Prove that a glider escaped

[didgogns]

A glider facing Northeast is at coordinate (N, N), and the board above the line x = -y is empty except that glider.

My conjecture is that if N is large enough, regardless of the state below x = -y, the glider is never interfered. This kind of proof could have been done by early life pioneers when they were dealing with R-pentomino and others, but I can't find it.

Similar proof of xWSSes need stronger condition because of long 1-cell thick line, but I think it can be proved for gliders. Thanks in advance.

[b-engine]

I don't know whatever you mean.
An easy way to see if a glider would never interfere with the pattern is to see if it would escape the bounding diamond of the main pattern.

[didgogns]

Is there a proof that a glider outside of a bounding diamond of rest of the pattern (finite or infinite) never interacts with rest of the pattern?

[b-engine]

Is there a proof that a glider outside of a bounding diamond of rest of the pattern (finite or infinite) never interacts with rest of the pattern?

No unless you evolve the pattern - a c/2 front end (such as B-heptomino or a xWSS) might (not always) emerge and crash onto the glider.

[HartmutHolzwart]

Is there a proof that a glider outside of a bounding diamond of rest of the pattern (finite or infinite) never interacts with rest of the pattern?

No unless you evolve the pattern - a c/2 front end (such as B-heptomino or a xWSS) might (not always) emerge and crash onto the glider.

It depends on the direction of the glider. If the glider is moving away from the bounding diamond, it can never be caught, as the bounding diamond maximally grows with the same speed.

[b-engine]

It depends on the direction of the glider. If the glider is moving away from the bounding diamond, it can never be caught, as the bounding diamond maximally grows with the same speed.

There are counterexamples.
For example, in both patterns the glider escapes the bounding diamond (marked with state 2), but then B crashes with the glider:

Code: Select all

x = 12, y = 12, rule = LifeHistory
7.B$6.3E$5.BE3B$4.3BE3B$3.9B$2.9B$.9B$3BC5B$.B3C3B$2.CB2CB$3.3B$4.B!

Code: Select all

x = 16, y = 16, rule = LifeHistory
11.B$10.3E$9.BE3B$8.3BE3B$7.9B$6.9B$5.9B$4.9B$3.9B$2.9B$.9B$3BC5B$.B3C
3B$2.CB2CB$3.3B$4.B!

[HartmutHolzwart]

In these cases, the glider does not move away from the bounding diamond, but perpendicular. Therefore I wrote”it depends”.

[g0t0]

Code: Select all

x = 13, y = 7, rule = LifeSuper
13A$.11B$2.9B$3.7B$4.5BA$5.3B2.A$6.B.3A!

I think the glider escaped.

Code: Select all

x = 36, y = 30, rule = LifeSuper
3$16.B$15.B.B$14.B3.B$13.B5.B$12.B7.B$11.B.B5.B.B$10.B3.B3.B3.B$9.B5.
B.B5.B$8.B7.B7.B$7.B.B5.3B5.B.B$6.B3.B3.4BA3.B3.B$5.B5.B.5B2A.B5.BA.A
$4.B7.7B2A7.B2A$5.B5.B.5B2A.B5.B.A$6.B3.B3.4BA3.B3.B$7.B.B5.3B5.B.B$
8.B7.B7.B$9.B5.B.B5.B$10.B3.B3.B3.B$11.B.B5.B.B$12.B7.B$13.B5.B$14.B
3.B$15.B.B$16.B!

[dvgrn]

A glider facing Northeast is at coordinate (N, N), and the board above the line x = -y is empty except that glider.

My conjecture is that if N is large enough, regardless of the state below x = -y, the glider is never interfered.

Cartesian coordinates, right? The picture doesn't work with traditional [down|south]-is-positive-Y Golly coordinates.

Is there a proof that a glider outside of a bounding diamond of rest of the pattern (finite or infinite) never interacts with rest of the pattern?

John Conway's proof from 1970 is about spaceship speeds. But doesn't it generalize to any pattern of live cells behind the defined line? You can't turn on cell X (in the linked diagram) or any of its cousins at T=2, whether it's part of a spaceship or not.

(?) (I'm not seeing why spaceship-ness is important to that speed proof, but I could be missing something obvious I suppose.)

Nathaniel's article says "Notice that this result doesn’t only apply to spaceships, but also to other configurations that are (initially) finite and travel across the grid, such as puffers and wickstretchers." But it seems like the same speed limit should apply even to unbounded-population initial configurations, as long as they are entirely on or below the key line.

-------------------

N doesn't have to be very big -- the exact value of N depends on the phase of the glider and what you mean by being "at coordinate (N, N)".

Code: Select all

x = 73, y = 74, rule = LifeHistory
B$.B$2.B$3.B$4.B$5.B$6.B$7.B$8.B$7.2AB$6.4AB$6.2A.2AB$8.2A2.B$13.B$14.
B$13.2AB$12.4AB$12.2A.2AB$14.2A2.B$19.B$20.B$19.2AB$18.4AB$18.2A.2AB$
20.2A2.B$25.B$26.B5.2A$25.2AB5.2A$24.4AB3.A$24.2A.2AB$26.2A2.B$31.B$32.
B$32.AB$31.3AB$31.A.2AB$32.3A.B$32.2A3.B$38.B$38.AB$37.3AB$37.A.2AB$38.
3A.B$38.2A3.B$44.B$44.AB$43.3AB$43.A.2AB$44.3A.B$44.2A3.B$50.B$50.AB$
49.3AB$49.A.2AB$50.3A.B$50.2A3.B$56.B$56.AB$55.3AB$55.A.2AB$56.3A.B$56.
2A3.B$62.B$62.AB$61.3AB$61.A.2AB$62.3A.B$62.2A3.B$68.B$68.AB$67.3AB$67.
A.2AB$68.3A.B$68.2A!

[I6_I6]

Is it possible for the frontend of an initially infinite pattern travel diagonally in a speed faster than c/4 forever, or has this also been disproved as part of the spaceship speed limit proof? If it was possible, you could place it behind the glider and wait until it catches up, because didgogns never specified that the part behind the NE glider had to be finite.

[HartmutHolzwart]

There was an example by Amling in a topic called “Level wave speed limit?“

https://conwaylife.com/forums/viewtopic ... ve#p160383

[g0t0]

A glider facing Northeast is at coordinate (N, N), and the board above the line x = -y is empty except that glider.

My conjecture is that if N is large enough, regardless of the state below x = -y, the glider is never interfered.

Cartesian coordinates, right? The picture doesn't work with traditional [down|south]-is-positive-Y Golly coordinates.

Is there a proof that a glider outside of a bounding diamond of rest of the pattern (finite or infinite) never interacts with rest of the pattern?

John Conway's proof from 1970 is about spaceship speeds. But doesn't it generalize to any pattern of live cells behind the defined line? You can't turn on cell X (in the linked diagram) or any of its cousins at T=2, whether it's part of a spaceship or not.

(?) (I'm not seeing why spaceship-ness is important to that speed proof, but I could be missing something obvious I suppose.)

Nathaniel's article says "Notice that this result doesn’t only apply to spaceships, but also to other configurations that are (initially) finite and travel across the grid, such as puffers and wickstretchers." But it seems like the same speed limit should apply even to unbounded-population initial configurations, as long as they are entirely on or below the key line.

-------------------

N doesn't have to be very big -- the exact value of N depends on the phase of the glider and what you mean by being "at coordinate (N, N)".

Code: Select all

x = 73, y = 74, rule = LifeHistory
B$.B$2.B$3.B$4.B$5.B$6.B$7.B$8.B$7.2AB$6.4AB$6.2A.2AB$8.2A2.B$13.B$14.
B$13.2AB$12.4AB$12.2A.2AB$14.2A2.B$19.B$20.B$19.2AB$18.4AB$18.2A.2AB$
20.2A2.B$25.B$26.B5.2A$25.2AB5.2A$24.4AB3.A$24.2A.2AB$26.2A2.B$31.B$32.
B$32.AB$31.3AB$31.A.2AB$32.3A.B$32.2A3.B$38.B$38.AB$37.3AB$37.A.2AB$38.
3A.B$38.2A3.B$44.B$44.AB$43.3AB$43.A.2AB$44.3A.B$44.2A3.B$50.B$50.AB$
49.3AB$49.A.2AB$50.3A.B$50.2A3.B$56.B$56.AB$55.3AB$55.A.2AB$56.3A.B$56.
2A3.B$62.B$62.AB$61.3AB$61.A.2AB$62.3A.B$62.2A3.B$68.B$68.AB$67.3AB$67.
A.2AB$68.3A.B$68.2A!

Code: Select all

x = 18, y = 9, rule = B3/S23Super
$2.13A$3.11B$4.9B$5.7B.A$6.5B3.A$7.3B2.3A$8.B!

I think the glider must do not touch the bounding diamond.
But a pattern can move with speed >c/2o temporarily.
Can a pattern move with speed >c/4d temporarily?

[dvgrn]

... a pattern can move with speed >c/2o temporarily.
Can a pattern move with speed >c/4d temporarily?

Yes, depending on your definitions -- or even >c/4d indefinitely. See Hartmut's linked example above.

However, I'm not sure that that wave is relevant to the original question about gliders escaping. It seems as if Conway's proof safely rules out starting a finite chunk of such a wave entirely behind a given diagonal line, and having it move past the diagonal line at a speed greater than c/4. And of course if the chunk isn't finite (in the "forward" direction, where it matters for catching up to a glider) then it can't be entirely behind any given diagonal line.

[SecondTypist]

[…]
I think the glider must do not touch the bounding diamond.
[…]

Close but no cigar.
A glider would be considered “escaped” If it were outside the bounding diamond, facing in such a way that it were moving away from the bounding diamond both vertically and horizontally, and had sufficient separation between itself and the bounding diamond, to such a degree where the diagonally connected bit does not border a cell that borders two or more cells that are inside the bounding diamond of the rest of the pattern.
The final is the only additionally needed criterion, as cells within the bounding diamond interfering with any other part of the evolution of the glider also requires the aforementioned criterion to be true. One could further extend this definition if they wanted to detect such an occurrence earlier, but I’m not going to.

Code: Select all

 x = 12, y = 6, rule = LifeHistory
.B6.B$2.A6.B$3.A4.A.A$.A2.B2.A3.A$A6.3A$3A!

Edit: I also seem to be wrong. I guess it must not be bordering a cell that borders any cells within the bounding diamond.

Code: Select all

 x = 6, y = 6, rule = LifeHistory
2.B$3.2A$4.A$.A3.B$A$3A!