ConwayLife.com

Hello everyone,

I am struggling to find something nice to investigate for my exploration, a task in which I am allowed to explore anything math-related. I encountered the game of life, and fell in love with it immediately, and would love to make it the focus of this project. Does anyone know of a good pattern to look at / investigate a rule for? The maths level must not be too easy, as the requirements are to show quite a high level of maths...

I have thought of a few things, but none really convince me:

1. Deriving the R-Pentomino, starting from a certain point and looking at the generations, to make sense of how such a small pattern can grow out to become so big.
2. Finding a rule to predict if a cell will stay alive or die, according to what it has on its perimeter (a bit like defining the rules of the game, but in mathematical notation)
3. Finding a new pattern? Describing how to come up with it?

I am very much open to ideas, and also I am no expert, I conducted very basic research, and know about the basic patterns, and not much more.

If you could point me in a basic topic to research, I would be very appreciative...

Take a look if these topics might interest you.

• (13,1)c/31 spaceship
• Quadratically growing replicator
• Strange loop

I think these problems are hard enough and solving them would involve in particular a lot of discrete maths and modular arithmetic. Let me know if they're too hard and you would like something simpler.

If I correctly understand what Noaminski means by "Maths SL IB (11th grade)", then those problems are definitely way too hard.

I'm not sure what constitutes an acceptable level of math in this context. Does there have to be new results? Or rather, is it a survey of interesting things that have already been done, or is being studied?

It would be a good idea to check the LifeWiki for ideas as well.

An alternative is to look at Wireworld and its variants. While it's not the Game of Life, it does at least fall under the same category of cellular automata, and might be easier to understand. Helpfully, many of the patterns run algorithms based on mathematics (e.g. binary multiplication). I've even constructed a binary phi calculator in a variant rule I also came up with. (While there is a phi calculator in CGoL, it seems relatively difficult for me to understand, but that might just be because I built the other one!)

[/shameless self-promotion]

I have sent you a PM regarding this.

FractalFusion wrote:I'm not sure what constitutes an acceptable level of math in this context. Does there have to be new results? Or rather, is it a survey of interesting things that have already been done, or is being studied?

There are umpteen research projects on my current backlog that have at least some mathematical component. It would be great if someone wanted to work on one of them:

-- Build the smallest-known glider guns for all periods from 14 to 999. The glider gun GitHub project has everything pretty well organized, but the 'confirmed' folder doesn't actually contain an entry for each gun period. Quite a few still have to be constructed based on some simple adjustments to a template gun in the 'variable' folder.

-- Come up with a "one-time turner" toolkit equivalent to Chris Cain's stable duplicator script. Stable duplicators are re-usable, but also fairly large and awkward; sometimes you just want to build a constellation of small still lifes that can be hit with a single glider to construct... well, any glider-constructible object, like a c/10 copperhead spaceship for example. Michael Simkin has a good collection of one-time turners and splitters, but the sorting and labeling hasn't been done to make it really easy to pick out a constellation with exactly the right timing for a specific purpose.

Like the previous item, this mostly just takes a lot of modulo-8 math, with the addition of some slightly brain-bending parity issues. To be specific: if you reflect a glider 180 degrees using a 90-degree color-preserving one-time turner combined with a 90-degree color-changing one-time turner, you can get two different output timings depending on which turner comes first (because the glider travels a half-diagonal farther in one case than the other).

-- The quadratically growing replicator made an appearance on codeholic's list. Of course that's a fairly large project (though it's a lot smaller than it used to be before simeks' single-channel construction toolkit showed up.) It's fairly easy to break the project up into manageable pieces now, and just tackle one or two pieces.

For example, it's probably a good idea to start by finding four stable reflector circuits arranged in an adjustable-size diamond, so that 1) the total time around the loop is a power of two, or a small multiple of a power of two, and 2) loops in the same orientation will be separated by an exact power-of-two distance (or again, maybe a small multiple of a power of two, if it turns out to be too difficult to get an exact power of two both spatially and temporally).

The power-of-two obsession is just because Golly can simulate pure power-of-two repeating patterns a lot faster using the HashLife algorithm. Maybe that's kind of an arbitrary mathematical limitation to place on a replicator design -- but it sure will be impressive to watch it run, if it works out the way I hope it will!

If anyone is interested in picking up any of these projects, just let me know and I can provide more details. Or a longer list of ideas would be easy to generate, if that's any use...!

For small and fun project I would suggest to take some design concept that you liked in complex CA, and make a rule that demonstrate that design in simplistic and fun way (any design including pi-calculator, caterpillar, tape reader, slow salvo constructor etc. can be investigated).

As a demonstration here is my project, that demonstrates Gemini design in small and compact rule called GeminoidParticles.

Alternatively building smaller or more optimal Gemini in GeminoidParticles rule can be fun as well.