HighLife
HighLife | ||
| ||
View animated image | ||
Rulestring | 23/36 B36/S23 | |
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Rule integer | 6216 | |
Character | Chaotic |
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; it's only when patterns get complex that their behavior differs. 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.
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 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. This contrasts with the standard Game of Life, where all known sawtooths are complex, precisely-engineered 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 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 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 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 the only non-standard spaceships that are known to work in HighLife are the turtle, 86P9H3V0, and some flotillae of the standard spaceships. There are also several known spaceships that are specific to HighLife^{[2]}, the most well-known of which is the bomber.
Elementary
Speed | Direction | Smallest known | Minimum # of cells | |
---|---|---|---|---|
c/2 | orthogonal | lightweight spaceship | 9 | |
2c/5 | orthogonal | 164P5H2V0 | 164 | |
c/3 | orthogonal | turtle | 44 | |
c/4 | orthogonal | 157P4H1V0 | 157 | |
c/5 | orthogonal | 52P5H1V0 | 52 | |
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/24 | diagonal | basilisk | ? |
c/32 | diagonal | basilisk | ? |
c/63 | diagonal | basilisk | ? |
c/69 | diagonal | basilisk | ? |
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.
Universality
There is a proof sketch of this rule's universality. It is on conwaylife forums^{[3]}, which goal is to yield a proof-scheme 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 Turing-complete totalistic life-like 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