ConwayLife.com

Posted: **January 15th, 2021, 11:41 am**

I believe that both the Catagolue numbers and the near-infinite (2048×2048) torus numbers should be on the wiki page. There are some significant differences. The first column is the pattern name, the second is the inverse frequency on the torus, the third is the inverse frequency on Catagolue, and the last is how much more common it is on Catagolue (excluding spaceships and switch engines, otherwise all numbers would be about 9% less). The 80 or so most common still lifes are included, then a decent number of oscillators.

Even if this isn't added to the wiki, there are still interesting observations here. Most patterns are more common on the torus, with the block and the ship (the latter due to Herschel creation) making up for it. The few other patterns that seem to be more common on Catagolue are typically the symmetrical ones (look at dead spark coil for an extreme example, and ship-tie, pulsar, and pentadecathlon for lesser ones). The most common Catagolue objects that don't make the top 100 on a torus (and therefore aren't in this list) are up dove on dove and up wing and wing, which are both symmetric.

T- and C- are trans- and cis-, done to save space.

Code: Select all

Block			3.094	3.24	4.8%
Blinker			3.038	3.49	-4.5%
Beehive			5.23	6.11	-6.1%
Glider			N/A	11.38	N/A
Loaf			16.79	20.71	-11.0%
Boat			18.27	22.41	-10.6%
Ship			132.9	32.52	348.2%
Tub			81.71	103.0	-13.0%
Pond			86.78	105.7	-9.9%
Long boat		279.9	319.4	-3.9%
Toad			373.6	457.4	-10.4%
Ship-tie		730	630.1	27.1%
Beacon			1221	1463	-8.5%
Barge			1234	1516	-10.7%
Half-bakery		1943	1857	14.7%
Mango			3148	4130	-16.4%
Eater 1			5756	6401	-1.4%
LWSS 			N/A	7620	N/A
Long barge		8116	11395	-21.9%
Aircraft carrier	13077	13170	8.9%
Pulsar			19584	14210	51.2%
Paperclip		20597	22662	-0.3%
MWSS			N/A	29571	N/A
Long ship		24259	33266	-20.0%
Integral sign		35129	36188	6.5%
Shillelagh		32987	41951	-13.7%
Boat-tie		37688	42846	-3.5%
Snake			37880	47819	-13.1%
Big S			39770	47962	-9.0%
Bi-pond			49455	55025	-1.4%
Trans-boat with tail	87374	100421	-4.6%
Boat tie ship		109739	120807	-0.4%
Hat			90686	127518	-22.0%
Very long ship		161862	155993	13.8%
HWSS			N/A	170180	N/A
Very long boat		128043	174266	-19.4%
Tub with tail		139675	184682	-17.0%
Table and table		181444	246810	-19.4%
Dead spark coil		1397161	261257	486.6%
Canoe			219181	297543	-19.2%
Cis-mirrored R-bee	346884	403702	-5.7%
Moose antlers		382224	448808	-6.6%
Beehive on dock		629542	550251	25.5%
Block on table		414201	552142	-17.7%
Block on dock		422961	606314	-23.5%
Scorpion		810964	754334	17.9%
Beehive with tail	648130	912916	-22.1%
Twin hat		460605	956388	-47.2%
Loop			810964	1115786	-20.3%
Long snake		1318008	1131502	27.8%
Fourteener		959835	1147666	-8.3%
Pentadecathlon		1.63E6	1.32E6	35.8%
Bookends		1.42E6	1.41E6	10.4%
Cis-boat with tail	1.18E6	1.74E6	-25.6%
Cis-rotated hook	1.53E6	1.77E6	-4.8%
Elevener		1.44E6	1.81E6	-13.0%
Mirrored dock		1.95E6	1.96E6	8.7%
Block on cap		1.50E6	2.03E6	-19.3%
Trans-loaf with tail	1.52E6	2.09E6	-20.3%
Cis-shillelagh		1.86E6	2.12E6	-3.7%
T-mirrored R-bee	1.39E6	2.19E6	-30.1%
Clock			1.69E6	2.26E6	-17.9%
T-block on long hook	1.68E6	2.33E6	-20.8%
Block-laying SE		N/A	2.53E6	N/A
Prodigal		2.61E6	2.58E6	10.7%
Broken snake		2.08E6	2.86E6	-20.1%
T-hook and R-bee	2.09E6	3.09E6	-25.7%
Eater siamese eater	2.32E6	3.17E6	-19.8%
Block and two tails	2.17E6	3.17E6	-25.0%
Cis-boat on dock	2.31E6	3.18E6	-20.3%
C-block on long hook	2.42E6	3.39E6	-21.9%
Very long snake		2.43E6	3.43E6	-22.1%
Boat with long tail	3.02E6	3.69E6	-10.0%
Long shillelagh		2.96E6	3.84E6	-15.3%
Beehive at loaf		3.37E6	3.86E6	-4.4%
T-R-bee and R-loaf	3.02E6	4.61E6	-28.2%
Long integral		3.86E6	4.79E6	-11.6%
Tub with long tail	3.43E6	4.81E6	-21.8%
Cis-hook and R-bee	3.23E6	4.82E6	-26.5%
Hook with tail		3.70E6	4.83E6	-15.8%
Loaf siamese loaf	3.66E6	4.93E6	-18.5%
Long canoe		5.13E6	6.01E6	-6.3%
11 loop			3.90E6	6.10E6	-30.0%
Ortho-loaf on table	8.01E6	6.12E6	43.5%
Cis-loaf with tail	4.79E6	6.15E6	-14.5%
Symmetric scorpion	4.77E6	6.15E6	-15.0%
Claw with tail		4.61E6	6.41E6	-21.0%
Bee hat			5.06E6	6.75E6	-17.9%
Trans-rotated R-bee	5.10E6	6.88E6	-18.7%
Glider-producing SE	N/A	7.01E6	N/A
Great on-off		1.16E7	1.75E7	-27.6%
Figure eight		3.33E7	3.68E7	-0.9%
Spark coil		3.68E7	5.36E7	-24.8%
Mold			4.57E7	6.43E7	-22.1%
Quadpole on ship	1.10E8	1.57E8	-23.3%
Tripole			2.11E8	3.18E8	-27.4%
Mazing			2.24E8	3.38E8	-27.3%
Blocker			3.66E8	4.85E8	-17.1%
Jam			4.08E8	6.41E8	-30.2%
Trans-QB shuttle	5.70E8	1.12E9	-44.1%
Cis-QB shuttle		5.70E8	1.14E9	-45.2%
T-beacon on table	9.29E8	1.35E9	-24.4%
Cis-beacon on table	8.96E8	1.37E9	-28.3%
Cis-beacon on dock	6.97E8	1.44E9	-46.9%
Test tube baby		1.29E9	1.73E9	-18.3%
Achim's p4		1.00E9	1.78E9	-38.3%
Cis-beacon on anvil	1.11E9	1.85E9	-33.9%
Octagon 2		1.67E9	2.15E9	-14.5%
Unix			1.67E9	2.17E9	-15.6%
1 beacon		3.34E9	2.22E9	65.1%
Tub test tube baby	2.39E9	2.37E9	10.6%
T-beacon on dock	2.51E9	3.14E9	-12.3%
Tumbler			2.39E9	3.28E9	-20.2%

Posted: **January 15th, 2021, 3:48 pm**

I can explain some of these. Many of the larger or rarer symmetrical objects that form in Catagolue soups occur when the pattern becomes symmetric early on, which forces it to stay in that symmetry (or go into a higher symmetry) throughout its evolution. I don't think that the pentadecathlon and ship-tie would be as affected by this as the dead spark coil and pulsar because orthogonal lines of reflection symmetry tends to form more often than diagonal lines of reflection symmetry, orthogonal lines of D2_+1 reflection symmetry tend to form more often than orthogonal lines of D2_+2 reflection symmetry, and orthogonal lines of D2_+1 often develop gutter symmetry; however I could be wrong. A potential alternative explanation for why pentadecathla appears much more frequently in Catagolue is that the relatively small soup size results in a higher tendency for the pattern to coalesce into a bottleneck early on, and one of these bottlenecks, a pi-sequence interacting with a block in a certain way, results in a pentadecathlon. However, a pi-sequence interacting with a block in a different way will result in an integral, and integrals only occur 6.5% more frequently on Catagolue, compared to 35.8% for pentadecathla. One possibility is that because the pi+block collision that produces an integral doesn't take as much time or space as the pi+block collision that produces a pentadecathlon, the pi+block collision that produces an integral could still produce an integral on a torus that has other objects elsewhere, while this would be much less feasible for the pi+block collision that produces a pentadecathlon. Alternatively, you may be correct that the pentadecathlon appears more often with smaller initial soups due to its symmetry, or there could be another reason that neither of us have considered.
The only other object with a major frequency different that I can explain (or at least attempt to explain) is the ship. On a practically infinite plane with a small starting configuration, it won't be uncommon for a B- or R-sequence to form near the edge and edgeshoot Herschel ash, which includes a ship, beyond the rest of the pattern, where it is less likely to be destroyed. However, on a torus, it's impossible to edgeshoot Herschel ash "beyond the edge of the pattern", and the B-sequence or Herschel sequence may even crash into something before creating a ship. This may also explain why blocks are slightly more common in Catagolue, but half-blockades are also edgeshot, and a B-sequence or blockade sequence could also occur as a bottleneck that a relatively small soup coalesces into early on.

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