Glider synthesis

Glider synthesis (or glider construction) is the construction of an object by means of glider collisions. It is generally assumed that the gliders should be arranged so that they could come from infinity - that is, gliders should not have had to pass through one another to achieve the initial arrangement (or else it is considered “not fully functional”). LWSSes, MWSSes and HWSSes can also be used in syntheses; these spaceships can themselves be easily synthesized from gliders at any point along their trajectory, so this conversion is often left as an implicit step.

<html><div class="rle"><div class="codebox"><div style="display:none;"><code></html>x = 34, y = 31, rule = B3/S23 33bo$31b2o$32b2o$9bo$bo8bo$2bo5b3o$3o3$5bo$6bo$4b3o$24bobo$25b2o$25bo 2$27bobo$27b2o$28bo$31b3o$31bo$32bo7$5b2o$6b2o$5bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ ZOOM 8 ]]<html></code></div></div><canvas width="200" height="300" style="margin-left:1px;"><noscript></html>
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An 8-glider synthesis of a loafer
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Features of syntheses

Four main characterizing features of a synthesis are the geometry, construction time, glider cost, and number of stages.

The geometry is the number of directions of incoming gliders:

  • four-directional: gliders collide from all four directions
  • three-directional: gliders collide from all directions but one
  • two-directional; further divisible in head-on and 90° syntheses. All two-glider syntheses are necessarily two-directional.
  • unidirectional, which assumes the initial presence of a target (usually a still life or an oscillator) to be hit with gliders.

Since gliders are themselves glider-constructible, any multidirectional synthesis can be technically downgraded to a fewer-directional one, usually at the cost of increasing the construction time, cost, and/or number of stages needed for the synthesis. More challenging is finding a two- or three-directional synthesis for a particular object where few or no parts of the synthesis reactions extend outside the final pattern's bounding box in a particular direction. This is especially important for the synthesis of temporary bait objects, which will need to be placed sometimes quite close to other components without perturbing them. For especially tight locations, sometimes it will be useful to construct an LWSS (or another standard c/2 spaceship) some distance away from the synthesis nexus and let that collide with a glider in the final stages; this allows synthesis at a 45° angle, rather than a 90° angle as required for synthesis by gliders from separate directions.

The construction time is simply the number of generations it takes to complete a synthesis. For multi-stage syntheses, each stage has its own construction time.

The number of stages is a count of how many separate operations a synthesis can be divided into, with pauses of arbitrary length between the stages. Often a particular synthesis operation cannot be achieved by a direct collision of gliders, and a synthesis procedure instead requires first synthesizing a number of bait objects, and then hitting these with gliders to produce the final result.

The cost is the number of gliders expended over the course of the synthesis. Similar to the construction time, it can be defined also for individual synthesis stages. A *WSS is considered to cost 3 gliders. The discovery of the reverse caber tosser in 2018 proved that there is a universal constant upper bound on the cost to synthesise any synthesisable object; currently, the best known upper bound is 32 gliders.

Of particular interest is slow salvo synthesis: unidirectional synthesis where every stage has a glider cost of one. Perhaps surprisingly, anything that is glider synthesizable is also slow salvo synthesizable; a result that crucially depends on the existence of movable targets, one-time turners, and splitters.

Still life syntheses

In the 1990s, glider syntheses for all still lifes and known oscillators with at most 14 cells were found by David Buckingham. Almost all of these were successfully reduced to a synthesis cost of less than 1 glider per ON cell, or "1 glider per bit".[1]

A collaborative effort ending in May 2014 completed glider syntheses of all still lifes with 17 or fewer cells.[2][3][4] A second, longer effort claimed to have completed all the 18-bit still lifes in November 2014,[5][6] but it was later found that some of these syntheses were erroneous.[citation needed] The project was finally completed for real in October 2019.[7] The syntheses for 19-bit still lifes were completed in February 2020.[8] Later optimization projects reduced the maximum cost of construction for 15-bit,[9][10] 16-bit,[11][12] and 17-bit[13] still lifes to less than one glider per bit, in November 2016, May 2017, and September 2019 respectively.

The following table displays the minimum, average, and maximum costs for strict still lifes with up to 20 cells as of February 8, 2020.[14]

Live cellsCount
( A019473 )		Min. costAvg. costMax. cost
4222.5003
5122.0002
6523.2004
7422.7504
8923.5564
91034.0005
102544.3605
114644.9137
1212145.9268
1324046.5719
1461937.36010
151,35348.51712
163,28639.58514
177,773410.83816
1819,044412.42030[n 1]
1945,759414.09876[n 2]
20112,24341230 unsynthesized[n 3]
  1. All but 788 18-bit still lifes can be constructed with strictly less than one glider per bit as of August 15, 2020.
  2. All but 3,710 19-bit still lifes can be constructed with strictly less than one glider per bit as of August 15, 2020.
  3. All but 15,477 20-bit still lifes can be constructed with strictly less than one glider per bit as of August 15, 2020.

Spaceship syntheses

Perhaps the most interesting glider syntheses are those of spaceships, because these can be used to create corresponding guns and rakes. Many of the c/2 spaceships that are based on standard spaceships have been synthesized, mostly by Mark Niemiec. In June 1998, Stephen Silver found syntheses for some of the Corderships (although it was not until July 1999 that Jason Summers used this to build a Cordership gun). Many larger Corderships also have known glider syntheses, and others could easily be generated using the same techniques. In general, larger Corderships have declined in importance after the discovery of four-, three- and two-engine versions.

In May 2000, Noam Elkies suggested that 60P5H2V0, a 2c/5 spaceship found by Tim Coe in May 1996 might be a candidate for glider synthesis. Initial attempts to construct a synthesis for this spaceship got fairly close, but it was only in March 2003 that Summers and Elkies managed to find a way to perform the crucial last step. Summers then used the new synthesis to build a c/2 forward rake for the 2c/5 spaceship; this was the first example in Life of a rake which fires spaceships that travel in the same direction as the rake but more slowly.

After the loafer was discovered and synthesized in 2013, a number of new spaceship syntheses were found during a short period of time in late 2014 and early 2015, including the dart, crab, 25P3H1V0.2, 30P5H2V0, x66, and weekender. Most of this was due to the work of Martin Grant.

NameSpeedFirst synthesisBest current synthesis
DateDiscovererFewest gliders
60P5H2V02c/5 orthogonal2003-03-17Noam Elkies46
loaferc/7 orthogonal2013-02-17Adam P. Goucher8
dartc/3 orthogonal2014-12-02Martin Grant25
crabc/4 diagonal2014-12-26Martin Grant14
Parallel HBK(6,3)c/2459122014-12-31Michael Simkin38,380
30P5H2V02c/5 orthogonal2015-01-01Martin Grant47
25P3H1V0.1c/3 orthogonal2015-01-06Martin Grant47
x66c/2 orthogonal2015-01-11Martin Grant12
weekender2c/7 orthogonal2015-01-25Martin Grant59
puffershipc/2 orthogonal2015-02-11Chris Cain60
Gemini(2560,512)c/168497932015-02-16Dave Greene173,449
half-X66 with HWSSc/2 orthogonal2015-03-08Chris Cain9
B29c/4 diagonal2015-04-06Tanner Jacobi25
Pushalong 1c/2 orthogonal2015-06-12Martin Grant58
30P4H2V0.4c/2 orthogonal2015-09-10Tanner Jacobi85
0hd Demonoid65c/438852 diagonal2015-12-06Chris Cain12,016
copperheadc/10 orthogonal2016-03-05Aidan F. Pierce13
fireshipc/10 orthogonal2016-03-21Nico Brown18
25P3H1V0.2c/3 orthogonal2017-12-15Martin Grant26
Orthogonoid16c/217251 orthogonal2017-12-30Dave Greene37,625
2-engine Cordershipc/12 diagonal2017-12-31Dave Greene9
46P4H1V0c/4 orthogonal2019-02-04Tanner Jacobi79
spiderc/5 orthogonal2019-03-07Martin Grant237
camelship(3,1)c/39482642019-05-04Dave Greene26,614
27P4H1V1c/4 diagonal2019-10-28Goldtiger99731
loopship1000130c/20003511 orthogonal2020-01-08Dave Greene56,643
56P6H1V0c/6 orthogonal2020-03-25Martin Grant294
58P5H1V1c/5 diagonal2020-04-03Goldtiger997100
31P8H4V0c/2 orthogonal2020-04-10Goldtiger99737
70P2H1V0.1c/2 orthogonal2020-07-18Martin Grant203
44P5H2V02c/5 orthogonal2020-08-29Goldtiger99740

Other syntheses of note
A 3-glider synthesis of a pentadecathlon.

A 3-glider synthesis of a pentadecathlon was found in April 1997 by Heinrich Koenig, which came as a surprise because it was widely assumed that such a small synthesis would already be known.

Along similar lines, a 3-glider synthesis of an infinite growth pattern was found in October 2014 by Michael Simkin,[15] and a 3-glider synthesis of a clean switch engine was discovered in March 2017 by Luka Okanishi.[16]

2-glider syntheses
Main article: 2-glider collision

There are 71 distinct 2-glider collisions, of which 28 produce nothing, six produce a block, five produce a honey farm, three produce a B-heptomino, three produce a pi-heptomino, three produce a blinker, three produce a traffic light, two produce a glider, two produce a pond, two produce a loaf and a blinker, one produces a boat, one produces a beehive, one produces a loaf, one produces an eater 1, one produces lumps of muck, one produces a teardrop, one produces an interchange, one produces a traffic light and a glider, one produces an octomino, one produces a bi-block, one produces four blocks, one produces two blocks, one produces a blinker, loaf, tub and block, and one produces the so-called two-glider mess, a methuselah stabilizing after 530 generations and consisting of four gliders, eight blinkers (including a traffic light), four blocks, a beehive and a ship.

All 71 such syntheses can be seen below in a pattern put together by Jason Summers on January 29, 2005:

<html><div class="rle"><div class="codebox"><div style="display:none;"><code></html>x = 379, y = 369, rule = B3/S23 154bo36bo18bo$58bo18bo19bo16bo19bo18bo19bo16bo18bo36bo$57bo18bo19bo16b o19bo19b3o16bo17b3o16b3o15bo17bo$57b3o16b3o17b3o14b3o17b3o36b3o51bo18b 3o$226b3o4$190bo$58b3o17b3o89bo18b2o55bo$58bo19bo20bo17bo20bo13b2o15b 2o18bobo14b2o20bo16b2o$59bo19bo18b2o16b2o19b2o13bobo14bobo34bobo18b2o 16bobo$98bobo15bobo18bobo12bo53bo20bobo$bo4bo3bobo3bobobo2bo3bo3bo3bo 4bo4bobo2$bobo2bo2bo3bo4bo4bo3bo3bo3bobo2bo2bo2$bo2bobo2bo3bo4bo4bobob o3bo3bo2bobo2bo2bobo2$bo4bo2bo3bo4bo4bo3bo3bo3bo4bo2bo4bo266bo$269bo 22bo21bo62bo$bo4bo3bobo5bo4bo3bo3bo3bo4bo4bobo182bo15bo21bo22bo22b3o 18bo22bo17bo$229bo15bo22b3o20b3o40bo22bo18b3o$86bo16bo17bo16bo16bo15bo 17bo16bo22b3o13b3o86b3o20b3o$66bo18bo16bo17bo16bo16bo15bo17bo16bo$65bo 19b3o14b3o15b3o14b3o14b3o13b3o15b3o14b3o$65b3o4$224b3o15b3o59b2o$76b3o 16b3o16b3o15b3o15b3o14b3o16b3o15b3o19bo17bo11b2o23b2o20bobo20b2o40b2o$ 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THUMBLAUNCH ]] #C [[ WIDTH 1200 HEIGHT 1200 ZOOM 3 ]]<html></code></div></div><canvas width="200" height="300" style="margin-left:1px;"><noscript></html>
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All 71 distinct 2-glider collisions, arranged by what they synthesize.
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gollark: No, cobalt gets better speed.
gollark: Ardite? Faster?
gollark: Manyulllun?
gollark: Ardite?
gollark: I think they just strip out proprietary code.

See also

References
  1. Mark D. Niemiec (June 20, 2015). "Re: 4 glider syntheses". ConwayLife.com forums. Retrieved on February 21, 2018.
  2. Dave Greene (May 23, 2014). "Constructions Known for All Still Lifes up to 17 Bits". Game of Life News. Retrieved on September 17, 2014.
  3. Martin Grant (January 6, 2014). "17-bit SL Syntheses (100% Complete!)". ConwayLife.com forums. Retrieved on September 17, 2014.
  4. Martin Grant (May 17, 2014). Re: 17-bit SL Syntheses (discussion thread) at the ConwayLife.com forums
  5. Martin Grant (October 2, 2014). "18-bit SL Syntheses (100% Complete!)". ConwayLife.com forums. Retrieved on February 21, 2018.
  6. Martin Grant (November 12, 2014). Re: 18-bit SL Syntheses (discussion thread) at the ConwayLife.com forums
  7. Ian07 (October 9, 2019). Re: 18-bit SL Syntheses (100% Complete!) (discussion thread) at the ConwayLife.com forums
  8. Martin Grant (February 8, 2020). Re: 19-bit still life syntheses (discussion thread) at the ConwayLife.com forums
  9. BlinkerSpawn (October 27, 2016). "15 in 15: Efficient 15-bit Synthesis Project (DONE!)". ConwayLife.com forums. Retrieved on February 21, 2018.
  10. Martin Grant (November 19, 2016). Re: 15 in 15: Efficient 15-bit Synthesis Project (2 SLs remain) (discussion thread) at the ConwayLife.com forums
  11. Bob Shemyakin (December 20, 2016). "16 in 16: Efficient 16-bit Synthesis Project". ConwayLife.com forums. Retrieved on February 21, 2018.
  12. Goldtiger997 (May 24, 2017). Re: 15 in 15: Efficient 15-bit Synthesis Project (2 SLs remain) (discussion thread) at the ConwayLife.com forums
  13. Tanner Jacobi (September 9, 2019). Re: 17 in 17: Efficient 17-bit synthesis project (discussion thread) at the ConwayLife.com forums
  14. Adam P. Goucher. "Syntheses". Catagolue. Retrieved on February 8, 2020.
  15. Michael Simkin (October 24, 2014). "Re: Making switch-engines". ConwayLife.com forums. Retrieved on February 21, 2018.
  16. Luka Okanishi (March 12, 2017). "Re: Thread For Your Accidental Discoveries". ConwayLife.com forums. Retrieved on February 21, 2018.

Forum threads
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