Thread for basic questions

[biggiemac]

The issue with the caterpillar isn't that things have to be a multiple of 4. The waterbear uses *WSS for a p158 helix, which last I checked is 2*79. Successive spaceships in the helix are just in different phases, like this.

Code: Select all

x = 37, y = 71, rule = B3/S23
4b3o$6bo$5bo2$16b3o$15bo2bo$18bo$18bo$15bobo3$24b3o$24bo2bo$24bo$18bo
5bo$3bo13b3o5bobo$2b3o12bob2o$b2obo13b3o$b3o14b2o$2b2o7$b3o$o2bo$3bo$
3bo$obo3$2b3o$2bo2bo$2bo$2bo$3bobo20b3o$26bo2bo$26bo$26bo$27bobo3$34b
3o$33bo2bo$36bo$28bo7bo$13bo13b3o3bobo$12b3o11b2obo$12bob2o10b3o$13b3o
11b2o$13b2o7$11b3o$11bo2bo$11bo$11bo$12bobo3$12b3o$11bo2bo$14bo$14bo$
11bobo!
#C [[ TRACK 5/79 -23/79 ]]

The issue is one of spacing, as well as one of helix technology. First spacing, because it can set a hard lower bound. The spacing between the ships has to be such that after one full period, the front is advanced by the ship's speed. Let's work out what that means for the caterpillar. At 17c/45, a p45 helix would need to move forward by 17 cells in 45 generations, while a *WSS moves 22.5 cells in that time. This means there is only 5.5 cells between ships; 5 cells on some phases and 6 on others.

Code: Select all

x = 5, y = 11, rule = B3/S23
2bo$b3o$2obo$3o$b2o2$b3o$bo2bo$bo$bo$2bobo!

You can see why that won't work.

With p90, we get 11 cells. Now we get workable spacing, even for HWSS.

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x = 20, y = 18, rule = B3/S23
b3o11b3o$o2bo10bo2bo$3bo13bo$3bo9bo3bo$obo10bo3bo$17bo$14bobo5$b3o11b
3o$bo2bo10bo2bo$bo13bo$bo13bo3bo$2bobo10bo3bo$15bo$16bobo!

So p90 is the absolute minimum period for a caterpillar running on burning spaceships.

But the current way helices are built is by having a glider reaction burn them, left to right and then right to left, repeating. If we use that technology, there needs to be room for a signal to pass between the consecutive ships to get back to the other side. That limits us further.

Lastly, there are only a handful of known glider reactions that help build a real working helix. codeholic made a helix searcher script to assemble helices for a given velocity using all the collisions known.

The fastest a signal can propagate upwards is 11c/24, and that reaction spits the glider out in the same direction it came in.

Code: Select all

x = 19, y = 9, rule = B3/S23
10bo$9b3o$2b3o3b2obo4b2o$bo2bo3b3o5bobo$4bo4b2o5bo$o3bo$o3bo$4bo$bobo!
#C [[ TRACK 0 -11/24 STOP 40 ]]

Reactions that turn the glider around are slower. The p270 helix for the caterpillar uses the 11c/24 straightaway reaction a bunch in the middle to compensate for the slow turns. The best found with all the present-day collisions is p225. If there was a way to burn the helix not using gliders, it could reach the minimum of p90, but we really have no idea how to search for it.


Edit: If I recall correctly, "oblique" ran a search specific to the waterbear trying to find a non-glider helix to support the reaction. Despite very nearly succeeding, the reaction was too messy and couldn't work. If it hadn't been so messy, and a 2-ship salvo could sustain the front end, a construction cluster could have been built with only 4 total helix spaceships, cutting off 60% or so of the ship's size.

That demonstrates the other likely problem in building minimum-period helices. Whatever burns the front end cannot interfere with the next ship in line until the full period has passed.

[Rhombic]

What are the average densities used for Sparse Life soup? (aka early universe)

[dvgrn]

What are the average densities used for Sparse Life soup? (aka early universe)

Sparse Life density is more of a range than a specific number. Non-zero but arbitrarily low. So low that you don't get any qualitatively different behavior by lowering it further. Low enough that, even though in an infinite grid there will still be an infinite number of ON cells, no infinitely large cluster will be present anywhere. Lowering the density beyond that point basically just moves the individual clusters farther and farther apart, without really changing What Happens If You Wait Long Enough.

Anybody have a better link to a complete PDF of a Sparse Life paper, or are they all hiding behind previews?

[Goldtiger997]

Several times recently when I have been trying to find nice natural syntheses for objects, I have come across this component which I have been unable to synthesise.

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x = 4, y = 4, rule = B3/S23
2bo$b3o$o2bo$3o!

Does anyone know a synthesis?

[praosylen]

Has anyone ever done a systematic search to find all chaotic outer-totalistic rules?

[muzik]

Are there any fuses that burn a boat/barge slower than c/4?

[shouldsee]

what will be the long-run result of this soup?

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x = 150, y = 150, rule = B1237/S02346:T150,150
obobobo10bo2bo2bo2bo2bo2bobo2bobo2bo3b2ob3obobo3b5obob3ob2obobobo4b5o
5b2ob3obo3b5obobobobob2o18bobobo2bobobo$b2obobob4ob4ob2obo2bobob5ob5ob
obob2o2b4obobob2obo2bobo4b2o2bobobob4o4b5obo5b3o6b2obobob2ob2ob5ob9obo
bo2bobobo$bob2obobo2bobob3obo2bobobob2o6b2ob2obo3bo2b2o2bob2o2bobobob
6obobobobobo3b4o3bob2obobo4b4ob3obobobo3bob3o5bo2b2ob3obo2bobob3o$bobo
bobob2o6bob2obobob9ob5obobob5o2bobobo2b2obo2b2o2bobobobob2ob2o5b2obob
2obo2bobo2bobo2bob2obobobo3bob3ob3obo3b3ob3o4bobo$bobobobo2b8o4bobob2o
8bob2o2b3obo6bob3obo2bobob3obobobob2obobob5obobob4o2bob5ob2ob2o2bobobo
b3o4b2obo3bobobob3o2b2ob2o$bobo3bobo

[Rhombic]

I have seen quite a few of these naturally

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x = 4, y = 4, rule = B3/S23
o$3o$2b2o$b2o!

and I was wondering whether they are a source of natural toads (when their evolution is changed by a glider or else) because they seem rather common

[muzik]

That is "the other octomino", a predecessor of "the" octomino:

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x = 4, y = 4, rule = B3/S23
2b2o$2b2o$3o$bo!

[shouldsee]

Has anyone ever done a systematic search to find all chaotic outer-totalistic rules?

I guess this depends on the definition of chaos. This paper[1] made another attempt to characterise stability by introducing defect branching, whereas damage-based attempts have been made previously.

[1]:Baetens, J.M. & Gravner, J., 2014. Stability of cellular automata trajectories revisited: branching walks and Lyapunov profiles. Journal of Nonlinear Science, pp.1–39.

[Rhombic]

Is there any chance that any other spaceship (apart from the xq12) can exist in this rule? I don't mind whether this is naturally or zfind created:
https://catagolue.appspot.com/census/b2 ... s2a3-i4/C1

[Rocknlol]

Is there any chance that any other spaceship (apart from the xq12) can exist in this rule? I don't mind whether this is naturally or zfind created:
https://catagolue.appspot.com/census/b2 ... s2a3-i4/C1

I was able to find two:

Code: Select all

x = 23, y = 7, rule = B2-a3i4aijk/S2a3-i4
2bo2bo$o5bo4bo9bo$2bo7bobo7bobo$bobo6bo4bobo4bo$2bobobo6bo5bo$3bo8b3o
3b3o$5bo9bobo!

[muzik]

How can I find reactions akin to those usable in the waterbear and other engineered oblique spaceships? Of course not manually.

[biggiemac]

Fuse finder can get you some options. The original caterpillar started from a reburnable fuse sparky enough to be made into rakes. I imagine oblique fuses can be easily searched for, though for all fuses only a low fraction will be reburnable.

To get glider trails like the waterbear or (27,1)c/72 runs from, I guess just collide glider with object to see if you get glider and object back again in same orientation. Not sure what automated utilities work on that sort of search.

[Caenbe]

Is there a phoenix in which at least one cell dies of overpopulation?

[dvgrn]

Is there a phoenix in which at least one cell dies of overpopulation?

Probably not in B3/S23. It may be possible to produce an airtight proof by enumerating cases, though higher periods might be a problem I suppose.

A cell has to be surrounded by at least 4 ON cells to die of overpopulation. This forms at least a 5-cell polyplet, containing all cells that have to have been OFF at tick T-1, and have to be OFF again at tick T+1 -- so any cells that don't die of overpopulation, will have to die of underpopulation.

Seems to me the only case that's even vaguely likely is this one (and it doesn't look too promising):

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x = 3, y = 3, rule = LifeHistory
A.A$.C$A.A!
#C [[ THUMBNAIL ]]

If any two of the cells causing overpopulation are next to each other, then they will fail to die of underpopulation at T=1... which means they also will have to die of overpopulation. That means turning ON more cells at T=0. Pretty quick there are going to be not enough OFF cells left in the hypothetical phoenix, that can turn on at T+1 (and off again at T+2) to keep an active reaction going.

[BlinkerSpawn]

Is there a phoenix in which at least one cell dies of overpopulation?

Across all neighborhoods in which a live cell has 4 or more neighborhoods, for all configurations except one at least one neighbor for the cell in question has 2 neighbors, the exception being 4c.
(I will refer to the 3x3 pattern of this neighborhood as A in what follows)
But since the corner cells themselves cannot survive, A must either occur in isolation (i.e. the 5x5 centered on A includes only A) or in a dense field such that all four corner cells also die of overpopulation.
But if A occurs in isolation, then immediately outside A there must be sparks such that the tub that forms under A's uninhibited evolution dies immediately.
Some spark patterns actually do this quite well:

Code: Select all

x = 9, y = 9, rule = B3/S23
4bo$2bobobo$bo5bo$3bobo$2o2bo2b2o$3bobo$bo5bo$2bobobo$4bo!

But since a tub is stable, each cell in it has 2 or 3 (exclusively 2, in this case) live neighbors, so each cell in the tub must die of overpopulation.
This problem operates under the opposite of induction: instead of the solution to a higher-level case following for a proof for a lower-level case, here we have the solution for a single death relying on very specific solutions for multiple deaths, which (by similar strategies to those applied above) can be shown to rely on proofs for larger death patterns.
Since the requisite pattern must be an oscillator whose necessary period tends to infinity upon repetition of the above, it cannot exist.
(Yes, dvgrn, I'm aware of your post, but this is a bit more rigorous in my opinion)

[Gamedziner]

A toroidal solution exists in which all cells die from overpopulation:

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x = 8, y = 4, rule = B3/S23:T8,4
2o2b2o$2o2b2o$2o2b2o$2o2b2o!

[Caenbe]

Given my original wording, this would be a solution:

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x = 11, y = 10, rule = B3/S23
4bo$6bo$4bobo$2bobo2$2o3bo2b2o$5bob4o$bo5b4o$3bo4b2o$3bobo!

So I suppose I should say that the phoenix must be an oscillator, just to be rigorous.

Still, just because one cell dying of overpopulation requires more around it, I don't see why that has to run away to infinity. For example, why couldn't the 12-omino in my "solution" appear during the phoenix's period?

EDIT: I see now, the cells in the 12-omino must have previously been off, so there's no way for its central cells to be born. I wonder, could the proof rely solely on looking at the phases immediately before and after the one with the overpopulated cell? Do we really have to consider the phoenix's entire cycle?

[dvgrn]

EDIT: I see now, the cells in the 12-omino must have previously been off, so there's no way for its central cells to be born. I wonder, could the proof rely solely on looking at the phases immediately before and after the one with the overpopulated cell? Do we really have to consider the phoenix's entire cycle?

That was along the lines of what I was hoping for in my previous post -- that when you boringly enumerate all the relevant cases in a 5x5 or 7x7 square around the overpopulated T=0 cell, it would turn out that there's a cell that has to stay on for more than one tick, in every case.

If there's some spark pattern like BlinkerSpawn's spark pattern, above, that has predecessors coming in from far away with no cells that are ON for more than one tick, then you'd have to look at the entire cycle. But I think it can be shown that BlinkerSpawn's example doesn't have a phoenix-compatible T-2 -- you can't turn on the central cell in the X shape and the four corner cells, without having some supporting cell stay on for two ticks. E.g., the yellow cell can't be made to die of underpopulation, and if you make it die of overpopulation then some of the red cells will fail to turn ON:

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x = 9, y = 9, rule = LifeHistory
4.D$2.D.D.D$.D5.D$3.DED$2D.CDC.2D$3.D.D$.D2C.2CD$2.D.D.D$4.D!

It might take a search utility to run through all the cases, but it seems very possible that a contradiction can be found by T=-2 in every case, somewhere in the eight cells around the overpopulated cell.

[shouldsee]

Is there any quantity conserved under B3/S23?

[Bullet51]

Is there any quantity conserved under B3/S23?

In principle, yes. One can assign a quantity to a universe ,where two universes share the same value if and only if one can evolve into the other.
In practice, it's hard to say. A pattern with its glider destruction shares the same value with the empty universe.

A much easier question:
Is there any quantity that monotonically increases during the evolution of a infinite 50% soup?

[praosylen]

Is there any quantity conserved under B3/S23?

Does symmetry count?

A much easier question:
Is there any quantity that monotonically increases during the evolution of a infinite 50% soup?

Fraction of cells that have ever been on is the only example I can think of right now.

Edit: I suppose fraction of cells that have ever been off also works.

[biggiemac]

Conservation laws are really difficult to come up with.

Candidates:

The least useful conserved quantity - a constant. F(universe) = C is conserved for all universes but cannot distinguish one universe from another.

More useful but not generally calculable. F(universe) = average(G(that universe at t->infinity)), where G is any well-behaved function, preferably finite for any universe. For example, the integral of a 2D Gaussian envelope windowed by living cells. The average is taken over the period of the universe. The issue with this is that it requires knowledge of the state of the universe as t->infinity, so while it is necessarily conserved it isn't much more helpful. Probably also someone can create a pattern that breaks this approach, maybe by making some aperiodic pseudorandom device that remains near where it starts, so that its average can't be computed nor neglected.

What's needed to produce a true conservation law is some feature invariant under the evolution operator. Symmetry works, as A for Awesome pointed out.

The evolution operator is isotropic (totalistic as well but that doesn't help). That at least means that any definition of angular momentum is conserved. The only patterns that I can associate angular momentum with are rotating oscillators.

The evolution operator is also the same for all space, so linear momentum is conserved. This means spaceships are viable.

The evolution operator is also the same for all time, so energy can be considered conserved. This means oscillators and spaceships go on forever in an isolated universe.

In a random soup though, standard definitions of angular momentum, momentum or energy are pretty meaningless, since particles are springing in and out of existence before acquiring any specific attributes.

[praosylen]

The evolution operator is also the same for all space, so linear momentum is conserved. This means spaceships are viable.

Is this really a valid conclusion? This pattern would seem to give a counterexample:

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x = 4, y = 6, rule = B3/S23
2o$2o2$2bo$2b2o$bobo!