Difference between revisions of "OCA:HighLife"

From LifeWiki
Jump to navigation Jump to search
 
(32 intermediate revisions by 11 users not shown)
Line 5: Line 5:
 
|b              = 36
 
|b              = 36
 
|s              = 23
 
|s              = 23
 +
|ruleinteger    = 6216
 +
|reversal        = B0123478/S0134678
 
|animated        = true
 
|animated        = true
 
}}
 
}}
'''HighLife''' is a [[cellular automaton#Well-known Life-like cellular automata|Life-like cellular automaton]] in which [[cell]]s survive from one generation to the next if they have 2 or 3 [[neighbours]], and are born if they have 3 or 6 neighbours. It was named by [[:Category:Patterns found by John Conway|John Conway]] and was first considered in 1994 by Nathan Thompson. It is mainly of interest due to a simple [[replicator]] that it allows.
+
'''HighLife''' is a [[cellular automaton#Well-known Life-like cellular automata|Life-like cellular automaton]] in which [[cell]]s survive from one generation to the next if they have 2 or 3 [[neighbours]], and are born if they have 3 or 6 neighbours. It was named by [[John Conway]] and was first considered in 1994 by Nathan Thompson. It is mainly of interest due to a simple [[replicator]] that it allows.
  
Because its [[rulestring]] is so similar to that of [[Conway's Game of Life]], many simple patterns exhibit the same behavior in both rules; it's only when patterns get complex that their behavior differs. Nonetheless, it exhibits such rich structure that [[:Category:Patterns found by John Conway|John Conway]] himself stated
+
Because its [[rulestring]] is so similar to that of [[Conway's Game of Life]], many simple patterns exhibit the same behavior in both rules; only when patterns get complex do their behavior differ. Nonetheless, it exhibits such rich structure that [[John Conway]] himself stated
  
<blockquote>''"It seems to me that 'B36/S23' is really the game I should have found, since it's so rich in nice things."'' <ref>[http://www.tip.net.au/~dbell/articles/HighLife.zip HighLife - An Interesting Variant of Life] by David Bell (.zip file)</ref></blockquote>
+
<blockquote>''"It seems to me that 'B36/S23' is really the game I should have found, since it's so rich in nice things."''<ref>[http://www.tip.net.au/~dbell/articles/HighLife.zip HighLife - An Interesting Variant of Life] by David Bell (.zip file)</ref></blockquote>
  
 
==Notable patterns==
 
==Notable patterns==
 
[[File:Highlifefourboats.png|frame|right|The "four boats" constellation.]]
 
[[File:Highlifefourboats.png|frame|right|The "four boats" constellation.]]
 
All of the most common [[still life]]s, [[oscillator]]s and [[spaceship]]s from the standard Life rules behave the exact same under the HighLife rules, including the [[block]], [[beehive]], [[blinker]], [[toad]], [[beacon]], [[glider]], [[lightweight spaceship]], [[middleweight spaceship]], and [[heavyweight spaceship]]. On the other hand, even though [[traffic light]]s and [[honey farm]]s themselves behave the same in both rules, they do not occur naturally in HighLife with any sort of regularity due to their common [[predecessor]]s being unstable. Conversely, there is a new familiar four composed of four boats in HighLife which evolves from one honey farm predecessor.
 
All of the most common [[still life]]s, [[oscillator]]s and [[spaceship]]s from the standard Life rules behave the exact same under the HighLife rules, including the [[block]], [[beehive]], [[blinker]], [[toad]], [[beacon]], [[glider]], [[lightweight spaceship]], [[middleweight spaceship]], and [[heavyweight spaceship]]. On the other hand, even though [[traffic light]]s and [[honey farm]]s themselves behave the same in both rules, they do not occur naturally in HighLife with any sort of regularity due to their common [[predecessor]]s being unstable. Conversely, there is a new familiar four composed of four boats in HighLife which evolves from one honey farm predecessor.
 +
 +
Certain patterns act differently from their Life counterparts. For example, the [[dead spark coil]] will act extremely similarly to its living counterpart, with a single cell oscillating on and off inside (a rotor impossible in regular Life). Also, blinkers can be placed against one or two houses and will oscillate normally.
 +
 +
An infinitely-long line will replicate according to [[Rule 54]] due to the presence of B6, which is absent in regular Life which causes such lines to follow [[Rule 22]] instead.
  
 
===The replicator===
 
===The replicator===
By far the most notable pattern in HighLife is the simple [[replicator]], shown to the right. It is by far the most well-known replicator in any [[Life-like cellular automaton]]. It repeatedly copies itself along a diagonal line. It copies itself the first time after 12 [[generation]]s, then produces another two copies after another 24 generations, followed by another four copies after another 48 generations, and so on. In general there are 2<sup>n</sup> copies of the replicator at generation 12(2<sup>n</sup> - 1) and their centers are evenly spaced 4 cells apart. The two ends of the replicator line expand at a [[speed]] of c/6.
+
By far the most notable pattern in HighLife is the simple [[replicator]], shown to the right. It is by far the most well-known replicator in any [[Life-like cellular automaton]]. It repeatedly copies itself along a diagonal line according to [[Rule 90]]. It copies itself the first time after 12 [[generation]]s, then produces another two copies after another 24 generations, followed by another four copies after another 48 generations, and so on. In general there are 2<sup>n</sup> copies of the replicator at generation 12(2<sup>n</sup> - 1) and their centers are evenly spaced 4 cells apart. The two ends of the replicator line expand at a [[speed]] of c/6.
  
Because of the way the replicator duplicates itself, it can be considered a [[sawtooth]] with expansion factor 2 and a minimum repeating population of 22. Because the replicator is so small, it often occurs naturally from [[soup]]. This contrasts with the standard Game of Life, where all known sawtooths are complex, precisely-engineered patterns.
+
Because of the way the replicator duplicates itself, it can be considered a [[sawtooth]] with expansion factor 2 and a minimum repeating population of 22. Because the replicator is so small, it often occurs naturally from [[soup]] (the Life equivalent is [[butterfly]]). This contrasts with the standard Game of Life, where all known sawtooths are complex, precisely-engineered patterns.
  
 
{| style="margin-left:auto;margin-right:auto;"
 
{| style="margin-left:auto;margin-right:auto;"
 
|-
 
|-
|[[Image:replicator_predecessor.png|framed|right|A 3-generation predecessor of the replicator<br />{{JavaRLE|replicatorpredecessor|brief}}]]
+
|{{EmbedViewer
|[[Image:replicator.png|framed|right|The replicator itself<br />{{JavaRLE|replicator|brief}}]]
+
|pname = replicator
 +
|position = center
 +
|caption = The replicator
 +
}}
 +
|{{EmbedViewer
 +
|pname = replicatorpredecessor
 +
|position = center
 +
|caption = A 6-cell, 3-generation predecessor of the replicator<br>
 +
}}
 
|[[Image:replicator_gen12.png|framed|right|The replicator after 12 generations]]
 
|[[Image:replicator_gen12.png|framed|right|The replicator after 12 generations]]
 
|[[Image:replicator_gen36.png|framed|right|The replicator after another 24 generations]]
 
|[[Image:replicator_gen36.png|framed|right|The replicator after another 24 generations]]
 
|}
 
|}
 +
 +
A natural [[period]] 96 [[oscillator]] based on the replicator and a pair of [[block]]s functioning as [[eater]]s exists:
 +
 +
{{EmbedViewer
 +
|pname = highlifereplicatorxp96
 +
|position = center
 +
|caption = The period 96 replicator oscillator. ({{LinkCatagolue|xp96_33y3gokezyasa62zzytcc|rule=b36s23|style=brief}})<br>
 +
}}
  
 
===Still lifes===
 
===Still lifes===
Because the only difference between the HighLife rules and the standard Life rules is that there is another way for cells to be born (when they have exactly six alive [[Moore neighbourhood|neighbours]]), all [[still life]]s in the HighLife rule are necessarily still lifes under Conway's rules as well. Also, very few still lifes under the standard Life rules have dead cells with six alive neighbours, so the list of still lifes for the two rules are almost identical for small cell counts. The smallest patterns that are still lifes in the standard Life rules but not in HighLife are [[ship]] (with [[:Category:Strict still lifes with 6 cells|6]] cells) and [[hat]] (with [[:Category:Strict still lifes with 9 cells|9]] cells). Also, any pattern involving a [[bun]] or a [[cap]] that is a still life under the standard rules is not a still life in HighLife.
+
Because the only difference between the HighLife rules and the standard Life rules is that there is another way for cells to be born (when they have exactly six alive [[Moore neighbourhood|neighbours]]), all [[still life]]s in the HighLife rule are necessarily still lifes under Conway's rules as well. Also, very few small still lifes under the standard Life rules have dead cells with six alive neighbours, so the list of still lifes for the two rules are almost identical for small cell counts. The smallest patterns that are still lifes in the standard Life rules but not in HighLife are [[ship]] (with {{slcells|6}} cells) and [[hat]] (with {{slcells|9}} cells). Also, any pattern involving a [[bun]] or a [[cap]] that is a still life under the standard rules is not a still life in HighLife.
  
 
{| class="wikitable" style="margin-left:auto;margin-right:auto;"
 
{| class="wikitable" style="margin-left:auto;margin-right:auto;"
Line 96: Line 118:
 
|}
 
|}
  
===Change in Frequency===
+
===Change in frequency===
 +
{{main|List of common still lifes in HighLife}}
 
From the same random starting conditions, HighLife usually settles into fewer objects than in [[Conway's Game of Life|Life]].  This chart shows the change in frequency of common or notable objects in Life and in HighLife.  In the chart, objects are ranked by their formation density (the number of objects per cell of empty space) rather than total frequency out of all objects. This is because the frequency of total objects also changes between Life and HighLife.
 
From the same random starting conditions, HighLife usually settles into fewer objects than in [[Conway's Game of Life|Life]].  This chart shows the change in frequency of common or notable objects in Life and in HighLife.  In the chart, objects are ranked by their formation density (the number of objects per cell of empty space) rather than total frequency out of all objects. This is because the frequency of total objects also changes between Life and HighLife.
  
Line 165: Line 188:
 
| 4.55&times;10<sup>-9</sup>
 
| 4.55&times;10<sup>-9</sup>
 
| +5395%
 
| +5395%
| Eleveners appear 55 times as often in HighLife, because of a predecessor involving a [[pi-heptomino]] and a [[blinker]].
+
| Eleveners appear 55 times as often in HighLife, because of a predecessor involving a [[pi-heptomino]] and a [[blinker]].<!--please specify-->
 
|-
 
|-
 
| [[Ship]]
 
| [[Ship]]
Line 175: Line 198:
  
 
===Spaceships===
 
===Spaceships===
All of the [[standard spacehip]]s from the standard Life rules work in HighLife, but the only non-standard spaceships that are known to work in HighLife are the [[turtle]], [[117P9H3V0|86P9H3V0]], and some [[flotilla]]e of the standard spaceships. There are also several known spaceships that are specific to HighLife<ref>{{cite web|url=http://fano.ics.uci.edu/ca/rules/b36s23/ |title=HighLife (B36/S23) |publisher=David Eppstein|accessdate=April 15, 2009}}</ref>, the most well-known of which is the bomber.
+
All of the [[standard spacehip]]s from the standard Life rules work in HighLife, but only a few non-standard Life spaceships are known to work in HighLife, notably [[turtle]], [[crab]], and [[117P9H3V0|86P9H3V0]]. There are also several known spaceships that are specific to HighLife<ref>{{cite web|url=http://fano.ics.uci.edu/ca/rules/b36s23/ |title=HighLife (B36/S23) |publisher=David Eppstein|accessdate=April 15, 2009}}</ref>, the most well-known of which is the bomber.
 +
 
 +
Currently the speeds known for elementary Life spaceships but not elementary HighLife spaceships are (1,0)c/7, (3,0)c/7, (1,1)c/7 and (2,1)c/6, and speeds known for HighLife and not Life are (1,0)c/8 and (1,0)c/98.
 +
 
 +
<!--Someone add known degenerate speeds here-->
  
 
====Elementary====
 
====Elementary====
Line 189: Line 216:
 
| orthogonal
 
| orthogonal
 
| [[lightweight spaceship]]
 
| [[lightweight spaceship]]
| [[:Category:Patterns with 9 cells|9]]
+
| {{cells|9}}
|-
 
! [[2c/5 orthogonal|2c/5]]
 
| orthogonal
 
| [http://fano.ics.uci.edu/ca/rules/b36s23/g9.html 164P5H2V0]
 
| [[:Category:Patterns with 164 cells|164]]
 
 
|-
 
|-
 
! [[c/3 orthogonal|c/3]]
 
! [[c/3 orthogonal|c/3]]
 
| orthogonal
 
| orthogonal
 
| [[turtle]]
 
| [[turtle]]
| [[:Category:Patterns with 44 cells|44]]
+
| {{cells|44}}
 
|-
 
|-
 
! [[c/4 orthogonal|c/4]]
 
! [[c/4 orthogonal|c/4]]
 
| orthogonal
 
| orthogonal
| [http://fano.ics.uci.edu/ca/rules/b36s23/g8.html 157P4H1V0]
+
| [https://conwaylife.com/forums/viewtopic.php?p=59428#p59428 47P4H1V0]
| [[:Category:Patterns with 157 cells|157]]
+
| {{cells|47}}
 
|-
 
|-
 
! [[c/5 orthogonal|c/5]]
 
! [[c/5 orthogonal|c/5]]
 
| orthogonal
 
| orthogonal
 
| [http://fano.ics.uci.edu/ca/rules/b36s23/g5.html 52P5H1V0]
 
| [http://fano.ics.uci.edu/ca/rules/b36s23/g5.html 52P5H1V0]
| [[:Category:Patterns with 52 cells|52]]
+
| {{cells|52}}
 
|-
 
|-
! [[c/98 orthogonal|c/98]]
+
! [[c/6 orthogonal|c/6]]
 +
| orthogonal
 +
| [http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2332&p=59378#p59423 310P6H1V0]
 +
| {{cells|310}}
 +
|-
 +
! [[c/8 orthogonal|c/8]]
 +
| orthogonal
 +
| [http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2332&p=59378#p59363 138P8H1V0]
 +
| {{cells|138}}
 +
|-
 +
! c/98
 
| orthogonal
 
| orthogonal
 
| [https://catagolue.appspot.com/object/xq98_mgw4b6z31zsoz6x2d6/b36s23 24P98H1V0]
 
| [https://catagolue.appspot.com/object/xq98_mgw4b6z31zsoz6x2d6/b36s23 24P98H1V0]
| [[:Category:Patterns with 24 cells|24]]
+
| {{cells|24}}
 +
|-
 +
! [[2c/5 orthogonal|2c/5]]
 +
| orthogonal
 +
| [http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2332&start=25#p59382 106P5H2V0]
 +
| {{cells|106}}
 +
|-
 +
! [[2c/7 orthogonal|2c/7]]
 +
| orthogonal
 +
| [http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2332&p=64436#p64436 28P7H2V0]
 +
| {{cells|28}}
 
|-
 
|-
 
! colspan="5" |
 
! colspan="5" |
Line 221: Line 263:
 
| diagonal
 
| diagonal
 
| [[glider]]
 
| [[glider]]
| [[:Category:Patterns with 5 cells|5]]
+
| {{cells|5}}
 
|-
 
|-
 
! [[c/5 diagonal|c/5]]
 
! [[c/5 diagonal|c/5]]
 
| diagonal
 
| diagonal
| [[28P5H1V1]]
+
| [http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2332&p=59424#p59424 28P5H1V1]
| [[:Category:Patterns with 28 cells|28]]
+
| {{cells|28}}
 
|-
 
|-
 
! [[c/6 diagonal|c/6]]
 
! [[c/6 diagonal|c/6]]
 
| diagonal
 
| diagonal
 
| [[#Bomber|bomber]]
 
| [[#Bomber|bomber]]
| [[:Category:Patterns with 19 cells|19]]
+
| {{cells|19}}
 
|-
 
|-
 
|}
 
|}
Line 244: Line 286:
 
! Minimum # of [[cell]]s
 
! Minimum # of [[cell]]s
 
|-
 
|-
! [[c/24 diagonal|c/24]]
+
! [[c/12 diagonal|c/12]]
 
| diagonal
 
| diagonal
| [[basilisk]]
+
| [http://conwaylife.com/forums/viewtopic.php?f=11&t=2332&start=25#p52577 unnamed]
| ?
+
| 760
 +
|-
 +
! c/18
 +
| diagonal
 +
| [http://conwaylife.com/forums/viewtopic.php?f=11&t=2332&start=25#p53404 unnamed]
 +
| 280
 +
|-
 +
! c/24
 +
| diagonal
 +
| [http://pentadecathlon.com/lifeNews/2013/01/new_highlife_velocities.html basilisk]
 +
| 151971
 +
|-
 +
! c/30
 +
| diagonal
 +
| [http://conwaylife.com/forums/viewtopic.php?f=11&t=2332&start=25#p52577 unnamed]
 +
| 353
 
|-
 
|-
! [[c/32 diagonal|c/32]]
+
! c/32
 
| diagonal
 
| diagonal
| [[basilisk]]
+
| [http://pentadecathlon.com/lifeNews/2013/01/new_highlife_velocities.html basilisk]
| ?
+
| 117483
 
|-
 
|-
! [[c/63 diagonal|c/63]]
+
! c/63
 
| diagonal
 
| diagonal
| [[basilisk]]
+
| [http://pentadecathlon.com/lifeNews/2013/01/new_highlife_velocities.html basilisk]
 
| ?
 
| ?
 
|-
 
|-
! [[c/69 diagonal|c/69]]
+
|-
 +
! c/69
 
| diagonal
 
| diagonal
| [[basilisk]]
+
| [http://pentadecathlon.com/lifeNews/2013/01/new_highlife_velocities.html basilisk]
| ?
+
| 859 billion
 
|-
 
|-
 
|}
 
|}
Line 268: Line 326:
 
====Bomber====
 
====Bomber====
 
{{disambiglink|name=bomber|link=Bomber|text=For other uses of the term}}
 
{{disambiglink|name=bomber|link=Bomber|text=For other uses of the term}}
The '''bomber''' is a replicator-based spaceship that occurs naturally and was discovered by Nathan Thompson. It can be formed by placing a [[blinker]] in the path of the replicator as shown below. The spaceship itself has a [[period]] 48 and travels diagonally at [[speed]] c/6. The blinker reacts with one of the spawned replicators such that it destroys itself and the spawned replicator while leaving another blinker on the other side of the spaceship. It is thus a [[glide symmetric spaceship]] with [[mod]] equal to 24.
+
The '''bomber''' is a replicator-based spaceship that occurs naturally and was discovered by Nathan Thompson. It can be formed by placing a [[blinker]] in the path of the replicator as shown below. The spaceship itself has a [[period]] 48 and travels diagonally at [[speed]] c/6. The blinker reacts with one of the spawned replicators such that it destroys itself and the spawned replicator while leaving another blinker on the other side of the spaceship. It is thus a [[glide symmetric spaceship]] with [[mod]] equal to 24. It is the 3rd most common spaceship in HighLife, being slightly more common than the [[middleweight spaceship]].<ref>{{CiteCatagolue|page=census/b36s23/C1|title=Census|accessdate=July 3, 2019}}</ref>
  
 
{| style="margin-left:auto;margin-right:auto;"
 
{| style="margin-left:auto;margin-right:auto;"
|-
+
|{{EmbedViewer
|[[Image:Bomber_predecessor.png|framed|left|A predecessor of the bomber<br />{{JavaRLE|bomberpredecessor|brief}}]]
+
|pname = bomber
|[[Image:Bomber.gif|framed|left|The bomber itself]]
+
|position = center
 +
|caption = The bomber ({{LinkCatagolue|xq48_07z8ca7zy1e531|rule=b36s23|style=brief}})<br>
 +
|viewerconfig = #C [[ AUTOSTART GPS 8 TRACKLOOP 48 1/6 1/6 ]]
 +
}}
 +
|{{EmbedViewer
 +
|pname = bomberpredecessor
 +
|position = center
 +
|caption = A 9-cell predecessor of the bomber
 +
}}
 
|}
 
|}
 +
 +
==Universality==
 +
There is a proof sketch of this rule's universality as well as an explicit [https://en.wikipedia.org/wiki/Rule_110 Rule 110] [[unit cell]].<ref>{{cite web|url=http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2597&start=25#p62446|title=List of the Turing-complete totalistic life-like CA|author=Peter Naszvadi|work=ConwayLife.com forums|date=August 7, 2018|accessdate=May 17, 2019}}</ref> In the same topic on ConwayLife forums,<ref>{{cite web|url=http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p38167|title=List of the Turing-complete totalistic life-like CA|author=Peter Naszvadi|work=ConwayLife.com forums|date=December 12, 2016|accessdate=May 17, 2019}}</ref> there is a proof-scheme covering all rules in the [[outer-totalistic]] [[rulespace]] between [[B3/S23]] and B3678/S23678.
 +
 +
An explicit [https://en.wikipedia.org/wiki/Rule_110 Rule 110] [[unit cell]] construction<ref>{{cite web|url=https://www.conwaylife.com/forums/viewtopic.php?f=11&t=2597&start=25#p62217|title=List of the Turing-complete totalistic life-like CA|author=Peter Naszvadi|work=ConwayLife.com forums|date=July 29, 2018|accessdate=Jan 7, 2020}}</ref> proves its Turing-completeness; the pattern itself is a polyglot working in three other life-like cellular automata rules between [[B36/S23]] and [[B368/S238]]; the last rule is also known as [[LowDeath]], in which the native replicator - of which several components of the unit cell based on - has a slightly different evolution sequence.
 +
 +
==See also==
 +
*[[List of common oscillators in HighLife]]
 +
*[[List of common spaceships in HighLife]]
  
 
==References==
 
==References==
Line 283: Line 358:
 
{{LinkCatagolueRule|b36s23}}
 
{{LinkCatagolueRule|b36s23}}
 
{{LinkEppsteinRule|b36s23}}
 
{{LinkEppsteinRule|b36s23}}
 +
{{LinkForumThread|f=11|t=2332|title=HighLife}}

Latest revision as of 02:13, 8 January 2020

HighLife
HighLife rule
Rulestring 23/36
B36/S23
Rule integer 6216
Character Chaotic
Black/white reversal B0123478/S0134678

HighLife is a Life-like cellular automaton in which cells survive from one generation to the next if they have 2 or 3 neighbours, and are born if they have 3 or 6 neighbours. It was named by John Conway and was first considered in 1994 by Nathan Thompson. It is mainly of interest due to a simple replicator that it allows.

Because its rulestring is so similar to that of Conway's Game of Life, many simple patterns exhibit the same behavior in both rules; only when patterns get complex do their behavior differ. Nonetheless, it exhibits such rich structure that John Conway himself stated

"It seems to me that 'B36/S23' is really the game I should have found, since it's so rich in nice things."[1]

Notable patterns

The "four boats" constellation.

All of the most common still lifes, oscillators and spaceships from the standard Life rules behave the exact same under the HighLife rules, including the block, beehive, blinker, toad, beacon, glider, lightweight spaceship, middleweight spaceship, and heavyweight spaceship. On the other hand, even though traffic lights and honey farms themselves behave the same in both rules, they do not occur naturally in HighLife with any sort of regularity due to their common predecessors being unstable. Conversely, there is a new familiar four composed of four boats in HighLife which evolves from one honey farm predecessor.

Certain patterns act differently from their Life counterparts. For example, the dead spark coil will act extremely similarly to its living counterpart, with a single cell oscillating on and off inside (a rotor impossible in regular Life). Also, blinkers can be placed against one or two houses and will oscillate normally.

An infinitely-long line will replicate according to Rule 54 due to the presence of B6, which is absent in regular Life which causes such lines to follow Rule 22 instead.

The replicator

By far the most notable pattern in HighLife is the simple replicator, shown to the right. It is by far the most well-known replicator in any Life-like cellular automaton. It repeatedly copies itself along a diagonal line according to Rule 90. It copies itself the first time after 12 generations, then produces another two copies after another 24 generations, followed by another four copies after another 48 generations, and so on. In general there are 2n copies of the replicator at generation 12(2n - 1) and their centers are evenly spaced 4 cells apart. The two ends of the replicator line expand at a speed of c/6.

Because of the way the replicator duplicates itself, it can be considered a sawtooth with expansion factor 2 and a minimum repeating population of 22. Because the replicator is so small, it often occurs naturally from soup (the Life equivalent is butterfly). This contrasts with the standard Game of Life, where all known sawtooths are complex, precisely-engineered patterns.

x = 5, y = 5, rule = B36/S23 2b3o$bo2bo$o3bo$o2bo$3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
The replicator (click above to open LifeViewer)
RLE: here Plaintext: here
x = 4, y = 4, rule = B36/S23 b3o$o$o$o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
A 6-cell, 3-generation predecessor of the replicator
(click above to open LifeViewer)
RLE: here Plaintext: here
The replicator after 12 generations
The replicator after another 24 generations

A natural period 96 oscillator based on the replicator and a pair of blocks functioning as eaters exists:

x = 35, y = 19, rule = B36/S23 2o33b$2o10bo22b$11b2o22b$10bobo22b$9b3o23b$35b$15b3o17b$14bobo18b$14b2o19b$ 14bo20b$35b$35b$35b$35b$35b$35b$35b$33b2o$33b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
The period 96 replicator oscillator. (Catagoluehere)
(click above to open LifeViewer)
RLE: here Plaintext: here

Still lifes

Because the only difference between the HighLife rules and the standard Life rules is that there is another way for cells to be born (when they have exactly six alive neighbours), all still lifes in the HighLife rule are necessarily still lifes under Conway's rules as well. Also, very few small still lifes under the standard Life rules have dead cells with six alive neighbours, so the list of still lifes for the two rules are almost identical for small cell counts. The smallest patterns that are still lifes in the standard Life rules but not in HighLife are ship (with 6 cells) and hat (with 9 cells). Also, any pattern involving a bun or a cap that is a still life under the standard rules is not a still life in HighLife.

Size Count Image Links
≤3 0
4 2 Highlife4cellstilllifes.png Download RLE: click here
5 1 Highlife5cellstilllifes.png Download RLE: click here
6 4 Highlife6cellstilllifes.png Download RLE: click here
7 4 Highlife7cellstilllifes.png Download RLE: click here
8 9 Highlife8cellstilllifes.png Download RLE: click here
9 9 Highlife9cellstilllifes.png Download RLE: click here
10 25 Highlife10cellstilllifes.png Download RLE: click here
11 44 Highlife11cellstilllifes.png Download RLE: click here
12 111 Highlife12cellstilllifes.png Download RLE: click here
13 218 Highlife13cellstilllifes.png Download RLE: click here

Change in frequency

Main article: List of common still lifes in HighLife

From the same random starting conditions, HighLife usually settles into fewer objects than in Life. This chart shows the change in frequency of common or notable objects in Life and in HighLife. In the chart, objects are ranked by their formation density (the number of objects per cell of empty space) rather than total frequency out of all objects. This is because the frequency of total objects also changes between Life and HighLife.

Object Density in HighLife Density in Life Change in Density Notes
All Objects 4.30×10-3 6.64×10-3 -35% The overall decrease of objects and increase of sparks causes HighLife to stabilize over twice as fast on average.
Block 1.83×10-3 2.11×10-3 -13% Although the block is less common in HighLife, it is still the most common object.
Beehive 7.46×10-4 1.25×10-3 -40% Beehives are also less common, but more common than the blinker.
Blinker 6.88×10-4 2.15×10-3 -68% Blinkers are much less common, since the t-tetromino and related patterns evolves in HighLife into a large spark (see the bomber below), rather than traffic light.
Loaf 4.19×10-4 3.89×10-4 +8% Loaves are slightly more common in HighLife, but still far behind the beehive.
Boat 4.14×10-4 3.58×10-4 +16% In HighLife, a hat, as well as other predecessors, will evolve into a very common formation of four boats.
Tub 9.73×10-5 8.00×10-5 +21% Tubs experience a 21% increase - the largest of the top ten most common objects.
Pond 3.49×10-5 7.53×10-5 -54% Ponds, which are almost as common as tubs in Life, are almost three times rarer than tubs in HighLife. Life's four-cell Prepond dies out in HighLife.
Aircraft carrier 1.30×10-5 5.00×10-7 +2516% Aircraft carriers are 26 times more common. A common heptaplet Pi1.png evolves into two aircraft carriers and a blinker (in Life it is a parent of the pi heptomino sequence).
Elevener 2.50×10-7 4.55×10-9 +5395% Eleveners appear 55 times as often in HighLife, because of a predecessor involving a pi-heptomino and a blinker.
Ship 0 4.92×10-5 -100% The center cell of a ship has six living neighbors and is born in HighLife. This birth causes it all to die.

Spaceships

All of the standard spacehips from the standard Life rules work in HighLife, but only a few non-standard Life spaceships are known to work in HighLife, notably turtle, crab, and 86P9H3V0. There are also several known spaceships that are specific to HighLife[2], the most well-known of which is the bomber.

Currently the speeds known for elementary Life spaceships but not elementary HighLife spaceships are (1,0)c/7, (3,0)c/7, (1,1)c/7 and (2,1)c/6, and speeds known for HighLife and not Life are (1,0)c/8 and (1,0)c/98.


Elementary

Speed Direction Smallest known Minimum # of cells
c/2 orthogonal lightweight spaceship 9
c/3 orthogonal turtle 44
c/4 orthogonal 47P4H1V0 47
c/5 orthogonal 52P5H1V0 52
c/6 orthogonal 310P6H1V0 310
c/8 orthogonal 138P8H1V0 138
c/98 orthogonal 24P98H1V0 24
2c/5 orthogonal 106P5H2V0 106
2c/7 orthogonal 28P7H2V0 28
c/4 diagonal glider 5
c/5 diagonal 28P5H1V1 28
c/6 diagonal bomber 19

Engineered

Speed Direction Smallest known Minimum # of cells
c/12 diagonal unnamed 760
c/18 diagonal unnamed 280
c/24 diagonal basilisk 151971
c/30 diagonal unnamed 353
c/32 diagonal basilisk 117483
c/63 diagonal basilisk ?
c/69 diagonal basilisk 859 billion

Bomber

For other uses of the term 'bomber', see Bomber (disambiguation).

The bomber is a replicator-based spaceship that occurs naturally and was discovered by Nathan Thompson. It can be formed by placing a blinker in the path of the replicator as shown below. The spaceship itself has a period 48 and travels diagonally at speed c/6. The blinker reacts with one of the spawned replicators such that it destroys itself and the spawned replicator while leaving another blinker on the other side of the spaceship. It is thus a glide symmetric spaceship with mod equal to 24. It is the 3rd most common spaceship in HighLife, being slightly more common than the middleweight spaceship.[3]

x = 9, y = 14, rule = B36/S23 bo$bo$bo3$3bo$2b2o$bobo$3o2$6b3o$5bobo$5b2o$5bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART GPS 8 TRACKLOOP 48 1/6 1/6 ]]
The bomber (Catagoluehere)
(click above to open LifeViewer)
RLE: here Plaintext: here
x = 10, y = 6, rule = B36/S23 b3o6b$o9b$o9b$o8bo$9bo$9bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
A 9-cell predecessor of the bomber (click above to open LifeViewer)
RLE: here Plaintext: here

Universality

There is a proof sketch of this rule's universality as well as an explicit Rule 110 unit cell.[4] In the same topic on ConwayLife forums,[5] there is a proof-scheme covering all rules in the outer-totalistic rulespace between B3/S23 and B3678/S23678.

An explicit Rule 110 unit cell construction[6] proves its Turing-completeness; the pattern itself is a polyglot working in three other life-like cellular automata rules between B36/S23 and B368/S238; the last rule is also known as LowDeath, in which the native replicator - of which several components of the unit cell based on - has a slightly different evolution sequence.

See also

References

  1. HighLife - An Interesting Variant of Life by David Bell (.zip file)
  2. "HighLife (B36/S23)". David Eppstein. Retrieved on April 15, 2009.
  3. Adam P. Goucher. "Census". Catagolue. Retrieved on July 3, 2019.
  4. Peter Naszvadi (August 7, 2018). "List of the Turing-complete totalistic life-like CA". ConwayLife.com forums. Retrieved on May 17, 2019.
  5. Peter Naszvadi (December 12, 2016). "List of the Turing-complete totalistic life-like CA". ConwayLife.com forums. Retrieved on May 17, 2019.
  6. Peter Naszvadi (July 29, 2018). "List of the Turing-complete totalistic life-like CA". ConwayLife.com forums. Retrieved on Jan 7, 2020.

External links

  • HighLife (discussion thread) at the ConwayLife.com forums