The key question is: what is the smallest set of tracks necessary to reconstruct a (hypothetical) helix? Most self-supporting spaceships start with a small set of tracks that they then use to build new ones.
1. In order to build new tracks, we have two options: either we use a kickback (so we need to be able to generate a forward glider) or we use an OTT (actually we do have one - the clean boat laying pair produces the boats in the right orientation). Since we have good sideways glider capabilities I suppose they can be fairly dirty.
2. We also need to be able to generate tracks in the right phase and location. Consider the following three vectors in (x,y,t) coordinates: Here the coordinate system is such that the gliders as going (1,1)c/4 and the climber is (-28, -3)c/84.
(1, 1, 4) - this is the displacement a glider undergoes after 4 steps. We can shift a glider stream by this displacement just by waiting.
(49, 24, 0) - this is the displacement between the gliders in a glider stream. We can shift a glider stream by this displacement by doing nothing.
(13, 4, 3) - this is the displacement the climber shifts a glider. We can shift a glider stream by this displacement by sending a crawler on it.
The determinant is 539. This means there are 539 different track positions modulo these transformations. Assign each displacement an index mod 539 by mapping (x, y, t) to 13x-49y+9t.
Fortunately, if we use the ott method we have extra degrees of freedom. Clearly we don't care about the timing of the boats, so we can image we are also allowed a displacement of (0, 0, 1) in addition. Its index is 9 which is relatively prime to 539, so we can get any boat position.
We also have extra degrees of freedom in the boat-hitting gliders. Note that if we shift the boat-hitting glider by (-1, 1, -4) the output glider is shifted by (0, 0, 0), so we essentially get to adjoin (-1, 1, -4) to our displacements list. Its index is 441 which is unfortunately NOT relatively prime to 539 and they have a fairly large common factor of 49.
EDIT: I realized there actually is a way to make arbitrary tracks. We can make any p2-slow salvo that doesn't take up too much space and we can use the boat layer to make targets.
EDIT2: 8 tracks suffices to be able to have a choice between fire and rephase without firing. Specifically, observe that in the boat-layer+dirty rake combination, we can adjust the boat layer without changing the track so that the gliders cleanly destroy the boats, leaving an output stream of blocks. So 4 tracks suffices to have an option between laying blocks vs not laying blocks. Then we just run a rake on the remaining 4 tracks. This can probably be reduced to 6 tracks.
As a corollary, 9 tracks suffices to give a choice between leave a boat and rephase, so 17 tracks suffice to build everything we need. This can surely be improved.
EDIT3: An explicit demonstration of junk producer, rephaser, and rake running on 7 tracks, together with the start of an unknown slow salvo synthesis
Code: Select all
x = 1256, y = 759, rule = B3/S23
35b2o$34b2o$36bo6$27bo$26b2o$26bobo14$84b2o$83b2o$85bo2$32b2o$31b2o$
33bo2$76bo$75b2o$75bobo14$133b2o$132b2o$134bo2$81b2o$80b2o$82bo2$125bo
$124b2o$124bobo14$182b2o$181b2o$183bo2$130b2o$129b2o$131bo2$174bo$36b
2o135b2o$35b2o136bobo$37bo6$28bo$27b2o$27bobo5$231b2o$230b2o$232bo$
242bo$179b2o60bobo$178b2o57b2obo4bo$180bo59bo2b3o$241bo3bo$223bo12bo6b
o$85b2o135b2o12bo6bo$84b2o136bobo11b2o$86bo151b2o2$207bo$206b3o$205bo
3bo$204b2o2bo$77bo126b2o$76b2o127bobo34bo5b3o$76bobo127bobo33b2o3b5o$
206bobo32bo4bo2b2obo$196bo11bo12bo24b5ob2o$196bobo5bobobo13b2o22bob5o$
196b2o6bobob2o32b2ob3o3bo$208bo34bo2b2o$203bobo28b2o7b3o$202bo2bo28b2o
$201bo37b2o2b2o$201bo26b2o$201b2ob2o5b2o14b2o64b3o$205bo5b2o16bo63bo$
203b3o88bo$241b2o$134b2o72b2o31b3o$133b2o72bobo29b2obobo$135bo72bo29bo
bo2b3o$242b3o$239bo2b2o41b2o$233b2o50bobo$147bo85bo51bo$147bobo$126bo
20b2o$125b2o$125bobo10$342b3o$342bo$343bo2$98bo84b2o105b3o$98bobo81b2o
106bo$98b2o84bo106bo2$334b2o$9b2o323bobo$8b2o324bo$10bo$175bo$174b2o$
174bobo2$9bo$8b2o$8bobo4$49bo$49bobo$49b2o340b3o$391bo$392bo2$232b2o
105b3o$231b2o106bo$233bo106bo2$383b2o$58b2o323bobo$57b2o324bo$59bo$
224bo$223b2o$223bobo2$o57bo$obo54b2o$2o55bobo6$440b3o$440bo$441bo2$
281b2o105b3o$280b2o106bo$282bo106bo2$432b2o$107b2o323bobo$106b2o324bo$
41b2o65bo$41b2o230bo$272b2o$69b2o201bobo26bo$69b2o230bo3bo$107bo192bo
4b2o4b4o$97b2o7b2o192b2o3bo5bo2b2o$97b2o7bobo195b2o6bo2b2o$305bo6bo2bo
$125b2o176bob2o6b2o$125b2o175bo2b2o$303bo2bo$153b2o150b2o$57b2o94b2o
142b2ob2o2b2o183b3o$57bobo236bobo190bo$57bo123b2o113bo193bo$181b2o111b
3o2bo3bo$292bo5bobo136b3o$209b2o81bo2bobo3b2o134bo$209b2o81bo2bobob3o
136bo$300bo$237b2o57bobo2bo179b2o$156b2o79b2o58b2o2bo6b2o8bo24b3o135bo
bo$155b2o142b3o6b2o7b2o24bo137bo$157bo107b2o32b2o4bo11b3o24bo$265b2o
30b2o2bo3bo13bo2bobo$298b3o4bo13b2obobo$293b2o4b3o20b3o$293b2o$156bo$
155b2o178b2o$155bobo177bobo$246bo88bo$246bobo$246b2o3$106b2o430b3o$
106bobo429bo$106bo432bo2$486b3o$486bo$487bo2$530b2o$205b2o185b3o135bob
o$204b2o186bo137bo$206bo186bo2$197bo$197bobo$197b2o$205bo$204b2o178b2o
$204bobo177bobo$225bo158bo$225bo3bo$224bo2bob2o4b4o$223bo5b2o4bo2b2o$
224b2o2b3o5bo2b2o$155b2o79bo2bo347b3o$155bobo71bobo5b2o348bo$155bo69b
3o360bo$218b2o4bo5b2o$218b2o5bo2bo306b3o$223b2o3bo306bo$225b2o4bo304bo
2$179b2o45bo4bo347b2o$179bobo45b4o210b3o135bobo$180bo260bo137bo$207b2o
29b2o8b2o2b2o188bo$207bobo18b3o7b2o8b2o2b3o$208bo19b3o4bo11b3o2b2o$
229bo5bo12b2o2bo14b3o$235bo31bo$223b2o43bo$223b2o6b3o199b2o$433bobo$
433bo2$267b2o$267bobo$267bo$636b3o$636bo$637bo$648bo$584b3o60b2o$584bo
60b5obo$585bo59b3o3b2o$648b2obo$628b2o18bobo$490b3o135bobo10b2o$490bo
137bo13b3o$491bo151b2o2$612b3o$316b3o293b3o$316bo293b2o3bo$317bo294bo$
482b2o171bo$482bobo125b2obo34b2o3bo3bo$482bo128b2obo32b3o3bo$614b2o32b
o3bo5b2o$316b2o285bo7bo2b2o12bo22b2o6bo$316bobo282b2o11bo13bo30bo$316b
o285b2o10b2o32b4o3bob2o$610b2ob3o37bo$610bo29b2o7b4o$608bobo29b2o9bo$
607b2o$606b2o25b3o64bo$607b2ob2o5b2o14bo65b2o$608bo3bo4b2o15bo64bobo$
610b2o$610bo36bobo$539b3o72b2o$539bo73bobo29b2o4bo$540bo73bo31bo4bo$
645bobo3bo$648bobo40b2o$365b3o271b2o49b2o$365bo188bo84b2o51bo$366bo
185b2o$531b2o20b2o$531bobo$531bo2$365b2o$365bobo$365bo5$749bo$748b2o$
748bobo2$697bo$505bo82b3o105b2o$503b2o83bo107bobo$504b2o83bo2$740b2o$
414b3o322b2o$414bo326bo$415bo$580b2o$580bobo$580bo2$414b2o$414bobo$
414bo4$456bo$454b2o342bo$455b2o340b2o$797bobo2$746bo$637b3o105b2o$637b
o107bobo$638bo2$789b2o$463b3o322b2o$463bo326bo$464bo$629b2o$629bobo$
629bo2$407bo55b2o$405b2o56bobo$406b2o55bo5$847bo$846b2o$846bobo2$795bo
$686b3o105b2o$391b2o293bo107bobo$391b2o294bo2$419b2o417b2o$419b2o91b3o
322b2o$512bo326bo$447b2o64bo$447b2o229b2o$678bobo$475b2o201bo$475b2o
229b2o3b2o5b2o$512b2o192bo3b3o4b5o$503b2o7bobo191b2o9bo4bo$503b2o7bo
197b3o4b3o2bo$718bo2b2o$531b2o186b2o$531b2o175b2o3bo$710bo2bo$559b2o
151bo183bo$559b2o142b3o4b3o182b2o$702bob2o189bobo$587b2o113bo$587b2o
112b3obo138bo$699b2o2b6o134b2o$615b2o80b3o3bo4bo134bobo$615b2o87b2o2bo
$702b3o$643b2o59b4o42bo136b2o$561b3o79b2o58b2o2b2o5b2o7b2o24b2o135b2o$
561bo145bo6b2o33bobo136bo$562bo108b2o50bobo$671b2o30bo3bo2b3o14bo$703b
o21b2obob2o$699b2o3bo2bo19b2obo$699b2o5bo22bo$561b2o$561bobo177b2o$
561bo178b2o$653bo88bo$651b2o$652b2o2$945bo$944b2o$944bobo2$893bo$892b
2o$892bobo3$799bo136b2o$610b3o185b2o135b2o$610bo187bobo136bo$611bo2$
604bo$602b2o$603b2o$610b2o$610bobo177b2o$610bo178b2o$631bo4bo154bo$
629b2ob3o2bo3b2o$630b2obo7bobo$632bo10bo$631b2o2bo5b3o350bo$631b4o358b
2o$631bobobo357bobo$623b2o9b2o$623b2o6b2ob2o306bo$627b3o3bo307b2o$627b
3obobo307bobo$629bo2bo2b2o$584b2o47b2o2bo$584bobo46b2o2bo210bo136b2o$
585bo70bo190b2o135b2o$612b2o17bo5bo5b2o9b2obo189bobo136bo$612bobo17bo
3bo6b2o8bo4bo$613bo43bo$633b3o3b3o11b4o17bo$673b2o$628b2o7bo35bobo$
628b2o7bo$637bo201b2o$838b2o$840bo2$673b2o$672b2o$674bo368bo$1042b2o$
1042bobo2$991bo$990b2o$990bobo3$897bo136b2o$896b2o135b2o$896bobo136bo
3$723bo$722b2o$722bobo2$888b2o$887b2o$889bo2$722b2o$721b2o$723bo368bo$
1091b2o$1091bobo2$1040bo60b2o$1039b2o57bobobobobo$1039bobo55b2obo6bo$
1098bobo2bobo$1096bo5bo$946bo136b2o11b2obo2bo$945b2o135b2o14b2obo$945b
obo136bo3$772bo293b3o$771b2o293b3o$771bobo291bo3bo$1063b3o$937b2o126bo
bo34bo$936b2o129bo34bo4b4o$938bo118bo10bo13bobo16bobo3b2ob2o$1056bo11b
o13b2o19bo3bo3b2o$771b2o283b3o5bo17b2o17b3o6b2o$770b2o292b5o32b2o2b2o$
772bo295bo31b2o4b3o$1064b3o27b2o4b2o3bob2o$1063bo30b2o3b2ob3o$1062bo5b
o31b4o$1062bo5bo20bo11b2o$1062bobobo5bo15b2o64b2o$1063b3o5b2o15bobo63b
obo$1154bo$1099b2o$995bo72b2o$994b2o71bobo27bo3b4o$994bobo71bo28bo5b2o
$1098bob2ob2o$1100bobo$821bo186bo84b2o50b3o$820b2o185bo85bo2bo48bo$
820bobo184b3o84bobo49bo2$986b2o$985b2o$987bo2$820b2o$819b2o$821bo5$
1203b2o$1203bobo$1203bo$959bo$958bo85bo106b2o$958b3o82b2o106bobo$1043b
obo105bo3$870bo323b3o$869b2o323bo$869bobo323bo2$1035b2o$1034b2o$1036bo
2$869b2o$868b2o$870bo2$910bo$909bo$909b3o$1252b2o$1252bobo$1252bo2$
1093bo106b2o$1092b2o106bobo$1092bobo105bo3$919bo323b3o$918b2o323bo$
918bobo323bo2$1084b2o$1083b2o$861bo223bo$860bo$860b3o55b2o$917b2o$919b
o9$1142bo106b2o$845b2o294b2o106bobo$845b2o294bobo105bo2$873b2o$873b2o
93bo$967b2o$901b2o64bobo$901b2o$1133b2o$929b2o201b2o$929b2o203bo26b4o$
1160bo3b2o6b2o$957b2o8b2o191b2o3bo6bobo$957b2o7b2o195b2o7bob2o$968bo
204b2o$985b2o175bob2o7bo$985b2o175bo3bo$1163bo$1013b2o142bo6bobo$1013b
2o140bo2bobo4bo$1158b3o$1041b2o113b2o4bo$1041b2o113bob2o2bo$1156bo5bo$
1069b2o81b6o3bo$1069b2o81bo4bo3bo$1153bo3bo$1097b2o55bo3b3o18b2o2b2o$
1017bo79b2o57b2o3bo6b2o8b3o23b2o$1016b2o139b2o9b2o15bo18bobo$1016bobo
106b2o30b3obo3bo10bobo25bo$1125b2o30b2obo4bo11b4obo2bo$1159b3o3bo12b3o
b3o$1153b2o$1153b2o6b3o2$1016b2o$1015b2o90bo87b3o$1017bo88bo88bo$1106b
3o87bo13$1066bo186b2o$1065b2o186bobo$1065bobo185bo$1058bo$1057bo$1057b
3o3$1065b2o$1064b2o178b3o$1066bo177bo$1086bo3bo154bo$1085b2o3bo5bo$
1084bo2bo2b2o4bo2b2o$1084bobo8bo5bo$1085bo4bo5bobo2b2o$1088bobo5bo3b2o
$1086b3o9b3o$1086b3o$1079b2o4bo$1079b2o3bobob2o$1084bo2bo$1085bo2bo$
1085b6o$1040b2o49bo$1040bobo$1041bo46bo$1068b2o21bo7b2o9b3obo$1068bobo
19bo8b2o8bo3bob2o$1069bo19bobo4bo12b2o4bo12b2o$1090bo5bo12bob4o13bobo$
1090bo5bo31bo$1084b2o$1084b2o6b3o4$1127b3o$1127bo$1128bo14$1177b2o$
1177bobo$1177bo6$1176b3o$1176bo$1177bo14$1226b2o$1226bobo$1226bo6$
1225b3o$1225bo$1226bo!
#C [[ RLE tracks 27b2o$27bobo$27bo7$18b3o$18bo$19bo17$24b2o$24bobo$24bo51$28b2o$28bobo$28bo7$19b3o$19bo$20bo66$b2o$bobo$bo6$3o$o$bo! 1225 590 ]]
#C [[ PASTET EVERY 84 PASTEDELTA 28 3 PASTE tracks ]]