Pulsar

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Pulsar
2b3o3b3o2$o4bobo4bo$o4bobo4bo$o4bobo4bo$2b3o3b3o2$2b3o3b3o$o4bobo4bo$o 4bobo4bo$o4bobo4bo2$2b3o3b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ]] #C [[ HEIGHT 600 THUMBSIZE 3 ZOOM 30 GPS 2 ]]
Pattern type Oscillator
Number of cells 48
Bounding box 15 × 15
Frequency class 12.1
Period 3
Mod 3
Heat 42.7
Volatility 0.73
Strict volatility 0.73
Rotor type Pulsar
Discovered by John Conway
Year of discovery 1970

Pulsar (rarely referred to as Cambridge pulsar CP 48-56-72[note 1]) is a large but surprisingly common period-3 oscillator. It was found by John Conway in March 1970.[1]

Extensions and interactions

The rotor of a pulsar consists of four mutually stabilizing quadrants; alternate arrangements exist for any odd multiple of 4 (for the version with 12 copies, see quasar). A closely related oscillator — the pulsar quadrant — includes just the external "horns" of the rotor and can be stabilized on its own.

Two other oscillators, the bracket pulsar and cross, can be extended in much the same way.

x = 42, y = 15, rule = B3/S23 32bo3bo$2b3o3b3o21bo3bo$32bo3bo$o4bobo4bo18b2o3b2o$o4bobo4bo5bo2bo8bob o3bobo$o4bobo4bo5bo2bo5b5o5b5o$2b3o3b3o6b2o2b2o$15b3o4b3o$2b3o3b3o$o4b obo4bo14b5o5b5o$o4bobo4bo2b3o4b3o5bobo3bobo$o4bobo4bo4b2o2b2o8b2o3b2o$ 18bo2bo10bo3bo$2b3o3b3o7bo2bo10bo3bo$32bo3bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 4 LOOP 3 ZOOM 12 ]]
The pulsar (left), cross (center) and bracket pulsar (right)
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RLE: here Plaintext: here

Arms

There exists a heptomino, that acts as a p3 oscillator partial but can mutually stabilise with a quadrant of the pulsar's nucleus, replacing a standard horn in its blinker phase to comprise an "arm" and causing it to undergo a different period-3 evolutionary sequence. Due to its interference with the quadrant's nucleus, only a single arm may exist per quadrant. Despite an arm requiring the flipping of only six cells' states in this phase, no arm variant has occurred naturally, assumedly due to the pulsar's fragility. However, the two variants with two arms in D2_+1 symmetry,[note 2] and the variants with four arms in D4_+1 and C4_1 symmetry,[note 3] have occurred seminaturally.

The same heptomino can also mutually support a quadrant from a different orientation, replacing another quadrant as one of its two supports, allowing two distinct two-quadrant half-pulsars with arms.

x = 99, y = 78, rule = B3/S23 9b2o$10b2o$4b3o3b2o$11bo$2bo4bobo4bo$2bo4bobo4bo$2bo4bobo4bo$4b3o3b3o 2$4b3o3b3o$2bo4bobo4bo$2bo4bobo4bo$2bo4bobo4bo2$4b3o3b3o6$6b2ob2o15b2o 60b2o$5b2o3b2o13b2o60b2o$5b2o3b2o13b2o3b3o11b3o3b3o13b3o3b3o12b2o3b3o$ 5bo5bo13bo61bo$2bo4bobo4bo7bo4bobo17bobo14bo4bobo12bo4bobo4bo$2bo4bobo 4bo7bo4bobo3b3o5b3o3bobo3b3o8bo4bobo3b3o6bo4bobo4bo$2bo4bobo4bo7bo4bob o4b3o3b3o4bobo4b3o7bo4bobo4b3o5bo4bobo4bo$4b3o3b3o11b3o3b3o3bo3bo3b3o 3b3o3bo9b3o3b3o3bo7b3o3b3o2$4b3o3b3o11b3o3b3o11b3o3b3o9bo3b3o3b3o11b3o 3b3o3bo$2bo4bobo4bo7bo4bobo4bo7bo4bobo4bo7b3o4bobo4bo7bo4bobo4b3o$2bo 4bobo4bo7bo4bobo4bo7bo4bobo4bo8b3o3bobo4bo7bo4bobo3b3o$2bo4bobo4bo7bo 4bobo4bo7bo4bobo4bo14bobo4bo7bo4bobo2$4b3o3b3o11b3o3b3o11b3o3b3o13b3o 3b3o11b3o3b3o6$6b2ob2o15b2ob2o18b2o17b2o$5b2o3b2o13b2o3b2o18b2o15b2o$ 5b2o3b2o13b2o3b2o12b3o3b2o15b2o3b3o$5bo5bo13bo5bo19bo15bo$2bo4bobo4bo 7bo4bobo4bo12bobo4bo9bo4bobo$2bo4bobo4bo7bo4bobo4bo6b3o3bobo4bo9bo4bob o3b3o$2bo4bobo4bo7bo4bobo4bo5b3o4bobo4bo9bo4bobo4b3o$4b3o3b3o11b3o3b3o 7bo3b3o3b3o13b3o3b3o3bo2$o3b3o3b3o11b3o3b3o11b3o3b3o13b3o3b3o$3o4bobo 4bo7bo4bobo4bo7bo4bobo4bo9bo4bobo4bo$b3o3bobo4bo7bo4bobo4bo7bo4bobo4bo 9bo4bobo4bo$7bobo4bo7bo4bobo4bo7bo4bobo4bo9bo4bobo4bo$25bo19bo21bo$4b 3o3b3o12b2o3b3o12b2o3b3o14b2o3b3o$25b2o18b2o20b2o$26b2o18b2o20b2o4$6b 2ob2o15b2o$5b2o3b2o13b2o41b2o$5b2o3b2o13b2o3b3o11b3o3b3o14b2o$5bo5bo 13bo41b2o3b3o$2bo4bobo4bo7bo4bobo17bobo17bo$2bo4bobo4bo7bo4bobo3b3o5b 3o3bobo3b3o8bo4bobo$2bo4bobo4bo7bo4bobo4b3o3b3o4bobo4b3o7bo4bobo3b3o$ 4b3o3b3o11b3o3b3o3bo3bo3b3o3b3o3bo7bo4bobo4b3o$66b3o3b3o3bo$4b3o3b3o 11b3o3b3o11b3o3b3o$2bo4bobo4bo7bo4bobo4bo7bo4bobo4bo7bo3b3o3b3o$2bo4bo bo4bo7bo4bobo4bo7bo4bobo4bo7b3o4bobo4bo$2bo4bobo4bo7bo4bobo4bo7bo4bobo 4bo8b3o3bobo4bo$5bo5bo13bo5bo13bo5bo17bobo4bo$5b2o3b2o13b2o3b2o13b2o3b 2o21bo$5b2o3b2o13b2o3b2o13b2o3b2o14b3o3b2o$6b2ob2o15b2ob2o15b2ob2o21b 2o$71b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 4 LOOP 3 ZOOM 6 HEIGHT 600 ]]
All pulsar arm variants
(click above to open LifeViewer)
x = 31, y = 13, rule = B3/S23 26b2o$25b2o$4bo5bo9bo4b2o$4bo5bo9bo4bo$4b2o3b2o9b2o2b3o$24bobobo$3o2b 2ob2o2b3ob3o2b2ob2o2b3o$2bobobobobobo5bobobo$4b3ob3o9b3o2b2o$5bo3bo11b o4bo$4b2o3b2o9b2o4bo$4b2o3b2o9b2o$3b2o5b2o7b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 4 LOOP 3 ZOOM 12 ]]
The cis- and trans- forms[note 4] of the half-pulsar with arms
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Dependent oscillators

x = 28, y = 28, rule = B3/S23 2b3o3b3o2$o4bobo4bo$o4bobo4bo$o4bobo4bo$2b3o3b3o2$2b3o3b3o$o4bobo4bo$o 4bobo4bo$o4bobo4bo2$2b3o3b3o3$9b2o6b3o3b3o$10bo$6b4ob2o2bo4bobo4bo$10b obo2bo4bobo4bo$10bo4bo4bobo4bo$10bo6b3o3b3o$10bo$17b3o3b3o$15bo4bobo4b o$15bo4bobo4bo$15bo4bobo4bo2$17b3o3b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ GPS 4 LOOP 3 ZOOM 12 HEIGHT 600 ]]
Two can support a highly volatile small period-3 oscillator that contains a tub in one phase (as seminaturally occurred in tetramer form[note 5])
(click above to open LifeViewer)
Catagoluehere

Occurrence

See also: List of common oscillators

Despite its size, pulsar is the fourth most common oscillator (and the most common of period greater than 2) in Achim Flammenkamp's census; the only oscillators more common are blinker, toad, and beacon.[2] It is by far the most common period-3 oscillator, being about 45,000 times more common than jam.[3] Overall, the pulsar is the twenty-first most common object on Adam P. Goucher's Catagolue.[4]

Compared to Catagolue's typical 16 × 16 soups, the pulsar is 30% more common in 8 × 8 soups and 14% more common in 10 × 10 soups; this is probably because a pulsar is more likely to survive if there is nothing around it.

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.[5][4]

The smallest pattern by population that evolves into a pulsar has seven cells; there are several of these.[note 6]

Construction

Pulsar can be constructed with 3 gliders; there are multiple known syntheses.[6][note 7]

A direct Herschel-to-pulsar factory was found by yoleo in October 2023.[7]

x = 41, y = 33, rule = B3/S23 30bo5bo$30bo5bo$30b2o3b2o2$26b3o2b2ob2o2b3o$bo7b2o17bobobobobobo$b2o6b obo18b2o3b2o$obo6bo$30b2o3b2o$28bobobobobobo$26b3o2b2ob2o2b3o2$30b2o3b 2o$30bo5bo$30bo5bo16$24b3o$24bo$25bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 450 THEME Book ZOOM 10 X -2 Y -4 AUTOSTART GPS 12 T 0 PAUSE 3 T 8 PAUSE 1 T 87 PAUSE 1 T 152 PAUSE 1 T 209 PAUSE 1 T 230 PAUSE 1 LOOP 231 ]]
A three-glider synthesis[6]
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x = 30, y = 14, rule = B3/S23 19b3o3b3o2$17bo4bobo4bo$bo15bo4bobo4bo$2bo14bo4bobo4bo$3o16b3o3b3o2$ 19b3o3b3o$17bo4bobo4bo$17bo4bobo4bo$17bo4bobo4bo2$2o2b2o13b3o3b3o$2o2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 450 THEME Book ZOOM 10 X -4 Y 0 AUTOSTART GPS 12 T 0 PAUSE 3 T 16 PAUSE 1 T 50 PAUSE 1 T 54 PAUSE 1 T 78 PAUSE 1 T 90 PAUSE 1 LOOP 91 ]]
A Blockic one-glider seed for the pulsar[6][8]
(click above to open LifeViewer)
x = 41, y = 46, rule = B3/S23 8b2o$8b2o6b2o$16bo$14bobo$14b2o2$11b2o$10bo2bo$2b2o7b2o$bobo$bo$2o5$5b o$5bobo$5b3o$7bo2$39b2o$39b2o10$23b2o$22bobo$22bo$21b2o7$27b2o$26bobo$ 26bo$25b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 2 ZOOM 10 X 4 Y 0 THEME Book AUTOSTART T 0 PAUSE 3 T 299 PAUSE 1 LOOP 300 ]]
H-to-pulsar factory[7]
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In other rules

In Pedestrian Life (B38/S23), the pulsar is about 400 times less common than in Life because the pre-pulsar does not evolve into it. However, there are new predecessors, making it still the most common period-3 oscillator.[9]

x = 12, y = 5, rule = B38/S23 2bo$bobo5b3o$o3bo4bo$bobo5b3o$2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 450 THEME Book ZOOM 20 X 2 Y 0 AUTOSTART GPS 8 T 0 PAUSE 3 T 44 PAUSE 1 LOOP 45 ]]
A pulsar evolving from a pre-traffic light and a pi heptomino in Pedestrian Life
(click above to open LifeViewer)

As a more extreme example, the pulsar is about 1500 times less common in Grounded Life (B35/S23) than in Life because the pre-pulsar does not evolve into it, instead evolving to a common alien oscillator. Furthermore, there are no new predecessors, accounting for the extreme increase in rarity. Despite all of these, it still the most common period-3 oscillator.

x = 5, y = 3, rule = B35/S23 o3bo$2ob2o$o3bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 500 WIDTH 600 THUMBSIZE 3 ZOOM 36 GPS 2 AUTOSTART ]]
A 8-cell pre-pulsar predecessor from Life is a p4 oscillator in Grounded Life
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On the other hand, in EightLife (B3/S238), the pulsar is 170 times more common than in Life because the pi-heptomino evolves into it.[10]

x = 3, y = 3, rule = B3/S238 3o$obo$obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 450 THEME Book ZOOM 15 AUTOSTART GPS 15 T 0 PAUSE 3 LOOP 135 ]]
A pulsar evolving from a pi heptomino in EightLife
(click above to open LifeViewer)

See also

Notes

  1. The numbers refer to the populations of the three phases. The Life pulsar was indeed discovered at Cambridge, like the first real pulsar a few years earlier.
  2. variant 1, variant 2
  3. D4_+1 variant, C4_1 variant
  4. cis- form, trans- form
  5. tetramer form
  6. An example is 3bo$o5bo$b2ob2o!, which evolves into another 7-cell predecessor after one tick.
  7. There are 77 matches for the pulsar in the octo3g database (out of 464746 lines). There are also 404 matches in the octo3obj database (out of 3421907 lines), and 99 matches in the octohash database (out of 455380 lines). If all collisions with at least one occurrence of pulsar in the ash are included, then there are 632 results in the octo3g database (including three collisions with two pulsars in the ash), 4833 results in the octo3obj database (including 53 results with two pulsars in the ash) and 551 results in the octohash database (including two collisions with two pulsars in the ash).

References

  1. Dean Hickerson's oscillator stamp collection. Retrieved on March 14, 2020.
  2. Achim Flammenkamp (September 7, 2004). "Most seen natural occurring ash objects in Game of Life". Retrieved on January 15, 2009.
  3. Adam P. Goucher. "Census". Catagolue. Retrieved on October 27, 2018.
  4. 4.0 4.1 Adam P. Goucher. "Statistics". Catagolue. Retrieved on March 1, 2023.
  5. gameoflifeboy (April 20, 2015). Re: Soup search results (discussion thread) at the ConwayLife.com forums
  6. 6.0 6.1 6.2 The 1 period 3 oscillator buildable from 3 gliders at Mark D. Niemiec's Life Page (download pattern file: lg/48pu.rle)
  7. 7.0 7.1 yoleo (October 17, 2023). Re: Thread For Your Useless Discoveries (discussion thread) at the ConwayLife.com forums
  8. knightlife (June 23, 2013). Blockic Seeds (discussion thread) at the ConwayLife.com forums
  9. Adam P. Goucher. "census/b38s23/C1". Catagolue. Retrieved on March 19, 2023.
  10. Adam P. Goucher. "census/b3s238/C1". Catagolue. Retrieved on March 26, 2023.

External links