Here's something else interesting: There are two two-glider collisions that result in a loaf and blinker with the same relative position. It turns out that this is because each of those two collisions results in a traffic light predecessor placed so that its center is four cells away (orthogonally) from the nearest cell in the block. Since the block and the traffic light are each common individually, it's conceivable that some patterns could be common primarily due to arising from some combination of the two. Here is a list of what patterns arise from an interaction of a block and a traffic light predecessor by the displacement between what the center of the traffic light would be and the closest cell in the block:
- (3, 0): Depending on the orientation of the traffic light predecessor, either a block in the same place as it started or nothing
- (4, 0): Depending on the orientation of the traffic light predecessor, either a loaf and blinker, as discussed earlier, or a block and Beethoven
- (5, 0): Diehard
- {6, 0): A beehive (Although the beehive is only alone after a symmetrical sequence lasting for about 50 generations)
- (3, 1): Depending on the orientation of the traffic light predecessor, either nothing, a beehive, a traffic light displaced one cell orthogonally from where it would have formed without the block, or a constellation of one block, one beehive (which is from a honey farm that got mostly destroyed), one boat, and Herschel ash and two gliders (including the Herschel's SNG) in the same direction with a relative displacement of (52.5, 1.5)
- (4, 1): Depending on the orientation of the traffic light predecessor, either a blockade or a glider escaping a constellation of two blocks, two loaves, and three beehives, including one from a teardrop
- (5, 1): A block and a blinker
- (6, 1): Three-quarters of a traffic light (because the block and one of the blinkers mutually annihilated)
- (2, 2): Depending on the orientation of the traffic light predecessor, either nothing or a blinker
- (3, 2): Depending on the orientation of the traffic light predecessor, either nothing, a blinker, a blinker and a traffic light, or ash of the octomino
- (4, 2): Three-quarters of a traffic light (because the block and one of the blinkers mutually annihilated)
- (5, 2): Half of a traffic light (because the block and two of the blinkers mutually annihilated)
- (3, 3): Half of a traffic light (because the block and two of the blinkers mutually annihilated)
- (4, 3): A blinker, block, loaf, and toad
There are other possible interactions that would likely be common, like a traffic light predecessor interacting with a blinker or beehive, but this list already contains some interesting data, specifically concerning the formation of toads, which was one of the objects mentioned. A toad results from the interaction with a displacement of (4, 3). In addition, the octomino creates a toad, although, like with the century, it is later destroyed. In addition to an interaction between a block and a traffic light predecessor, an octomino is formed by one of the two-glider collisions, giving at least two potentially common predecessors of this toad-creating sequence.
Code: Select all
x = 12, y = 13, rule = TripleB3S23
2B$2B9.A$9.2A$3.B6.2A$3.B$2.B.B5$2.A$2.2A$.A.A!
#C [[ T 35 GPS 40 T 166 GPS 10 T 167 GPS 30 ]]
It's possible that the octomino sequence cut off before it could destroy the toad or the interaction between a block and a traffic light predecessor with displacement (4, 3) could account for a significant proportion of toads.
Another interesting result from looking at two-glider collisions is the fact that the pond and fishhook have two-glider syntheses despite being less common than the ship and tub, which don't. About ships, I agree with Dave Greene that the reason for ships is most likely due to the fact that they occur in Herschel ash. There are actually four two-glider collisions that make a ship, but none of them only make a ship. Instead, the ship is placed near two blocks with positioning characteristic of a Herschel, and on the other side of those two blocks lies another block (in three out of four collisions that produce a ship) or a beehive, eight blinkers (from one complete traffic light and three partially destroyed traffic lights), and two blocks. I'm not sure what the reason is for the tub, though. There's a two-glider collision that creates a tub as well as a loaf, block, and blinker (which do not have the same relative position as in the interaction between the block and traffic light predecessor with a displacement of (4, 3)), but if we count that, then we also need to count the six ponds produced from the three two-gliders collisions that result in a ∏-heptomino, so I still don't understand why the tub's greater commonness than the pond is not reflected in the ash of two-glider collisions. Granted, two-glider collisions are not mandated to have the same object frequency as 16×16 soups, but it seems unlikely that one object would appear eight times as much as a less common object.
In general, how common a particular pattern is depends more on how common its predecessors are than how simple it is (although some objects' predecessors are common because those predecessors are small). The traffic light and honey farm are common because they have small predecessors.
Code: Select all
x = 27, y = 21, rule = TripleB3S23
.B18.B5.B$3.A17.2A2D$.EGFA16.BCDCF$DBC.D18.FAE$24.D10$2.D$24.CE$.D.D20.DB$.CGC21.G$2.D$.ABA19.2G$22.ADBA!
Another example is that the pulsar and pentadecathalon, despite being relatively large and complex oscillators, are more common than simpler oscillators, like the clock and bipole. The reason that the pulsar is so common is because it has relatively small (and therefore likely relatively common) predecessors, like this eight-cell predecessor, which has eight eight-cell descendants that other patterns could converge to.
Code: Select all
x = 6, y = 4, rule = B3/S23
5bo$obo2bo$bo2bo$4b2o!
The pentadecathlon also results from a symmetric arrangement of two common traffic light predecessors.
Code: Select all
x = 3, y = 10, rule = B3/S23
bo2$3o5$3o2$bo!
There are probably many examples of patterns that occur much more often than their size would suggest. One particularly impressive example due to its sheer size is
Lidka ash.
If you're going to attempt to determine how likely a particular object is to have formed in a particular way (which I think would provide interesting and possibly useful data), then I would also like to see results from seeing how commonly different patterns are
formed, as opposed to how likely they are to form and survive. For example, the block occurs about 7.7% more frequency than the blinker
according to Catalogue, but since blocks can serve as eaters, is it possible that the blinker is actually formed more often than the block but is simply less likely to survive before everything settles? Likewise, the fishhook might be overtaken by the LWSS for the same reason as well as the fact that LWSSes that aren't created near the edge of the ash (which is more likely than it may seem because LWSSes (and objects in general) are more likely to be spawned by soups that last longer before stabilizing, which tend to have larger groups of ash, so not all of the ash is near the edge) would likely crash into a still-life or oscillator instead of escaping. Also, if many toads are formed by centuries or octominoes, the toad will probably be more common as a proportion of the total number of objects, possibly even passing the long boat.