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1% Density Soups

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1% Density Soups

Postby Electro_blob » June 22nd, 2018, 12:57 pm

Hello, I was experimenting with toroidal soups in Golly, and tried a 1% density soup in a 100,000 by 100,000 toroidal universe. By generation 100, these are the objects that emerged in order of approximate commonness: block, blinker (these two were the vast majority), bee hive, pond, traffic light, (unknown pattern), loaf, pi heptimino, blockade, and tub. Of these, the one's that interest me the most are the pi heptimino and the unknown pattern, which were the only two large Methuselah like patterns I could find, and were fairly common. I wanted to know if anyone could identify the unknown pattern and its seed.
Code: Select all
x = 32, y = 55, rule = B3/S23
4$24b3o$15bo8b2ob2o$13b4o5bo6bo$12bob4obo2b2o5bo$11b2o7bo7b2o$12b3obo
3bobo$13b2o2bobo2bo4bo$24b2o6$24b2o$24b2o8$10bobo$10b2o$11bo2$16b2o$
16b2o3$11b2o5b3o$11b2o2$16b3o$8b2o5bo3bo$8b2o5b2ob2o$15bo4bo$16b2ob2o$
16bo3bo$17b3o3$7b2o8b2o$6bo2bob2o3bo2bo$6bo2bob2ob5o$7b2o5b4o$11b2obob
o$12bo3bo$14b2o$14b2o!

I also know this type of soup is known as "sparse life," but the wiki doesn't give much more information on it then that and the fact that it has many blocks and blinkers at first, and infinite growth patterns later on. I have not observed any block-laying switch engines, but perhaps this is something that occurs when the escaping gliders collide with other still lives.
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Re: 1% Density Soups

Postby Macbi » June 22nd, 2018, 1:18 pm

The unknown pattern is the evolution of the r-pentomino.
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Re: 1% Density Soups

Postby Electro_blob » June 22nd, 2018, 2:11 pm

Macbi wrote:The unknown pattern is the evolution of the r-pentomino.


Thank you! :D
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Re: 1% Density Soups

Postby NickGotts » June 28th, 2018, 11:38 am

A 1% density soup, although certainly interesting in itself, isn't really "sparse life". The entry in the lexicon (http://www.conwaylife.com/ref/lexicon/lex_s.htm) is:
sparse Life This refers to the study of the evolution of a Life universe which starts off as a random soup of extremely low density. Such a universe is dominated at an early stage by blocks and blinkers (often referred to collectively as blonks) in a ratio of about 2:1. Much later it will be dominated by simple infinite growth patterns (presumably mostly switch engines). The long-term fate of a sparse Life universe is less certain. It may possibly become dominated by self-reproducing patterns (see universal constructor), but it is not at all clear that there is any mechanism for these to deal with the all junk produced by switch engines.

1% can't really be called "extremely low". A more precise definition could say that sparse life is what you get as the density of an initial infinite random field approaches zero (or as the height and breadth of an initial finite random field, the expected number of "on" cells, and the ratio of the total number of cells to that expected number, all approach infinity). I've written articles on what happens in such fields, and am currently in the early stages of working on another (but that might be a long time coming!). There's some discussion on this thread: viewtopic.php?f=2&t=1406&start=25 ("Making switch engines").
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Re: 1% Density Soups

Postby KittyTac » June 28th, 2018, 10:00 pm

You could set it to 0.01% or something to emulate Sparse Life. Because you can't really have an infinite field on a computer.
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Re: 1% Density Soups

Postby calcyman » June 29th, 2018, 4:33 am

To emulate Sparse Life on a random N-by-N grid, you want the density to tend to zero in such a way that N^2 density^c tends to infinity for every positive integer c (as otherwise there will whp be no (c+1)-cell clusters).

A density of 1/log2(N)^2 would work quite well: for a 1024-by-1024 grid, this gives a density of 1%, and for a 1048576-by-1048576 grid, this gives a density of 0.25%.

I was thinking about creating a Golly script which creates an initial sparse life universe, minus any 1- and 2-cell clusters (so that the population remains small). For the 0.25%-populated 1048576-by-1048576 grid, there will be about 2.5 billion cells including the 1- and 2-cell clusters, but less than 1 million cells if you exclude them. There should be a way to efficiently create this setup in time proportional to the number of cells after exclusion.
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Re: 1% Density Soups

Postby NickGotts » July 2nd, 2018, 5:06 am

It's true you can't have an infinite field on a computer, but you can work out a lot about what would happen on such a field using a computer - as I have in fact done. Basically, after a certain number of generations (expressed as a negative power of the initial density) the dynamics will be dominated by infinite growth patterns that were not present at the start, but result from sequences of collisions starting with a glider or r-pentomino, and otherwise involving only blocks or blinkers. So an adequate finite, toroidal representation of Sparse LIfe should at least replicate this, but I think we'll have to wait some time before our hardware is up to the job, or else devise some special-purpose software to do it.

None of this is to say that low-denisty toroidal fields that don't meet this condition (such as 1% or .25% or .01% density) are not interesting in themselves. However, there's an additional wrinkle with such fields. If the density is low enough relative to the size, the exact relationship between the x and y dimensions becomes important, because gliders will travel right round the torus and collide with the remnants of the pattern that produced them (e.g., and most commonly, r-pentomino remnants). It's also possible, although I don't know how easy it is in Golly, to twist the torus, so that when a glider exits from one side of the field, it doesn't appear directly opposite on the other side.
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Re: 1% Density Soups

Postby KittyTac » July 2nd, 2018, 11:58 am

Or use :P instead of :T to kill off gliders that come near the edge.
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Re: 1% Density Soups

Postby benetnasch85 » July 2nd, 2018, 11:46 pm

NickGotts wrote: It's also possible, although I don't know how easy it is in Golly, to twist the torus, so that when a glider exits from one side of the field, it doesn't appear directly opposite on the other side.

Twisting is not necessary. If the sides of the toroidal domain are two consecutive fibonacci numbers, or at least have the same ratio, a single glider will cross the domain on different paths until it has crossed everywhere, so a single glider is guaranteed to interact with a single fixed target.
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Re: 1% Density Soups

Postby NickGotts » July 3rd, 2018, 11:24 am

Twisting is not necessary. If the sides of the toroidal domain are two consecutive fibonacci numbers, or at least have the same ratio, a single glider will cross the domain on different paths until it has crossed everywhere, so a single glider is guaranteed to interact with a single fixed target.


Sure, an untwisted torus with any ratio between dimensions without a common factor will give the result you describe. But a twisted torus might be interestingly different. For example, *WSSs would also not just go straight round to their starting point in the direction the twist was applied. (I think you could apply a twist in both directions.)
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Re: 1% Density Soups

Postby benetnasch85 » July 4th, 2018, 8:49 am

NickGotts wrote:Sure, an untwisted torus with any ratio between dimensions without a common factor will give the result you describe.

Yes, the path will cover the domain, but if the ratio of sides is near a ratio of small numbers, a glider's path will drift across the domain in small steps, and a large target will be hit near its edge. The golden ratio, as approximated by consecutive fibonacci numbers, is as "far" as possible from ratios of small numbers, and gives the best chance for a glider to hit a target of any size anywhere across its cross section.
For example, *WSSs would also not just go straight round to their starting point in the direction the twist was applied.

Good point, although *WSS become rarer relative to gliders as the domain is made more sparse.
(I think you could apply a twist in both directions.)

For a domain to be twisted in both directions, it would need to either include an extra small rectangle or have a rectangular cutout in one corner to fit together seamlessly. The latter would be a good option to add to Golly for this purpose. In Golly's help file it could be described like this:
:T30+5,20-2 -- torus with shifts of +5 on the horizontal edges and -2 on the vertical edges and with a 5x2 rectangle removed from one corner.

Edit: We want to specify the twist in both directions in such a way that both gliders and *WSS are equally likely to hit a target anywhere across its cross section. I think we can achieve this by using three or four consecutive fibonacci numbers as in these examples: T:89-55,55-34 (larger cutout) or T:89-34,55-21 (smaller cutout).
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Re: 1% Density Soups

Postby calcyman » July 4th, 2018, 1:36 pm

A torus twisted in both directions is equivalent to a rectangular torus twisted in only one direction (because you can choose one of the two edges of the fundamental parallelogram to be exactly horizontal), so Golly already supports these.
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Re: 1% Density Soups

Postby benetnasch85 » July 4th, 2018, 3:39 pm

I think that rotating the fundamental parallelogram is not equivalent to rotating the grid to eliminate skew in one direction.

Golly's methods for twisting grids on a torus cause a *WSS along one axis to vary its path across the rectangle, but a *WSS along the other axis retraces its path. We are trying to twist the grid so that a *WSS moving in either direction will not retrace its path. Since we have to add or subtract a small rectangle from the domain to twist the grid in both directions in this way, we're no longer talking about projecting the surface of a torus onto a rectangle.
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Re: 1% Density Soups

Postby calcyman » July 4th, 2018, 5:40 pm

I didn't mean rotating 'the' fundamental parallelogram. My point is that there are lots of choices of parallelogram giving rise to the same lattice, parametrised by PSL(2, Z), and at least one of them has a horizontal edge.

Specifically, draw a horizontal line from the origin until it hits a point on the sublattice L. Then choose another lattice point of minimal positive y-coordinate. These two vectors form a basis of L.
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Re: 1% Density Soups

Postby NickGotts » July 6th, 2018, 7:08 am

I think you're both wrong! Or we are somehow at cross purposes. Consider the following. It's a 4-by-4 field twisted in both directions, implemented in a normal 16-by-16 toroidal field. Whatever state cell A1 in the 4-by-4 is in would be represented by all the A1s in the 16-by-16 being in that state, and so on for all other cells.

A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 D1 D2 D3 D4
B1 B2 B3 B4 C1 C2 C3 C4 D1 D2 D3 D4 A1 A2 A3 A4
C1 C2 C3 C4 D1 D2 D3 D4 A1 A2 A3 A4 B1 B2 B3 B4
D1 D2 D3 D4 A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4
A4 A1 A2 A3 B4 B1 B2 B3 C4 C1 C2 C3 D4 D1 D2 D3
B4 B1 B2 B3 C4 C1 C2 C3 D4 D1 D2 D3 A4 A1 A2 A3
C4 C1 C2 C3 D4 D1 D2 D3 A4 A1 A2 A3 B4 B1 B2 B3
D4 D1 D2 D3 A4 A1 A2 A3 B4 B1 B2 B3 C4 C1 C2 C3
A3 A4 A1 A2 B3 B4 B1 B2 C3 C4 C1 C2 D3 D4 D1 D2
B3 B4 B1 B2 C3 C4 C1 C2 D3 D4 D1 D2 A3 A4 A1 A2
C3 C4 C1 C2 D3 D4 D1 D2 A3 A4 A1 A2 B3 B4 B1 B2
D3 D4 D1 D2 A3 A4 A1 A2 B3 B4 B1 B2 C3 C4 C1 C2
A2 A3 A4 A1 B2 B3 B4 B1 C2 C3 C4 C1 D2 D3 D4 D1
B2 B3 B4 B1 C2 C3 C4 C1 D2 D3 D4 D1 A2 A3 A4 A1
C2 C3 C4 C1 D2 D3 D4 D1 A2 A3 A4 A1 B2 B3 B4 B1
D2 D3 D4 D1 A2 A3 A4 A1 B2 B3 B4 B1 C2 C3 C4 C1
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Re: 1% Density Soups

Postby NickGotts » July 6th, 2018, 7:49 am

No - I'm obviously wrong! Different copies of the same cell have different neighbours.
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Re: 1% Density Soups

Postby benetnasch85 » July 6th, 2018, 12:38 pm

In a variation of your notation, with capital letters for cells in one copy of the domain and lower-case letters for cells in six surrounding copies of it, a doubly-twisted toroidal domain can look like this:

e3 e4 e5 h1 h2 h3 c1
f3 A1 A2 A3 A4 A5 d1
g3 B1 B2 B3 B4 B5 e1
h3 C1 C2 C3 C4 C5 f1
a5 D1 D2 D3 D4 D5 g1
b5 E1 E2 E3 E4 E5 h1
c5 F1 F2 F3 a1 a2 a3
d5 G1 G2 G3 b1
e5 H1 H2 H3 c1
a2 a3 a4 a5 d1

Edit: I'm not sure we can cover a torus with a single copy of this domain without distorting the grid. We have done in two dimensions what is done to a shifted toroidal domain in one dimension, but the term "toroidal" may no longer be strictly correct.
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