Let’s have a spaceship for every possible speed in a nonrelativistic INT rule!

[pzq_alex]

(By “speed”, I mean simplified speed. The problem for unsimplified speeds would much much more harder. )

It seems that this question is a simple and basic one, and it’s currently unsolved.

The speed limit

Apparently, the speed limit for nonrelativistic (with none of B012a) is c/2 orthogonal and c/3 diagonal. More generally, the maximum speed a slope x/y spaceship may travel at is (x,y)c/(2x+y), assuming x>=y. For example, the maximum speed of a knightship is (2,1)c/5. (Note that without S4w5a, the speed limit is (x,y)c/(2x+2y). ) One nice way of visualizing this is by taking the convex hull of all speeds, which is a octagon, and is generated by this:

Code: Select all

x = 3, y = 3, rule = B3ai4a/S3ai4a5ai6ac7c8
3o$3o$3o!

Speed Ortho-Demonoids

The way to prove that those speeds can be achieved is certainly via the universal constructor / slow salvo / wickstretcher and fuse lighter method (does this have a name?) as in speed [Orthog|Dem]onoids. To achieve slopes other than 0 and 1 though, we will need two separate stages, involving c/2o and c/3d wickstretchers taking turns. (Turns out that this design was also considered by Macbi. )The ship would progress like this:

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x = 109, y = 41, rule = LifeHistory
99.D.D$100.D$99.D.D$98.D$97.D$96.D$95.D$94.D$93.D$90.D.D$68.24D$65.D.
D22.D$66.D35.3D3.D$65.D.D36.D3.D$64.D12.3D14.3D3.D.3D.3D$63.D15.D14.D
4.D4.D.D.D$62.D7.2D3.D.3D.3D10.3D.D3.3D.3D$61.D8.D3.D2.D3.D.D$60.D9.2D
.D3.3D.3D$59.D$56.D.D$34.24D$31.D.D22.D$32.D35.3D3.D$31.D.D36.D3.D$30.
D12.3D14.3D3.D.3D.3D$29.D15.D14.D4.D4.D.D.D$28.D7.2D3.D.3D.3D10.3D.D3.
3D.3D$27.D8.D3.D2.D3.D.D$26.D9.2D.D3.3D.3D$25.D$22.D.D$24D$22.D$34.3D
3.D$36.D3.D$9.3D14.3D3.D.3D.3D$11.D14.D4.D4.D.D.D$2.2D3.D.3D.3D10.3D.
D3.3D.3D$2.D3.D2.D3.D.D$2.2D.D3.3D.3D!

The reason that these two stages have to be c/2o and c/3d is that for example, if you take both stages to be c/2o (and orthogonal to each other), you would arrive at a speed limit of (x,y)/(2x+2y).

Still, this method cannot reach the speeds on the boundary, i.e. those of form (x,y)c/(2x+y).

The Pesky Boundary

The key point here is to consider these knight-sloped lines:

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x = 8, y = 9, rule = LifeHistory
BDF$BDF$.BDF$.BDF$2.BDF$2.BDF$3.BDF$3.BDF!

If at some tick, the leading edge of the spaceship falls behind one of those lines, then it cannot reach the next line next generation, and this is how we derive the speed limit. Thus, to keep itself going at the speed limit, it will have to touch the line every generation.

I’m imagining a design like this: a line of replicators aligned along the knight-sloped lines (the blocks represent replicators):

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x = 15, y = 10, rule = LifeHistory
13.F$12.3F$.2A7.2F.F$.2A5.2F$2.2B2.2F$2.2B$3.2C$3.2C$4.2D$4.2D!

The replicators would travel ENE-ward. Any speed on the speed limit boundary now amounts to any speed in the 1D CA the replicators emulate. Note that the 1D CA will have to be either multi-state or higher-range.

[toroidalet]

I have found rules where a block is a splitter for a c/2 spaceship and there is a c/3d wickstretcher, as well as rules where a block is a splitter for both a c/2 and c/3d ship (we need some way to turn the c/3 ships into c/2s), but I'm having trouble finding a rule with all 3:

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x = 27, y = 10, rule = B2in3acijq4ckqw5-ceiq6en7e8/S1e2in3-cek4acnqr5-jkn6-c7c
2bo$obo16b2o$2bo15b4o$5b2o11b3o$5b2o12bobo$22bo$23bobo$24b3o$23b4o$24b
2o!

(interestingly, it can stretch a barberpole wick as well)

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x = 39, y = 13, rule = B2n3-cnqy4cikwz5ai6n7e/S1e2cin3-eky4aenw5eijn6n
26b2o$2bo7bo15b2o$obo5bobo$2bo7bo$13b2o17b2o$13b2o16b2o$30b2o$30bo2bo
2$37b2o$36b2o$35b2o$35bo2bo!

Because there are a lot of rules, I am pretty sure that one exists, perhaps with a different c/2 ship or position. Does anybody want to look for such a rule?
I might try looking for a c/3d-and-c/2o replicator later.