3/S0123,B2/S0123,B23/S0,B1}
- Rules with B0234567 without S123456
Also:
- With B0/S6 and without either of B1/B2, or with B0/S67 without B2, all patterns expand at (2,1)c/2.
- With B0123 and without any of S567, patterns can never leave their bounding box.
The slowest known orthogonal elementary non-adjustable spaceship in any Life-like rule is a c/5648 orthogonal spaceship in the rule Gems, followed by a c/2068 orthogonal spaceship in Gems Minor. The slowest known (and highest-period) such spaceship in a range-1 isotropic non-totalistic rule is a 14c/63503454 ship in B2n3-ce4e5iy6ae/S2aei3cejk4eijkrw5cnry6-e78.[16]
Currently known speeds
There is currently an ongoing tabulation at the 5s project[17] cataloging the smallest known spaceships for each speed across different rules.
Alongside these, certain "series" of speeds can be proved to all exist in range-1 isotropic cellular automata:
- All orthogonal spaceships with the unsimplified speed c/n are known to exist; true-period c/1, c/2 and c/3 spaceships are known.
- All orthogonal spaceships with the unsimplified speed mc/n are known to exist where n/m ≥ 9, in B2ck3ae4arw5n6k7c/S1e2-cn3aekry4-cnqw5jnry.[18]
Notes
See also
General
Conway's Life
- List of spaceships
- List of natural spaceships
- Subsections of LifeWiki:Game of Life Status page, including velocities and periods
- LifeWiki:Spaceship Search Status Page
- Spaceships (category)
Other cellular automata
- Day & Night#Spaceships
- HighLife#Spaceships
- Grounded Life#Spaceships
- LifeWiki:Smallest Spaceships Supporting Specific Speeds
References
- ↑ "Glider". The Life Lexicon. Stephen Silver. Retrieved on April 18, 2009.
- ↑ "Fish". The Life Lexicon. Stephen Silver. Retrieved on April 18, 2009.
- ↑ "Gliders in Life-Like Cellular Automata". David Eppstein. Retrieved on April 18, 2009.
- ↑ 4.0 4.1 Nathaniel Johnston (October 30, 2009). "Spaceship Speed Limits in "B3" Life-Like Cellular Automata". Retrieved on December 12, 2016.
- ↑ 5.0 5.1 vyznev (February 4, 2018). Re: B0 hyper-relativistic speeds, in which it was noted that INT rules have a speed limit of x+y ≤ 32*p
- ↑ Jason Summers' jslife pattern collection. Retrieved on January 17, 2019.
- ↑ Bullet51 (November 17, 2022). Re: Suggested LifeWiki edits (discussion thread) at the ConwayLife.com forums
- ↑ Alexey Nigin (March 7, 2016). "New Spaceship Speed in Conway’s Game of Life". Retrieved on June 11, 2016.
- ↑ Tim Coe. "c/5 Orthogonal spaceship". Paul's Page of Conway's Life Miscellany. Retrieved on April 18, 2009.
- ↑ David Bell. "New c/5 spaceship". Paul's Page of Conway's Life Miscellany. Retrieved on April 18, 2009.
- ↑ David Eppstein (April 26, 2000). "Searching for Spaceships (PDF)". Retrieved on December 4, 2023.
- ↑ Oblique Life spaceship created at Game of Life News. Posted by Adam P. Goucher on May 19, 2010.
- ↑ LaundryPizza03 (August 15, 2023). Re: Speed limits and theorems about the existence of periodic objects (discussion thread) at the ConwayLife.com forums
- ↑ David Eppstein, comment in glider.c
- ↑ LaundryPizza03 (April 10, 2020). Re: Spaceships in Life-like cellular automata (discussion thread) at the ConwayLife.com forums
- ↑ NimbleRogue (September 12, 2024). Re: Foam Rules, edit 2
- ↑ "AforAmpere/sssss/tree/master/data".
- ↑ silversmith (December 31, 2021). Re: Re: Rules with small adjustable spaceships (discussion thread) at the ConwayLife.com forums
Further reading
- Carter Bays, Gliders in Cellular Automata, in: Robert A. Meyers (ed). Encyclopedia of Complexity and Systems Science, Springer 2009, pp. 4240–4249.
External links
- Spaceship at Wikipedia
- Spaceship at the Life Lexicon
- Spaceship Discussion Thread (discussion thread) at the ConwayLife.com forums (For Conway's Life)
- Spaceships in Life-like cellular automata (discussion thread) at the ConwayLife.com forums (For outer-totalistic rules)
- Miscellaneous Spaceship (collections) in Other Cellular Automata (discussion thread) at the ConwayLife.com forums