Pentadecathlon
| Pentadecathlon | |||||||||
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| Pattern type | Oscillator | ||||||||
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| Number of cells | 12 | ||||||||
| Bounding box | 16 × 9 | ||||||||
| Frequency class | 18.6 | ||||||||
| Period | 15 | ||||||||
| Mod | 15 | ||||||||
| Heat | 22.4 | ||||||||
| Volatility | 1.00 | ||||||||
| Strict volatility | 1.00 | ||||||||
| Discovered by | John Conway | ||||||||
| Year of discovery | 1970 | ||||||||
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- For the website named "Pentadecathlon", see Game of Life News.
Pentadecathlon (or PD; plural pentadecathlons[note 1]) is a period-15 oscillator that was found in 1970 by John Conway[2] while tracking the history of short rows of cells (see one-cell-thick pattern); indeed, an orthogonal row of 10 cells evolves into this object. It is the only known oscillator that is a polyomino in more than one phase (besides the blinker), and is the smallest oscillator with a period greater than its minimum population.
Uses of the pentadecathlon
- See also: p30 technology
The pentadecathlon is so called because it has a period of 15 generations. This, being a factor of 30, means that it can be elegantly used in combination with period 30 devices (based on the queen bee shuttle). Firstly, it can reflect a glider 180° as in p60 glider shuttle, and a pair of perpendicular pentadecathlons can rotate a glider 90° or 180° as in 6 bits and 106P135.
A pentadecathlon can interact with another pentadecathlon to form a larger non-trivial oscillator, with bi-pentadecathlon 1 being a notable such example.
Hassling capabilities
The pentadecathlon is classified as a pulsating oscillator, since it undulates throughout its cycle. During this process, the pentadecathlon throws off multiple accessible sparks. More specifically, the oscillator produces horizontal T-nose sparks and horizontal V sparks in the form of phi sparks, as well as vertical domino sparks. Two copies of these domino sparks can be used to hassle toads in two distinct ways. The V sparks elegantly convert blocks into gliders, which forms the basis of the p30 xWSS-to-glider converter and aforementioned glider reflectors. This property is also exploited in numerous oscillators.
| Hassling a long barge; p15 (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
| Hassling an object in the butterfly sequence; p15 buddleia (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
| Hassling a toad; p60 toadflipper (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
| Hassling a toad; p60 toadsucker (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
Phase changing
The pentadecathlon can undergo phase-changing reactions. The first two in the gallery below are in the form 14 + 15N, while the last two are in the form 6 + 15N.
A related pattern, the snacker, involves a pentadecathlon being forced to oscillate at period 9.
| Period 14 (see tetradecathlon) (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
| Period 29 (nonaisocathlon) (see p29 pentadecathlon hassler) (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
| Period 21 (monoicosathlon)[3][4] (using p7 pipsquirters) (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
| Period 36[5] (hexatriacontathlon) (see traffic jam; sparker found by James Pascua[6][7]) (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
Occurrence
- See also: List of common oscillators
Pentadecathlon is the most common natural oscillator of period greater than three (and indeed, the second most common natural oscillator of period greater than 2) in Achim Flammenkamp's census. In fact, it is the fifth or sixth most common oscillator overall in this census, being about as frequent as the clock, but much less frequent than the blinker, toad, beacon or pulsar.[8] It is also the most common oscillator with a volatility of 1. The pentadecathlon is also the fifty-second most common object on Adam P. Goucher's Catagolue and by far the most common period 15 oscillator, with all other natural oscillators of that period featuring it combined with some other object.[9]
The frequency of the pentadecathlon significantly depends on soup parameters; they are more common in smaller soups and lower densities, while they are less common on a torus.[note 2]
The pulsar-on-pentadecathlon I is the largest object to have occurred in the B3/S23/C1 census as of July 2023, with 100 cells in its maximum phase, and with first known natural occurrence on Catagolue in April 2015.[10][9]
| pulsar-on-pentadecathlon I (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
There are 35 results in the octohash database with the final pattern containing a pentadecathlon.[note 3] There are also 173 results in the octo3obj database and 6 results in the octo3g database[note 4] with a pentadecathlon in the ash.
Construction
A 5-glider synthesis was reported by Douglas G. Petrie in June 1973.[11] Soon Peter Raynham found a 4-glider recipe.[12]
On April 11, 1997, Heinrich Koenig found a three-glider collision that produced a clean pentadecathlon. This was a surprising result at the time. Four-glider pentadecathlon recipes had been known and used for many years, so this was an unlooked-for improvement, very similar to Luka Okanishi's discovery of a three-glider synthesis of a switch engine almost twenty years later. This also makes the pentadecathlon the rarest object in Catagolue for which a 3-glider synthesis is known.
Below are shown animations of several ways to construct a pentadecathlon, with visible construction envelopes.
| A 3G synthesis for the pentadecathlon (click above to open LifeViewer) |
| A 4G synthesis[12] (click above to open LifeViewer) |
| A 5G synthesis[11] (click above to open LifeViewer) |
| A three-object 1G seed[13] (click above to open LifeViewer) |
| A two-object 1G seed from the octohash database (click above to open LifeViewer) |
| A period-2 1G seed from the octo3obj database (click above to open LifeViewer) |
In other rules
See also
- Polypentadecathlon
- Karel's p15
- Loaflipflop
- Mold on pentadecathlon
- Pentadecathlon on 37P7.1
- Pentadecathlon on thumb 1
Notes
- ↑ The correct Ancient Greek dual and plural forms, pentadecathlo and pentadecathla,[1] are not in common use.
- ↑ 12 times as common in an 8 × 8 soup, 5.5 times as common in a 10 × 10 soup, 1.8 times as common in a 16 × 16 soup at 25% density, and 0.75 times as common on a torus.
- ↑ There is one clean 1G seed (headerless RLE: 7b2o$2b2o3bobo$2bo5bo2b3o$obo8bo$2o10bo!). There are two other "exceptional" results: b2o$o2bo$o2bo$b2o2$5b2o$5bo$3bobo$3b2o2$2bo$b2o$bobo! creates 7 escaping gliders and has final population 271 excluding the pentadecathlon; 8b2o$8bobo$9bobo$10bo5$b2o$o2bo$obo2b2o$bo3bobo$5bo! creates 30 escaping gliders and has final population 1593 excluding the pentadecathlon. Eight other results converge to a sequence (bo$2o$2bo2$bo2bo$2o2b2o$o4bo4$3b2o$3b2o!) which evolves to an escaping glider and a pi+block collision. All other results converge to the pi+block sequence.
- ↑ One 3G collision is the clean synthesis (headerless RLE: 3bo$4bo$2b3o3$5b2o$5bobo$o4bo$b2o$2o!). Two other collisions (9bo$8bo$8b3o$o$b2o$2o2$5bo$3b2o$4b2o! and bo$2bo$3o$18bo$19bo$17b3o3$17b2o$18b2o$17bo!) converge to a pi+block collision. Three more collisions evolve to a B+glider collision (bo$2bo$3o25$21b3o$21bo2bo$21bob2o!) via 2G syntheses of the B-heptomino sequence.
References
- ↑ "ἆθλον". English Wiktionary. Retrieved on 2016-06-16.
- ↑ Dean Hickerson's oscillator stamp collection. Retrieved on March 14, 2020.
- ↑ hotcrystal0 (January 2, 2022). Re: Symbiosis (discussion thread) at the ConwayLife.com forums
- ↑ iNoMed (January 2, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ David Raucci (January 2, 2022). Message in #cgol on the Conwaylife Lounge Discord server
- ↑ James Pascua (June 10, 2021). Re: JP21's Lonely Freedomical Room (discussion thread) at the ConwayLife.com forums
- ↑ James Pascua (June 11, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ Achim Flammenkamp (September 7, 2004). "Most seen natural occurring ash objects in Game of Life". Retrieved on January 15, 2009.
- ↑ 9.0 9.1 Adam P. Goucher. "Statistics". Catagolue. Retrieved on September 22, 2020.
- ↑ gameoflifeboy (April 20, 2015). Re: Soup search results (discussion thread) at the ConwayLife.com forums
- ↑ 11.0 11.1 Robert Wainwright (June 1973). Lifeline, vol 10, page 6.
- ↑ 12.0 12.1 Robert Wainwright (September 1973). Lifeline, vol 11, pages 18, 19.
- ↑ Life Period-15 Oscillators at Mark D. Niemiec's Life Page (download pattern file: 12/12pc.rle)
External links
- Pentadecathlon at the Life Lexicon
- Pentadecathlon at Adam P. Goucher's Catagolue
- Pentadecathlon at Heinrich Koenig's Game of Life Object Catalogs
- Life Period-15 Oscillators at Mark D. Niemiec's Life Page (download pattern file: 12/12pc.rle)
- Pentadecathlon Crane at Game of Life News. Posted by Heinrich Koenig on January 11, 2009.
- Patterns
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- Natural periodic objects
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- Periodic objects with minimum population 12
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