Pentadecathlon

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Pentadecathlon
2bo4bo2b$2ob4ob2o$2bo4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ]] #C [[ THUMBSIZE 3 GPS 5 ZOOM 24 HEIGHT 300 WIDTH 600 ]]
Pattern type Oscillator
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
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.

x = 15, y = 15, rule = B3/S23 7b8o$7bob4obo$7b8o5$3o7bo$obo6bobo$3o5bobo$3o4bobo$3o5bo$3o$obo$3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 12 GPS 15 THEME Book STARTFROM 15 ]]
Hassling a long barge; p15
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere
x = 21, y = 21, rule = B3/S23 13bo4bo$5b2o4b2ob4ob2o$4b2o7bo4bo$4bo2bo$2b2ob2o$b2obo$bo2bo$3bo4$bo$b o$obo$bo$bo$bo$bo$obo$bo$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 12 GPS 15 THEME Book STARTFROM 15 ]]
Hassling an object in the butterfly sequence; p15 buddleia
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere
x = 18, y = 10, rule = B3/S23 bo14bo$bo14bo$obo12bobo$bo14bo$bo6bo7bo$bo6b2o6bo$bo6b2o6bo$obo6bo5bob o$bo14bo$bo14bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 12 GPS 15 THEME Book STARTFROM 60 ]]
Hassling a toad; p60 toadflipper
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere
x = 18, y = 11, rule = B3/S23 bo$bo14bo$obo13bo$bo13bobo$bo6bo7bo$bo6b2o6bo$bo6b2o6bo$obo6bo6bo$bo 13bobo$bo14bo$16bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 12 GPS 15 THEME Book STARTFROM 60 ]]
Hassling a toad; p60 toadsucker
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere


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.

x = 40, y = 15, rule = B3/S23 b2o5b2o20b2o5b2o$b2o5b2o20b2o5b2o2$17b6o$bo7bo6bo6bo6bo7bo$3o5b3o4bo8b o4b3o5b3o$o9bo5bo6bo5bo9bo$b3o3b3o7b6o7b3o3b3o2$4bobo26bobo4$b2o5b2o 20b2o5b2o$b2o5b2o20b2o5b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 660 THUMBSIZE 2 ZOOM 10 GPS 15 THEME Book STARTFROM 14 ]]
Period 14 (see tetradecathlon)
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere
x = 58, y = 22, rule = B3/S23 10bo7bo20bo7bo$9bobo5bobo18bobo5bobo$10bo7bo20bo7bo4$23bo2bob2obo2bo$10b3o3b 3o3b2o2bo4bo2b2o3b3o3b3o$3b2o6bo5bo5bo2bob2obo2bo5bo5bo6b2o$3bo50bo$2obo50bob 2o$o2b3o46b3o2bo$b2o3bo14b2o12b2o14bo3b2o$3bob2o14bobo10bobo14b2obo$3bo2bo16b o10bo16bo2bo$4b2ob3o6b2o5b2o8b2o5b2o6b3ob2o$6bobobo5b2o22b2o5bobobo$6bo3bo2bo 30bo2bo3bo$7b2ob4o30b4ob2o$9bo38bo$9bobo34bobo$10b2o34b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 660 THUMBSIZE 2 ZOOM 10 GPS 15 THEME Book STARTFROM 29 ]]
Period 29 (nonaisocathlon) (see p29 pentadecathlon hassler)
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere
x = 54, y = 22, rule = B3/S23 6b2o18b2o18b2o$6bobo15bo4bo15bobo$8bo14bo6bo14bo$3bo2bob2o3b3o6bo8bo6b 3o3b2obo2bo$3b4o8b3o4bo8bo4b3o8b4o$7b2o2b2o9bo8bo9b2o2b2o$3b4o16bo6bo 16b4o$2bo3bob2o14bo4bo14b2obo3bo$o2b2obobo5bo11b2o11bo5bobob2o2bo$2ob o4bo4bobo22bobo4bo4bob2o$3bobobob2o2b2o24b2o2b2obobobo$3bobobobo34bob obobo$b2o4bobo34bobo4b2o$2bob2obo2b2o30b2o2bob2obo$2bobo3bobo32bobo3b obo$b2o2bob2obo32bob2obo2b2o$2bobo4bo34bo4bobo$2bo3b3o36b3o3bo$3b2obo 40bob2o$8b3o32b3o$b7o2bo32bo2b7o$bo2bo2bo38bo2bo2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 660 THUMBSIZE 2 ZOOM 10 GPS 15 THEME Book STARTFROM 21 ]]
Period 21 (monoicosathlon)[3][4] (using p7 pipsquirters)
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere
x = 72, y = 19, rule = B3/S23 22bo12b2o12bo$13bo8bo10bo4bo10bo8bo$13bo8bo9bo6bo9bo8bo$13bo17bo8bo17b o$18b3o3b3o4bo8bo4b3o3b3o$5bo3b3o3b3o13bo8bo13b3o3b3o3bo$4b2o16bo9bo6b o9bo16b2o$5bo7bo8bo10bo4bo10bo8bo7bo$b2obo8bo8bo12b2o12bo8bo8bob2o$o4b o3bo3bo44bo3bo3bo4bo$4bo3bobo50bobo3bo$2o68b2o$4b2o4bo50bo4b2o$bo7b2o 50b2o7bo$o4bo3b5o44b5o3bo4bo$bob2o4b2ob2o44b2ob2o4b2obo$o9bo50bo9bo$2o 68b2o$o70bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 660 THUMBSIZE 2 ZOOM 8 GPS 15 THEME Book STARTFROM 36 ]]
Period 36[5] (hexatriacontathlon) (see traffic jam; sparker found by James Pascua[6][7])
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere

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]

x = 22, y = 15, rule = B3/S23 2b2o5b2o$3b2o3b2o$o2bobobobo2bo$3ob2ob2ob3o$bobobobobobo$2b3o3b3o5b3o$ 15bo3bo$2b3o3b3o3bo5bo$bobobobobobo$3ob2ob2ob3o8bo$o2bobobobo2bo8bo$3b 2o3b2o$2b2o5b2o3bo5bo$15bo3bo$16b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 380 THUMBSIZE 2 ZOOM 12 THEME Book STARTFROM 15 ]]
pulsar-on-pentadecathlon I
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagoluehere

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.

x = 18, y = 18, rule = B3/S23 8bobo$8b2o$9bo5bo$15bobo$15b2o11$3o$2bo$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 14 THEME Book AUTOSTART GPS 12 T 20 PAUSE 3 T 79 PAUSE 1 LOOP 80 ]]
A 3G synthesis for the pentadecathlon
(click above to open LifeViewer)
x = 22, y = 18, rule = B3/S23 bo18bo$2bo16bo$3o16b3o13$2b2o14b2o$bobo14bobo$3bo14bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 14 THEME Book AUTOSTART GPS 12 T 20 PAUSE 3 T 79 PAUSE 1 LOOP 80 ]]
A 4G synthesis[12]
(click above to open LifeViewer)
x = 30, y = 22, rule = B3/S23 27bobo$14bo12b2o$12bobo13bo$13b2o13$16b2o3b2o$15b2o4bobo$17bo3bo$3o$2bo$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 14 THEME Book AUTOSTART GPS 12 T 20 PAUSE 3 T 79 PAUSE 1 LOOP 80 ]]
A 5G synthesis[11]
(click above to open LifeViewer)
x = 18, y = 10, rule = B3/S23 10b2o$10b2o3$b2o7b2o$obo6bo2bo$2bo7b2o4bo$15bobo$14bo2bo$15b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 14 THEME Book AUTOSTART GPS 12 T 20 PAUSE 3 T 99 PAUSE 1 LOOP 100 ]]
A three-object 1G seed[13]
(click above to open LifeViewer)
x = 20, y = 10, rule = B3/S23 7b2o$2b2o3bobo$2bo5bo$obo$2o3$17b2o$17bobo$17bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 14 THEME Book AUTOSTART GPS 12 T 20 PAUSE 3 T 79 PAUSE 1 LOOP 80 ]]
A two-object 1G seed from the octohash database
(click above to open LifeViewer)
x = 12, y = 16, rule = B3/S23 5b2o$4bo2bo$5b2o2$bo6b3o$obo$obo$bo6$10b2o$9b2o$11bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 400 THUMBSIZE 2 ZOOM 14 THEME Book AUTOSTART GPS 12 T 20 PAUSE 3 T 79 PAUSE 1 LOOP 80 ]]
An 1G seed from the octo3obj database
(click above to open LifeViewer)


In other rules

  • In EightLife, it still functions as a period 15 oscillator, but it generates extra internal sparks.

See also

Notes

  1. The correct Ancient Greek dual and plural forms, pentadecathlo and pentadecathla,[1] are not in common use.
  2. 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.
  3. 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.
  4. 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

  1. "ἆθλον". English Wiktionary. Retrieved on 2016-06-16.
  2. Dean Hickerson's oscillator stamp collection. Retrieved on March 14, 2020.
  3. hotcrystal0 (January 2, 2022). Re: Symbiosis (discussion thread) at the ConwayLife.com forums
  4. iNoMed (January 2, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  5. David Raucci (January 2, 2022). Message in #cgol on the Conwaylife Lounge Discord server
  6. James Pascua (June 10, 2021). Re: JP21's Lonely Freedomical Room (discussion thread) at the ConwayLife.com forums
  7. James Pascua (June 11, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  8. Achim Flammenkamp (September 7, 2004). "Most seen natural occurring ash objects in Game of Life". Retrieved on January 15, 2009.
  9. 9.0 9.1 Adam P. Goucher. "Statistics". Catagolue. Retrieved on September 22, 2020.
  10. gameoflifeboy (April 20, 2015). Re: Soup search results (discussion thread) at the ConwayLife.com forums
  11. 11.0 11.1 Robert Wainwright (June 1973). Lifeline, vol 10, page 6.
  12. 12.0 12.1 Robert Wainwright (September 1973). Lifeline, vol 11, pages 18, 19.
  13. Life Period-15 Oscillators at Mark D. Niemiec's Life Page (download pattern file: 12/12pc.rle)

External links