Strangest CA-related mathematical coincidences?

[confocaloid]

What are the strangest known CA-related mathematical coincidences?
E.g. two or more apparently unrelated reactions with the same effect, without any obvious explanation for why that would be the case.

LoM-based 8c/96d reaction
(posted in "Spaceship Discussion Thread")

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#C https://conwaylife.com/forums/viewtopic.php?p=160357#p160357
#C https://conwaylife.com/forums/viewtopic.php?p=160345#p160345
#C https://conwaylife.com/forums/viewtopic.php?p=143269#p143269
x = 171, y = 171, rule = B3/S23
28bo$20bo7bo$20b3o5bo$22bobo3b2o$20bo2bo$20b2o2bo$8b2o$8b2o7$2o$2o$14b
2o3b2o$13bobo3b4o3bo$13bo3bo2bo2b2obo$13b2o3bo4b2obo$18bo3bo$17bo3$27b
o6b3o$27bob2o3b2o$27bobo5b2o$27bob3o$27b2o14$28bo$29b2ob2o12bo$29b2obo
12bobo$31bo12bo3bo$44bo3bo$42b3obobo$41bo2bob2o$41bobo$42b2o8$24b2o$
20bo$22bo$20bobo$65b2o6bo$64bo3bo3bobo$35b3o26b2o3bobo3bo$34bo4bo8bo
15b2obobo2bobo$38b2o8b4o13b2o2bo3bo$34bo3bo25bo4bo$34bo30bo2bo$34bobo
29bo$34bo11bobo$32bo3b2o9bo$36b2o10b2o$35bobo8bo2bo$30bo5bo9b2ob2o2$
30b2ob2o$32b3o24bo$57bo2bo7b3o$56bo3bo5b2obobo$56bo3bo5b2obobo$56bo3b
2o5bo3bo$57bo2bo7bobo$69bo4$65bo$64bobo$64bobo$65bo5$57bo48bo$56bobo
46bobo$56bobo45bo3bo$57bo47bobo$89b3o14bo$88bo3bo$87bo2b3o10b4o$88bo3b
o9bo4bo$49bo39b3o13bo$48bobo30b2o7b2o9bo2bo3bo$48bobo29bo2bo18bob3obo$
49bo31b2o20bob3o3$89bo$88b5o$93bo$73b2o$40bo31bo2bo12bo3bo$40bo32b2o
14b3o2$38b3o$40bo$21bo3b2o10bob2o$21b2o13b2o66bo$37b2o26b2o29bo7bo$37b
ob2o23bo2bo28b3o5bo$24bo12b5o23b2o31bobo3b2o17b3o$23bobo10bobo2b2o53bo
2bo22bo3bo$23bo12b2o3b3o52b2o2bo21b2o3bo$24bo2bo13b2o83bo$13b2o9bo3bo
11b2o80b4o$13b2o9b2obo12bo5bo73b2obo$24bo20b3o72bo2bo$24b3o17b5o72b2o$
14b2o27b2o3b2o2b3ob2o$14b2o32b2o2bobo49b2o$47b2obo3bo48b2o$46bob5o44b
3o6bo$46b2o3bo44bob2obo4bo37bobo$97bo8bo37b3o$93b2o5b4obo39b2o$93b2o
32bo$93bo5bo26b2o$94bo30bo17bo$94bobo29b2o15bobo$126b2o15b3o21bo$44bo
83bo13bo24b4o$43bobo96b5o$41bo10bo98b3o$41bobo3bo4bo59bo$41b5o2bo63bo
38b2o12bobo$46b2o103b2o13bo$51bo57b2o39bo2bo13b2o$51b2o100bo11bo2bo$
109bobo40b3o10b2ob2o$153b2o$150b2o$149bo2bo2$39b2o112b2o$39b2o2b2o106b
o2bo$43b2o106b3o4$163b2o$163b2o7$155b2o$155b2o!

F117 + L156_1 and F176 + L112
(posted in "Herschel conduits and other stable circuits")
F117 + L156_1 and F176 + L112 perform the same relocation of a Herschel, only with 15-tick time difference:

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x = 77, y = 69, rule = TripleLife
62.2F$61.F2.F$62.2F8$67.2F$67.F$65.3F6$73.2F$73.F.F$75.F$75.2F7$65.2F
$56.2F7.2F$57.F$57.F.F$52.2B4.2F$53.B20.2B$31.2B20.B.B18.B$32.B6.2B
13.2B16.B.B$32.B.B4.2B31.2B$33.2B$29.B27.D$29.3B25.3D$21.2B9.B15.D11.
D$21.2B4.2D2.2B15.3D8.2D14.D$27.D.D21.D22.D.D$17.D11.D20.2D23.D$17.3D
9.2D12.2B$20.D22.2B$7.2F10.2D$8.F57.2B4.2D$8.F.F55.2B2.2BD.D$9.2F41.B
3.2B12.B.B.D$32.2D17.B.B3.B14.B.2D$32.2D16.B.B3.B13.2D2B$51.B3.B14.D$
46.2B4.4B.B5.2D6.3D$46.B.B7.B.B4.D9.D$48.B3.2B2.B.B5.3D$9.F38.2B2.2B
3.B8.D$9.F.F$9.3F30.2D$11.F11.2D18.D$23.D16.3D$21.2B.3D12.BF$21.2B2.B
F13.B$25.B.B12.B.2B$2.2F23.B7.B2.B.B2.B$3.F23.2B6.4B.2B$3F36.B$F36.B.
B$37.2B!

SE38T127
(posted in "Thread For Your Useless Discoveries")
There are two different Herschel-to-glider converters with systematic description "SE38T127", i.e. two different ways to produce an output glider on the same lane with the same timing:

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x = 44, y = 52, rule = TripleLife
15.2B$16.B$16.B.B$17.2B6$33.2B$33.B$31.B.B$10.D20.2B8.D$9.D.D27.3D$9.
2D27.D$29.2B4.2B.2D$9.2D18.2B5.B$9.D.D21.3B$10.D22.B$35.2D$35.2D3$12.
2D$11.D.D26.2D$11.D28.D$10.2D29.3D$43.D2$7.2F$8.F$8.F.F$9.2F17.2F.2F$
28.2F.F.F$33.F$33.2F4$9.F$9.F.F18.2F$9.3F18.2F$11.F5$2.2F$3.F19.2F$3F
20.F$F23.3F$26.F!
#C [[ T 75 PAUSE 1 T 162 PAUSE 1 STOP 257 ]]

2c/11d in B2n3-e/S234c

(edit: it turns out there are two different natural 2c/11d spaceships. Are these mechanisms related in any way, or this is a coincidence?)

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#C [[ ZOOM 8 TRACK 2/11 2/11 ]]
x = 25, y = 5, rule = B2n3-e/S234c
2o2bobo14bobo$2o2bo15bo2b2o$3bo$2b3o15b2o2bo$3bo17bob2o!

[H. H. P. M. P. Cole]

There is absolutely no reason I have come across that explains why there are 40 4-cell objects in Factorio. This is probably not a coincidence. Here's why I posted it in this thread:

- The rulestring is relatively short.
- All the objects are, remarkably, mod 1.

The mystery of the treasure trove of 4-cell pbjects is still intriguing to me despite investigating Factorio for quite a long time.

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x = 149, y = 80, rule = R3,C2,S2,B3,N+
2o13bobo13bo2bo$2o13bobo13bo2bo9$15bobo13bo2bo2$15bobo13bo2bo9$31bo2b
o3$31bo2bo17$2b2o11b2o14b2o12bobo15bobo15b2o15b2o10bobo9b2o12b2o8bobo
$bo2bo9bo3bo10bo4bo9bo3bo13bobo14b2o16bobo9bo2bo7bo2bo11b2o8b2o9$2b2o
11b2o14b2o12bobo15bobo15b2o15b2o10bobo9b2o12b2o8bobo2$bo2bo9bo3bo10bo
4bo9bo3bo13bobo14b2o16bobo9bo2bo7bo2bo11b2o8b2o9$2b2o11b2o14b2o12bobo
15bobo15b2o15b2o10bobo9b2o12b2o8bobo3$bo2bo9bo3bo10bo4bo9bo3bo13bobo14b
2o16bobo9bo2bo7bo2bo11b2o8b2o14$bo2b3o!

[Haycat2009]

I do not know wheteher this counts, but there are 4 F Herschel-accepting conduits with step between 110 and 117: HF110B, F114, F116 and F117

[confocaloid]

I do not know wheteher this counts, but there are 4 F Herschel-accepting conduits with step between 110 and 117: HF110B, F114, F116 and F117

Maybe there are no particular reasons why that could not count for someone else. But for me at least, that does not count. First, it is not a coincidence. All four mentioned conduits have different steps. Second, it is not very strange. Even for Herschel-to-Herschel conduits specifically, there are different conduits for each of steps 156, 198, 200 e.g., listed in the old collection "h-to-h-collection-26Aug2017.rle".

[confocaloid]

There are several different period-46 gun mechanisms, and reactions at multiples of 46. This may not be the strangest one, but it is certainly a coincidence, acknowledged by others --

I am utterly speechless at the moment... Behold, the new smallest period-46 glider gun with a 555-cell bounding box (37x15):

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x = 37, y = 15, rule = B3/S23
24b2o$24b2o$36bo$10b2o22b3o$2o7bobo13b2o6bo$2o7bo15b2o6b2o$9b3o4$9b3o
21b3o$2o7bo20b2o$2o7bobo13bo4b2o$10b2o11bobo4bobo$24b2o!

I'm utterly speechless at the moment... Behold, a brand-new period-46 glider gun without any Snarks, twin-bees shuttles or Tanner's P46s:

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x = 30, y = 31, rule = B3/S23
13b2o$8b2o3bo2bobo$2o5b2o6b2ob3o$bo7bo11bo$bobo11b2ob3o$2b2o11b2obo2$
20b2o$20bo$18bobo$2o5b2o9b2o$2o5bobo$7bobo$8b2o4$20b2o$20bobo$20bobo5b
2o$10b2o9b2o5b2o$9bobo$9bo$8b2o2$11bob2o11b2o$9b3ob2o11bobo$8bo11bo7bo
$9b3ob2o6b2o5b2o$11bobo2bo3b2o$15b2o!

Is the allusion intentional?

It is indeed intentional; both posts involve absolutely-enormous discoveries pertaining to period-46 glider guns.

I find it very interesting that the two earliest known simple guns (p30 GGG and p46 twin bees gun) both have comparably simple but unrelated guns at multiples of their period, 30 x 4 = 120 for the Simkin gun and 46 * 4 = 184 for this gun. Furthermore, while the originals are based upon shuttles, the latter are based upon Herschel conduits.

Also the p138 gun based on Gabriel Nivasch's p138 is 46 x 3. Many a strange coincidence.

[confocaloid]

There is a composite F154 (made out of two Fx77) and an elementary L154, where the active reactions converge at T = 135 ignoring a dying noninteracting spark:

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#C https://conwaylife.com/forums/viewtopic.php?p=170678#p170678
#C https://conwaylife.com/forums/viewtopic.php?p=170750#p170750
#C [[ STOP 135 ]]
x = 70, y = 78, rule = LifeHistory
42.2A$43.A$43.A.A$44.2A$45.3B$45.3B.B$47.4B.B$47.8B$49.6B$49.2B3C2B3.
2B3.2A$49.2BC4B3.4BA2BA$49.B3C4B.5B2A.A$49.14B.B.A.2A$49.14B2A2.A.A$
49.13BA2.2A$50.12B.2A$51.10B3.A$51.11B2.A.A$49.12B2.2A.3A$48.11B5.B4.
A$41.2A6.9B4.B2AB3A$42.A6.12B.B2A.A$42.A.AB2.11B2A2B$43.2AB.10B.BA.AB
$45.12B3.2A$45.12B$46.9B.B2A$45.12BA.A$45.11B3.A$46.10B3.2A$46.9B$46.
9B$47.7B$46.9B.2B$46.13BD$44.15BDBD$43.5B2E9B3DB$43.5B2E11BD$44.16B$
47.6B.B$49.B7$51.A$49.3A$25.A22.A$25.3A20.2A$28.A17.4B$27.2A16.3B$27.
4B14.4B$8.A20.3B12.5B$8.3A12.B.3B.4B11.6B$11.A10.18B2.8B$10.2A3.B5.2B
2E11BD14B$10.8B3.2B2E9B3D13B$12.8B.13BDBD4B.7B$12.22BD15B$11.15B.2B2.
19B$9.17B8.17B$7.18B11.15B.2B2.2B$7.2BC15B12.22BD$6.3BCBC4B.7B13.8B.
13BDBD$7.2B3C13B10.8B3.2B2E9B3DB$6.5BC14B9.2A3.B5.2B2E11BD$5.10B2.8B
11.A10.13B$4.4B11.6B8.3A12.B.3B.B$4.3B12.5B9.A$2.4B14.4B$2.2A16.3B$3.
A17.4B$3A20.2A$A22.A$24.3A$26.A!

[confocaloid]

[...]
LoM-based 8c/96d reaction
[...]
2c/11d in B2n3-e/S234c
[...]

Along the same lines, I didn't see a reason why the following two reactions (found in a rule located midway between Life-like rules B3/S23 and B35/S23) must have the same simplified speed (orthogonal 4c/31 in both cases):

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#C An orthogonal 8c/62 spaceship and an orthogonal 16c/124 puffer
x = 39, y = 25, rule = B35cejkr/S23
2bo23b2o$bobo21b2obo$o2bo22bob2o8bo$3bo22bobo9bo$obo25bo$bo36bo$38bo$
30b2o$30b2o7$23b2o$23b2o7$30b2o$30b2o!

Is there a common mechanism that I failed to notice? Or this is an example of a mathematical coincidence?