What on earth is this shape? [LTL Moore Majority]

[Extrementhusiast]

WARNING: This gets relatively heavy into the math side of things.

So I was messing around with Larger than Life majority rules using the Moore neighborhood, and was wondering what the minimally-sized survivors would be for each range. It turns out that the minimal survivors were not converging to a perfect circle as I had first expected (based on my experience with range-1 cellular automata), but were instead converging to some other shape, which resembled a rounded square rotated by 45°. Here is the minimal survivor for the r=20 case:

r20.rle

Displayed in regular Life to shave off some of the loading time.

(63.7 KiB) Downloaded 190 times

Since this shape wasn't a perfect circle, but was still the target of a relatively simple convergence, I naturally wanted to find out exactly what shape it was, so I came up with a few apparent properties (assuming center at (0, 0) with radius 1):

• boundary passes through (2/3, 2/3)
• at x=1/3, dy/dx=-1/2 at the boundary (y≈.915)
• (shape area)/(bounding box area)≈.738

I was first looking for a result in terms of a fourth-degree implicit equation, and (assuming Occam's razor) found the equation 9*x^2*y^2+16*(x^2+y^2-1)=0. (The equation must be symmetric between the x's and y's, and must have only terms of even degree in both x and y, due to the D8 symmetry of the shape.) However, this equation fails the second test, as dy/dx=-50/(17*sqrt(34))≈-.5044076. Nevertheless, there is a very slight but nonzero chance that this is the actual solution, as some or all of the properties that I listed could be slightly off (which would change everything), as trying to find the properties of a shape that you don't know the exact shape of gets...weird.

Or maybe I'm looking at this wrong, as there seem to be slight cusps present at each of the four "corners" of the rounded square. (It's not all that obvious of an effect, but the shape does seem to curve more strongly than expected when it meets the bounding box.) Perhaps each quadrant is actually part of a different larger curve; it could also be that the proper way to analyze this would be to extend three of the corners of the original square to infinity, before the cellular automaton is ever run. I honestly don't know; that's why I'm asking the board, after all!

[gameoflifemaniac]

By the way, is there a LTL rule, whose smallest survivors are perfect circles? but don't forget about Extremeenthusiast's question

[ntdsc]

My counterclockwise and clockwise rotation pattern converges to that same shape

Code: Select all

x = 588, y = 588, rule = B3/S12456
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ob6obo2b4o$282b4o4bo6bo4b3o$281bob2o4bobo4bobo4b2o$280bo2bo5bo2bo2bo2b
o5bo$279b2o2bo5bo3b2o3bo5b5o$279b5o5bo3b2o3bo5bo2b2o$283bo5bo2bo2bo2bo
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o$117bo354bo$116bo356bo$115bo358bo$114bo360bo$113bo362bo$112bo364bo$
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In my last post, I showed a hexagonal grid of them, and the center would be clockwise surrounded by counterclockwise, and they would alternate. But the effect of the hexagonal grid maybe is a transformation like in the reflection, rotation mentioned in D8 symmetry: https://en.wikipedia.org/wiki/Dihedral_group
I'm wondering if a larger grid of a hexagonal lattice (and pointwise members that make a hexagonal lattice don't look exactly like a honeycomb, but a pointwise member is shared with a pointwise member of a neighboring hexagon, and you can search for a hexagonal lattice and see) could be used as computing a transformation matrix, because having them in a hexagonal grid alters the "rotation" arms further from the center where neighbors of the clockwise/counterclockwise expand and touch and then alter the flow.

[BlinkerSpawn]

By the way, is there a LTL rule, whose smallest survivors are perfect circles?

Doubtful, considering we're dealing with square neighborhoods.
That's why the pattern is (roughly) diamond-shaped.

[Gamedziner]

By the way, is there a LTL rule, whose smallest survivors are perfect circles?

Doubtful, considering we're dealing with square neighborhoods.
That's why the pattern is (roughly) diamond-shaped.

Depending on your definition of "perfect," there may be no pattern in any (square-neighborhood) cellular automaton which can be considered a perfect circle.

[calcyman]

By the way, is there a LTL rule, whose smallest survivors are perfect circles?

Doubtful, considering we're dealing with square neighborhoods.
That's why the pattern is (roughly) diamond-shaped.

Depending on your definition of "perfect," there may be no pattern in any (square-neighborhood) cellular automaton which can be considered a perfect circle.

It could be asymptotically perfectly circular -- i.e. the scaling limit is an exact circle. This often happens on square grids:

https://cp4space.wordpress.com/2014/10/ ... lar-mazes/

I think certain lattice gases also produce asymptotically circular vortices.