This post is about 8-cell patterns that produce *WSSs.
There are no patterns with fewer than eight cells that produce any form of spaceship other than a glider. With 8 cells, of course you can produce an LWSS (just remove the spark from the LWSS’s recurring 9-cell pattern). But it turns out both the MWSS and HWSS can emerge from an 8-cell starting pattern, although in all those cases there will also be static debris and gliders.
The rle code below is for a field containing examples of all the possible velocity-class sets involving *WSSs, when starting with 8 cells. In fact, these patterns, together with some minor variants as described below, and rotations and reflections, are all those 8-cell patterns which produce a *WSS. The gaps in the rows are where patterns that do not produce a *WSS, but were selected by the scripts I used, were removed. The scripts all looked, among some subset of 8-cell patterns, for those that did not, after 8192 steps, consist solely of a stationary pattern repeating on a cycle that is a factor of 12, plus retreating gliders that will not interact with anything in future. Some of those removed patterns are “methusalehs” that just take more than 8192 steps to stabilise (discussed in an earlier post). Others, to be discussed in a later post, do actually consist of a stationary pattern plus retreating gliders at 8192, but include a pentadecathlon – so their stationary portion is not the same after s+12 steps as it is after s steps.
x = 2040028, y = 300076, rule = B3/S23
o39999bo40000bo39998b3o39997b3o39997b4o39996b4o40001bo39994bo40001bo
39999bo39997bo39999bo39999bo$o39999bo39999b3o39997bo40002bo39996bo3bo
39995bo39999bo40003bo39995b2obo39996b2obo40001bo39995bo5bo39993bo$obo
39997bobo39997bo2b2o39998b2o39998bo39996bo39999bo3bo39995bobob2o39994b
2ob2o39997bo40001bo39995b2obobo39996bo39999bo4bo$4bo40000bo39996bo
40002bo39998b2o39995bo39999bo40001bo40001bo39998b3o39994bo2bo39997bo
40000b4o39996b4o$2b3o39997b3o79999bo40001bo119995bo39999bo80003bo
39997bo59996$o3bo39995bo39999bo39999b3o2bo39994b3o2bo39994bo39999bo
39999bo39999bo39999bo2b3o39994bob2o39996bo2b3o39996bo39999bo39999bo
39999bo39999bo39999b2o39999bo39997bo3bo39995bo39999bo39999bo39999bo
39999bo$obo39997bobo39997bobo39997bo2bo39996bo2bo39997b2o2bo39994bobo
39997bobo39997bobo39997bo3bo39999bo39997bobo39995bobo39997bobo39997bob
o39997bobo39997bobo39997bo39999bobo39998bobo39996bobo39997bobo39997bob
o39997bobo39997bobo$o2bo40002bo79994bobo39999bo39996bo2bo159998b2o
39996bo2b3o39994bo2bo40002bo199995bo39999bo39997bo2bo160002bo$3bo
39998bobo39997bobobo79994bo39999bobo39998bob2o39996bobobo39995bobo
159997bobo39997bobobo39995bob2o39996bobobo39995bobo39997bo80001bo
39997bob2o39996bobobo39995bobo39997bobo39997bobobo$bo40003bo39999bo
160000bo39998bo39999b2o159998bo39999bo40000bo39998bo39999b2o39997b3o
39995bob3o39995bo40003bo39998bo39999b2o39998bo39999bo$40004bo39999bo
159998bo39999bo39999bo160000bo39999bo39998bo39999bo39999bo159999bo
39999bo39999bo40000bo39999bo59995$24b3o39983b3o39987b2o39998b2o80001b
2o79999b2o39994b3o39997b3o39997b2o40015bo39982b2o40006bo39993b2o39997b
o120008b2o39989bo$2o22bo2bo39982bo2bo39986bo39999bo80002bo80000bo
39995bo2bo39996bo2bo21b2o39973bo40015b2obo39980bo40006b2obo39991bo
39997b2obo120006bo39989b2obo$o360024bo39991bo39999bo79990bo79992bo
159999bo$400016b2obo39980b2o40006bo$400017bo39982bo40006b2obo39989b2o$
480008bo39991bo3$80003b3o199994b3o$40000b2o40001bo2bo199993bo2bo40007b
2o$40000bo280010bo399989bo$720000b2obo40006b2o$720001bo40008bo5$
120002b3o79995b3o$120002bo2bo79994bo2bo6$560001bo$560000b2obo39998b2o$
560001bo40000bo59974$3o39997b3o40007b3o39996b3o39988b3o40003b3o39991b
3o40001b3o39993b3o40005b3o39989b3o39997b3o40009b3o40001b3o39997b3o
39981bo40012bo39986b3o39997b3o39997b3o39997b3o39997b3o40004b3o40009b3o
40000b3o39975bo40004bo$80010bo2bo39995bo2bo79993bo2bo119990bo2bo79996b
o2bo39996bo2bo120012bo2bo39980bo40012bo39986bo2bo319996bo40004bo$
600000bo40012bo359986bo40004bo$560000bo120017bo39999bo$560000bo120017b
o39998b2obo$560000bo120017bo39999bo159982bo$880000b2obo$200000b3o
120003b3o480002bo79989bo$16b3o519981b3o280007b2obo159987bo$16bo2bo
519980bo2bo280007bo119989bo39998b2obo$160008b3o199989b3o559997b2obo
39997bo$160008bo2bo199988bo2bo400009bo159987bo$40012b3o439985b3o
280009b2obo239997bo$40012bo2bo439984bo2bo280009bo239998b2obo$600011b3o
39986b3o360010bo$600011bo2bo39985bo2bo200005bo$840008b2obo$240004b3o
39993b3o560006bo199991bo$240004bo2bo39992bo2bo759996b2obo$80000b3o
360007b3o599988bo3$120000b3o280006b3o59978$120012bo39987b2o39998b2o
39998b2o39998b2o120008b2o39988bo80020b2o120005b2o40001b2o39999b2o
40011b2o$120012b3o39985bo39999bo39999bo39999bo120009bo39989b3o80018bo
120006bo40002bo40000bo40012bo$120000b2o11bo319987bo$120000bo8$440011b
2o$440011bo279988bo$720000b3o$720001bo14$160011bo79998bo$160011b3o
79996b3o$160012bo79998bo2$520000bo$520000b3o$520001bo5$400000bo$
400000b3o$400001bo$200010bo439989bo$200010b3o439987b3o$200011bo439989b
o4$760000bo$760000b3o$760001bo2$280009bo$280009b3o$280010bo7$680000bo$
680000b3o$680001bo59937$80000b3o79997b3o39997b3o39997b3o40006bo39990b
3o39997b3o39998b3o79996bo39999bo80011b3o39985bo40018b3o40005b3o40000b
3o39999b3o80001b3o39998b3o40002b3o39955bo80051b3o39983bo39993bo39967bo
40011bo39987bo39999bo40008bo39990bo40004bo39994bo79999bo40003bo79995bo
119999bo40018bo80001bo39999bo80005bo$280009b3o199988b3o39997b3o119997b
3o359997b3o120035b3o39991b3o39965bo40011b3o39985bo39999bo40008b3o
39988bo40004b3o39992bo79999b3o40001bo79995b3o119997b3o40016bo80001bo
39999bo80005bo$200010bo79999bo199990bo39999bo119999bo359999bo120037bo
39993bo39966bo40012bo39986bo39999bo40009bo39989bo40005bo39993bo80000bo
40002bo79996bo119999bo40017bo80001bo39999bo80005bo$200010b3o$200011bo
319997b3o1280004bo$1000042b3o799971bo$1560002bo240013bo$1560002bo$
1560002bo$400000bo$400000b3o$400001bo$1680008bo$1680008bo$1680008bo2$
800000bo1239999bo$800000b3o79997bo39999bo1119999b3o$800001bo79998b3o
39997b3o1119998bo$880001bo39999bo$80028bo$80028b3o1279969bo$80029bo
1279970bo$1360000bo2$1240000bo$240009bo79994bo919995bo$160011bo79997b
3o79992b3o919993bo$160011b3o79996bo79994bo1000004bo$160012bo439987bo
680009bo39999b3o$600000b3o600010bo79996b3o39998bo$600001bo600011b3o
79995bo$1200014bo239985bo$480008b3o160007b3o319979bo440004bo39994bo$
960000b3o440002b3o39992bo$280000b3o679998bo440004bo199993bo$1600000b3o
$1600001bo4$680000bo$680000b3o$680001bo399998bo400002bo$760000bo
319999b3o400000b3o$760000b3o319998bo79998bo320003bo$760001bo399998bo$
1160000bo679999bo$1840000b3o$1840001bo119998bo$1960000b3o$360002bo
759997bo840000bo$360002b3o759995bo$360003bo759996bo9$720000bo$720000b
3o$720001bo9$1920000bo$1920000b3o$1920001bo!
The top row (row 1) consists of 14 single-cluster patterns that increase their cell-count on the first step. The full set of these (not counting rotations and reflections) was tested. The row includes six patterns that develop quickly into a single LWSS (velocity-class set {N/2} once rotated), seven that leave static debris and send out gliders in all four directions plus an LWSS (so their velocity-class set is {ZV, N/2, NE/4, SE/4, SW/4, NW/4}, of which five develop into the same stable state (columns 1, 2, 4, 8, 12). Column 3’s pattern produces gliders in three directions along with an LWSS, if rotated so the LWSS goes north its velocity-class set would be {ZV, N/2, NE/4, SE/4, SW/4}. There are two different ways a set of 3 glider-directions can relate to the direction of a *WSS: either the missing direction can be 45° away from the *WSS direction of travel, as in the current case, or 135° away from it (which would be symbolised {ZV, N/2, NE/4/, SE/4, NW/4}).
Row 2 consists of 25 single-cluster patterns that maintain the same cell-count (8) at step 1. Since all patterns up to 7 cells have been tested and none produce a *WSS, an 8-cell patterns that does so cannot have a cell-count below 8 at any stage: it may go through a number of 8-cell steps, but must eventually develop through one of the patterns that increases cell-count at step 1. Hence if there had been no *WSS producers among the latter, there would have been no need to test those having 8-cell immediate successors. As it is, we know there will not be any patterns that end up looking different from all those in row 1.
Row 3 contains 16 patterns that start as a 5-cell pi-predecessor and a preblock. There are two distinct 5-cell pre-pi patterns, not counting rotations and reflections; one of the two has reflectional symmetry on one orthogonal axis, which makes a difference if we want to start counting patterns, and comparing the numbers leading to paraticular results – see below. Row 3 includes examples of patterns producing a *WSS and gliders in two directions. There are four ways a pair of glider directions of travel can relate to a *WSS travel direction: the two glider directions may be opposite each other, or at a right angle; and in the latter case, the *WSS direction may be between the two glider directions (at a 45° angle with both), at a 135° angle with both, or at a 45° angle with one and a 135° angle with the other – the case here, symbolised {ZV, N/2, NE/4, SE/4}.
Each of the patterns in row 3 actually represents more than one distinct pattern, because the preblock can have any of four orientations. The sixteen patterns constitute four groups of four giving the same final pattern (modulo rotations/reflections), all producing a single LWSS. However, the first eight of the 16 patterns each represent four distinct possibilities (four different orientations of the preblock), while the other eight, in which the pi-predecessor has an axis of symmetry, each represent just two possibilities (or to put it another way, pairs of sets of 4 form mirror images). So in total, the patterns shown represent (8*4)+(8*2) = 48 distinct LWSS-producers.
Row 4 is of 27 patterns including a pi-predecessor and a blinker. It consist of three sets of nine patterns that give the same final nine results (three sets because one of the two pi-predecessors has an axis of symmetry which it shares with the blinker); all 27 produce a single LWSS plus debris. New velocity-class sets in this row are {ZV, N/2, NE/4, SE/4, NW/4} – first appearing in column 1, once reflected left-right; {ZV, N/2, SE/4} (appearing first in column 3, after rotation), and {ZV, N/2} (appearing first in column 7).
Row 5 is of 12 patterns consisting of an r-pentomino and pre-block. The pattern in column 6 (the third pattern in the row as columns 1-3 are empty) produces an HWSS, the other 11 an LWSS each. No new velocity-class sets appear. Each pattern in this row represents no fewer than 36 distinct staerting patterns, since any orientation of the preblock can be combined with any of nine 5-cell patterns: the r-pentomino itself (as shown), its six 5-cell parents and two 5-cell grandparents.
Finally, row 6 is of 39 patterns each consisting of an r-pentomino and blinker. In each case, any of the r-pentomino’s eight 5-cell predecessors could be substituted for it, so each pattern shown represents a set of nine.
• Patterns in columns 3, 5, 6, 7, 8, 10, 11, 13, 14, 18, 19, 20, 21, 25, 26, 28, 29, 30, 33, 34, 35, 36, 38, 41, 43, 46, 47, 49 and 52 produce an LWSS. So 29*9=261 in all.
• Those in columns 9, 16, 17, 23, 24, 31, 37 and 50 produce an MWSS, so 8*9=72 in all. Column 17 yields a new velocity-class set: {ZV, N/2, SE/4, SW/4}.
• Number 40 produces an HWSS. Note that the r-pentomino and blinker are quite close, but it has been checked that all the 5-cell r-pentomino predecessors could be paired with a blinker to produce the pattern. So 9 in all. The velocity-class set is novel: {ZV, N/2, NE/4, SW/4}.
• Number 32 produces two LWSSs travelling in opposite directions. Again, representing 9 patterns. The velocity-class set for the pattern is {ZV, N/2, NE/4, SE/4, S/2, SW/4, NW/4}.
Summing up, if my search strategy, Python script coding and arithmetic are correct, there are 771 distinct 8-cell patterns that produce a single LWSS, or 771*8 = 6168 counting rotations and reflections (none of the patterns have any symmetry when considered as a whole), and 9 (72 with rotations and reflections) that produce two LWSSs. There are 73 that produce an MWSS (584 with rotations and reflections), and 45 that produce an HWSS (360 with rotations and reflections). In Sparse Life (see above), *WSSs of all three types (LWSS, MWSS, HWSS) produced from 8-cell starting patterns will hugely outnumber all others of the same type, once the initial few thousand steps are complete. The LWSS, MWSS and HWSS population ratios to be expected in a sufficiently large chunk of the field are 789::73::45.
Finally for this post, two copies of the two-LWSS producing pattern, arranged at right angles, can readily make a 16-cell pattern that sends spaceships in all either cardinal and semi-cardinal directions (and leaves static debris) – velocity-class set {ZV, N/2, NE/4, E/2, SE/4, S/2, SW/4, W/2, NW/4}. But we can do at least a little better: since a glider can hit a beehive in a way that results in an r-pentomino (or rather, the r-pentomino’s fifth successor), and the beehive has 4-cell predecessors, a blinker and an accompanying straight row of 4 (which becomes a beehive) can be placed so a glider from one copy of the two-LWSS pattern hits the blinker, and the combination develops as a second copy of the two-LWSS pattern, at right angles to the first. An example pattern is shown below:
x = 181, y = 312, rule = B3/S23
178bo$178b3o$179bo23$166bo$166bo$166bo273$25bo$25bo$25bo6$o$o$o$o!