熠熠种花 - Glimmering Garden (B3-jknr4ity5ijk6i8/S23-a4city6c7c)

[GUYTU6J]

Earlier this year I published the construction of x-in-a-row (3,12) oblique spaceships using copies of cloverleaf. Below is a review of two examples with annotation:

Code: Select all

#C [[ ARROWTRACK 3/1275 12/1275 ARROW 143.5 24.5 111.5 16.5 3.2 ]]
#C [[ ARROWTRACK 3/1275 12/1275 ARROW 111.5 16.5 82.5 16.5 3.2 ]]
#C [[ ARROWTRACK 3/1275 12/1275 ARROW 82.5 16.5 49.5 16.5 3.2 ]]
#C [[ ARROWTRACK 3/1275 12/1275 ARROW 49.5 16.5 16.5 19.5 3.2 ]]
#C [[ LABELTRACK 3/1275 12/1275 LABEL 165 0 3.2 "(3,12)c/1275" ]]
#C [[ LABELTRACK 3/1275 12/1275 LABEL 127.5 0 3.2 "(-32,8,-65)" ]]
#C [[ LABELTRACK 3/1275 12/1275 LABEL 97 0 3.2 "(-29,0,-41)" ]]
#C [[ LABELTRACK 3/1275 12/1275 LABEL 66 0 3.2 "(-33,0,-12)" ]]
#C [[ LABELTRACK 3/1275 12/1275 LABEL 33 0 3.2 "(-33,-3,-41)" ]]
#C [[ ARROWTRACK 3/1035 12/1035 ARROW 143.5 84.5 111.5 79.5 3.2 ]]
#C [[ ARROWTRACK 3/1035 12/1035 ARROW 111.5 79.5 79.5 79.5 3.2 ]]
#C [[ ARROWTRACK 3/1035 12/1035 ARROW 79.5 79.5 46.5 79.5 3.2 ]]
#C [[ LABELTRACK 3/1035 12/1035 LABEL 165 60 3.2 "(3,12)c/1035" ]]
#C [[ LABELTRACK 3/1035 12/1035 LABEL 127.5 60 3.2 "(-32,5,-61)" ]]
#C [[ LABELTRACK 3/1035 12/1035 LABEL 95.5 60 3.2 "(-32,0,-19)" ]]
#C [[ LABELTRACK 3/1035 12/1035 LABEL 63 60 3.2 "(-33,0,-19)" ]]
x = 186, y = 111, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7cSuper
44.B9.B22.B9.B18.B9.B$43.3B5.B.3B.BM17.3B5.B.3B.2B13.3B5.B.3B.2B$39.
2B.5B3.6B.2MB12.2B.5B3.6B.3B8.2B.5B3.6B.3B$11.B9.B16.9B2.6B2M5B9.9B2.
13B5.9B2.13B$10.3B5.B.3B.2B11.10B2.6BMBM5B7.10B2.14B3.2BM7B2.14B$6.2B
.5B3.6B.3B9.10B2.16B5.10B2.16B.2B3M5B2.16B$5.9B2.13B7.19BM4BM4B4.30B
2MB2M23B$4.10B2.14B5.20BMBM3BM3B3.31BM2B2M23B$3.10B2.16B3.22B2M2B2MBM
3.30B2M4B2M21B12.B9.B$3.29B2.24B6MB2.20B2M9B3MBMBM22B10.3B5.B.3B.2B$
2.30B4.23B2M5B3.18B2MBM6B.BM2B3M23B5.2B.5B3.6B.3B$.30B3.22BM2BM25B3M
2B2M37B3.9B2.13B$.31B.23B3M26BM2B3M37B3.10B2.14B$3.30B.23BM3B.3B2.18B
3MBM4B2.27B.3B3.10B2.16B$.33B.26B7.26B3.26B7.29B$33B.28B5.27BM.28B5.
30B$.27B.3B3.14B2D13B4.14B2D5B2M20B2D13B2.30B$2.26B8.13B2D14B4.13B2D
4B2MB2M18B2D14B.31B4.B9.B$.20BM7B9.28B5.18B2M8B.28B2.30B2.3B5.B.3B.2B
$2.14B2D2B3M8B8.26B7.18BM7B3.26B.38B3.6B.3B$3.13B2DB2M2BM8B3.3B.27B2.
3B.28B.65B2.13B$5.15B4M9B.121B.10B2.14B$6.10B2MBMB2M9B.95B.36B2.16B$
2.3B.8B5M14B.30B3.59B2.56B$.13BM19B.31B2.60B.14B2D39B$15BM2BM14B3.30B
3.59B2.13B2D38B$.14BMBM13B4.30B3.59B5.52B$2.14BM4BM2BM8B2.29B4.58B7.
52B$3.18BM2BM8B3.16B2.10B5.16B2.10B.16B2.10B3.3B.53B$2.18B2M2BM7B5.
14B2.10B7.14B2.10B3.14B2.10B3.8BM48B$2.19BMBM7B7.13B2.9B9.13B2.9B5.
13B2.9B3.8B3M42B.3B$3.16B2.3M7B9.3B.6B3.5B.2B12.3B.6B3.5B.2B8.3B.6B3.
5B.2B5.6B2M44B$4.14B2.10B11.2B.3B.B5.3B17.2B.3B.B5.3B13.2B.3B.B5.3B
10.6BM45B$5.13B2.9B16.B9.B22.B9.B18.B9.B12.3M3BM3BM42B$7.3B.6B3.5B.2B
62.2M37.M2BM3B2M46B$8.2B.3B.B5.3B65.3M37.M5BMB2M46B$12.B9.B67.2M38.3B
2M11B2.24B3M9B$131.3B3MBM6B2.10B.14BMBM10B$132.2B2MB2M6B2.25BMB3M9B$
134.B2M.6B3.21BM4BMB2M8B$135.2B.3B.B5.3B2.14B3M2B2M9B$139.B9.B4.12B2M
BM3B3M9B$155.13BM5BM10B$154.14BM15B$154.29B$155.15BM2.10B$156.14B2.
10B$157.13B2.9B$159.3B.6B3.5B.2B$160.2B.3B.B5.3B$164.B9.B13$41.B9.B
22.B9.B21.B9.B$40.3B5.B.3B.2B17.3B5.M.3B.2B16.3B5.B.3B.2B$36.2B.5B3.
6B.3B12.2B.5B3.BM4B.3B11.MB.5B3.6B.3B$35.9B2.13B9.9B2.B2M10B8.B2M6B2.
13B$34.10B2.14B7.10B2.2M12B6.2MB2M5B2.14B$33.10B2.16B5.10B2.B2M13B4.B
4M5B2.16B12.B9.B$33.29B4.29B3.MBM26B10.3B5.B.3B.2B$32.30B3.30B2.3BM
26B6.2B.5B3.6B.3B$31.30B3.30B2.B3MBM2B3M19B6.9B2.13B$31.23B3M6B.15BM
15B.BMBM2B2M2BM20B4.10B2.14B$33.14B2M8B2M5B2.11BMBM16B2.M3BMB3M21B2.
10B2.16B$31.15BMB2M8BM10B2M6BM20B2M2B3M24B.29B$30.15B2MB2M4B5M10BM4B
5M23B3M23B.30B$31.15B3M7BMB.3B2.5B3MB4MBMB2M9B.4B.26B.3B.30B$32.15BMB
M8B7.11B2MB3M9B6.26B5.31B4.B9.B$31.19BMBM6B5.8B2M3B3M12B4.28B6.30B2.
3B5.B.3B.2B$32.14B2DB2M2BM7B4.13BM2D13B3.14B2D13B2.38B3.6B.3B$33.13B
2DB2M3BM7B4.12BM2D14B3.13B2D53B2.13B$35.15BMB2M9B5.28B4.55B.10B2.14B$
36.14B2M10B7.26B6.26B2.36B2.16B$32.3B.27B2.3B.27B.3B.83B$31.112B2D39B
$30.98B2.13B2D38B$31.30B3.30B2.30B6.52B$32.31B2.31B.31B5.52B$33.30B3.
30B2.30B.3B.53B$32.30B3.30B2.30B.8BM48B$32.29B4.29B3.29B.8B3M42B.3B$
33.16B2.10B5.16B2.10B4.16B2.10B2.6B2M44B$34.14B2.10B7.14B2.10B6.14B2.
10B4.6BM45B$35.13B2.9B9.13B2.9B8.13B2.9B6.3M3BM3BM42B$37.3B.6B3.5B.2B
12.3B.6B3.5B.B2M10.3B.6B3.5B.2B6.M2BM3B2M46B$38.2B.3B.B5.3B17.2B.3B.B
5.3B2.3M11.2B.3B.B5.3B10.M5BMB2M46B$42.B9.B22.B9.B4.2M15.B9.B12.3B2M
11B2.24B3M9B$131.3B3MBM6B2.10B.14BMBM10B$132.2B2MB2M6B2.25BMB3M9B$
134.B2M.6B3.21BM4BMB2M8B$135.2B.3B.B5.3B2.14B3M2B2M9B$139.B9.B4.12B2M
BM3B3M9B$155.13BM5BM10B$154.14BM15B$154.29B$155.15BM2.10B$156.14B2.
10B$157.13B2.9B$159.3B.6B3.5B.2B$160.2B.3B.B5.3B$164.B9.B!

Here comes some simple mathematical modelling. A pair of cloverleaf capable of pushing is annotated by a vector that starts from the center of "follower", ends at the center of "driver", and has three components (Δx, Δy, Δt) describing the spatio-temporal differences between them. Any x-in-a-row spaceship can then be expressed as an ordered set of vectors. By program enumeration of cloverleaf interactions, a group of vectors have been collected. On the other hand, it is possible to write down necessary conditions for a chosen set of vectors to be a spaceship in the form of constraint formulae.

The two spaceships shown above have the same terminals for westbound T-ship, which gives two constraints:

ΣΔy = 5
2ΣΔx - ΣΔt = -95

With generous assistance from FWCamelship in private correspondence, an affirmative answer can be given to the question "Are there 4-in-a-row spaceships?". FWCamelship (EDIT: corrected attribution) found these 16 new spaceships:

Code: Select all

#C [[ TRACK 3/795 12/795 ]]
x = 955, y = 60, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
949bo$348bo119bo359bo119b3o$39bo67bo51bo67bo51bo67b3o50bo66b3o237bo
119b3o117bob2o$40b2o64b3o49bobo65b3o51b2o64b2o2bo48bobo64b2o2bo235b3o
117b2o2bo116b2o2bo$41bo65b3o50b2o65b3o51bo69bo49b2o68bo48b2o63bo54b2o
65b3o50b2o69bo48b2o70b2o$38bo2bo64b2ob2o47b3o65b2ob2o47bo2bo69b2o46b3o
69b2o48b2o61b3o54b2o63b2ob2o50b2o68b2o48b2o70b2o$7bo30b2ob2o63b2obo17b
o31bo66b2obo17bo30b2ob2o67b2o15bo32bo69b2o15bo30b2obo61b2obo20bo30b2ob
o64b2obo17bo30b2obo68b2o15bo30b2obo70b2o$6b3o30b5o36bo26bo18b3o32b3o
36bo26bo18b3o30b5o82b3o33b3o81b3o29bo4bo59bo22b3o29bo4bo36bo26bo18b3o
29bo4bo82b3o29bo4bo68bo$5b2o31b2o4bo80b2o31b4o2bo80b2o31b2o4bo80b2o32b
4o2bo79b2o31bo4bo59bo21b2o31bo4bo81b2o31bo4bo81b2o31bo4bo$6bo29b2o2b2o
2b2o80bo30bo2bo3bo81bo29b2o2b2o2b2o80bo31bo2bo3bo80bo31bo5b2o30bobo25b
2o20bo31bo5b2o80bo31bo5b2o80bo31bo5b2o34b3o3bo$b3o3bo28bo6b2o18bo15bo
3bo37b3o3bo28b2o5b2o18bo15bo3bo37b3o3bo28bo6b2o34b3o2b2o35b3o3bo29b2o
5b2o33b3o2b2o35b3o3bo28b2o2bo4bo16b2o12b4o41b3o3bo28b2o2bo4bo17bo15bo
3bo37b3o3bo28b2o2bo4bo33b3o2b2o35b3o3bo28b2o2bo4bo34bobo3b2o$o2bo3bob
2o25b3o35bo4b2o2b2o35bo2bo3bob2o26b2o35bo4b2o2b2o35bo2bo3bob2o25b3o40b
obo3b2o33bo2bo3bob2o27b2o39bobo3b2o33bo2bo3bob2o24b2o6b3o15bobo15b2o
25bo13bo2bo3bob2o24b2o6b3o28bo4b2o2b2o35bo2bo3bob2o24b2o6b3o33bobo3b2o
33bo2bo3bob2o24b2o6b3o34bobo4b2o$o61b2o8b2obo3bob3o36bo61b2o8b2obo3bob
3o36bo77bobobo2bo34bo77bobobo2bo34bo35b2o23bo10b3o2bobo22bo17bo35b2o
24b2o8b2obo3bob3o36bo35b2o40bobobo2bo34bo35b2o41bobobo2b2o$4b2o2bo53bo
9b2ob2o5bo41b2o2bo53bo9b2ob2o5bo41b2o2bo66bo3bo2b3o39b2o2bo66bo3bo2b3o
39b2o2bo28bo23b2o9bo5bo27bo17b2o2bo28bo24bo9b2ob2o5bo41b2o2bo28bo37bo
3bo2b3o39b2o2bo28bo42bo2bobo$5b2o65b2o3b2o46b2o65b2o3b2o46b2o69bo6bo
41b2o69bo6bo41b2o65bo2bo25bo2b3o18b2o65b2o3b2o46b2o69bo6bo41b2o72bo2b
4o$74bo3bo115bo3bo116b3o2bo114b3o2bo112b3o38bo79bo3bo116b3o2bo117bo3bo
bo$12bo62bo2bo53bo62bo2bo53bo63bo2b2o51bo63bo2b2o51bo119bo62bo2bo53bo
63bo2b2o51bo$75b3o117b3o118bobo117bobo155b2o79b3o118bobo117bo2bo$316b
3o117b3o156bo200b3o117bo2bo$917b3o21$760b2o120bo66bo$39bo119b2o120bo
66bo51bo67bo169b3o121bo65bo54b2o63b3o$39b2o66bo53bo65bo52bobo64b3o51b
2o64b3o170bo66bo51bo2bo64b3o54bo62bob2o$37b2ob2o64b3o49bo2bo64b3o53b2o
62b2o2bo51bo63b2o2bo45b3o117b2ob2o65b3o51b3o63b2o2bo50bo2bo62b2o2bo$
39b2o66b3o49b3o65b3o50b3o68bo47bo2bo68bo44bobo66bo50b2o69b3o121bo49b2o
b2o66b2o$35bo70b2ob2o115b2ob2o50bo69b2o46b2ob2o67b2o41bo2b2o65b3o49b2o
68b2ob2o53bo66b2o49b5o66b2o$7bo98b2obo17bo35bo62b2obo17bo35b3o64b2o15b
o32b5o65b2o15bo26bob2o2bo62b2obo20bo34b2o62b2obo17bo32b2ob3o64b2o15bo
33b2o4bo64b2o$6b3o31bob3o35bo26bo18b3o30b2ob3o35bo26bo18b3o31b4o2bo79b
3o30b2o4bo80b3o27bob3o62bo22b3o30b3ob2o35bo26bo18b3o30bo2bo2b2o79b3o
30b2o2b2o2b2o63bo$5b2o32b3o2bo80b2o31bo2bo2b2o79b2o32bo2bo3bo78b2o30b
2o2b2o2b2o78b2o27b2o3bo63bo21b2o32bo3bo81b2o32bo5bo79b2o32bo6b2o$6bo
31b2o2b3o81bo31bo5bo81bo31b2o5b2o79bo30bo6b2o80bo69bobo25b2o20bo32b4o
83bo32b2o85bo32b3o38b3o3bo$b3o3bo31bob2o20bo15bo3bo37b3o3bo30b2o23bo
15bo3bo37b3o3bo31b2o38b3o2b2o35b3o3bo29b3o39b3o2b2o35b3o3bo54b2o12b4o
41b3o3bo33bo21bo15bo3bo37b3o3bo71b3o2b2o35b3o3bo72bobo3b2o$o2bo3bob2o
30b2o31bo4b2o2b2o35bo2bo3bob2o63bo4b2o2b2o35bo2bo3bob2o68bobo3b2o33bo
2bo3bob2o68bobo3b2o33bo2bo3bob2o50bobo15b2o25bo13bo2bo3bob2o63bo4b2o2b
2o35bo2bo3bob2o68bobo3b2o33bo2bo3bob2o69bobo4b2o$o61b2o8b2obo3bob3o36b
o61b2o8b2obo3bob3o36bo77bobobo2bo34bo77bobobo2bo34bo60bo10b3o2bobo22bo
17bo61b2o8b2obo3bob3o36bo77bobobo2bo34bo78bobobo2b2o$4b2o2bo53bo9b2ob
2o5bo41b2o2bo53bo9b2ob2o5bo41b2o2bo66bo3bo2b3o39b2o2bo66bo3bo2b3o39b2o
2bo52b2o9bo5bo27bo17b2o2bo53bo9b2ob2o5bo41b2o2bo66bo3bo2b3o39b2o2bo71b
o2bobo$5b2o65b2o3b2o46b2o65b2o3b2o46b2o69bo6bo41b2o69bo6bo41b2o65bo2bo
25bo2b3o18b2o65b2o3b2o46b2o69bo6bo41b2o72bo2b4o$74bo3bo115bo3bo116b3o
2bo114b3o2bo112b3o38bo79bo3bo116b3o2bo117bo3bobo$12bo62bo2bo53bo62bo2b
o53bo63bo2b2o51bo63bo2b2o51bo119bo62bo2bo53bo63bo2b2o51bo$75b3o117b3o
118bobo117bobo155b2o79b3o118bobo117bo2bo$316b3o117b3o156bo200b3o117bo
2bo$917b3o!

The vectors and apgcodes are:

First row
[(-32, 2, -14), (-31, 3, -17)] - xq795_y9o88gyzy827erb2zos0ogy3g04u38s6k8y98c350156ogie4z2731y4131yl12bf1
[(-32, 2, -15), (-31, 3, -16)] - xq795_8sogy4gogykgg9e4z37031y3104fo27c52ya7okcqgc319ezy93221yzy88serq8
[(-33, 2, -16), (-31, 3, -17)] - xq795_2bre72yzy8ggc4szym4eigo651053c8yb63qggg4y4go8zyp1fb21ym3631y2124c62
[(-32, 2, -17), (-32, 3, -16)] - xq795_2bre72yzy8ggc4szyoeigo61b653syb63qggg4y4go8zyr4ei11yk3631y2124c62
[(-30, 1, -9), (-31, 4, -18)] - xq795_yzytogndrgsydgogzg88oyz8qres8yk1yg46502naz3620kgke4zy11
[(-32, 1, -13), (-31, 4, -18)] - xq795_21i7is8zw31yveqi8oykgzyy274111yiejh3y9gogzyzylg40ggqy7an20564
[(-33, 1, -15), (-31, 4, -18)] - xq795_3ogh97yzy9gc6c8zyl8smxc84026yd7dqhooy5o88zyo1togyl1107553y1112c8a6
[(-34, 1, -17), (-31, 4, -18)] - xq795_yi9y2475hezzzzgo4mcyz0sii6xgyce2mcq36z0pg9q44r17yw1w5c531yd111zx101
Second row with the vectors in reversed order
[(-31, 3, -17), (-32, 2, -14)] - xq795_y4gs6rmozgo8y1323011yo7i4s8yee2mcq36z0tmj570c28yq12131yi111zy42447
[(-31, 3, -16), (-32, 2, -15)] - xq795_o422eyzyk4e9ggz11022ypo8oyoe913cgqcko7ylo31hiszyn2c89ru3c01zys1263
[(-31, 3, -17), (-33, 2, -16)] - xq795_go42364yzyj4e9ggz011ysocogyne913cgqcko7ym8qres8zypcob1114zys11647
[(-32, 3, -16), (-32, 2, -17)] - xq795_y4gs6rmoyzy7oazgo8y1323011yo2npcksyfehgo7z0tmj570c28ys3022zy42447
[(-31, 4, -18), (-30, 1, -9)] - xq795_21t9ey462icwsigcyaosi72zya4c72ye25c65e897yhgg862ez02w2yzyi346
[(-31, 4, -18), (-32, 1, -13)] - xq795_y4gs6rmozgo8y1323011yn9hhieyfe2mcq36z0tmj570c28yo433ym111zy42447
[(-31, 4, -18), (-33, 1, -15)] - xq795_y4gs6rmoyzy7oazgo8y1323011yn9hhieyhehgo7z0tmj570c28yo433zy42447
[(-31, 4, -18), (-34, 1, -17)] - xq795_y4gs6rmoyzy8s026zgo8y1323011yn9hhieyi1eaa6z0tmj570c28yo433zy42447

In summary, a simple modelling aids spaceship building effectively.

This post is dedicated to a senior electrical engineer with profound foresight into Chinese IT industry.

[GUYTU6J]

I found a 5-in-a-row (6,24)c/870 spaceship:

Code: Select all

#C [(-33, 15, -106),(-33, 4, -114),(-33, -2, -42)]
#C [[ X 12 TRACK 6/870 24/870 ]]
x = 129, y = 23, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
115bo$114b3o$113bo2b2o$69b2o42b4obo$56bo11bob2o42b2ob3o$55b4o8b2ob2o
47b2o$54bo4bo8b3o42b2ob2ob2o$48b2o3bo3bobo9bo4b2o36bo2bo3bo$49b2o6b3o
14b2o36bo3bo2bo5bo$7bo65b2o37bo2bobo6b3o$6b3o64bo39b2ob3o4b2o2bo$5b2o
66bo50b4o$6bo118b2o$b3o3bo3bo$o2bo3b2o$o5bob2o$4b2o39b2o76b2o$5b3obo
35b2o76b2o$5b2ob2o115bo$6b2o119bo$125bob2o$123b2ob2o$123b2o!

The displacement is (6,24), twice of the usual (3,12), because a period involves two T-ships instead of only one. It gives these two constraints:

ΣΔy = 17
2ΣΔx - ΣΔt = 64

There are 38 sets of vectors satisfying the formulae, the first half of which are:

Code: Select all

[(-31, 18, -109), (-28, -1, -73)]
[(-33, 12, -114), (-29, 5, -74)]
[(-32, 12, -118), (-30, 5, -70)]
[(-31, 11, -120), (-32, 6, -70)]
[(-34, 10, -100), (-33, 7, -98)]
[(-33, 10, -114), (-29, 7, -74)]
[(-34, 10, -115), (-28, 7, -73)]
[(-34, 10, -115), (-29, 7, -75)]
[(-33, 10, -116), (-29, 7, -72)]
[(-34, 10, -116), (-29, 7, -74)]
[(-35, 10, -116), (-28, 7, -74)]
[(-31, 10, -118), (-31, 7, -70)]
[(-32, 10, -120), (-31, 7, -70)]
[(-29, 9, -74), (-33, 8, -114)]
[(-29, 9, -74), (-34, 8, -116)]
[(-34, 9, -100), (-34, 8, -100)]
[(-34, 9, -113), (-28, 8, -75)]
[(-34, 9, -116), (-28, 8, -72)]
[(-33, 9, -119), (-30, 8, -71)]

However, only the first one and its reverse imply working 4-in-a-row spaceships:

Code: Select all

#C [[ Y -5 TRACK 3/315 12/315 ]]
x = 107, y = 69, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
47bo$46b3o$46bo2bo15b2o$7bo39bob2o15bo$6b3o32b3o3bobo14b3o$5b2o34bobo
3b3o13bob3o$6bo34bo24b3o31bo$b3o3bo35b3o12b2o6bo2bo29b3o$o2bo3bob2o32b
3o13bobo2b2o3b2o30b2o$o59bo4bo2b2o31bo$4b2o2bo51b2ob2o31bo3bo$5b2o92b
3o$103b3o$12bo90bo2bo$102bo3bo$101b2o2b2o$100bo3bo$101b3o$101b3o9$99bo
bo$101b2o13$50bo$49b3o13b2o$7bo40bob2o14bo$6b3o39b2o2bo11b3o$5b2o46b2o
8bob3o$6bo47b2o10b3o31bo$b3o3bo45b2o3b2o6bo2bo29b3o$o2bo3bob2o42bo5bob
o2b2o3b2o30b2o$o59bo4bo2b2o31bo$4b2o2bo51b2ob2o31bo3bo$5b2o92b3o$103b
3o$12bo90bo2bo$102bo3bo$101b2o2b2o$56bo43bo3bo$56b2obo41b3o$49b3o4bo
44b3o$49bobo$49b2o$50bo$50b2obo$51b3o$52bo3$99bobo$101b2o!

[(-31, 18, -109), (-28, -1, -73)] - xq315_y5ggc4syxmvoe4zy3oo3iaks8yx1ya6j2sogggz071jw30a201yzybsgp6364012zi9k0ez11444
[(-28, -1, -73), (-31, 18, -109)] - xq315_0o88xg2h17z8tj657012yysi2ekc6k8y5g8gogzy4eq6ocsyw8s973y8s9472zy51363

Their speed (6,24)c/630 can be simplified to (3,12)c/315, reaching the lowest possible period of the series. The westbound T-ships inside both of them are just lucky enough to pass the second cloverleaf from left. Steric hindrance of cloverleaf is exactly the reason why other vector sets do not yield practical spaceships. Therefore, the constraint on vectors is a necessary but not sufficient condition for spaceship construction.

On a side note, the fifth xq795 in the last post ([(-30, 1, -9), (-31, 4, -18)]) and the sixth ([(-32, 1, -13), (-31, 4, -18)]) share a minimum population of 69 cells, but the latter is slightly wider. Both xq315 in this post share a minimum population of 86 cells and bounding box, but the latter has a smaller maximum population.

[GUYTU6J]

I am attempting to find the relationship between (unsimplified) period and displacement of an x-in-a-row oblique spaceship and the parameters ΣΔy and 2ΣΔx - ΣΔt. The displacement depends on the number of T-ships, with one T-ship corresponding to a (3,12) displacement, which implies an additional 12 to the ΣΔy. So it is easy to write

Number of T-ships is n_T = (7+ΣΔy)/12
Displacement is (3n_T, 12n_T)

Meanwhile, the period is linearly related to both the number of vectors n_V and number of T-ships n_T. There are examples posted above where n_V = 2, n_T = 1, p = 795; n_V = 2, n_T = 2, p = 630 (not simplified to 315); n_V = 3, n_T = 1, p = 1035; n_V = 3, n_T = 2, p = 870. Intuitively the period would then be p = 480 + 240n_V - 165n_T.

But how can I be sure that 2ΣΔx - ΣΔt is not involved? I modified the constraint to

ΣΔy = 5
2ΣΔx - ΣΔt = -95 + 240 = 145

There are 14×3!=84 sets of vectors satisfying the formulae, with the first 14 of which are:

Code: Select all

(-29, 5, -73),(-25, 0, -115),(-25, 0, -115)
(-30, 3, -98),(-26, 2, -94),(-25, 0, -115)
(-30, 3, -98),(-27, 2, -96),(-25, 0, -115)
(-30, 3, -98),(-28, 2, -109),(-25, 0, -104)
(-30, 3, -98),(-30, 2, -113),(-25, 0, -104)
(-30, 3, -98),(-31, 2, -115),(-25, 0, -104)
(-34, 3, -112),(-28, 1, -106),(-31, 1, -113)
(-26, 2, -94),(-28, 2, -109),(-28, 1, -106)
(-26, 2, -94),(-30, 2, -113),(-28, 1, -106)
(-26, 2, -94),(-31, 2, -115),(-28, 1, -106)
(-26, 2, -96),(-31, 2, -112),(-31, 1, -113)
(-27, 2, -96),(-28, 2, -109),(-28, 1, -106)
(-27, 2, -96),(-30, 2, -113),(-28, 1, -106)
(-27, 2, -96),(-31, 2, -115),(-28, 1, -106)

The first spaceship is (3,12)c/795; instead of p = 480 + 240×3 -165×1 = 1035, its period is smaller by 240.

Code: Select all

#C [(-29, 5, -73),(-25, 0, -115),(-25, 0, -115)]
#C [[ TRACK 3/795 12/795 ]]
x = 129, y = 34, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
48bo$47b2o$46b2ob2o$47bo2b2o$51b2o$46b2ob3o$46bob3o$46b3o8$57b2o$57b2o
3$7bo$6b3o$5b2o$6bo119b3o$b3o3bo48b2o62b2o6bo$o2bo3bob2o44bobo61b2o2b
2o2b2o$o54bobo45bo2b2o12bo4b2o$4b2o2bo46b3o39b3o2bo2bobo13b5o$5b2o51bo
2b2o34bo4b2o4b2o12b2ob2o$62bo34b2o2bo6b2o13bo2bo$12bo86bo23bo$59b3o37b
o23b2o$103bo21bo$100b2o$101b2o!

From ΣΔy = 5, 2ΣΔx - ΣΔt = -95; ΣΔy = 17, 2ΣΔx - ΣΔt = 64, there is an linear equation (2ΣΔx - ΣΔt) = (53*ΣΔy - 645)/4. To fix the period equation, introduce a difference d = (2ΣΔx - ΣΔt)-(53*ΣΔy - 645)/4, and deduct it from period to get p = 480 + 240n_V - 165n_T - d = 240n_V - (2ΣΔx - ΣΔt) - ΣΔy/2 + 445/2. It seems to be done here...

... except it is not. See excerpt of another batch of results:

Code: Select all

(-14, 19, 55),(-33, 4, -64),(-25, -18, -40)
(-14, 19, 55),(-34, 4, -64),(-24, -18, -40)
(-14, 19, 55),(-28, 4, -76),(-17, -18, -2)
(-14, 19, 55),(-33, 3, -61),(-22, -17, -37)
(-14, 19, 55),(-30, 3, -70),(-10, -17, 2)
(-14, 19, 55),(-32, 2, -66),(-26, -16, -38)
(-14, 19, 55),(-29, 2, -71),(-14, -16, -3)
(-14, 19, 55),(-29, 2, -72),(-15, -16, -4)
(-14, 19, 55),(-31, 1, -68),(-14, -15, -10)
(-14, 19, 55),(-30, 1, -69),(-14, -15, -7)
(-14, 19, 55),(-31, 0, -66),(-29, -14, -42)
(-31, 15, -50),(-29, 4, -15),(-21, -14, -2)
(-31, 15, -50),(-31, 3, -16),(-19, -13, -1)
(-31, 15, -50),(-31, 3, -16),(-21, -13, -5)
(-31, 15, -50),(-31, 3, -17),(-21, -13, -4)
(-31, 15, -50),(-31, 2, -14),(-22, -12, -9)
(-31, 15, -50),(-31, 2, -14),(-26, -12, -17)
(-31, 15, -50),(-32, 2, -14),(-20, -12, -7)
(-31, 15, -50),(-32, 2, -14),(-23, -12, -13)
(-31, 15, -50),(-32, 2, -15),(-21, -12, -8)
(-31, 15, -50),(-32, 2, -16),(-22, -12, -9)
(-31, 15, -50),(-32, 2, -16),(-26, -12, -17)
(-31, 15, -50),(-33, 2, -16),(-20, -12, -7)
(-31, 15, -50),(-33, 2, -16),(-23, -12, -13)
(-31, 15, -50),(-30, 2, -18),(-21, -12, -1)
(-31, 15, -50),(-33, 2, -18),(-22, -12, -9)
(-31, 15, -50),(-33, 2, -18),(-26, -12, -17)
(-31, 15, -50),(-31, 2, -20),(-21, -12, -1)
(-31, 15, -50),(-32, 1, -21),(-21, -11, -2)
(-31, 15, -50),(-33, 0, -12),(-24, -10, -19)
(-31, 15, -50),(-32, 0, -14),(-26, -10, -19)
(-31, 15, -50),(-32, 0, -20),(-28, -10, -17)
...

Having n_V = 3, ΣΔy = 5, 2ΣΔx - ΣΔt = -95, the period is expected to be p = 240×3 - (-95) - 5/2 + 445/2 = 1035. However, there are 11×3! = 66 sets of vectors with (-14, 19, 55), and at least one of them makes a period-795 spaceship:

Code: Select all

#C [(-14, 19, 55),(-33, 4, -64),(-25, -18, -40)]
#C [[ TRACK 3/795 12/795 ]]
x = 126, y = 42, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
76b3o$75bo3bo$77bobo$77b3o$77bo4$41bo$41bo$40b3o3b2o$38b3ob2o32bo2bo$
38b2obob4ob3o$41b5o4bo$42bo6b2o$40bobo$40bo4$39b2o$39b2o$38b2o$38bo$
38bo$119b2o$119b2o$117b2o$117bo$118bo2$7bo109bobo3bo$6b3o109b2o5bo$5b
2o113bo3b2o$6bo92b2o23bo$b3o3bo91bobo16bo2b2o$o2bo3bob2o89b3o15bobo$o
99b2o16b3o$4b2o2bo90bo$5b2o88b2obo$96b3o$12bo84bo!

Thus, some terms are still missing in the proposed period formula p = 240n_V - (2ΣΔx - ΣΔt) - ΣΔy/2 + 445/2.

[GUYTU6J]

Here comes some simple mathematical modelling. A pair of cloverleaf capable of pushing is annotated by a vector that starts from the center of "follower", ends at the center of "driver", and has three components (Δx, Δy, Δt) describing the spatio-temporal differences between them. Any x-in-a-row spaceship can then be expressed as an ordered set of vectors.

The order of vectors in a spaceship appears written right-to-left, because the east terminal that emits T-ship(s) is regarded as the center. This is also the reason why a vector starts from "follower" instead of "driver". Going from east to west, the vector component Δx is negative (ranging from -35 to -7 for known ones). A "driver" located northeast of "follower" is noted with a positive Δy. And with the phase of "driver" behind "follower", Δt is negative.

While I was explaining these, it occurred to me that the sign of Δt might be a key to the period formula problem. There are three vectors with a large positive Δt: (-15, 32, 85), (-14, 19, 55) and (-32, 10, 118). The second one, (-14, 19, 55), was mentioned in the last post. Here is a hightly curved 6-in-a-row (3,12)c/1035 spaceship with (-15, 32, 85):

Code: Select all

#C [(-15, 32, 85), (-31, 10, -70), (-21, -18, -44), (-21, -19, -52)]
#C [[ X 8 TRACK 3/1035 12/1035 ]]
x = 132, y = 63, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
60b3o$59bo2bo$59b4o$63b3o$65bo$60bob2obo$59b3ob3o$58b2o2bobo$59b3obo4b
o$60bob3o3b2o$69b2o$69bo6b3o$68bo6bo2bo$74bo$74bo4bo$74b3o2bo14$50bo2$
50b2o$51bo$44bo2bo$44bo3bo$44bo3bo$45bo2bo$45b3o10$128b3o$128bo2bo$
107bo20bo2bo$7bo98b3o$6b3o96b2o2bo13bobo3bo$5b2o99b4o12b4o2bo$6bo95b2o
bob2o13bobo2bo$b3o3bo92b5o15b2o2bobobo$o2bo3bob2o89bo18b2o4bobo$o100bo
2bo15b2o3bobo$4b2o2bo92bobo17bo3b3o$5b2o95bo4bo2bo$107bo2bo$12bo93b2o
2bo$107bobo$107b3o!

Another 6-in-a-row (3,12)c/1035 spaceship with (-32, 10, 118):

Code: Select all

#C [(-32, 10, 118), (-33, 14, -104), (-25, 0, -104), (-21, -19, -37)]
#C [[ X 8 TRACK 3/1035 12/1035 ]]
x = 154, y = 43, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
68b3o$61b2o8b2o$60bob2o8bo$59b2ob2o4b5o$60b3o7bo$61bobo$64bobo$63b2o2b
o$63b2o3bo$36b3o25bob2o$35bo3bo2b2o20b2o$35bo3bob2obo$36bobo2b2ob2o$
35bobobo2b3o$34b2o3bobobo$35bo5b2o$37b5o$39b2o2$35b2o$33bo2bo$33b3o2$
138bo$85b3o48b2o$85bo2bo47b2o$85b4o42bo2bo$81bo2bo46bo3bo$82bo48bo3bo$
79bo2bo49bo2bo8b3o$80b2o4b2o44b3o9bobo$7bo78bo56bo2b2o$6b3o134bo2bo$5b
2o136bo2bo4bo$6bo80b2o61bobo$b3o3bo79b2o60bo2bo$o2bo3bob2o142bo$o148b
5o$4b2o2bo136b2obob2o$5b2o137b4o$144bo2b2o$12bo132b3o$146bo!

The proposed period formula gives 1275 to both incorrectly.
For the following 8-in-a-row (3,12)c/1515 spaceship, Δy = 5 and 2Δx - Δt = -335:

Code: Select all

#C [(-15, 32, 85), (-14, 19, 55), (-26, -11, -16), (-32, -11, -44), (-21, -12, -1), (-21, -12, -2)]
#C [[ X 10 TRACK 3/1515 12/1515 ]]
x = 164, y = 70, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
124bo2b2o$118b3o2bo2bobo$118bo4b2o4b2o$118b2o2bo6b2o$120bo$120bo$124bo
$121b2o$101bo20b2o$101b2o$102b2o$102bo$98bo2bo$98b3o$99bo21bo$120b3o$
119bo2bo$118b3o2bo$119bo2b3o$118bo$48b3o68bo$48bobo$46b2o2bo$44bo$43b
2o3bo$43bo5b2o$45bo3bobo2$50bo$51bo$50b2o$48b2o$28bo19b2o$27b3o$26bo2b
o$25b2o$22b6o$22bob2o2b2o$24bo3bobo$25bo4bo102bo$132b3o6b3o$28bobo100b
2obo6b3o$29b2o101b3o2bo$27b2o104b4o$27bo107b2o$7bo$6b3o$5b2o$6bo$b3o3b
o$o2bo3bob2o$o$4b2o2bo$5b2o2$12bo6$160b3o$160bo2bo$163bo$156bo4b3o$
155b3o2bobo$156bo2bob2o$157b3o2b2o$158b3obo$161bo!

The proposed period formula gives 1995, two times 240 more than 1515, corresponding to the existence of (-15, 32, 85) and (-14, 19, 55). In these examples, each vector that is either (-15, 32, 85), (-14, 19, 55) or (-32, 10, 118) corresponds to a decrease of 240 in the period. Therefore, a straightforward fix for the formula p = 240n_V - (2ΣΔx - ΣΔt) - ΣΔy/2 + 445/2 would be not counting any of these three vectors in n_V.

[FlutterWithKSparkler]

A new spaceship with known c/4 orthogonal parts.

Code: Select all

#C completed spaceship: xq4_y183a28xgy6sogzgg088o31hwij5h11ry2e62y0ogz0l40eaaavll0v4044hv444a0llle4wtd5z110223oghw9pkhggry2ec8zy12oa82x1y6731
#C Catagolue URL: https://catagolue.hatsya.com/object/xq4_y183a28xgy6sogzgg088o31hwij5h11ry2e62y0ogz0l40eaaavll0v4044hv444a0llle4wtd5z110223oghw9pkhggry2ec8zy12oa82x1y6731/b3-jknr4ity5ijk6i8s23-a4city6c7c
#C breadth = 25
#CLL state-numbering golly
x = 25, y = 34, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
13bo$11bo3bo$11bobobo$10b2o3b2o$13bo3$5bo5b3o5bo$3bo3bo2bobobo2bo
3bo$3bobobo2bo3bo2bobobo$2b2o3b2ob5ob2o3b2o$5bo6bo6bo$11bobo$12bo$
12bo$11bobo2$5bo3b2obob2o3bo$5b2ob2o2bo2b2ob2o$6b2ob7ob2o$5b3o2b5o
2b3o$5bobo9bobo$5bo2b3obob3o2bo$4bo2b2ob5ob2o2bo$7b5ob5o$9b7o$2b3o
5bo3bo5b3o$b5o6bo6b5o$o2bo2bo3bo3bo3bo2bo2bo$obobobobo3bo3bobobobo
bo$8b3o3b3o$9b7o$11b3o$8b3obob3o!

It makes a puffer and rake with period 140.

Code: Select all

x = 38, y = 110, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
13bobo$12b2ob2o$11b3ob3o3$15bo$14b3o$7bobo3bobobo3bobo$6b2ob2o2bobobo
2b2ob2o$5b3ob3obobobob3ob3o2$14bobo$15bo$15bo$15bo$13b5o$8b2ob3o3b3ob
2o$8bo6bo6bo$8bo6bo6bo$8bo3bo5bo3bo$7b2obo4bo4bob2o$8b2o2b7o2b2o$9bo2b
o5bo2bo$13bobobo$6bo6bobobo6bo$4bo3bo3b7o3bo3bo$4bobobo5bobo5bobobo$3b
2o3b2o4bobo4b2o3b2o$6bo4b2obobob2o4bo$11bo2b3o2bo$11bo7bo$15bo$12b2obo
b2o$12bo5bo2$3bobo$2b2ob2o3b3o$b3ob3o3bo2$11bo14b5o$25bo2bo2bo$25bo2bo
2bo$26bo3bo3$2bobo23b5o$b2ob2o21bo2bo2bo$3ob3o20bo2bo2bo$28bo3bo23$10b
3o$11bo2$11bo4$29b2o$28bob2o3bo$28bob2o2bobo$30bo4bo2$22bo$22bobo10b2o
$22b3o9b4o$37bo$35b2o4$30b2o$13b2o15bo$13b3o14b3o$2bobo10b3o$b2ob2o$3o
b3o5$21b2o$21b2o4$10b3o$11bo2$11bo!

[GUYTU6J]

A new spaceship with known c/4 orthogonal parts.
...
It makes a puffer and rake with period 140.

Whoa, an extension to xq4_xnmkz0g0b98gxogz120l5531wtd5zxrrazx1 with two more escorting xq4_nmkz31 published at the end of year? Astounding!

That is in three ways I could see akin to the Life fireship: 1) a not quite "tagalong" to known spaceship; 2) a start to development of moving technology (puffers, rakes, etc) at the speed; 3) some resemblance to the 3c/20 technology. I built a backrake for xq315_0o88xg2h17z8tj657012yysi2ekc6k8y5g8gogzy4eq6ocsyw8s973y8s9472zy51363 in five minutes flat:

Code: Select all

#C [[ TRACK 0 -1/4 ]]
x = 145, y = 558, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
46bo$44bo3bo$44bobobo$43b2o3b2o$46bo3$38bo5b3o5bo$36bo3bo2bobobo2bo3bo
$36bobobo2bo3bo2bobobo$35b2o3b2ob5ob2o3b2o$38bo6bo6bo$44bobo$45bo$45bo
$44bobo2$38bo3b2obob2o3bo$38b2ob2o2bo2b2ob2o$39b2ob7ob2o$38b3o2b5o2b3o
$38bobo9bobo$38bo2b3obob3o2bo$37bo2b2ob5ob2o2bo$40b5ob5o$42b7o$35b3o5b
o3bo5b3o$34b5o6bo6b5o$33bo2bo2bo3bo3bo3bo2bo2bo$33bobobobobo3bo3bobobo
bobo$41b3o3b3o$42b7o$44b3o$41b3obob3o2$56bo$54bo3bo$54bobobo$53b2o3b2o
$56bo$31bobo$30b2ob2o$29b3ob3o3$49bo$29bobo16b3o$28b2ob2o15b3o$27b3ob
3o25$23b2o$24b2o$24bo$24b2o4$49bo$48b3o$48b3o10$45b3o$44bo2bo$44bo2bo
9bo$55bo3bo$46bo8bobobo$47bo6b2o3b2o$48bobo6bo$47bob3o$48bo$48bobo$48b
3o5$25b2o$24bobo$24b2o3$76bobo$75b2ob2o$74b3ob3o3$76bo$75b3o$68bobo3bo
bobo3bobo$67b2ob2o2bobobo2b2ob2o$66b3ob3obobobob3ob3o2$75bobo$76bo$76b
o$76bo$10b2o62b5o$9bo59b2ob3o3b3ob2o$9bo3bo55bo6bo6bo$9b3o2bobo52bo6bo
6bo$15bob3o49bo3bo5bo3bo$16bo2bo48b2obo4bo4bob2o$13b2o2b3o49b2o2b7o2b
2o$14b3o53bo2bo5bo2bo$13b2o59bobobo$14b2o51bo6bobobo6bo$16bob2o45bo3bo
3b7o3bo3bo$16b3o46bobobo5bobo5bobobo$17bo46b2o3b2o4bobo4b2o3b2o$67bo4b
2obobob2o4bo$72bo2b3o2bo$25b2o45bo7bo$24bobo49bo$24b2o47b2obob2o$73bo
5bo2$77b3o6bobo$77b3o5b2ob2o$76bo3bo3b3ob3o$74b2o2b2o$30bo30b5o8bo3bo$
29b2o29bo2bo2bo7bo2bo$28b2ob2o27bo2bo2bo8b3o$29bo2b2o27bo3bo$33b2o46b
2o$28b2ob3o47b2o$28bob3o26b5o16b2o$28b3o27bo2bo2bo15bo$58bo2bo2bo15bo$
59bo3bo19$79b3o$80bo2$80bo13$56b2o$55bobo$55b2o$14b2o49b2o7bo$16bo47b
2o2bobo3b2o$13bo2bo47b2o3b2o4b2o$14b3o47b2o8b2o$65b2o5b2o$18bo$14b2ob
3o39b3o10bo$13bo2bo2b2o37b5o8b2o$13bo5bo38bo3bo7b3o14bobo$13b2o71b2ob
2o$59bo25b3ob3o$60bo$59b2o5bo$65b2o$62bob3o$62bo40b5o$58b2obo17b3o20bo
2bo2bo$59b3o18bo21bo2bo2bo$60bo42bo3bo$80bo$103b3o2$95b5o2bobobo2b5o$
94bo2bo2bob5obo2bo2bo$94bo2bo2bo2bobo2bo2bo2bo$22b3o70bo3bo3b3o3bo3bo$
22bo2bo78bo$22bo2bo78bo$103b3o$17bobo3bo80bo$16b4o2bo79bobobo$16bobo2b
o75b2o2b7o2b2o$14b2o2bobobo33b2o38b2o3bobobobo3b2o$13b2o4bobo33bobo38b
3o11b3o$14b2o3bobo33b2o40bo13bo$15bo3b3o74bo3bo3bo3bo3bo$96bo3b9o3bo$
97b2o2bo2bo2bo2b2o$102b2ob2o$102bobobo$94bobo4bo5bo4bobo$93b2ob2o5bobo
5b2ob2o$92b3ob3o4b3o4b3ob3o$99b5ob5o$99b2o2b3o2b2o$104bo$101bo5bo$101b
2o3b2o$101b2o3b2o2$113b5o$112bo2bo2bo$112bo2bo2bo$90b3o20bo3bo$46b2o
41b5o$88bo2bo2bo$88bobobobo$96bo2bo$52b2o42b3o$51b2ob2o32b3o6bo$40b2o
13bo31b5o14bo$39b2o10bob3o30bo2bo2bo12b3o$40b3o3bo3b2o34bobobobo11bo2b
2o$41b3o7b2o50bobobo$43b3o56bo$103b3o$104bo$56b2o$55bobo$55b2o9$108bo$
29b3o2bo72b3o$29bobob3o71b3o$29b2o4b2o$30b2o2bo$30b2ob2o$31b3o7b3o7bo$
32bo7bo2bo$40bob3o$40bo10bo$41bo11$15b3o4bo79bo$15bo2bo2bo2bo76b3o$18b
o2bo2bo59b2o15bo2bo$17bo3b3o59bobo18b2o$18bo64b2o17bob5o$18b2o82bobo3b
o$16b3o84b3o3bo$17bo82bo2bob2o$100bobo$100b3o$114b5o$113bo2bo2bo$113bo
2bo2bo$114bo3bo$108bo$107b3o$107b3o3$88b3o$82b2o6bo$81b2o2b2o2b2o$82bo
4b2o$83b5o$84b2ob2o$85bo2bo41bo$85bo42bo3bo$85b2o41bobobo$87bo39b2o3b
2o$130bo3$57bo7b2o55bo5b3o5bo$56b2o3bobo2bo53bo3bo2bobobo2bo3bo$55b2o
4b2o57bobobo2bo3bo2bobobo$26bo29b2o61b2o3b2ob5ob2o3b2o$25b3o30b2o62bo
6bo6bo$24b2obo56b2o42bobo$28b2o29bo23bobo43bo$29b2o28b2o22b2o44bo$27b
3o29b3o66bobo$24b3o2b2o$24bo2bo94bo3b2obob2o3bo$24b3obo93b2ob2o2bo2b2o
b2o$27bobo2b3o88b2ob7ob2o$30bo3bo87b3o2b5o2b3o$34bo87bobo9bobo$32b2o
88bo2b3obob3o2bo$47bo73bo2b2ob5ob2o2bo$46b3o75b5ob5o$45b2obo77b7o$45b
2o72b3o5bo3bo5b3o$44b2o72b5o6bo6b5o$44bo72bo2bo2bo3bo3bo3bo2bo2bo$44bo
72bobobobobo3bo3bobobobobo$46bo7bo70b3o3b3o$54bo71b7o$128b3o$125b3obob
3o$46bo$45b3obo90bo$45bo3b2o87bo3bo$47bo2bo30b3o54bobobo$47bobo30bo3bo
2b2o48b2o3b2o$47b3o30bo3bob2obo50bo$81bobo2b2ob2o24bobo$80bobobo2b3o
24b2ob2o$79b2o3bobobo24b3ob3o$80bo5b2o43bo$82b5o43b3o$84b2o47bo$113bob
o12bobobo$80b2o2b2o26b2ob2o10b2o2bo$78bo2b3obo25b3ob3o10b3o$78b3o2b2o
44bo$138bo$120b2o15b3o$123b2o11bo2b2o$124b2o$123b2o$121b3o$122bo$70bo$
69b3o$56b3o9bobo$55bo3bo7b2o$13b3o41bobo7b3o63bo$11b3obo41b3o7bo2bo2b
2o57b3o$10b3ob2o41bo8bo2bobo2bo57b3o$9b2o56b5o2bo$10b2o2bo53bo2b3o$11b
2ob2o$13b2o$13bo2$56bo2bo12$29b2o96bo$24bo5bo95b3o$23b2o2bo2bo94b2o$
24b3ob2o96bo$25bo101bo3bo$126b3o$27b3o92b3o$27bo2bo90bo2bo$27bo93bo3bo
$28b2o79b2o10b2o2b2o$108bobo12bo3bo$108b2o14b3o$124b3o14bo$139bo3bo$
133bo5bobobo$132b3o3b2o3b2o$132b3o6bo5$126bobo$106b3o16b2o$105bo2bo$
85bo19bo3bo$84b3o18b2o3b2o$83bo2bo14bo5b2ob2o9bo$45bo36b2o16b3obo3bob
2o8b2o$44b3o32b5obo13b2o2b2o4bo$44bob2o31bo3bobo14bo3bo15bo$43bo34bo3b
3o$39b3obo37b2obo2bo$39b2o44bobo15bo$39bo5b2o38b3o$45bo$46bo2$47bo$46b
2o$b5o$o2bo2bo$o2bo2bo$bo3bo13bo$18b3o$17bo2bo88b2o$17b4o3bo83bobo$18b
2o5bo82b2o$23bo2bo$23bo2bo$24bob2o$23b4o$22bo37b3o$18b4o31b2o7b3o$18bo
2bo32b2o3bo3b3o$19b3o28b3obo10b2o$50bo13b2o$50b2ob2o12b2o$52b2o13bo6bo
$67bobo3b3ob2o$68b2o6bo2bo$72bo2bo3bo$58b2o12bo3bo2bo$71b2ob2ob2o$71b
2o$72b3ob2o$73bob4o$74b2o2bo$75b3o$76bo2$118b2o$118b3o$120b3o2$115b2o$
114b4o$114bo2b2o$115b3o$28b3o85bo$28bobo57bo$28bo58bob3o$26bo59b2o2b3o
$26bobo58b2obo2bo$28b2o57bobo2b3o$28b2o56b3o4bo$27b3o56bo$26b2ob3o54bo
2bo$24b3ob2o57b3o$26b2o3b2o3b2o$27b2o7b3o$27b2o9b3o$49b3o2$52bo47b2ob
3o$51b2o47b3o3bo$102b2o2bo$100b2o4bo$99b2o4bo$100b3o$101bo3$48b3o$48bo
2bo$46b3o2bo$46bo2bo$46b3o$51bo$51bo2bo$51bob2o$50bo2bo$50b3o$51bo14$
60bo$59b3o13b2o$17bo40bob2o14bo$5b3o8b3o39b2o2bo11b3o$4b5o6b2o46b2o8bo
b3o$3bo2bo2bo6bo47b2o10b3o31bo$3bobobobo2b2o3bo3bo41b2o3b2o6bo2bo29b3o
$13bo3b2o44bo5bobo2b2o3b2o30b2o$16bob2o50bo4bo2b2o31bo$14b2o54b2ob2o
31bo3bo$15b3obo89b3o$15b2ob2o93b3o$16b2o95bo2bo$112bo3bo$111b2o2b2o$
66bo43bo3bo$66b2obo41b3o$59b3o4bo44b3o$59bobo$59b2o$60bo$60b2obo$61b3o
$62bo3$109bobo$111b2o!

Can I expect more c/4 orthogonal research?

[GUYTU6J]

More notes on the oblique spaceship. I wrote a script to verify the collected group of vectors, which found a pair missing in the original post due to misidentification. The pairs that emits T-ships have the following vectors:

Code: Select all

#C North: (-34, -2, -13), (-33, 13, -103)
#C West: (-25, 9, -74), (-33, 6, -68)
#C East: (-8, -18, 1), (-20, -19, -38)
#C South: (-33, 7, -103), (-34, 6, -115), (-24, -8, -18), (-28, -7, -24), (-23, -12, -16), (-26, -9, -23), (-31, -13, -43)
#C [[ STOP 315 ]]
x = 665, y = 283, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7cHistory
45.B9.B$44.3B5.B.3B.2B$11.B9.B18.2B.5B3.6B.3B50.B9.B$10.3B5.B.3B.2B
13.9B2.13B47.3B5.B.3B.2B$6.2B.5B3.6B.3B11.10B2.14B42.2B.5B3.6B.3B$5.
9B2.13B8.10B2.16B40.9B2.13B$4.10B2.14B7.29B38.10B2.14B$3.10B2.16B5.
30B37.10B2.16B$3.29B3.30B38.29B$2.30B3.31B36.30B$.30B6.30B34.30B$.31B
3.33B33.31B$3.30B.33B36.30B$.33B.27B.3B35.33B$33B3.26B38.33B$.27B.3B
3.7B3A18B38.27B.3B12.B9.B$2.26B8.8BA20B37.26B15.3B5.B.3B.2B$.28B8.3B
2AB2A21B35.28B10.2B.5B3.6B.3B$2.29B8.B2A25B35.29B7.9B2.13B$3.29B8.2A
24B37.29B5.10B2.5BA8B$5.28B3.3B.6B2A19B38.28B3.10B2.5B3A8B$6.26B3.8B
3AB2A19B38.26B4.16B2AB2A8B$2.3B.27B.9BA3BA19B35.3B.27B2.17B4A9B$.8BA
24B.8B4A18B36.8BA41B2A11B$8B3A22B3.9BA21B33.8B3A22B.17B2A12B$.BA4B2A
22B6.30B34.BA4B2A22B5.17BA12B$2.6BA24B3.30B36.6BA24B.33B$3.3A3BA3BA
19B3.29B38.3A3BA3BA52B$2.A2BA3B2A21B5.16B2.10B37.A2BA3B2A21B2.27B.3B$
2.A5BAB2A19B7.14B2.10B38.A5BAB2A19B4.19B2A5B$3.3B2A11B2.10B8.13B2.9B
40.3B2A11B2.10B3.20B2A6B$4.3B3ABA6B2.10B11.3B.6B3.5B.2B42.3B3ABA6B2.
10B5.19B2A8B$5.2B2AB2A6B2.9B13.2B.3B.B5.3B47.2B2AB2A6B2.9B7.29B$7.B2A
.6B3.5B.2B18.B9.B50.B2A.6B3.5B.2B10.28B$8.2B.3B.B5.3B84.2B.3B.B5.3B
15.26B$12.B9.B89.B9.B12.3B.27B$134.33B$133.33B$134.30B$135.31B$136.
30B$135.30B$135.29B$136.16B2.10B$137.14B2.10B$138.13B2.9B$140.3B.6B3.
5B.2B$141.2B.3B.B5.3B$145.B9.B34$11.B9.B89.B9.B$10.3B5.B.3B.2B84.3B5.
B.3B.2B$6.2B.5B3.6B.3B79.2B.5B3.6B.3B$5.9B2.13B76.9B2.13B$4.10B2.14B
74.10B2.14B$3.10B2.16B72.10B2.16B$3.29B71.29B12.B9.B$2.30B70.30B11.3B
5.B.3B.2B$.30B70.30B8.2B.5B3.6B.3B$.31B4.B9.B54.31B6.9B2.13B$3.30B2.
3B5.B.3B.2B52.30B4.10B2.14B$.38B3.6B.3B49.33B2.10B2.16B$39B2.13B46.
33B3.29B$.27B.10B2.14B46.27B.3B3.30B$2.36B2.16B46.26B6.30B$.56B44.28B
5.31B$2.55B45.29B5.30B$3.53B47.29B2.33B$5.52B48.61B$6.52B48.26B2.27B.
3B$2.3B.53B43.3B.27B2.26B$.8BA48B43.8BA52B$8B3A42B.3B43.8B3A22B2.29B$
.BA4B2A44B48.BA4B2A22B5.29B$2.6BA45B48.6BA24B5.28B$3.3A3BA3BA42B47.3A
3BA3BA19B6.26B$2.A2BA3B2A46B45.A2BA3B2A21B3.3B.27B$2.A5BAB2A46B44.A5B
AB2A19B3.20BA12B$3.3B2A11B2.24B3A9B46.3B2A11B2.10B2.20B2A11B$4.3B3ABA
6B2.10B.14BABA10B46.3B3ABA6B2.10B4.17B2ABA9B$5.2B2AB2A6B2.25BAB3A9B
46.2B2AB2A6B2.9B6.15B2A14B$7.B2A.6B3.21BA4BAB2A8B49.B2A.6B3.5B.2B8.
15BABA12B$8.2B.3B.B5.3B2.14B3A2B2A9B52.2B.3B.B5.3B11.17B3ABA8B$12.B9.
B4.12B2ABA3B3A9B54.B9.B12.18BABA8B$28.13BA5BA10B78.16B2.10B$27.14BA
15B80.14B2.10B$27.29B82.13B2.9B$28.15BA2.10B84.3B.6B3.5B.2B$29.14B2.
10B86.2B.3B.B5.3B$30.13B2.9B91.B9.B$32.3B.6B3.5B.2B$33.2B.3B.B5.3B$
37.B9.B19$131.B9.B$19.B9.B100.3B5.B.3B.2B$18.3B5.B.3B.2B92.2B.5B3.6B.
3B$14.2B.5B3.6B.3B90.9B2.13B$13.9B2.13B87.10B2.14B$12.10B2.14B85.10B
2.16B$11.10B2.16B84.29B$11.29B82.30B$10.30B81.30B$9.30B82.31B$9.31B
83.30B$11.30B80.33B$9.33B78.33B$8.33B80.27B.3B$9.27B.3B82.26B$10.26B
85.28B$9.28B85.29B$10.29B84.15BA13B$11.29B85.12B3A13B$11.B.28B70.B9.B
4.10BABABAB3A7B$10.3B.26B70.3B5.B.16BA3BA4B2A6B$6.2B.32B65.2B.5B3.19B
6A4BA7B$5.37B63.9B2.19B2ABAB2A4BA6B$4.12B3A22B63.10B2.19B2A5BA2BA5B$
3.13BABA20B64.10B2.20B3A2BA12B$3.13BAB3A20B62.36BA3BA9B$2.10BA4BAB2A
20B61.40BA9B$.10B3A3B4A19B61.50B$.9B2ABA2BA22B62.35B2AB2.10B$3.9B4A
23B64.31B2ABA2.10B$.11B2A2B2ABA18B63.33B3AB2.9B$14BABAB2A17B63.35BAB
3.5B.2B$.15B2A15B.2B65.33B.B5.3B$2.26B.3B70.26B4.B9.B$.28B.B70.28B$2.
29B71.29B$3.29B71.29B$5.28B72.28B$6.26B74.26B$2.3B.27B69.3B.27B$.8BA
24B67.8BA24B$8B3A22B67.8B3A22B$.BA4B2A22B70.BA4B2A22B$2.6BA24B69.6BA
24B$3.3A3BA3BA19B70.3A3BA3BA19B$2.A2BA3B2A21B70.A2BA3B2A21B$2.A5BAB2A
19B71.A5BAB2A19B$3.3B2A11B2.10B72.3B2A11B2.10B$4.3B3ABA6B2.10B74.3B3A
BA6B2.10B$5.2B2AB2A6B2.9B76.2B2AB2A6B2.9B$7.B2A.6B3.5B.2B79.B2A.6B3.
5B.2B$8.2B.3B.B5.3B84.2B.3B.B5.3B$12.B9.B89.B9.B34$642.B9.B$434.B9.B
196.3B5.B.3B.2B$433.3B5.B.3B.2B188.2B.5B3.6B.3B$429.2B.5B3.6B.3B186.
9B2.13B$428.9B2.13B85.B9.B87.10B2.14B$235.B9.B181.10B2.14B83.3B5.B.3B
.2B82.10B2.16B$234.3B5.B.3B.2B89.B9.B76.10B2.16B78.2B.5B3.6B.3B81.29B
$230.2B.5B3.6B.3B87.3B5.B.3B.2B72.29B76.9B2.13B78.30B$229.9B2.13B81.
2B.5B3.6B.3B70.30B75.10B2.14B76.30B$228.10B2.14B79.9B2.13B67.30B75.
10B2.16B75.31B$227.10B2.16B77.10B2.14B66.31B74.29B76.30B$227.29B75.
10B2.16B67.30B72.30B74.33B$226.30B75.29B64.33B70.30B74.33B$11.B9.B89.
B9.B89.B9.B3.30B56.B9.B8.30B51.B9.B.33B55.B9.B5.31B53.B9.B10.27B.3B$
10.3B5.B.3B.2B84.3B5.B.3B.2B84.3B5.B.3B.32B54.3B5.B.3B.2B3.30B51.3B5.
B.3B.27B.3B55.3B5.B.3B.2B3.30B51.3B5.B.3B.2B7.26B$6.2B.5B3.6B.3B79.2B
.5B3.6B.3B79.2B.5B3.6B.33B49.2B.5B3.6B.3B2.31B46.2B.5B3.6B.9BA17B55.
2B.5B3.6B.36B46.2B.5B3.6B.3B5.28B$5.9B2.13B76.9B2.13B76.9B2.42B47.9B
2.13B2.30B44.9B2.16BABA17B53.9B2.43B46.9B2.13B4.29B$4.10B2.14B74.10B
2.14B74.10B2.41B47.10B2.46B42.10B2.18B2A18B50.10B2.38B.3B46.10B2.14B
4.29B$3.10B2.16B72.10B2.16B72.10B2.37B.3B47.10B2.46B42.10B2.17B3A20B
48.10B2.23BA15B49.10B2.16B5.19B2A7B$3.29B71.29B13.B9.B47.49B51.53B.3B
43.30BA22B47.33B2A17B48.29B5.18B4A4B$2.30B12.B9.B47.30B12.3B5.B.3B.2B
42.33B2A16B49.54B46.33B3A17B47.33BA21B45.30B.3B.17BA2B3A4B$.30B12.3B
5.B.3B.2B42.30B9.2B.5B3.6B.BAB40.35B2A17B46.37B2A17B44.31B4A2BA17B45.
57B43.30B.16B3A6BAB2A4B$.31B7.2B.5B3.6B.3B41.31B7.9B2.8B2A3B38.32B2AB
A19B45.58B42.30BA2BA3BA18B44.39B2A17B42.47B3A6B3A4B$3.30B5.9B2.13B41.
30B5.10B2.7B2AB2A2B39.30BA4BA18B46.57B43.27B2A5B2A17B47.37B4A14B45.
55BA3B$.33B3.10B2.5BA8B38.33B3.10B2.9BA2B2A2B36.32BA4BA17B45.60B40.
30B2A21B47.32B2A9BA14B42.63B$33B3.10B2.5B3A8B36.33B4.25B2A2B34.33BA5B
2A16B43.33B.10B2A14B40.56B44.30B2AB2A9B2A14B40.33B.30B$.27B.3B4.16B2A
B2A8B36.27B.3B4.21B2AB3A3B35.30B2A2BA4BA17B43.27B.4B.9B2AB2A13B40.55B
45.29BAB4A5BAB2A14B42.27B.3B.30B$2.26B7.17B4A9B37.26B7.22BAB3A3B37.
28B2A6B3A16B45.26B.3B2A13BA14B40.53B47.28B3AB2A5B2A14B45.26B5.29B$.
28B5.17B2A11B37.28B6.22B3A6B35.30B2A22B46.30B2A10BAB3A13B40.53B47.33B
2A5BA17B42.28B5.16B2.10B$2.29B3.17B2A12B37.29B6.30B35.30BA24B45.30B3A
3BA3B2A15B43.40B2.10B48.31B2A24B43.29B4.14B2.10B$3.29B4.17BA12B37.29B
3.33B35.54B46.30B3A7B2A16B42.38B2.10B50.55B45.29B4.13B2.9B$5.28B.33B
38.28B.33B38.51B49.30B3A23B44.36B2.9B53.52B48.28B5.3B.6B3.5B.2B$6.26B
.33B40.26B3.27B.3B40.49B51.54B46.27B.6B3.5B.2B55.39B2.10B49.26B7.2B.
3B.B5.3B$2.3B.27B.27B.3B37.3B.27B3.26B40.3B.37B2.10B47.3B.53B43.3B.
27B.3B.B5.3B55.3B.38B2.10B46.3B.27B10.B9.B$.8BA24B.19B2A5B40.8BA24B.
28B38.8BA32B2.10B47.8BA37B2.10B42.8BA24B.B9.B55.8BA34B2.9B46.8BA24B$
8B3A22B.20B2A6B38.8B3A22B3.29B35.8B3A31B2.9B47.8B3A35B2.10B42.8B3A22B
67.8B3A25B.6B3.5B.2B46.8B3A22B$.BA4B2A22B4.19B2A8B37.BA4B2A22B6.29B
35.BA4B2A25B.6B3.5B.2B49.BA4B2A22B2.13B2.9B44.BA4B2A22B70.BA4B2A22B3.
2B.3B.B5.3B51.BA4B2A22B$2.6BA24B3.29B37.6BA24B6.28B35.6BA25B.3B.B5.3B
54.6BA24B2.3B.6B3.5B.2B46.6BA24B69.6BA24B5.B9.B53.6BA24B$3.3A3BA3BA
19B5.28B37.3A3BA3BA19B7.26B37.3A3BA3BA19B3.B9.B56.3A3BA3BA19B3.2B.3B.
B5.3B51.3A3BA3BA19B70.3A3BA3BA19B70.3A3BA3BA19B$2.A2BA3B2A21B7.26B37.
A2BA3B2A21B4.3B.27B35.A2BA3B2A21B70.A2BA3B2A21B8.B9.B51.A2BA3B2A21B
70.A2BA3B2A21B70.A2BA3B2A21B$2.A5BAB2A19B4.3B.27B36.A5BAB2A19B4.33B
34.A5BAB2A19B71.A5BAB2A19B71.A5BAB2A19B71.A5BAB2A19B71.A5BAB2A19B$3.
3B2A11B2.10B3.33B36.3B2A11B2.10B3.33B36.3B2A11B2.10B72.3B2A11B2.10B
72.3B2A11B2.10B72.3B2A11B2.10B72.3B2A11B2.10B$4.3B3ABA6B2.10B3.33B38.
3B3ABA6B2.10B5.30B39.3B3ABA6B2.10B74.3B3ABA6B2.10B74.3B3ABA6B2.10B74.
3B3ABA6B2.10B74.3B3ABA6B2.10B$5.2B2AB2A6B2.9B5.30B41.2B2AB2A6B2.9B7.
31B38.2B2AB2A6B2.9B76.2B2AB2A6B2.9B76.2B2AB2A6B2.9B76.2B2AB2A6B2.9B
76.2B2AB2A6B2.9B$7.B2A.6B3.5B.2B7.31B41.B2A.6B3.5B.2B9.30B40.B2A.6B3.
5B.2B79.B2A.6B3.5B.2B79.B2A.6B3.5B.2B79.B2A.6B3.5B.2B79.B2A.6B3.5B.2B
$8.2B.3B.B5.3B12.30B42.2B.3B.B5.3B12.30B42.2B.3B.B5.3B84.2B.3B.B5.3B
84.2B.3B.B5.3B84.2B.3B.B5.3B84.2B.3B.B5.3B$12.B9.B12.30B47.B9.B13.29B
47.B9.B89.B9.B89.B9.B89.B9.B89.B9.B$35.29B73.16B2.10B$36.16B2.10B74.
14B2.10B$37.14B2.10B76.13B2.9B$38.13B2.9B79.3B.6B3.5B.2B$40.3B.6B3.5B
.2B81.2B.3B.B5.3B$41.2B.3B.B5.3B89.B9.B$45.B9.B!

There are two pairs emitting westbound T-ship. (-25, 9, -74) is the one that have been used in the constructions and calculations since November 30th, corresponding to the original constraints ΣΔy = 5, 2ΣΔx - ΣΔt = -95. The other one, (-33, 6, -68), corresponds to the alternative constraints ΣΔy = 13, 2ΣΔx - ΣΔt = -93; its larger positive ΣΔy makes it less productive than (-25, 9, -74), see for instance, no 4-in-a-row spaceship can be built from it. The constraint is also subject to the two adjustments seen so far:
1)Adding 12 to ΣΔy while adding 159 to 2ΣΔx - ΣΔt, therefore adding one T-ship to the cycle (affecting displacement and period);
2)Adding 240 to 2ΣΔx - ΣΔt, therefore deducting 240 to the period.

There are seven pairs emitting southbound T-ship, among which (-24, -8, -18) was the missing one. It has the same emission timing as (-28, -7, -24) due to a shared intermediate B-heptomino.

Interestingly, these are the only pairs that leave an extra object while moving. Some other pairs coming close to doing so are:

Code: Select all

#C Blinker: (-32, 0, -14), (-35, -2, -13), (-29, -5, -24)
#C Boat: (-32, 9, -67), (-32, 9, -69); Ship: (-27, -10, -10)
#C Other: (-30, -4, -22), (-33, -11, -43)
x = 223, y = 159, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7cHistory
200.B9.B$199.3B5.B.3B.2B$195.2B.5B3.6B.3B$126.B9.B57.9B2.13B$125.3B5.
B.3B.2B52.10B2.14B$11.B9.B21.B9.B37.B9.B19.2B.5B3.6B.3B29.B9.B10.10B
2.16B$10.3B5.B.3B.2B16.3B5.B.3B.2B32.3B5.B.3B.2B14.9B2.13B26.3B5.B.3B
.2B6.29B$6.2B.5B3.6B.3B11.2B.5B3.6B.3B27.2B.5B3.6B.3B12.10B2.14B21.2B
.5B3.6B.3B4.30B$5.9B2.13B8.9B2.13B24.9B2.13B9.10B2.16B19.9B2.13B.30B$
4.10B2.14B6.10B2.14B22.10B2.14B8.29B17.10B2.45B$3.10B2.16B4.10B2.16B
20.10B2.16B6.30B16.10B2.16B.30B$3.29B3.29B19.29B4.30B17.60B$2.30B2.
30B18.30B4.31B15.60B$.30B2.30B18.30B7.30B13.56B.3B$.31B.31B17.31B4.
33B12.56B$3.30B2.30B18.30B2.33B15.36B2A17B$.65B15.33B2.27B.3B14.59B$
65B15.33B4.26B17.61B$.27B.3B.27B.3B17.27B.3B4.7B3A18B17.27B.2B3.28B$
2.26B6.7BA18B22.26B9.8BA20B16.26B7.10B2A14B$.28B4.8B2A18B20.28B9.3B2A
B2A21B14.28B2.3B.9B2AB2A13B$2.29B3.5B2AB2A19B19.29B9.B2A25B14.31B2A
13BA14B$3.29B3.6B2A21B19.29B9.2A24B16.29B2A10BAB3A13B$5.28B4.A27B20.
28B4.3B.6B2A19B17.28B3A3BA3B2A15B$6.26B6.26B22.26B4.8B3AB2A19B17.28B
3A7B2A16B$2.3B.27B.3B.4BAB3A18B17.3B.27B2.9BA3BA19B14.3B.30B3A23B$.8B
A31B3A2BA19B15.8BA24B2.8B4A18B15.8BA51B$8B3A29B2A2B3A18B15.8B3A22B4.
9BA21B12.8B3A49B$.BA4B2A22B2.8BAB2A18B18.BA4B2A22B7.30B13.BA4B2A22B.
16B2.10B$2.6BA24B.9B2A20B17.6BA24B4.30B15.6BA38B2.10B$3.3A3BA3BA19B2.
30B18.3A3BA3BA19B4.29B17.3A3BA3BA19B.13B2.9B$2.A2BA3B2A21B2.30B18.A2B
A3B2A21B6.16B2.10B16.A2BA3B2A21B4.3B.6B3.5B.2B$2.A5BAB2A19B3.29B19.A
5BAB2A19B8.14B2.10B17.A5BAB2A19B6.2B.3B.B5.3B$3.3B2A11B2.10B4.16B2.
10B20.3B2A11B2.10B9.13B2.9B19.3B2A11B2.10B10.B9.B$4.3B3ABA6B2.10B6.
14B2.10B22.3B3ABA6B2.10B12.3B.6B3.5B.2B21.3B3ABA6B2.10B$5.2B2AB2A6B2.
9B8.13B2.9B24.2B2AB2A6B2.9B14.2B.3B.B5.3B26.2B2AB2A6B2.9B$7.B2A.6B3.
5B.2B11.3B.6B3.5B.2B27.B2A.6B3.5B.2B19.B9.B29.B2A.6B3.5B.2B$8.2B.3B.B
5.3B16.2B.3B.B5.3B32.2B.3B.B5.3B64.2B.3B.B5.3B$12.B9.B21.B9.B37.B9.B
69.B9.B17$198.B9.B$197.3B5.B.3B.2B$193.2B.5B3.6B.3B$192.9B2.13B$191.
10B2.14B$190.10B2.16B$190.29B$189.30B$188.30B$188.31B$11.B9.B69.B9.B
69.B9.B8.30B$10.3B5.B.3B.2B64.3B5.B.3B.2B64.3B5.B.3B.2B2.33B$6.2B.5B
3.6B.3B59.2B.5B3.6B.3B59.2B.5B3.6B.36B$5.9B2.13B56.9B2.13B56.9B2.39B.
3B$4.10B2.14B54.10B2.14B54.10B2.39B$3.10B2.16B52.10B2.16B52.10B2.41B$
3.29B51.29B51.33BA21B$2.30B50.30B50.33B3A21B$.30B50.30B50.30B.2B2AB2A
21B$.31B11.B9.B27.31B11.B9.B27.31B.BA2B2A20B$3.30B9.3B5.B.3B.2B25.30B
9.3B5.B.3B.2B25.32BA2B2A20B$.33B4.2B.5B3.6B.3B22.33B4.2B.5B3.6B.3B22.
33B2ABABA21B$33B4.9B2.13B19.33B4.9B2.13B19.34BA2B3A20B$.27B.3B4.10B2.
14B19.27B.3B4.10B2.14B19.57B$2.26B7.10B2.16B19.26B7.10B2.16B19.26B.
31B$.28B6.29B17.28B6.29B17.28B.30B$2.29B3.30B18.29B3.30B18.57B$3.29B.
30B20.29B.30B20.55B$5.59B21.59B21.41B2.10B$6.26B3.30B21.27B2.30B21.
39B2.10B$2.3B.60B16.3B.60B16.3B.39B2.9B$.8BA55B16.8BA55B16.8BA27B.6B
3.5B.2B$8B3A49B.3B16.8B3A49B.3B16.8B3A22B2.2B.3B.B5.3B$.BA4B2A22B3.
26B21.BA4B2A22B3.26B21.BA4B2A22B8.B9.B$2.6BA52B21.6BA52B21.6BA24B$3.
3A3BA3BA19B.29B20.3A3BA3BA19B.29B20.3A3BA3BA19B$2.A2BA3B2A21B3.29B18.
A2BA3B2A21B3.29B18.A2BA3B2A21B$2.A5BAB2A19B6.28B17.A5BAB2A19B6.28B17.
A5BAB2A19B$3.3B2A11B2.10B7.26B19.3B2A11B2.10B7.26B19.3B2A11B2.10B$4.
3B3ABA6B2.10B4.3B.27B19.3B3ABA6B2.10B4.3B.27B19.3B3ABA6B2.10B$5.2B2AB
2A6B2.9B4.33B19.2B2AB2A6B2.9B4.19B2A12B19.2B2AB2A6B2.9B$7.B2A.6B3.5B.
2B4.20B3A10B22.B2A.6B3.5B.2B4.21B2A10B22.B2A.6B3.5B.2B$8.2B.3B.B5.3B
9.17B2A11B25.2B.3B.B5.3B9.16B3ABA9B25.2B.3B.B5.3B$12.B9.B11.16BA14B
27.B9.B11.15BAB2A12B27.B9.B$35.15BA3BA10B50.14B2AB2A11B$34.17B3A10B
50.20BA9B$34.21B2A6B51.17B2ABA8B$35.16B2.2B2A6B52.16B2.10B$36.14B2.5B
A4B54.14B2.10B$37.13B2.9B56.13B2.9B$39.3B.6B3.5B.2B59.3B.6B3.5B.2B$
40.2B.3B.B5.3B64.2B.3B.B5.3B$44.B9.B69.B9.B7$124.B9.B$123.3B5.B.3B.2B
$119.2B.5B3.6B.3B$118.9B2.13B$117.10B2.14B$116.10B2.16B$116.29B$41.B
9.B63.30B$40.3B5.B.3B.2B58.30B$36.2B.5B3.6B.3B57.31B$35.9B2.13B57.30B
$11.B9.B12.10B2.14B31.B9.B12.33B$10.3B5.B.3B.2B7.10B2.16B29.3B5.B.3B.
2B7.33B$6.2B.5B3.6B.3B6.29B24.2B.5B3.6B.3B7.27B.3B$5.9B2.13B3.30B23.
9B2.13B6.26B$4.10B2.14B.30B23.10B2.14B4.28B$3.10B2.47B21.10B2.16B4.
29B$3.29B.30B20.29B4.29B$2.62B18.30B6.19B2A7B$.62B18.30B8.18B4A4B$.
57B.3B19.31B3.3B.17BA2B3A4B$3.28B.10B2A14B25.30B.16B3A6BAB2A4B$.39B2A
17B22.49B3A6B3A4B$39BA21B19.33B.26BA3B$.27B.2B2.7BA21B19.27B.3B3.31B$
2.26B7.8BABA17B19.26B8.30B$.28B7.8BAB2A14B19.28B6.30B$2.29B.3B.BA10BA
14B19.29B4.29B$3.32BAB2A9BA15B19.29B4.16B2.10B$5.29B2A2BA8B2A14B22.
28B4.14B2.10B$6.29B2ABA5B3A14B25.26B6.13B2.9B$2.3B.31B2A24B19.3B.27B
7.3B.6B3.5B.2B$.8BA28BA24B18.8BA24B7.2B.3B.B5.3B$8B3A51B18.8B3A22B12.
B9.B$.BA4B2A22B.29B20.BA4B2A22B$2.6BA40B2.10B21.6BA24B$3.3A3BA3BA19B.
14B2.10B23.3A3BA3BA19B$2.A2BA3B2A21B3.13B2.9B23.A2BA3B2A21B$2.A5BAB2A
19B6.3B.6B3.5B.2B24.A5BAB2A19B$3.3B2A11B2.10B7.2B.3B.B5.3B29.3B2A11B
2.10B$4.3B3ABA6B2.10B12.B9.B31.3B3ABA6B2.10B$5.2B2AB2A6B2.9B56.2B2AB
2A6B2.9B$7.B2A.6B3.5B.2B59.B2A.6B3.5B.2B$8.2B.3B.B5.3B64.2B.3B.B5.3B$
12.B9.B69.B9.B!

The extra product could not survive from the sparks. Testing also suggests that (-32, 9, -67), (-32, 9, -69) and (-35, -2, -13) are not suitable for two consecutive pushing reactions in 315 ticks as a stray butterfly woudl crash into cloverleaf.

This (-32, -4, -18) pair is even more unusual: it is a period-240 T-ship gun in ground state, only emitting no southbound T-ship upon pushing. However, its emission would have conflict with the only westbound T-ship that sets off displacement.

Code: Select all

x = 66, y = 52, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7cHistory
43.B9.B$42.3B5.B.3B.2B$38.2B.5B3.6B.3B$37.9B2.13B$11.B9.B14.10B2.14B$
10.3B5.B.3B.2B9.10B2.16B$6.2B.5B3.6B.3B8.29B$5.9B2.13B5.30B$4.10B2.
14B3.30B$3.10B2.16B2.31B$3.29B3.30B$2.30B.33B$.30B.33B$.59B.3B$3.57B$
.42B2A16B$44B2A17B$.27B.12B2ABA19B$2.26B2.11BA4BA18B$.28B.11BA4BA17B$
2.31B.7BA5B2A16B$3.36B2A2BA4BA17B$5.33B2A6B3A16B$6.33B2A22B$2.3B.27B.
6BA24B$.8BA24B.30B$8B3A22B.30B$.BA4B2A24B.29B$2.6BA24B2.16B2.10B$3.3A
3BA3BA19B3.14B2.10B$2.A2BA3B2A22B4.13B2.9B$2.A5BAB2A21B6.3B.6B3.5B.2B
$3.3B2A11B2.12B7.2B.3B.B5.3B$4.3B3ABA6B2.13B11.B9.B$5.2B2AB2A6B2.9B.
3B$7.B2A.6B3.5B.2B2.3B$8.2B.3B.B5.3B6.3B$12.B9.B7.3B$30.3B$30.3B$30.
3B$30.3B$30.3B$30.3B$30.3B$30.3B$30.3B$30.3B$30.BAB$30.3B$30.BAB$30.
3A!

...
Hold on. Is a (3,12)c/75 spaceship possible? All copies of cloverleaf in such a spaceship would be perturbed every 75 ticks, and spinning in place for more cloverleaf periods (240 ticks) is not allowed, so there would be significantly different interactions.

[FlutterWithKSparkler]

A c/4 orthogonal rake with period 844.

Code: Select all

x = 43, y = 133, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
28bobo$27b2ob2o$26b3ob3o3$30bo$29b3o$22bobo3bobobo3bobo$21b2ob2o2bobob
o2b2ob2o$20b3ob3obobobob3ob3o2$29bobo$30bo$30bo$30bo$28b5o$23b2ob3o3b
3ob2o$23bo6bo6bo$23bo6bo6bo$23bo3bo5bo3bo$22b2obo4bo4bob2o$23b2o2b7o2b
2o$24bo2bo5bo2bo$28bobobo$21bo6bobobo6bo$19bo3bo3b7o3bo3bo$19bobobo5bo
bo5bobobo$18b2o3b2o4bobo4b2o3b2o$21bo4b2obobob2o4bo$26bo2b3o2bo$26bo7b
o$30bo$27b2obob2o$27bo5bo3$25b3o$26bo2$26bo31$5b5o$4bo2bo2bo$4bo2bo2bo
$5bo3bo7$5bobo$4b2ob2o$3b3ob3o11$b5o$o2bo2bo$o2bo2bo$bo3bo18$35b3o$34b
5o$33bo2bo2bo$33bobobobo13$17bobo$16b2ob2o$15b3ob3o!

Hold on. Is a (3,12)c/75 spaceship possible? All copies of cloverleaf in such a spaceship would be perturbed every 75 ticks, and spinning in place for more cloverleaf periods (240 ticks) is not allowed, so there would be significantly different interactions.

Yes, puffer and rake too.

Code: Select all

x = 87, y = 84, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
o7bo63b3o$o10bo61b2o$o2bo2bo4b2o61bo$2b2o3b2o5bo$9b4obo59b2o$8b2o4bo
61b2o$11b2o39bo18b2o4b2o$11bo39b2o14bo2bobo3b2o$11bo38b2o15b2o7bo$11b
3o37b2o$12bo40bo2b2o$11bo42b2ob2obo$9b3o42b3o2b3o$10bo44bo3b3o2$2bo$2b
o$2bo27bo$29b3o$28bo2bo27b3o$27b3o2bo26b3o$28bo2b3o26bo6$33b2o$27b3o3b
3o31bo$27bob2o2b2o32bo$26bo2bo37bo$24bo$24bo2bo$24bo3bo$25bo2bo$26b3o$
77b2o$76b2o$77bo$78bo2b2o$79b3o$80bo$70b3o$57bo2$57bo$56b3o$65b2o$66b
2o$64bo$68bo$68bo$58b2o6bo2b2o$58b2o4b2o4bo$60bobo2bo2b3o$60b2o2bo2$
29b2o$28bobo$28bobo54b2o$28b3o51b2o$31bo2b3o45bo$82bo3bo$83bo2bo$32b3o
2bobo44b3o$36bo5$53b3o15bo$53b3o14bo$54bo14b2o$70b2o3b3obo$76bob3o$75b
obo2b2o$71bo2b3ob3o$65bo5bob2ob2obo$64b3o3bob3o$66b2o2b3ob3o$63b2obo4b
o5b4o$63b3obo9bo2bo$65bobo9b3o$65b3o!

[GUYTU6J]

Hold on. Is a (3,12)c/75 spaceship possible? All copies of cloverleaf in such a spaceship would be perturbed every 75 ticks, and spinning in place for more cloverleaf periods (240 ticks) is not allowed, so there would be significantly different interactions.

Yes, puffer and rake too.
...

Awesome! The triangular T-ship loop design is similar to the first (3,12)c/315 spaceship, but the upper right corner here is more complex and delicate as the timings between laying blinker, moving blinker and splitting T-ship are aligned perfectly with no room for adjustment.

A c/4 orthogonal rake with period 844.
...

It can be viewed as a much simpler analog to the Weekender distaff in Life without the need for a slow-salvo construction of the forward internal spaceship. I tried to find c/4 cloverleaf-moving reactions and got these:

Code: Select all

x = 512, y = 155, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
26b2o38b2o38b2o$27bobo37bobo37bobo66b2o38b2o38b2o$26b2o38b2o38b2o66bob
o37bobo37bobo$26bo39bo39bo69b2o38b2o38b2o$177bo39bo39bo4$26bo39bo39bo
79bo39bo39bo$25b3o37b3o37b3o77b3o37b3o37b3o$24bob2o36bob2o36bob2o76bob
2o36bob2o36bob2o$24b2o2bo35b2o2bo35b2o2bo75b2o2bo35b2o2bo35b2o2bo$29b
2o38b2o38b2o8bobo67b2o38b2o6b5o27b2o7bobo$30b2o38b2o38b2o6b2ob2o67b2o
38b2o4bo2bo2bo27b2o5b2ob2o$29b2o38b2o7b5o26b2o6b3ob3o65b2o7b3o28b2o5bo
2bo2bo26b2o5b3ob3o$29bo39bo7bo2bo2bo25bo79bo7b5o27bo7bo3bo27bo$77bo2bo
2bo112bo2bo2bo$78bo3bo113bobobobo5$38b3o$37b5o$36bo2bo2bo$36bobobobo
56$136b2o$134bobo$136b2o$137bo3$303b2o$26bo59bo59bo59bo59bo36bo22bo59b
o59bo59bo$25b3o57b3o57b3o57b3o57b3o35b2o20b3o57b3o57b3o34b2o21b3o$24bo
b2o56bob2o56bob2o56bob2o56bob2o36bo19bob2o56bob2o56bob2o32bobo21bob2o$
24b2o2bo50b2o3b2o2bo55b2o2bo55b2o2bo55b2o2bo55b2o2bo55b2o2bo55b2o2bo
33b2o20b2o2bo$29b2o49bo8b2o58b2o58b2o58b2o58b2o58b2o58b2o32bo25b2o$30b
2o47b2o9b2o58b2o58b2o58b2o58b2o58b2o58b2o58b2o$29b2o48bo9b2o58b2o58b2o
58b2o58b2o58b2o58b2o58b2o$29bo59bo59bo59bo59bo59bo47b2o10bo47b2o10bo
59bo$378bobo57bobo$377b2o58b2o$377bo59bo4$367bobo59bobo$366b2ob2o57b2o
b2o56bobo$32b2o331b3ob3o55b3ob3o54b2ob2o$33bobo451b3ob3o$32b2o$32bo2$
199b2o58b2o$197bobo57bobo$199b2o58b2o$200bo59bo29$3bo$bo3bo199b5o56bob
o$bobobo198bo2bo2bo54b2ob2o57bobo$2o3b2o197bo2bo2bo53b3ob3o55b2ob2o$3b
o88b3o110bo3bo115b3ob3o$91b5o$90bo2bo2bo$90bobobobo3$158b3o$157b5o$
156bo2bo2bo$156bobobobo!

They can give rise to a period-1464 spaceship. I noticed that the two cloverleaf-to-T converters in the patterns above have the output on the same lane, with the sideway-emitting one lagging behind by 21 ticks, which is unluckily not applicable due to parity:

Code: Select all

#C [[ TRACK 0 -1/4 ]]
x = 52, y = 421, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
36b5o$35bo2bo2bo$35bo2bo2bo$36bo3bo2$38b3o2$30b5o2bobobo2b5o$29bo2bo2b
ob5obo2bo2bo$29bo2bo2bo2bobo2bo2bo2bo$30bo3bo3b3o3bo3bo$39bo$39bo$38b
3o$39bo$37bobobo$32b2o2b7o2b2o$31b2o3bobobobo3b2o$31b3o11b3o$32bo13bo$
31bo3bo3bo3bo3bo$31bo3b9o3bo$32b2o2bo2bo2bo2b2o$37b2ob2o$37bobobo$29bo
bo4bo5bo4bobo$28b2ob2o5bobo5b2ob2o$27b3ob3o4b3o4b3ob3o$34b5ob5o$34b2o
2b3o2b2o$39bo$36bo5bo$36b2o3b2o$36b2o3b2o2$35bo$34b3o$34b3o53$8b5o$7bo
2bo2bo$7bo2bo2bo$8bo3bo91$41b5o$40bo2bo2bo$40bo2bo2bo$41bo3bo52$2b3o$b
5o$o2bo2bo$obobobo27$6bo$4bo3bo$4bobobo$3b2o3b2o$6bo28$8bobo$7b2ob2o$
6b3ob3o29$10b5o$9bo2bo2bo$9bo2bo2bo$10bo3bo27$14b3o$13b5o$12bo2bo2bo$
12bobobobo36$35bo$34b3o$34b3o14$25b5o$24bo2bo2bo$24bo2bo2bo$25bo3bo!

I will aim for rakes later.
---
Unrelated, this section was prepared some days ago. A review on diagonal Julia chains discovered some Julia shuttles. Julia itself is still too big to stack diagonally for a continuous support, unfortunately, so the shuttles are rather bounded and sparks unexposed. Using higher period endpoints increases the shuttle period; compare p140, p168 and p280 below:

Code: Select all

x = 189, y = 56, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
20bo69bo2$20b2o68b2o61b2o$21bo69bo60bo$152bo2b2o$152b7o$155bo2bo$145bo
9bo2bo$144b3o8b3o$20b3o67b3o50bo2b2o$19bo2bo66bo2bo49b2obo$19bo2b2o65b
o2b2o49b2o$19b5o65b5o$22bo2bo66bo2bo$22bo2bo66bo2bo56bo$22b3o67b3o52bo
bob3o$149bob2o7$34b2o68b2o68b2o$33bo69bo69bo$33bo2b2o65bo2b2o65bo2b2o$
33b7o63b7o63b7o$36bo2bo5b2o59bo2bo5b2o59bo2bo5b2o$b2o33bo2bo6bobo22b2o
33bo2bo6bobo22b2o33bo2bo6bobo$o35b3o31bo35b3o31bo35b3o$o2b2o65bo2b2o
65bo2b2o$7o63b7o63b7o$3bo2bo5b2o59bo2bo66bo2bo5b2o$3bo2bo6bobo57bo2bo
66bo2bo6bobo$3b3o67b3o67b3o$26bo69bo69bo$25b3o67b3o67b3o$24bo2bo66bo2b
o66bo2bo$23b2obo66b2obo66b2obo$24b2o68b2o68b2o$22bo2bo5bo60bo2bo66bo2b
o5bo2$22b2o7b2o59b2o16bo51b2o7b2o7bo$23bo69bo15b3o51bo15b3o$31b2o75bo
2b2o58b2o5bo2b2o$30b3o74b2obo59b3o4b2obo$31bo76b2o61bo6b2o3$22b3o67b4o
21bo44b4o21bo$21bo2bo66bo2bo17bobob3o42bo2bo17bobob3o$21bo2b2o65bo2b2o
18bob2o43bo2b2o18bob2o$21b5o65b7o63b7o$24bo2bo66bo2bo66bo2bo$24bo2bo
66bo2bo66bo2bo$24b3o67b3o67b3o!

[GUYTU6J]

FlutterWithKSparkler requested privately for further information on the higher-period oblique spaceships. So I am posting the vectors here, with group s for 660 clean cloverleaf pairs sorted by Δy and the group r for 13 T-ship making pairs:

Code: Select all

s = """
[(-15,32,85)]
[(-30,19,-108),(-14,19,55)]
[(-31,18,-109)]
[(-31,17,-107),(-31,17,-108),(-32,17,-109),(-31,17,-110)]
[(-32,16,-105),(-32,16,-106),(-32,16,-109),(-31,16,-113)]
[(-31,15,-50),(-32,15,-103),(-32,15,-104),(-33,15,-104),(-32,15,-105),(-33,15,-105),(-33,15,-106)]
[(-32,14,-48),(-30,14,-99),(-32,14,-101),(-32,14,-103),(-33,14,-103),(-33,14,-104),(-34,14,-105)]
[(-33,13,-102),(-34,13,-102),(-34,13,-105),(-32,13,-106),(-33,13,-109),(-32,13,-119),(-31,13,-120)]
[(-34,12,-100),(-33,12,-101),(-34,12,-101),(-32,12,-104),(-33,12,-105),(-33,12,-106),(-34,12,-107),(-33,12,-108),(-33,12,-114),(-33,12,-116),(-32,12,-118),(-32,12,-119)]
[(-34,11,-100),(-33,11,-101),(-34,11,-101),(-34,11,-102),(-33,11,-109),(-33,11,-113),(-33,11,-114),(-32,11,-115),(-34,11,-116),(-33,11,-117),(-34,11,-117),(-32,11,-118),(-33,11,-118),(-33,11,-119),(-31,11,-120)]
[(-32,10,118), (-30,10,-49),(-31,10,-66),(-31,10,-67),(-31,10,-68),(-31,10,-70),(-30,10,-71),(-33,10,-99),(-34,10,-100),(-33,10,-101),(-31,10,-102),(-33,10,-102),(-34,10,-102),(-33,10,-103),(-33,10,-104),(-34,10,-112),(-34,10,-113),(-33,10,-114),(-34,10,-115),(-33,10,-116),(-34,10,-116),(-35,10,-116),(-31,10,-118),(-32,10,-120),(-33,10,-120)]
[(-31,9,-64),(-31,9,-65),(-32,9,-65),(-31,9,-66),(-32,9,-66),(-31,9,-67),(-32,9,-67),(-32,9,-68),(-31,9,-69),(-32,9,-69),(-31,9,-70),(-31,9,-71),(-30,9,-72),(-27,9,-73),(-29,9,-74),(-34,9,-100),(-34,9,-101),(-33,9,-102),(-33,9,-103),(-33,9,-105),(-33,9,-106),(-34,9,-110),(-34,9,-113),(-34,9,-114),(-35,9,-114),(-33,9,-115),(-34,9,-116),(-35,9,-116),(-34,9,-117),(-32,9,-118),(-34,9,-118),(-33,9,-119),(-32,9,-120),(-33,9,-120)]
[(-31,8,-62),(-30,8,-63),(-31,8,-63),(-31,8,-64),(-32,8,-64),(-30,8,-65),(-32,8,-65),(-31,8,-66),(-32,8,-66),(-30,8,-67),(-33,8,-67),(-32,8,-68),(-33,8,-68),(-31,8,-69),(-32,8,-69),(-32,8,-70),(-30,8,-71),(-31,8,-71),(-28,8,-72),(-30,8,-73),(-28,8,-75),(-31,8,-96),(-31,8,-99),(-34,8,-100),(-31,8,-101),(-33,8,-101),(-33,8,-102),(-34,8,-102),(-33,8,-103),(-33,8,-104),(-33,8,-106),(-34,8,-111),(-34,8,-112),(-35,8,-112),(-35,8,-113),(-33,8,-114),(-34,8,-115),(-34,8,-116),(-32,8,-118)]
[(-30,7,-59),(-30,7,-61),(-31,7,-61),(-33,7,-63),(-31,7,-64),(-33,7,-64),(-32,7,-65),(-33,7,-66),(-31,7,-67),(-32,7,-67),(-33,7,-67),(-34,7,-67),(-31,7,-68),(-32,7,-68),(-30,7,-69),(-31,7,-69),(-32,7,-69),(-33,7,-69),(-31,7,-70),(-29,7,-71),(-30,7,-71),(-31,7,-71),(-29,7,-72),(-30,7,-72),(-28,7,-73),(-30,7,-73),(-31,7,-73),(-28,7,-74),(-29,7,-74),(-25,7,-75),(-29,7,-75),(-32,7,-98),(-33,7,-98),(-32,7,-99),(-33,7,-99),(-32,7,-100),(-33,7,-100),(-32,7,-101),(-32,7,-102),(-32,7,-103),(-32,7,-104),(-31,7,-111),(-34,7,-112),(-35,7,-112),(-34,7,-113),(-35,7,-113),(-34,7,-115),(-33,7,-116),(-33,7,-117),(-32,7,-118)]
[(-33,6,-61),(-33,6,-64),(-33,6,-65),(-34,6,-65),(-32,6,-66),(-32,6,-67),(-33,6,-67),(-34,6,-67),(-31,6,-69),(-33,6,-69),(-32,6,-70),(-31,6,-71),(-32,6,-71),(-30,6,-74),(-30,6,-94),(-32,6,-99),(-32,6,-100),(-33,6,-100),(-32,6,-102),(-32,6,-104),(-34,6,-110),(-32,6,-112),(-34,6,-112),(-34,6,-113),(-33,6,-114),(-34,6,-114),(-33,6,-115),(-32,6,-117),(-31,6,-118)]
[(-32,5,-61),(-33,5,-62),(-33,5,-63),(-34,5,-63),(-34,5,-64),(-32,5,-65),(-33,5,-66),(-33,5,-67),(-31,5,-68),(-32,5,-68),(-31,5,-69),(-30,5,-70),(-31,5,-70),(-30,5,-71),(-30,5,-72),(-29,5,-73),(-29,5,-74),(-31,5,-96),(-34,5,-112),(-35,5,-112),(-33,5,-114),(-34,5,-114),(-33,5,-115),(-33,5,-116),(-31,5,-120)]
[(-29,4,-15),(-31,4,-18),(-30,4,-62),(-33,4,-63),(-34,4,-63),(-33,4,-64),(-34,4,-64),(-32,4,-66),(-33,4,-66),(-32,4,-67),(-31,4,-69),(-31,4,-70),(-30,4,-71),(-30,4,-72),(-29,4,-73),(-28,4,-75),(-28,4,-76),(-29,4,-97),(-34,4,-110),(-31,4,-111),(-33,4,-112),(-34,4,-112),(-33,4,-114),(-31,4,-118)]
[(-30,3,0),(-31,3,-16),(-32,3,-16),(-31,3,-17),(-31,3,-21),(-33,3,-61),(-31,3,-63),(-33,3,-63),(-33,3,-64),(-32,3,-65),(-34,3,-65),(-32,3,-66),(-33,3,-66),(-31,3,-67),(-31,3,-68),(-32,3,-68),(-30,3,-69),(-30,3,-70),(-31,3,-70),(-29,3,-71),(-30,3,-72),(-30,3,-98),(-34,3,-112),(-32,3,-114),(-32,3,-115),(-33,3,-116)]
[(-31,2,-14),(-32,2,-14),(-32,2,-15),(-32,2,-16),(-33,2,-16),(-32,2,-17),(-30,2,-18),(-33,2,-18),(-31,2,-20),(-29,2,-44),(-33,2,-63),(-34,2,-63),(-32,2,-65),(-32,2,-66),(-31,2,-67),(-31,2,-68),(-26,2,-70),(-29,2,-71),(-29,2,-72),(-28,2,-73),(-29,2,-73),(-26,2,-94),(-26,2,-96),(-27,2,-96),(-28,2,-109),(-31,2,-112),(-30,2,-113),(-27,2,-114),(-31,2,-115),(-28,2,-117),(-29,2,-119)]
[(-29,1,4),(-30,1,-9),(-31,1,-12),(-32,1,-13),(-32,1,-14),(-32,1,-15),(-33,1,-15),(-33,1,-16),(-34,1,-17),(-34,1,-18),(-33,1,-20),(-31,1,-21),(-32,1,-21),(-30,1,-42),(-32,1,-63),(-33,1,-63),(-32,1,-65),(-31,1,-67),(-31,1,-68),(-32,1,-68),(-30,1,-69),(-31,1,-70),(-28,1,-74),(-28,1,-106),(-31,1,-113),(-30,1,-114),(-28,1,-118)]
[(-32,0,-11),(-33,0,-12),(-33,0,-13),(-32,0,-14),(-35,0,-17),(-33,0,-18),(-31,0,-19),(-32,0,-19),(-33,0,-19),(-34,0,-19),(-31,0,-20),(-32,0,-20),(-32,0,-21),(-30,0,-22),(-31,0,-22),(-31,0,-23),(-30,0,-40),(-30,0,-41),(-29,0,-41),(-31,0,-42),(-32,0,-63),(-33,0,-63),(-31,0,-66),(-32,0,-67),(-29,0,-71),(-25,0,-104),(-25,0,-115)]
[(-34,-1,-14),(-35,-1,-15),(-33,-1,-16),(-34,-1,-17),(-35,-1,-17),(-34,-1,-18),(-34,-1,-19),(-32,-1,-21),(-33,-1,-21),(-31,-1,-23),(-32,-1,-23),(-30,-1,-39),(-32,-1,-42),(-32,-1,-43),(-28,-1,-73)]
[(-35,-2,-13),(-34,-2,-14),(-35,-2,-14),(-34,-2,-15),(-35,-2,-15),(-33,-2,-16),(-34,-2,-16),(-33,-2,-17),(-33,-2,-18),(-32,-2,-19),(-31,-2,-20),(-32,-2,-20),(-31,-2,-21),(-30,-2,-22),(-31,-2,-22),(-30,-2,-23),(-31,-2,-38),(-31,-2,-40),(-32,-2,-40),(-31,-2,-42),(-33,-2,-42)]
[(-33,-3,-9),(-32,-3,-10),(-32,-3,-11),(-35,-3,-13),(-35,-3,-14),(-33,-3,-15),(-29,-3,-16),(-34,-3,-16),(-33,-3,-17),(-33,-3,-18),(-32,-3,-20),(-31,-3,-21),(-30,-3,-23),(-30,-3,-24),(-33,-3,-41),(-33,-3,-42)]
[(-34,-4,-11),(-32,-4,-13),(-34,-4,-13),(-34,-4,-14),(-35,-4,-14),(-33,-4,-15),(-34,-4,-16),(-32,-4,-17),(-31,-4,-20),(-32,-4,-20),(-30,-4,-21),(-30,-4,-22),(-31,-4,-22),(-29,-4,-23),(-30,-4,-24),(-28,-4,-26),(-29,-4,-26),(-33,-4,-38),(-32,-4,-39),(-33,-4,-39),(-33,-4,-46),(-32,-4,-47)]
[(-35,-5,-13),(-33,-5,-15),(-33,-5,-16),(-33,-5,-17),(-32,-5,-19),(-31,-5,-20),(-30,-5,-21),(-30,-5,-22),(-30,-5,-23),(-29,-5,-24),(-29,-5,-25),(-32,-5,-38),(-33,-5,-38),(-33,-5,-39),(-33,-5,-46),(-33,-5,-48)]
[(-34,-6,-14),(-35,-6,-14),(-33,-6,-15),(-28,-6,-16),(-32,-6,-17),(-27,-6,-20),(-30,-6,-21),(-31,-6,-22),(-29,-6,-23),(-30,-6,-24),(-28,-6,-26),(-29,-6,-26),(-32,-6,-38),(-32,-6,-39),(-33,-6,-39),(-32,-6,-43)]
[(-30,-7,-8),(-33,-7,-9),(-33,-7,-11),(-32,-7,-12),(-33,-7,-12),(-33,-7,-13),(-33,-7,-14),(-34,-7,-14),(-32,-7,-15),(-32,-7,-16),(-30,-7,-17),(-31,-7,-17),(-33,-7,-17),(-28,-7,-20),(-29,-7,-21),(-29,-7,-22),(-28,-7,-23),(-32,-7,-37),(-33,-7,-38),(-32,-7,-39)]
[(-29,-8,-9),(-29,-8,-10),(-30,-8,-12),(-28,-8,-13),(-26,-8,-15),(-31,-8,-15),(-29,-8,-17),(-28,-8,-18),(-27,-8,-18),(-30,-8,-18),(-25,-8,-19),(-29,-8,-20),(-27,-8,-21),(-27,-8,-24),(-32,-8,-38),(-33,-8,-43),(-32,-8,-44),(-33,-8,-44)]
[(-30,-9,-9),(-31,-9,-11),(-32,-9,-11),(-31,-9,-13),(-32,-9,-14),(-29,-9,-17),(-29,-9,-19),(-26,-9,-20),(-28,-9,-20),(-28,-9,-21),(-31,-9,-35),(-33,-9,-42),(-33,-9,-43),(-32,-9,-44),(-33,-9,-44)]
[(-28,-10,-8),(-27,-10,-10),(-28,-10,-12),(-28,-10,-17),(-23,-10,-19),(-24,-10,-19),(-26,-10,-19),(-28,-10,-31),(-33,-10,-43),(-32,-10,-43),(-33,-10,-44)]
[(-21,-11,-1),(-21,-11,-2),(-21,-11,-3),(-21,-11,-4),(-24,-11,-7),(-24,-11,-13),(-26,-11,-16),(-21,-11,-19),(-23,-11,-19),(-32,-11,-41),(-33,-11,-43),(-32,-11,-44)]
[(-21,-12,-1),(-21,-12,-2),(-21,-12,-3),(-21,-12,-5),(-20,-12,-7),(-22,-12,-7),(-21,-12,-8),(-22,-12,-9),(-18,-12,-13),(-23,-12,-13),(-21,-12,-14),(-22,-12,-16),(-26,-12,-17)]
[(-19,-13,-1),(-21,-13,-1),(-21,-13,-2),(-21,-13,-4),(-21,-13,-5),(-21,-13,-6),(-19,-13,-7),(-29,-13,-41),(-30,-13,-44)]
[(-21,-14,-1),(-17,-14,-2),(-21,-14,-2),(-20,-14,-3),(-19,-14,-5),(-20,-14,-5),(-15,-14,-8),(-19,-14,-9),(-29,-14,-42)]
[(-18,-15,3),(-20,-15,1),(-21,-15,-3),(-14,-15,-5),(-17,-15,-5),(-21,-15,-5),(-15,-15,-6),(-14,-15,-6),(-14,-15,-7),(-14,-15,-10),(-29,-15,-44)]
[(-17,-16,2),(-19,-16,0),(-19,-16,-1),(-19,-16,-2),(-14,-16,-3),(-19,-16,-3),(-20,-16,-3),(-15,-16,-4),(-26,-16,-38),(-26,-16,-41),(-26,-16,-42),(-27,-16,-44)]
[(-11,-17,8),(-10,-17,2),(-13,-17,1),(-10,-17,-1),(-19,-17,-1),(-15,-17,-2),(-15,-17,-3),(-22,-17,-37)]
[(-9,-18,7),(-13,-18,3),(-7,-18,-1),(-13,-18,-1),(-16,-18,-2),(-17,-18,-2),(-18,-18,-3),(-19,-18,-34),(-22,-18,-38),(-23,-18,-40),(-24,-18,-40),(-25,-18,-40),(-21,-18,-44)]
[(-12,-19,7),(-18,-19,-35),(-21,-19,-37),(-20,-19,-51),(-21,-19,-52)]
[(-7,-20,3)]
"""
r = """
[(-34,-2,-13),(-33,13,-103)]
[(-25,9,-74),(-33,6,-68)]
[(-8,-18,1),(-20,-19,-38)]
[(-33,7,-103),(-34,6,-115),(-24,-8,-18),(-28,-7,-24),(-23,-12,-16),(-26,-9,-23),(-31,-13,-43)]
"""

Meanwhile, the new friend offered some T-ship conversions at c/4 orthogonal...

Code: Select all

x = 212, y = 103, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
198bo$196bo3bo$196bobobo$195b2o3b2o$118bo79bo$116bo3bo$116bobobo$115b
2o3b2o70bo5b3o5bo$118bo71bo3bo2bobobo2bo3bo$190bobobo2bo3bo2bobobo$
189b2o3b2ob5ob2o3b2o$112bo5b3o5bo65bo6bo6bo$110bo3bo2bobobo2bo3bo69bob
o$110bobobo2bo3bo2bobobo70bo$109b2o3b2ob5ob2o3b2o69bo$112bo6bo6bo71bob
o$118bobo$119bo72bo3b2obob2o3bo$119bo72b2ob2o2bo2b2ob2o$118bobo72b2ob
7ob2o$192b3o2b5o2b3o$112bo3b2obob2o3bo65bobo9bobo$112b2ob2o2bo2b2ob2o
65bo2b3obob3o2bo$113b2ob7ob2o65bo2b2ob5ob2o2bo$112b3o2b5o2b3o67b5ob5o$
112bobo9bobo69b7o$112bo2b3obob3o2bo62b3o5bo3bo5b3o$111bo2b2ob5ob2o2bo
60b5o6bo6b5o$114b5ob5o62bo2bo2bo3bo3bo3bo2bo2bo$116b7o64bobobobobo3bo
3bobobobobo$109b3o5bo3bo5b3o65b3o3b3o$108b5o6bo6b5o65b7o$107bo2bo2bo3b
o3bo3bo2bo2bo66b3o$107bobobobobo3bo3bobobobobo63b3obob3o$115b3o3b3o$9b
2o58b2o45b7o26b2o36bobo19b2o$10bo59bo47b3o29bo35b2ob2o19bo$9b2o58b2o
37bobo4b3obob3o25b2o34b3ob3o3bo13b2o$9bo59bo37b2ob2o37bo44b3o12bo$106b
3ob3o81b3o$11bo58bo$9bo3bo54bo3bo42bo$9bobobo54bobobo41b3o$8b2o3b2o52b
2o3b2o40b3o$11bo58bo3$5bo5b3o5bo44bo5b3o5bo$3bo3bo2bobobo2bo3bo40bo3bo
2bobobo2bo3bo$3bobobo2bo3bo2bobobo40bobobo2bo3bo2bobobo$2b2o3b2ob5ob2o
3b2o38b2o3b2ob5ob2o3b2o$5bo6bo6bo44bo6bo6bo$11bobo56bobo$12bo58bo$12bo
58bo$11bobo56bobo2$5bo3b2obob2o3bo44bo3b2obob2o3bo$5b2ob2o2bo2b2ob2o
44b2ob2o2bo2b2ob2o$6b2ob7ob2o46b2ob7ob2o$5b3o2b5o2b3o44b3o2b5o2b3o$5bo
bo9bobo44bobo9bobo$5bo2b3obob3o2bo44bo2b3obob3o2bo$4bo2b2ob5ob2o2bo42b
o2b2ob5ob2o2bo$7b5ob5o48b5ob5o$9b7o52b7o$2b3o5bo3bo5b3o38b3o5bo3bo5b3o
$b5o6bo6b5o36b5o6bo6b5o$o2bo2bo3bo3bo3bo2bo2bo34bo2bo2bo3bo3bo3bo2bo2b
o$obobobobo3bo3bobobobobo34bobobobobo3bo3bobobobobo$8b3o3b3o50b3o3b3o$
9b7o52b7o$11b3o56b3o$8b3obob3o50b3obob3o4$82bo$80bo3bo$80bobobo$79b2o
3b2o$82bo15$8bo$7b3o$7b3o2$67bo$66b3o$66b3o!

Code: Select all

x = 82, y = 74, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
25bo39bo$23bo3bo35bo3bo$23bobobo35bobobo$22b2o3b2o33b2o3b2o$25bo39bo3$
19bo5b3o5bo25bo5b3o5bo$17bo3bo2bobobo2bo3bo21bo3bo2bobobo2bo3bo$17bobo
bo2bo3bo2bobobo21bobobo2bo3bo2bobobo$16b2o3b2ob5ob2o3b2o19b2o3b2ob5ob
2o3b2o$19bo6bo6bo25bo6bo6bo$25bobo37bobo$26bo39bo$26bo39bo$25bobo37bob
o2$19bo3b2obob2o3bo25bo3b2obob2o3bo$19b2ob2o2bo2b2ob2o25b2ob2o2bo2b2ob
2o$20b2ob7ob2o27b2ob7ob2o$19b3o2b5o2b3o25b3o2b5o2b3o$19bobo9bobo25bobo
9bobo$19bo2b3obob3o2bo25bo2b3obob3o2bo$18bo2b2ob5ob2o2bo23bo2b2ob5ob2o
2bo$21b5ob5o29b5ob5o$23b7o33b7o$16b3o5bo3bo5b3o19b3o5bo3bo5b3o$15b5o6b
o6b5o17b5o6bo6b5o$14bo2bo2bo3bo3bo3bo2bo2bo15bo2bo2bo3bo3bo3bo2bo2bo$
14bobobobobo3bo3bobobobobo15bobobobobo3bo3bobobobobo$22b3o3b3o31b3o3b
3o$23b7o33b7o$25b3o37b3o$22b3obob3o31b3obob3o4$22bo39bo$21b3o37b3o$21b
3o37b3o12b5o$75bo2bo2bo$75bo2bo2bo$76bo3bo29$2bobo$b2ob2o$3ob3o!

... as well as a double-barrelled p480 rake resembling the p140 and p844 ones.

Code: Select all

#C [[ TRACK 0 -1/4 ]]
x = 50, y = 156, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
28bobo$27b2ob2o$26b3ob3o3$30bo$29b3o$22bobo3bobobo3bobo$21b2ob2o2bobob
o2b2ob2o$20b3ob3obobobob3ob3o2$29bobo$30bo$30bo$30bo$28b5o$23b2ob3o3b
3ob2o$23bo6bo6bo$23bo6bo6bo$23bo3bo5bo3bo$22b2obo4bo4bob2o$23b2o2b7o2b
2o$24bo2bo5bo2bo$28bobobo$21bo6bobobo6bo$19bo3bo3b7o3bo3bo$19bobobo5bo
bo5bobobo$18b2o3b2o4bobo4b2o3b2o$21bo4b2obobob2o4bo$26bo2b3o2bo$26bo7b
o$30bo$27b2obob2o$27bo5bo2$18bobo$17b2ob2o3b3o$16b3ob3o3bo2$26bo14b5o$
40bo2bo2bo$40bo2bo2bo$41bo3bo2$18b3o$17b5o$16bo2bo2bo22b3o$16bobobobo
21b5o$43bo2bo2bo$43bobobobo38$7bo$5bo3bo$5bobobo$4b2o3b2o$7bo7$5b3o$4b
5o$3bo2bo2bo$3bobobobo9$3bo$bo3bo$bobobo$2o3b2o$3bo19$35bobo$34b2ob2o$
33b3ob3o7$25b3o$25bo2bo$24bo3bo$22b2o3b2o$22b2ob2o$22b2obo$17b3o4bo$
16b5o$15bo2bo2bo$15bobobobo$29b2o$29bo$29bobo$30b2o!

Oh my, it is already the 2200th post milestone? What a ride! But then, I am going to head for a grand new journey while Glimmering Garden is accelerating towards the Information Era.

[LaundryPizza03]

I took a look at the (48,12)c/1500, and discovered that it is part of an infinite family with period (48,12)c/(1500+480n), n≥0. In terms of the formula for cloverleaf-push ships, we have N_t = 4 and N_v = 7+2n. They can be constructed by increasing the distance between the two halves by 120N cells. To the best of my knowledge, this is the only known infinite family of cloverleaf-push ships.

Code: Select all

x = 658, y = 782, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7cSuper
Q7.Q113.Q.Q.Q117.Q117.Q.Q.Q115.Q.Q.Q119.Q2$Q.Q5.Q111.Q7.Q113.Q.Q115.Q
7.Q111.Q7.Q115.Q.Q2$Q3.Q3.Q3.Q.Q.Q.Q.Q99.Q5.Q.Q115.Q123.Q119.Q113.Q3.
Q2$Q5.Q.Q111.Q3.Q3.Q115.Q121.Q117.Q.Q113.Q5.Q2$Q7.Q3.Q.Q.Q.Q.Q99.Q.Q
5.Q115.Q119.Q123.Q111.Q.Q.Q.Q.Q2$Q7.Q111.Q7.Q115.Q117.Q117.Q7.Q117.Q
2$Q7.Q113.Q.Q.Q115.Q.Q.Q113.Q.Q.Q.Q.Q113.Q.Q.Q119.Q18$125.I119.I119.I
119.I119.I$124.3I117.3I117.3I117.3I117.3I$123.I2.I116.I2.I116.I2.I
116.I2.I116.I2.I$122.3I2.I114.3I2.I114.3I2.I114.3I2.I114.3I2.I$123.I
2.3I114.I2.3I114.I2.3I114.I2.3I114.I2.3I2$120.2I118.2I118.2I118.2I
118.2I$120.2I118.2I118.2I118.2I118.2I11$157.3I117.3I117.3I117.3I117.
3I$156.I2.I116.I2.I116.I2.I116.I2.I116.I2.I$156.I3.I115.I3.I115.I3.I
115.I3.I115.I3.I$156.I3.I115.I3.I115.I3.I115.I3.I115.I3.I$157.I2.I
116.I2.I116.I2.I116.I2.I116.I2.I$154.2I118.2I118.2I118.2I118.2I$154.
2I118.2I118.2I118.2I118.2I$153.I119.I119.I119.I119.I103$120.I119.I
119.I119.I119.I$119.3I117.3I117.3I117.3I117.3I$119.3I117.3I117.3I117.
3I117.3I75$163.3I$162.I3.I$166.I$165.2I$162.3I45$171.2I.2I$166.2I2.I
4.I$111.2I52.2I3.2I2.I.I$110.2I54.I2.I6.2I$111.I55.3I$112.I2.2I51.3I.
I$113.3I53.3I$114.I54.I$129.2I38.2I$123.2I5.2I$123.I3.I2.I$123.I2.4I$
124.3I$128.I$127.3I$126.2I$127.I.I$128.I54$283.3I$282.I3.I$286.I$285.
2I$282.3I45$291.2I.2I$286.2I2.I4.I$231.2I52.2I3.2I2.I.I$230.2I54.I2.I
6.2I$231.I55.3I$232.I2.2I51.3I.I$233.3I53.3I$234.I54.I$249.2I38.2I$
243.2I5.2I$243.I3.I2.I$243.I2.4I$244.3I$248.I$247.3I$246.2I$247.I.I$
248.I54$403.3I$402.I3.I$406.I$405.2I$402.3I45$411.2I.2I$406.2I2.I4.I$
351.2I52.2I3.2I2.I.I$350.2I54.I2.I6.2I$351.I55.3I$352.I2.2I51.3I.I$
353.3I53.3I$354.I54.I$369.2I38.2I$363.2I5.2I$363.I3.I2.I$363.I2.4I$
364.3I$368.I$367.3I$366.2I$367.I.I$368.I54$523.3I$522.I3.I$526.I$525.
2I$522.3I45$531.2I.2I$526.2I2.I4.I$471.2I52.2I3.2I2.I.I$470.2I54.I2.I
6.2I$471.I55.3I$472.I2.2I51.3I.I$473.3I53.3I$474.I54.I$489.2I38.2I$
483.2I5.2I$483.I3.I2.I$483.I2.4I$484.3I$488.I$487.3I$486.2I$487.I.I$
488.I54$643.3I$642.I3.I$646.I$645.2I$642.3I45$651.2I.2I$646.2I2.I4.I$
591.2I52.2I3.2I2.I.I$590.2I54.I2.I6.2I$591.I55.3I$592.I2.2I51.3I.I$
593.3I53.3I$594.I54.I$609.2I38.2I$603.2I5.2I$603.I3.I2.I$603.I2.4I$
604.3I$608.I$607.3I$606.2I$607.I.I$608.I!

The minpop is 121 for n = 0, and 119 for n ≥ 1.