OCA:HighLife
| HighLife | |
| View static image | |
| View animated image | |
| 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
All of the most common still lifes, oscillators and spaceships from the standard Life rules behave the exact same way 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 instead evolving differently. The T-tetromino and other common traffic light predecessors instead result in nothing but sparks, and while not all patterns that evolve into a honey farm in ConwayLife evolve the same way in HighLife, some of them (as well as some other patterns, such as a hat) evolve into a new familiar four that consists of four boats.
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 or toads 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
- Main article: OCA:HighLife/Replicator
By far the most notable pattern in HighLife is the simple replicator, shown below. It is by far the most well-known parity rule replicator (and indeed, replicator in general) in any Life-like cellular automaton. It repeatedly copies itself along a diagonal line according to Rule 6 in a one-dimensional 2-cell neighbourhood. 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.
| The replicator (click above to open LifeViewer) RLE: here Plaintext: here |
| A 6-cell, 3-generation predecessor of the replicator (click above to open LifeViewer) RLE: here Plaintext: here |
| The replicator after 12 generations (click above to open LifeViewer) |
| The replicator after another 24 generations (click above to open LifeViewer) |
A natural period 96 oscillator based on the replicator and a pair of blocks functioning as eaters exists:
| The period 96 replicator oscillator (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: 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 using a bun or a cap as an induction coil that is a still life under the standard rules is not a still life in HighLife.
Click on "Expand" to the right to view a list of still lifes 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
- Main article: OCA:HighLife/List of common still lifes
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.
Click on "Expand" to the right to view statistics of object densities.
| 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. Many common methuselahs in Conway's Game of Life, such as the R-pentomino, block and glider, and the stairstep hexomino, die without leaving any trace. Also, patterns that make traffic lights in Conway's Game of Life are simply sparks in HighLife; block and glider, the two-glider octomino, and many patterns that converge to the same evolutionary sequence as the I-heptomino (although not the I-heptomino itself) die this way in HighLife. |
| 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 spaceships 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, 31P8H4V0, 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, (1,1)c/8 and (2,1)c/6, and speeds known for HighLife and not Life are (1,0)c/8 and (1,0)c/98.
Elementary
| Direction | Speed | Period | Smallest known | Minimum # of cells |
|---|---|---|---|---|
| orthogonal | c/2 | 2 | 37P2H1V0 | 37 |
| 4 | lightweight spaceship[n 1] | 9 | ||
| 6 | 205P6H3V0[3] | 205 | ||
| 8 | 31P8H4V0[n 2] | 31 | ||
| c/3 | 3 | 27P3H1V0[4] | 27 | |
| 6 | 65P6H2V0[3] | 65 | ||
| c/4 | 4 | 32P4H1V0[n 3] | 32 | |
| 8 | 334P8H2V0[3] | 334 | ||
| c/5 | 5 | 52P5H1V0 | 52 | |
| c/6 | 6 | 179P6H1V0[6] | 179 | |
| c/8 | 8 | 138P8H1V0[7] | 138 | |
| c/98 | 98 | 24P98H1V0 | 24 | |
| 2c/5 | 5 | 106P5H2V0[8] | 106 | |
| 2c/7 | 7 | 28P7H2V0[9] | 28 | |
| diagonal | c/4 | 4 | glider[n 4] | 5 |
| c/5 | 5 | 28P5H1V1[n 5] | 28 | |
| c/6 | 6 | 311P6H1V1[n 6] | 311 | |
| 48 | bomber | 19 |
Engineered
| Direction | Speed | Smallest known | Minimum # of cells |
|---|---|---|---|
| diagonal | 2c/15 | unnamed | 22773 |
| c/9 | unnamed | 2151 | |
| c/10 | unnamed | 2850 | |
| c/12 | unnamed | 458 | |
| c/18 | unnamed | 280 | |
| c/22 | unnamed | 13326 | |
| c/24 | unnamed | 1194 | |
| c/30 | unnamed | 353 | |
| c/32 | basilisk[13] | 117483 | |
| c/42 | unnamed | 891 | |
| c/63 | basilisk[13] | ? | |
| c/69 | basilisk[13] | 859 billion |
| A collection of these spaceships (click above to open LifeViewer) |
Bomber
- Main article: OCA:HighLife/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.[14]
| 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.[15] In the same topic on ConwayLife forums, there is a proof-scheme covering all rules in the outer-totalistic rulespace between B3/S23 and B3678/S23678.[16]
An explicit Rule 110 unit cell construction proves its Turing-completeness;[17] 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.
Patterns with their own pages
See also
Notes
- ↑ the next smallest known not in Life (the second-thinnest glide-symmetric 2c/4, after the xWSS's) is a width-11 37-cell xWSS backend variation
- ↑ next smallest and smallest known not in life is 32P8H4V0[3]
- ↑ first found in [3] while attempting to find a short even-symmetric c/4; status as smallest was preceded by the 43-cell shortest odd-symmetric,[4] and the 47-cell shortest width-11 asymmetric[5] before that
- ↑ next smallest known are the crab, wings and a 29-cell ship not compatible with Life
- ↑ found by AforAmpere[10]
is the only c/5d known and 27 half-diagonals wide; it is known that a thinner symmetrical one must be at least 23 - ↑ first found in [11]
due to [12], it is the thinnest possible
the program used to find this wasn't stated, if it was ikpx2 it may not be the shortest example
References
- ↑ HighLife - An Interesting Variant of Life by David Bell (.zip file)
- ↑ HighLife (B36/S23) at David Eppstein's Glider Database
- ↑ 3.0 3.1 3.2 3.3 3.4 DroneBetter (August 3, 2024). Re: HighLife , in which (most pertinently) p6 and p8 c/2's, a p8 c/4, and the smallest c/4 and p6 c/3 were found
- ↑ 4.0 4.1 JP21 (September 26, 2020). Re: HighLife , in which new smallest-of-speed spaceships 27P3H1V0 and (now-superseded) 43P4H1V0 were found
- ↑ velcrorex (April 23, 2018). Re: HighLife , in which the smallest c/4 w10a and w11a were found
- ↑ velcrorex (November 19, 2015). Re: Thread for Your Accidental Discoveries that Aren't in CGOL , in which the first c/6 was found
- ↑ velcrorex (April 22, 2018). Re: HighLife , in which the first c/8 was found
- ↑ velcrorex (April 23, 2018). Re: HighLife , in which 106P5H2V0 was found
- ↑ wildmyron (October 6, 2018). Re: HighLife , in which the first 2c/7 was found
- ↑ Sokwe (April 23, 2018). Re: HighLife , in which AforAmpere's c/5d was noted
- ↑ LaundryPizza03 (September 1, 2023). Re: HighLife , in which the first true-period c/6d was found at w22glide
- ↑ apgoucher (May 31, 2021). Message in #naturalistic on the Conwaylife Lounge Discord server , in which c/6d's were disproven up to logical width 10
- ↑ 13.0 13.1 13.2 New HighLife velocities at Game of Life News. Posted by Adam P. Goucher on January 19, 2013.
- ↑ Adam P. Goucher. "Census". Catagolue. Retrieved on July 3, 2019.
- ↑ Peter Naszvadi (August 7, 2018). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
- ↑ Peter Naszvadi (December 12, 2016). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
- ↑ Peter Naszvadi (July 29, 2018). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
External links
- HighLife at Wikipedia
- HighLife at Adam P. Goucher's Catagolue
- HighLife at David Eppstein's Glider Database
- MCell built-in Life rules: HighLife at Mirek Wójtowicz's Cellebration page
- HighLife (discussion thread) at the ConwayLife.com forums
- New HighLife velocities at Game of Life News. Posted by Adam P. Goucher on January 19, 2013.