HighLife
HighLife  


View animated image  
Rulestring  23/36 B36/S23 


Rule integer  6216  
Character  Chaotic  
Black/white reversal  B0123478/S0134678 
HighLife is a Lifelike 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]}
Contents
Notable patterns
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 infinitelylong 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 wellknown replicator in any Lifelike 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 2^{n} copies of the replicator at generation 12(2^{n}  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, preciselyengineered patterns.
A natural period 96 oscillator based on the replicator and a pair of blocks functioning as eaters exists:
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  Download RLE: click here  
5  1  Download RLE: click here  
6  4  Download RLE: click here  
7  4  Download RLE: click here  
8  9  Download RLE: click here  
9  9  Download RLE: click here  
10  25  Download RLE: click here  
11  44  Download RLE: click here  
12  111  Download RLE: click here  
13  218  Download RLE: click here 
Change in Frequency
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 ttetromino 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 fourcell 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 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 piheptomino 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 nonstandard 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 wellknown of which is the bomber.
Elementary
Speed  Direction  Smallest known  Minimum # of cells  

c/2  orthogonal  lightweight spaceship  9  
2c/5  orthogonal  106P5H2V0  106  
c/3  orthogonal  turtle  44  
c/4  orthogonal  157P4H1V0  157  
c/5  orthogonal  52P5H1V0  52  
c/6  orthogonal  310P6H1V0  310  
2c/7  orthogonal  28P7H2V0  28  
c/8  orthogonal  138P8H1V0  138  
c/98  orthogonal  24P98H1V0  24  
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 replicatorbased 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.
Universality
There is a proof sketch of this rule's universality. It is on conwaylife forums^{[3]}, which goal is to yield a proofscheme covering all rules that support glider and their rulestring matches B3[678]*/S23[678]*.
References
 ↑ HighLife  An Interesting Variant of Life by David Bell (.zip file)
 ↑ "HighLife (B36/S23)". David Eppstein. Retrieved on April 15, 2009.
 ↑ "List of the Turingcomplete totalistic lifelike CA".
External links
 HighLife at Wikipedia
 HighLife at Adam P. Goucher's Catagolue
 HighLife at David Eppstein's Glider Database
 HighLife (discussion thread) at the ConwayLife.com forums